Category Theory [1, 3 ed.] 978-3-88538-001-6

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Sigma Series

in Pure Volume

Horst

George

Mathematics 1

Herrlich E. Strecker

Category Theory 3rd

edition

Heldermann

Verlag

to

the

Oklawahaand

the

,Ichetucknee

V)

-.-:w.x

3‘51}-

.

.

.

5‘.

it.

C. Escher Omwikkeling I (Esrlu'r Gemeenlemuseum— 7719 Hague) M.

“ -

bat-3'23“

Foundation—Collection

Hangs

Contents

Preface

ix

I.

Introduction

ll. 1

Sets, classes.

Foundations

and

lll.

conglomerates Categories

Concrete

categories categories categories from

Auto Abstract New

..............................

IV.

13 l3

.........................................

IS

..........................................

old

23

.....................................

and

Special Morphisms and

Sections, retractions,

Special Objects

isomorphisms Monomorphisms. epimorphisms, and bimorphisms Initial, terminal, and zero objects Constant morphisms. zero morphisms, and pointed categories.

”\IO‘UI V.

Functors

Functors

ll

Categories

of

l2

Properties

of functors

l3

Natural

Transformations

categories

67

and

natural of

VI.

l8

Products

and

Limits

coproducts

77

86

100 l00

..................................

factorizations

..............

93

Categories

coequulizers

and and

in

isomorphisms categories ...................

...........................................

Intersections

48

......................................

.......................................

lsomorphisms cquivalenccs Functor categories

Equalizers

..

61

and

l6

.

53

..............................................

transformations

l7

46

53

Hom-functors

l5

38

.............

.............................

...................................................

l0

l4

32

.......................

Natural

and

32

...............................

l07 ll5

.....................................

\ii

viii

C amen

I9

Sources

20

Limits

2I

Pullbacks

22

Inverse

23

Complete categories

24

Functors

25

Limits

and

sinks

and

and

direct

Universal

maps

27

Adjoint

functors

28

Existence

categories

30

3I 32

34 35

adjoints

Hom-functors

I71

I77 I77

I94 207

........................................

Functors

217 2l7

functors

Representable Free objects Algebraic categories

22]

......................................

23]

................................................

and

functors

algebraic

Subobjects,

(J. J!) categories mono) (Epi, extremal

236

.....................

Quotient Objects,

and

Factorizations

249 249

...........................................

extremal

(Generating,

and

(extremal epi, mono) categories mono) and (extremal generating. mono)........

Reflective

General

Characterization

subcategories generation Algebraic subcategories

of 6-reflective

......................................

Pointed

39

Normal

and

40

Additive

4]

Abelian

categories categories Foundations

Bibliography of

exact

...................................................

Symbols

.........................................................

...............................................

Categories

categories ..........................................

..........................................

.........................................

275

...............................

and

XI.

275

Subcategories

reflective

36

Appendix:

255

...............................................

37

Index

I66

........

..............................................

X.

Index

.

..................................

Set-Valued

factorizations

38

...............

............................................

1X.

33

limits

.............................................

VIII. 29

155 reflect

and

Adjoint Situations

26

of

IS]

......................................

.........................................

preserve

VII.

I38

......................................

limits

in f unctor

I33

.........................................

pushouts

that

I26

..........................................

colimits and

(5

.................................

subcategories

.....

281

288

293 294

305 3l8

328

Preface

Our

in purpose earliest moment

writing which

this

book the

is

reader

to

present

the

theory

of

categories

at

the

as he appreciate it, that is, as soon modern and reasonably acquainted theory, algebra, general more topology. It is hoped that such a presentation will help him to prepare for in advanced these in and adequately topics subjects algebraic topology. Thus the book is designed for normal use the during early stages of graduate study— or in honors courses for undergraduates. possibly However, this does not mathematicians who have preclude its use as “armchair reading" for mature not yet had formal to the exposure subject. The attempt is made to present category theory mainly as a convenient new notions, which puts many language—one which ties together earlier results in their and which for existing proper perspective. provides a means appreciation of the unity that exists in modern mathematics, despite the tendencies toward and increasing fragmentation specialization. Our approach is heavily dependent upon numerous examples and exercises drawn from set theory. algebra, and topology. By continually tying down new notions to well-known concrete examples. it is hoped that the relatively high level of abstraction that is embodied in category theory can be kept from becoming a high level of obfuscation. Throughout the book we have striven to achieve a pedagogical soundness that would make it appropriate for use even without the aid of an instructor. Some care has been taken to arrive at a flow of topics that would be “selfand to resist the to abstract for the sake of motivating" temptation very quickly For an efficient efficiency. example. approach would have been to define the more first. and then to specialize them. We have taken quite general concepts the opposite approach. various by first abstracting to a categorical context concrete notions within the reader's probable realm of experience and then, when common features these notions among begin to appear. abstracting them to even more Thus. for example. we have used the usual progeneral notions. ducts of sets. groups. and topological spaces modules. to motivate categorical of limits before products. have investigated several special cases introducing

becomes

at

with

can

set

ix

Preface

x

the

and

general notion, notion

important

of

delayed introduction

have

until

functor

adjoint

the

of

the

central

where

point

it

can

and

very

be

easily

appreciated. is divided

book

The

it is intended

of

topics, and Chapter VII). be

that

since

covered,

that

on

even

in category

sections

VIII

sequentially (at

and

X could

Chapters reading, Chapter theory the distinction first

II

least

be omitted.

(Foundations)

between

and

sets

through It should

should

be

classes

is

instance, that a category possess products for all for all objects is far different from asking the same To facilitate into references, each chapter is divided

families.)

sequentially throughout the book and all items sequentially throughout it. The symbol C] given has been used to designate the ends of proofs as well as to mark those instances where the proof is left to the reader (and is thus an implied exercise). The to be implied exercises, depending upon one’s examples may also be considered inclination and mathematical The exercises that at the end background. appear of each section have been designed both as an aid in the understanding of the material of the section as well as a means to begin to apply it elsewhere. They range widely in their difficulty. Category theory is a relatively young field without settled terminology and notation. For this reason we have in general tried to use terminology that has become “standard“ over the last several and have years, strayed from this we when felt that there was a to do so. principle only compelling reason For example, we have used the term “dense f unctor" where the more standard term is “representative functor", both because “dense” seems to be more between and representable descriptive and because the confusion representative within

that

be covered

of its

families

class-indexed

these

“clusters"

natural

represent

for

quite important. (To ask, set-indexed

that

chapters

is short,

If time

mentioned

eleven

into

section

f unctors that

numbered

are

a

is avoided.

when

the

connection

book

In the

index

is used

for

confess

we

“sink".

However.

lead

to

useful

.

.

numbered

are

we

reference.

that

believe

tried

to

these

include

confusion

introduced

having

to

have

should

the

notions

the standard

new

be

terms

unlikely. In of “source"

terms

fundamental

so

this and

quite often descriptions point apology is also needed regarding the “order" used when denoting the composition of morphisms. Certainly mankind will long be plagued by (or at least mathematiciankind) the regrettable historical accident that the value of a function f at a point x has been denoted f(x) rather than (x)f. Because of this notation, the value of .1. .9. at x is written g(f(x)) and, consequently. the composition offand g is denoted

and

we

and

are

visualizations.

At this

and

an

.

by

g

it would

of. This

be

far

form

of

preferable

designation both

constitutes

aesthetically

have

chosen

and

a

switch

in the

order,

the practically traditional adopt the more is practically universal and to

write

composition f 9. However. notation its use by mathematicians simply because because a consistent would have certainly led to totally change of notation undue confusion and alienation—[a sequence, for example, would have to be denoted rather than and the nth of a X by by (,,x) (x,), homology group space (X),,H rather than 1-1,,(X)]. as

o

we

to

xi

Preface

Throughout have

and There

made

the text

included

provide

to

attempt

no

several

are

have

we

a

historical

literature

the

to

of the

development

theory.

of all, category theory is too young to treatment of the subject is intended to be

for this.

reasons

few references

very

First

“history". Secondly our only on an introductory level. Also, many categorical results have been found Finally, many independently (often in different forms) by several individuals. is concerned) older than of the results are actually (as far as their “essence” the study of categories category theory itself. For those who wish to continue have

or

real

a

do research

to

in the area,

have

we

provided

fairly extensive

a

bibliography. H.H. G.E.S.

Preface first

the

Since

gratified by

the

field

of

believe vation

via

to

we

have

we

of

errata

some

We wish sions

efforts

and in

to

need

has

grown

for

an

wished

edition

in many

the

typographical nature is especially to thank those express

making this edition

exposure directions

new

opportunity

incorporates In this

a

our

detailed

text

that

to

the

since

relies

subject.

The

Yet

we

then.

heavily

moti-

on

and exercises.

that

make.

to

had

introductory level

pleasedby

indeed

second

a

present

that

are

a

previously

not

examples

numerous

Thus

had

theory

is still

there

of Category Theory appeared in 1973, we have been and professional that it has generated among both students

interest

category

Edition

Second

edition

who

mathematicians

the

to

appreciation

provided by Heldermann Verlag improvements and corrections

many connection it should

included who have to

on

page

that

a

list of

382.

contributed

Dr.Norbert

be noted

suggestions for

Heldermann

for

revi-

his many

3 success.

H.H.

G.E.S.

Preface The

second

due

to the

edition

of this

text

to the Third has been out

Edition

of

for several

However, years. categorical language has become indispensable in many areas of mathematics as well as in the rapidly growing field of theoretical demand for an elementary computer science, there has been an undiminished we introduction to category are theory. Therefore pleased that our publisher Heldermann has made this work available Prof. Dr. Norbert again. Further information is available from the home page of this book, which you find at the web-site of the publisher at www.heldermann.de. fact

Abstract written

the

Verlag has

Heldermann

advanced

that

print

also

made

available

electronic

an

edition

of

our

more

text:

Concrete

and

jointly

with

Jiri

Categories. Adamek.

The

of

Cats,

It is accessible

via:

Joy

http://www.heldermann.de/Ebooks/ebook3.htm Bremen Manhattan

H.H.

2007

2007

G.E.S.

Acknowledgements It is

pleasure

a

make

this Most

instances

to

express

our

gratitude

to

the

many

people who have

helped

book

possible. especially we would

like

of his encouragement, for lectures at Bowdoin College

to

thank

his

Saunders

Mac

for the many and for his stim-

Lane

helpful suggestions, of 1969. We also wish during the summer to express our to Horst Schubert for appreciation allowing us to use the manuscript of his lecture notes on category theory and for his interest in our work on Mac Lane and Schubert have recently categorical reflections. Both Professors written advanced treatises on that we recommend category theory highly. We also owe special thanks to the late Johannes de Groot for his keen interest in our work and for his inspiring discussions. We are who have read grateful to our many colleagues and students of the and have offered valuable portions manuscript many suggestions and criticisms. In this regard we particularly wish to acknowledge W. E. Clark, H. Lenzing, S. J. Maxwell, P. N. Nanzetta, G. Osius, and Ch. Siebeneicher. We are indebted, too, to Bernhard Banaschewski for providing the possibility for us to work together for several weeks at McMaster University in 1968 and ulating

for his interest

in this

book.

for which we are Irving Kaplansky has buoyed us by his encouragement, are also due to our Susan very appreciative. Special thanks Booth, and typist, and Marc for their proof readers, Cheryl Strecker, patient and competent service. Finally, we would like to extend our general thanks to all categorists (both latent and declared) whose results we have used so freely.

I Introduction

How

can

do “new

you

problems with

math"

an

“old math"

mind?

Charlie

of the

Much

abstraction.

beauty Not

only

is derived

mathematics

of

does

to

one

see

the

the

fact

forest

that

rather

it affords

than

the

of the entire

study of the structure forest, abstraction—comparing forests. preparation Consider the development group theory and topology. Many and were studied different individually over long topological spaces groups periods of time before the general and abstract concepts of “group” or “topological space” were defined. In these cases, by properly abstracting the “essence" consideration and by making the proper to objects under of what was common theories in sense more and a beautiful, wider, definitions, new, emerged. Category theory involves the next level of abstraction—Le, comparing forests. and homomorphisms with It allows the comparison of the class of all groups and continuous the class of all topological spaces functions, and further, the and structure-preof sets structured comparison of these with other classes serving functions. for the study of categories. In so Below we will present four main reasons of the important will also be provided for some doing, it is hoped that motivation individual

trees,

but

for the

in

abstract

it offers

it allow

from

Brown?

notions

first

to

the

possibility

next

be encountered

for

stage of of modern

later

in the

book.

studying categories is that, like other mathematical abstractions, category theory provides a new language—a language that afi'ords of thought and expression as well as allowing easier communication economy that a language areas; brings to the surface among investigators in different theorems and the common basic ideas underlying various ostensibly unrelated to which a new context in a that constructions; and, hence, provides language The

i' From

PEANUTS

reason

®

for

by Charles

M.

Schulz

('3 I964

United

Feature

Syndicate,

Inc.

Introduction

2

view

old

powerful, readily be

C Imp. I

and helps to determine are. The need for really after considering the similarities among

problems. classical seen

Thus,

it

results

delineate such the

what

the

deep,

language following statements: a

new

can

product A, x A, of sets A, and A, (together with projection A, A, and 21,: A, x A, —v A,) has the property that if C is any set and f,: C A, are functions, then there exists a A, and f,: C and n, of=f,, that is, unique functionf: C A, x A, such that n, of=f, such that the diagram

I. (a) The cartesian functions 7r,: A, x

->

—.

—#

—.

c

if

A.

;\i

A2 If:

A.xA commutes.

if P is any set together with functions p,: P A, and (b) Furthermore, in as that described universal (a) for property A, having the same p,: P A, x A, A, x A, together with n, and n,, then there exists a bijection g: P that is, such that the diagram such that n, g g p, and 7:, —»

-.

-o

o

=

o

=

p,,P

/:\ '\Ah1XA2! commutes.

(a) The direct product A, x A, of groups A, and A, (together with proA,) has the jection homomorphism n,: A, x A, -v A, and n,: A, x A, and C and C f,: A, are homoA, f,: property that if C is any group A, x A, such morphisms. then there exists a unique homomorphism f: C thatn, of=f,and7r,of=f,. if P is any group together with homomorphisms p, : P A, (b) Furthermore, in described universal as that the same and P property A, having p,: an exists isomorphism (a) for A, x A, together with 1r, and 7r,. then there x p,. n, og p, andn,og g2P—o A, A,suchthat x of 3. (a) The A, and A, A, topological spaces topological product A, x functions (with projection continuous A,) A2 A, and n,: A, x A, 71,: A, and C C is if that has the A, f,: any topological space property and f,: C functions, then there exists a unique continuous A, are continuous and n, of=f,. function]: C A, x A, such that n, of=f, P is if any topological space together with continuous (b) Furthermore, as universal P functions A, having the same property A, and [2,: P p,: 2.

-»

—v

—>

—*

—v

—>

=

=

-»

-

—»

—»

—v

—.

—’

Introduction

that

described

in

homeomorphism One

(a) for P

g:

A

immediately

x

notices

set

function

bijection cartesian product and

is

tempted

the

group

H

H

homomorphism isomorphism direct product

H

H H

and

n. =

n2, then

p1 and

there =

1:; 0g

exists

a

p,.

following correspondences:

H

above

the

replace

to

A2 together with A, such that n, og

x

I

A,

—>

3

columns

topological continuous

homeomorphism topological product

H H

by

space function

a

single “general"

column—

object morphism isomorphism product —which

is what

able

do after

and providing the proper foundations definitions. Thus, an adequate simultaneous description of all three types of of is made course, products (and, many more) possible. various Moreover, we will be able not only to describe products simulwill be but able to about them taneously, prove things simultaneously as well. For example, the “uniqueness“ of the “product" (part (b) of each statement as follows: above) is a categorical theorem, the proof of which goes somewhat Since each of (A, x .42, 1:1, 1:2) and (P, p], p2) is “universal", there exist P such that the diagrams morphisms f: P A, x AZ and g: AI x A; we

are

to

—.

—>

P

P

V V. I

Al

LR Al

V X

and

A2

A]

A2

TR /2 Al A:

1":

X

9

A2

X

commute.

Now

phismsfc

because g and

of

composition identity on

the

is associative, each the diagram

morphisms

A,

of the

mor-

A2 makes

x

AixA':

A,

LVN i Rifle V

.41

[3,

commute

property shown sets,

an

spaces).

a

in part to be the

(fog)

(n, cf)

=

is the

(a). fag

identity isomorphism

og

on

for

P.

A:

p, cg

identity

Thus,

groups,

=

X

g is

and

on

an

a

=

m]. Hence, by the uniqueness A, x A2. Similarly, g of can be

isomorphism (that is, a bijection for for homeomorphism topological

Introduction

4

I

Chap.

“proof” depends upon two properties that we will later of category—the associativity of morphism composition require of an identity morphism associated with every object (see 3.1 and the existence should also be noted that in the definition of the categorical product and 3.8). It of two objects, not only the new object, but the attached projection morphisms In fact, the projections have a dominant role—how must be considered. they with other is the essence of the term are morphisms “product”. The composed of the have an even worse fate than the elements product objects objects they are not considered at all—nor need be. Similarly in general they comprise. They is the morphisms and how they are category theory the main consideration the serve little other than to remind us of the objects purpose composed; domain and range of the morphisms; and elements of objects are not mentioned Note

this

that

in the definition

all.

at

1, 2, and

Statements

3 above

also

can

illustrate

to

serve

the second

study of category theory—the “two for the price of one" or principle (which will be discussed in more detail in §4). Briefly, it is this:

categorical concept, of all

direction sequence,

symmetry

every in the

is

there

morphisms categorical definition

Hence, for categories,

true.

and

a

dual

in the

concept

description

statement

of

or

of the

is true, concept is two

by

original

has

category)

every two

is obtained

theorem

a

dual

concepts;

As

one.

which that

provided

for every reversing the

the

every

a

con-

(because original

theorem,

of is

two

proofs.

every proof, example, the dual of the notion “coproduct”, and the concept of coproduct is “disjoint union”. Thus, we and functions

theorems;

that

reason

“duality”

for the

For

“product”

of

in the

have

is

above

of the category

case

the

discussed

of sets

following:

A, d: A 2 of sets A, and A 2 (together with injection functions Al A2 and 11,:A2 —> A, Q: A2) has the property that 11,: A, if C is any set and f,: A, C and f2: A 2 —> C are functions, then there exists a (b f, andfo 112 f2, that is, unique functionf: A, A; —» C such thatfo p,

4.

(a)

The

union

disjoint

d7

—»

—»

=

that

such

the

=

diagram

[mi/z commutes

.

if P is (b) Furthermore, —» P having the and v2 2A2 for d; (a) A, A2 together with A

z

—»

P such

that

g

o

u,

=

v,

set

any

universal

same

u, and

together

g

o

112

=

functions

property

112, then

and

with

there

v1, that

exists

that

as a

is, such

v,: A, described

bijection that

—»

P

in

A, the diagram g:

Introduction

5

.5 s

3*

5

\

_-----> ‘Q

E

I}

..

3* Ct 3* commutes.

Statement

4(b) is

special

of the dual

of the

uniqueness theorem for the categorical product, Thus, by the duality principle, proved we know that 4(b) is true. If we specialize the notion of coproduct to groups, modules, or topological spaces, we obtain the concepts “free product”, “direct sum”, and “topological sum”, respectively. All of these are likewise essentially unique by the dual of the categorical theorem given above. The third reason for studying categories is that by using categorical techdifiicult areas of mathematics can in certain cases be niques, problems in some translated into easy problems in other areas. for the Brouwer Consider, example, function from the unit fixed-point theorem, which states that every continuous disc into itself has a fixed point. The essential lemma for this theorem—and a result that is difficult to prove in a purely topological setting—is the following: a

which

There circle to

exists

S that

leaves

answering Does

no

the

there

case

above.

was

continuous

function

point of the following question each

exist

continuous

a

from

the

unit

fixed.

circle

in the

function

Proving negative: h: D

—’

disc

D

this lemma

S such

the

onto

is

that

the

category

of

unit

equivalent diagram

.90

(9

\~‘

é’

S .—.—>S

identity commutes? There

exist

suitable

translation

the category spaces shall later call functors

of groups.

of the above

the

to

Does

becomes there

exist

These

translations

(see Chapter V). For a

a

following: group homomorphism

diagram

identity commutes?

from

processes

the are

special

cases

particular functor, g

from

0 into

topological of what

we

the translation 2 such

that

the

C ltap. I

Introduction

6

to (Where 0 is the trivial group and Z is the group of integers.) The answer that the ansxver to the latter question is clearly no, from which we can conclude the lemma can be established the former question is also no. Thus, topological via group theory. reason for the study of the theory of categories is that with it Our fourth notions—such as the concepts of make precise certain hitherto can one vague “universality” and “naturality”. “universal” Consider, for example, the following four similar type

theorems: THEOREM

A

For any

and

X, there exists

set

ftmetion f

a

there exists

a

X

:

a

X (called the free group

group

X such that for

—>

any group

homomorphism g: X

unique group

-+

generated by X)

Y and any function g: X -> Y such that g of = g, that

Y, is,

such that the diagram 1'

X——>X : :

A

5

g~

.u'r Y commutes. B

THEOREM

completely regular Hausdorfl' space X, there exists a compact Hausdorfl' space X (called the Stone-Cecil compactification of X) and a continuous X such that for any compact Hausdorff space Y and any continuous fitnctionf: X Y such that X Y, there exists a unique continuous fimction g : X flotation g: A commutes. g‘ f g, that is, such that the diagram of Theorem For

any

->

—r

—»

=

a

C

THEOREM

X

X, there exists

For any group by its commutator

abelian

group

morphism g : Theorem

group

Y and any homomorphism g: X —> X -* Y such that g o f = g, that

Y, there exists is, such

that

of any

unique homothe diagram of a

D

A and

Let

B be modules

Then

product.

cartesian

B)

a

A commutes.

THEOREM

and

subgroup)

X (which is the factor group homomorphism f z X —> X such that for

abelian

an

and

and

a

bilinear

bilinear

function such that g of To definition

g: =

g,

there

over

a

exists X

function f: X

—>

that

explain exactly of universality.

->

Y there

exists

is, such

that

what

these

Category

ring R and let X denote

commutative an

their

set

R-moduleX (called the tensor product of A X such that for any R-tnodule Y and any a unique linear transformation 9: X a Y the

diagram of

theorems

have

theory allows

in

Theorem common

this (see

A commutes.

requires a precise Chapter VII); in fact,

Introduction

will

all of the

that

above

7

theorems

of one merely special cases very functor theorem. general, yet very powerful, adjoint Making precise the notion of “naturality” was actually the original reason for the definition of a category by Eilenberg and Mac Lane. Consider the following example that motivated their work: we

see

are

theorem—the

Let V be we

form

can

finite dimensional

a

Vof all linear

the set

definition

pointwise

vector

of addition

the real

over

space

functionals

from

of functionals

numbers

V into

R. From

R.

Using multiplication by

and

the

V

usual

constant

V also becomes a vector with the same dimension as V. functionals, space from the theory of finite dimensional vector we know that V and V Thus, spaces, are V since the vector of R-Iinear f unctionals Likewise, is, (that isomorphic. space over V) has the same dimension as Vand V, we know that V and V are isodifference between these two situamorphic. However, there is a fundamental tions.

There

natural

is

V'

V and

F: V’

between

V defined

-»

V

at

V: for

—.

any

g

e

vector

them,

each

x

’.

-»

=

there F ;

o

g

We

is

V is g(x); that

is, (71y(x))(g)

there

is

induced

an

functional the

finite dimensional

way:

if F: V

V'

—v

vector

is any

which

space,

linear

linear

isomorphism V whose

on

then

transformation, F

V fly

the

value

following:

“connected"

are

if

linear

a

transformation

natural

no

that

V' is

->

(1) 11,, is defined without resorting to choosing a basis for V. (2) There is simultaneously defined an entire class of isomorphisms, each

is

Recall

linear

the

Notice

g(x).

=

V

F:

there

define

now

V, let 11,,(x) be that linear

e

by

induced

likewise.

will

natural?

R and

over an

V, but

V and

is meant

spaces

then

by F(g)

F: V

transformation

V. What

V and

finite dimensional

are

between

isomorphism

between

isomorphism

transformation

"V:

“natural”

a

in the

for

one

following

diagram

I

—>

V» C. When we

h

=

g

morphisms f and through

that

say

that

statement

o

f

the

or

that

statement

g exist such that the above B. Similarly, the statement

[1 factors

the square I A——>B h

9

CT“ commutes,

that

means

f

o

g

sometimes

morphisms

may in many

the fact that

k

=

c

It. This be

to

seem

order

of

writing

backwards.

the

composition it

However,

of

from

comes

of the

important examples (e.g., all concrete categories), the law is the law for functions. Notice that because of the composition composition h of composrtion, the notation A L) B i» C D IS unambiguous. assocnatmty .

.

.

.

.

.

.

.

—>

3.2

PROPOSITION

Let

be

g

a

‘6-morphisme: i.e., such

category A

A

—§

’6—objeet.Then there exists exactly one satisfying the properties 3(a) and 3(b) of Definition 3.1; and

e

o

o

f,

=

f

defined, and defined.

g is

e

each

=

is

e

o

9. whenever

=

g

whenever

Proof: Suppose that éce éandby(b),éoe

of

e

and

e is such

é.

e;hence.e=

=

a

morphism.

Then

by (a),

E]

DEFINITION

For

each

satisfying (a) 3.4

a

that

(a) f (b) e

3.3

A be

and

of

A

object

(b) above

a

‘6, the

category

is denoted

by

A ’6-morphism e: A called the ’6-identity of A.T

unique

l A and

is

—»

DEFINITION

A category

(1) small

Z is said

provided

(2) discrete t0ccasionally merely by l.

that

provided (when

the

to

’6‘ is

that

all

domain

bc: a

set;

of its morphisms is

well-known

or

arc

identities;

unimportant)

an

identity

is

denoted

C Imp. l I I

C aregories

18

for

that

(3) connected provided [wind/1, B) sé Q.

each

pair

of

(A, B)

g-objects,

given, it should be pointed out "6’ that a (0, all, dam. cod, c) is usually presented in the form category (0, (Itom(A, 3))“.3)” a 0). Note that if one is given the latter form for ‘6. then the original form can easily be recovered by letting .ll be the disjoint union of the 6? and cod: J/ 0 as follows: morphism sets and by defining dam: all Before

list of

a

examples

of

is

categories

=

,

—.

dom(f)

the

unique object

A such

that

for

the

unique object

B such

that

for

=

cod(f)

=

->

be defined

Actually, categories could

in

some

some

general by

ItomM, B);

B,

f

A,

fe ImmM, B).

e

of

means

classes

object

0,

the families

8))“.5,“

0me,

t

K

composition laws. If this is done, however, the a mormust be required to be pairwise disjoint, for otherwise and a unique codomain. not necessarily have a unique domain (Whether or not a given morphism f would be an identity might then depend and codomain off.) However, upon not f alone, but also upon the chosen domain if such a triple of

morphism morphism sets phism would

and

sets,

the

(0. “10",”, fails

to

then

the

be

category

a

difficulty

can

Enhance

°)

(9

x

only because its morphism sets are not by replacing each easily be overcome

pairwise disjoint, liom(A, B) by

set

set

a

117nm,B) “disjointifying trick"

This well

in

as

will be

some

expected

be

later

apply

applied Since

on.

in several the trick

whenever

it himself,

its

use

of the

examples below

is “standard", is appropriate.

the

as

reader

CATEGORIES

ABSTRACT

OF

EXAMPLES

3.5

should

constructions to

{(A,f, B) [[6 liom(A, 3)}.

=

with any given concrete category naturally associated category ‘6' (61, U. ham): whose class of objects is 6'; whose morphism sets Iinnt,,;(A, B) are the sets liomM, B); and whose composition law is the usual composition of on if no confusion seems From now functionsn‘ likely. we will not distinguish

(l) The =

between

a

(2) The

category

whose

sets:

and

whose

given

*

Notice not

that

category

of sets

morphisms composition

(3) The category are

concrete

of

for many

and

and

whose

relations:

IzomM. B)

sets

the category

are

the

class

naturally of objects

concrete

pairwise disjoint. Thus,

is the

categories the "trick"

usual

(e.g.. Top and mentioned

is the

of all relations

sets

composition of topological bundles, TopBun: whose law

associated

with

class

from

of all

A to

B;

relations. class

of

objects

Grp). the morphism sets should be applied in

above

it.

consists homM.

these

cases.

B)

Sec.

3

Abstract

of all

triples (X,

continuous

pairs of

the

and p’

r

o

p,

map. form s

=

o

B) where The

X and

B

p. For

each

X

r:

p'

.

(4) For all

n

B

B'

—’

continuous

are

associated

continuous.

object Morphisms in

continuous

r:

maps

X

a

of all maps

category

class this X'

a

B is

consists

category for which

r-

whose objects given commutative ring R, the category of R-matrices: and where each set positive integers; Itom(m, n) is the set of morphism a

the

are

s:

space B there is an with base space B, whose

—’

=

and

topological

TopBunB of all topological bundles of all pairs (X, p) where p: X B is from (X, p) to (X', p’) are all those in

—.

X'

_.

19

topological spaces and p: X (X, p, B) to (X ’, p', B’) consist

are

from

morphisms

(r, s) where

C aregoricx

x

with coefficients

matrices

m

in R.

is the usual

Composition

multiplication

of matrices.

(5) A chain Z

integers morphism

complex

of abelian

such

that

for

in

Ab,

and

is

groups

each

Z, G,- is

[6

indexed

family (Gi, (mid

a

by the

0‘1)

abelian

(1]-! is (liqodi complexes of abelian is the object class of a category ‘6’. A fi-morphism f from the groups chain complex (0,, (195:2 to the chain complex (GE, (1})‘52is an indexed family L, such that for each i e isa in Ab and the Z, G‘ f (LL-£2 G; morphism square a

an

0. The

=

group, of all chain

class

=

1.

Ci

‘—>G

i-l

fi-l V r

Gs' 7’0

{-1

l

commutes.

‘6 is

Composition is commonly called

defined the

in

the

category

obvious

of chain

complexes

(6) If ((4. S) is a quasi-ordered class (i.e.. relation S on 9'). then (9, s) gives rise to elements

of g and

such

that

morphism

i.e.,

way:

class

a

a

g

category

(fl)

o

(g)

of abelian

with a

a

=

(f,- gi). c

groups.

reflexive. transitive

whose

B) contains

objects are exactly

the

horrid/l, Conversely, any category ‘6' with the property that each morphism set hound/l, B) contains at most one member can be obtained in this way. By abuse of the language, we also call these classes. Likewise a ‘6 is called a categories quasi-ordered category partiallyordered class (resp. totally-ordered class) if and only if for each pair (A, B) of at most ‘6-objects hound/i, B) u liom.‘(B, A) contains (resp. exactly) one if A S

element

B. and

a

is empty

set

one

otherwise.

member.

(7) "‘0 is any monoid

(i.e., semigroup with identity), then G

be

regarded as a exactly one object, where the morphisms are precisely the members of G, and the composition law is the semigroup composition operation. Conversely, any category having exactly one object can be regarded as a monoid. with

category

(8) If

can

a

category

has

only

a

few

morphisms,

it is sometimes

expressed

in terms

Chap. III

Categories

20

shows

diagram that

of

a

as

arrows.

all of the

objects

dots and the

as

non-identity morphisms

Thus

:3-

.-1

\,/ 2 4=o-———->-1

\2/ l 3 etc.

It is

there with

interesting is

some

to

observe

concrete

that

categories g) given above, category naturally associated categories are called eoncretizable.

for each of the abstract

category

‘2’(35(1)) is “isomorphic"

@ such with

that

‘6. Such

the

Sec. 3

Abstract

(The relationship between

roughly analogous Most

of the

Indeed

concretizable

Definition

Because

of

the

of

metric

categories difficult

exhibit

(see Exercise

A Second

between that

it is somewhat

2]

categories and

concrete

that

to

abstract

Categories

to

categories is metrizable spaces.)

and spaces consider will

will

we

concretizable

example of

an

For any

( l)

is

e

(2) f

(3)

o

e

o

e

=

=

g

is not

Category A

correspondence

(CHARACTERIZATION

morphism

of

e

fi-identity; f iii/tenerer f

a

that

l2L).

one-to-one

PROPOSITION

concretizable.

category

a

IA between

H

‘6-identitymorphisms in any category ‘6, we will be able “object-free" definition of category which is equivalent to 3.6

be

g whenever

o

e

a

g is

o

our

Z-objects and to provide an definition.

earlier

IDENTITIES)

if, the following

category

is

e

OF

below

equivalent:

are

defined; defined.

Proof: By the definition of @—identity, (I) implies (2) and (3). Suppose that of an identity (2) is true. By the definition of category, we know the existence It: cod(e) cod(e). Hence, by (2) and the fact that It is an identity, we have —o

6

It

=

e

o

I1.

é

Thus, (2) implies (1). Similarly (3) implies (l). 3.7

DEFINITION

A

partial operation on a class .II is into .II. a( g, f) is usually denoted by gaf. to the operation provided that for all g of

(i.e.. whenever

a =

eag

that

for

any

defined)

is

of .l/

SECOND

with

is

on

to OF

=

and

J!

respect

DEFINITION

A category

Q5

category

partial operation

a

identities 3.8

gae

function

a

from

all is called

subset

a

of .II

.l/

x

identity with respect .ll, whenever (g, e) is in the domain and whenever is defined, g eag

e 6

a

6

an

=

gee

g.

Notice is

E]

(C, .II, dam, cod, s). the composition the

(by 3.6)

%‘-identities

are

law

precisely

the

c.

CATEGORY

pair (.II, a) where following conditions:

is

.l/

class

and

is

partial operation satifying For all f, g, It :— ./l. iffc g and g It are defined. then (I) Matching Condition: fa (g It) is defined and (fo g) h is defined. For all f. g, I: e ./1,fo (2) Associativity Condition: (g It) is defined if and only if (fog) h is defined. and when they are defined, they are equal. Condition: For every [6 .ll, there exist morphisms ec (3) Identity Existence and e” which are identities with respect to such that ec ofandfo e,, are definedil on

.I/

the

a

a

o

a

a

o

o

=~

o

e

’t Observe

that

(2) and

(3) together

imply that

the

identities

of (3)

are

unique.

o

22

Categories

(4) Smallness

of

Class

Morphism

Chap.

Condition:

For

all identities

cc and

11]

0,, in «II,

the class

{fe is

a

J!

|

9c

and

of

fo

defined}

are

9,,

set.

The

definitions

of

if category are equivalent in the following sense: (0, J1, dam, cod, 0) is a category according to the first definition, then (.11, o) is a category according to the second definition, and if (.11, o) is a category according to the second definition, then there exists an “essentially unique” category according to the first definition whose morphism class is J] and whose two

a

that composition law is (see Exercise 3H). Here “essentially unique” means two the are in the sense of any categories satisfying property (514. isomorphic Because it seems closer to the motivating examples (and thus closer to onc's 0

first definition

the

intuition),

However, since it use

can

of category will be used most sometimes simplify matters, we will

often

in the

reserve

the

sequel. right to

definition

the alternate

when the

(3.8) when it seems appropriate. Also, from now on, is written, it will usually mean that the composition makes or is as well as standing for the g cod(g) defined")

symbol f g (i.e., dom( f ) result of the composition. o

sense

=

“

o

EXERCISES Determine

3A.

which

which

of

the

following

be

can

considered

categories

as

and

cannot:

(a) (c)

(b)

‘

(d)

.

'

.:_—__—s.

(g)

°

g

‘1:

'

'5' Show

38.

that

there

is essentially

8 will be

a

category.

'

-—->

—>

'

.

b...

\./ 0?)

only

6——

V

one

(0

.Q.

(h)

-——>o

way

.

to

define

-(_—>. compositions

so

that

Sec.

4

New

3C.

Determine and

discrete, 3D.

which

which

Prove

of the

categories of 2.2, 3.5. and 3A

‘6 is discrete

category

a

hom(A,

B)

if and

‘3

=

Determine

3F.

Let

statements

‘6

all of the

(0,.//, equivalent:

are

small, which

are

are

categories that a

B.

=

both

are

0b“),

6

B

A

dam, cod. s) be

=

only if for all A. B

# .11If A

MIA} 3E.

23

connected.

are

that

Old

Categories From

connected

and

Prove

that

of

objects is

category.

discrete. the

following

g is small.

(a) (b) (c) (d)

0 is all

set.

a

is

dam

set.

a

is

set.

a

(e) cod is a set. (i) o is a set. 30.

Give

3H. then

according is

a

small

a

whose

category

class

not

finite.

if (0, all, dam. cod, a) is a category according to Definition 3.1. 3.8. and if (.II, o) is a mtegory category according to Definition dam and cod such that (I, -ll, dam, cod. 0) 3.8, then there exist functions J to 3.l denotes the class of all identities with respect according (where

Prove

that

c) is

(.II.

example of

an

to

category

a

o).

to

Let .1!

31.

if and

group

identity g f o

=

be

such

that

for

Show

that

the

e

with

together

set

a

only if it is

a

every

binary operation 0. Prove that (.11, o) is a category (according to Definition 3.8) with exactly one (ti-morphism I. there exists a ‘6—morphism9 such that a

e.

3].

Matching

Condition

(3.80))

Show

if A and

SK.

contains 3L.

that

without B

Construct

two

class. and

h is defined” (fag) changing the definition o

distinct

are

(in the

identity morphisms

no

morphism

“and

words

different

objects ofa

sense

categories

the domain

and

be

can

deleted

from

the

of category.

’6",then llama/1, B)

category

of 3.7).

’6 and

codomain

f/

such

functions

that of ‘6

the

object class, the

are

same

as

the

those

of .9.

§4

NEW

CATEGORIES

FROM

OLD

Subcategories 4.1

DEFINITION

A category

at

is said

the

following

conditions

(I)

0b(.59)

0b(‘6).

(2) M0438)

c

c

(3) The domain, the

to are

be

a

subcategory

of the

category

functions

of J6

‘6

provided

that

satisfied:

Mor(‘6). codomain

and

composition

functions

of '6’.

is

‘K-identity.

corresponding (4) Every :Jd-identity

a

are

restrictions

of

Notice

observe

that

(see Exercise

48).

Also

of

fi-objects

follow

not

the

from

other

conditions‘l‘

DEFINITION

subcategory Q of provided that for all A. B

(1)

pair (A, B)

hom.‘(A, B).

c

(4) does

condition

A

4.3

for each

(2) and (3) imply that

conditions

that

homQM, B)

4.2

C hop. I I I

Categories

24

‘6 is said

a

category

e

0b(.c’8).,l0"1$(A, B)

to =

be

of (6

full

subcategory IzomgM. B). a

EXAMPLES

Each

is

category

a

full

of itself.

subcategory

(2) The category of finite sets is a full subcategory of Set. is (3) The category of sets and injective (resp. surjective, bijective) functions subcategory of Set that is not full. is not a subcategory of Set. (4) The category of sets and relations (5) BooAlg is a subcategory of Lat and Lat is a subcategory of POST’r Neither

a

is full.

subcategory of Grp, Grp is a full subcategory of Mon, and Mon is a subcategory of SGrp, which is not full. (7) Bansz is a subcategory of BanSp,, which is not full, BanSp, is a full subcategory of NLinSp, but NLinSp is not a subcategory of LinTop. (8) None of the categories Grp, Top. pSet, POS, or Lat is a subcategory of Set. is “isomorphic” category (Why not?) [However, it is true that each concretizable

(6)

Ab is

with

some

a

full

subcategory

(l4.2(lO)).]

of Set

Quotient Categories 4.4

DEFINITION

An

called

equivalence

a

relation on

congruence

(1) every equivalence A, B 6 0b“). and

(2) wheneverf compositions 4.5

~

are

on

~

of

the class

morphisms

of

a

’6 is

category

‘6’ provided that:

under

class

1" and 9 meaningful). ~

is

~

contained

that

g' it follows

g

cf

in homM.

B) for

g’ of',

(whenever

~

some

the

PROPOSITION

is a congruence If of morphisms together ~

on

with

a

the

’d, then the class

category

law

composition

8

of equivalence classes defined by: 9

~

.‘75f

(where 57denotes Definition 3.8).

the

equivalence

class

=

of

g

°f

g under

~)

is

a

category

(in the

sense

of

E]

with the corresponding authors do) would identify each object A of a category 1' If we (as some (I) and (4) would obviously be equivalent. identity morphism l A. then conditions definition of he has a different it A caviling person might say that this is false (only because boolean algebras or of lattices).

Sec.

4.6

4

New

Old

75

DEFINITION

If

‘6/~

is

~

a

the

a

on

congruence

is called

above

‘6, then

category

of ‘6 with

quotient category

the

respect

to

and

~,

3) described

(9,

category

is denoted

objects objects

that

every

quotient category of a category ’6 has essentially a quotient does not result when category by an equivalence relation.

the

‘6’; in particular,

as

identified

are

EXAMPLES

dom(g)

=

~

Let

‘6

Top and

=

for

all A, B e to g. Then

only if f is homotopic topological spaces and is denoted and

example homotopy point-preserving category of topological spaces ’6

only if

there

is called

the

by hTop. =

~

~

=

=

that

Recall

0b(’6) andf, g e liomfim, B) letf~ g if ‘6’/~ is called the homotopy category of

that there is a baseg means pTop, and if f between I and y, then %/~ is called the homotopy with base point. g if and Grp and for all A, B e 0bl‘6) andf, g e Itom(A. B) letf is some I) e B such that [(a) bg(a)b" for all a e A. Then %’/~ category of groups and conjugacy classes of homomorphisms.

(3) If in the above

(4) Let

’6’

broadened

we

the

of

concept

categories

from

concrete

some natural constructions applied mainly because that are not (resp. concretizable) categories yield “categories" for quotients. For example. Freyd has concretizable. This is true the quotient category is even hTop is not. though Top concretizable, surprising is the fact (due to Kucera) that every (abstract) category suitable concretizable (isomorphic to) the quotient category of some

abstract

Products

of

it is

Below

concrete

to

necessarily shown

that

Even is

more

actually

category.

seen

such

that

a

the

as

and

pairs of objects objects and morphisms

consider

to

indeed

can

category

pairs

ofa

new

be obtained.

DEFINITION

“(6,. ’62,

.

.

.

,

(6,, are Mar

together

with

the

the

(i.e.. whenever the

is denoted

two

sides

categories,

then

‘6',

’6:

"

n)

~~~~~~

is defined is called

the x

-

product '

-

defined

operation

(gim‘lz

right side are equal)

Mar

x

composition

29---sf;.)

(fr and

to

ones

Categories

happens occasionally that one wishes morphisms from two given categories

category. 4.8

ones

of

of

sense

=

It

same

different

and is defined byzf~g ifand only ifdom(f) (l) lf‘6 is acategory is and cor!( f) a cod(g), then ‘6’l~ quasi-ordered class (in the Example 35(6)).

(2)

by

.

Observe

4.7

From

Categories

=

Mar

x

x

’62

x

’6’"

(ft '.‘li~f2'!12a---~fn‘ 9n)

only then). product category

by ‘6‘.

classes

morphism

by:

(and

the

of the

x

7,".

the of?)

left .,

is defined

side

’62,

.

.

.

.

’6" and

26

Chap. III

Categories

4.9

PROPOSITION

Every product category

3.8).

a

(in the

category

of Definition

sense

E]

Sums

of

4.10

is

of categories

Categories

DEFINITION

ll'

(6,, (62,

.

.

.

‘6’,are

,

the

then

categories,

of the

disjoint unionT

morphism

classes

Mor‘é’l

d9

Mor‘rS’2 ea“-

d:

Mar?”

together with the composition operation defined by: (f, i)o(g,j) is called

the

(6., (6’2,

of

category

sum

ifand

(fog,i)

=

.

.

‘6’,Iand

.,

if

only

i

=j

is denoted

by

‘6’, U‘d’ZLI-"U‘gn. PROPOSITION

4.11

Every

sum

is

of categories

category

a

C]

category.

Opposite Categories 4.12

DEFINITION

For any category ‘6’ = (0, .11, dam, cod, 0), the opposite (or dual) category of ‘6’ is the category we (0, all, cod, darn, it), where t is defined by f t g g o f. (Thus, ‘6 and ‘6" have the same objects and morphisms, but the domain =

and

codomain

sites"

switched

are

The 4.14

the

composition

laws

are

the

“oppo-

other.)

of

opposite category

category

any

is

a

I]

category.

PROPOSITION

For THE

4.15

The

opposite We will

DUALITY

last or

not

a

property P” property T The

@, (‘6”)"'

category

any

dual

’6.

=

C]

PRINCIPLE

allows

proposition

for any explicitly define

to

will

we

give

concerning is, roughly

an

define

for any “categorical concept” a dual statement. “categorical statement"

to

one

and

concept

bother

However,

are.

what

of dual

idea

categorical of

morphisms objects speaking, the corresponding

means

family (Al. A2.

.

.

.

,

A.) of classes

(A,

x

a; An

is the class

{|})U(Az

x

{2})u~-u(A.

of

an

statements

examples.

If P

W, the dual category of W" phrased property a

disjoint union Az‘b-H

and

concepts

by

concepts

and

A10 ofa

and

PROPOSITION

4.13

is

functions

of each

=

x

{n}).

Sec.

4

as

property

a

New

of W ; in other

Categories From

Old

words, the property

27

obtained

from

P

by reversing all

arrows.

Take

for

example

For

any

The

corresponding

For

any

object

For

Y

object

Translating

this

of

following property

’6’ there

exists

0176””there

of

a

exactly

exists

’6 there

exactly

exists

an

X in Q‘I’:

object

‘6’-morpht'smf :

Y

X.

—o

be:

for ‘6‘, we

property

P(X) of

one

for ‘6” would

property

into Y

object

any

Y

the

r6"”’-morphi.mt f:

one

‘6’-morphismf:

one

—>

X.

P°'(X):

obtain

exactly

Y

X

-r

Y.

In the category

if X is

a

Set, for example, the above property P(X) holds if and only singleton set and P""(X) holds if and only if X is the empty set. Quite

often, the dual concept P” of a concept P is denoted by “co-P” (cf. and coconstants and coseparators (§8), separators (§l2), equalizers and and equalizers (§l6), products coproducts (§l8)). A concept

and

P is called

is the

important

If S is

statements.

self-dual

fact a

that

if P one

=

P”. What “dualize”

can

the

concerning

statement

category, by definition, the dual statement holds in ‘6‘”. This, together with the fact that then

the so-called

“duality principle

If S is a categorical for all categories. We will have

Arrow 4.16

and

makes

duality so interesting only concepts, but also morphisms and objects of a S" holds in ‘6 if and only if S (‘6"")"', immediately implies =

to

use

this

then

S"

also holds

principle.

Categories

DEFINITION

If ‘6’ is any category, category whose class of which B

co-

categories":

occasions

Triangle

and

not

which holds for all categories,

statement

numerous

Categories

for

‘6

constants

a

B’

—»

then

objects

’éz-morphismfrom are

the

A

IS

precisely B to A'

—»

’6’-morphismssuch

for %’ (denoted the class of morphisms

category

arrow

that

1—»B’

is

a

Hi [It ,

,

commutes.

Composition

in $3 is defined

(a,

i.c.. by pasting the squares

by:

5) (a, b) ~

together

and

=

(a

e

a,

5

,,

of ‘6 and

pair (a, b) where

the square

1' A——>B

by ‘6") is the

12):

forgetting the middle

arrow.

A

—->

for

A’,

Chap.

C aregories

28

DEFINITION

4.17

by ‘6”)is triangles

then the triangle category for ‘6’ (denoted If ‘6 is any category, the category whose class of objects is precisely the class of commutative from of W and for which a ‘63-m0rphism

——>B’

A'

\/

\/ is

Ill

triple (a, b, c),

ordered

an

‘fi-morphismssuch

each

that

A

where

in the

square

BL

A’,

—)

CL)

B’, and

C’

are

diagram

M\ c\Aj—:->B. commutes.

Composition

in is” is defined

by:

(d. 5. E)°(a.b.c)

=

Bohéoc);

(the,

face.

i.e., by pasting prisms together and forgetting the middle Arrow

important

triangle categories are both concept of “functor categories", and

categories useful

and

special which

cases

will

of the

be defined

in §15. Comma

Categories

DEFINITION

4.18

If g is any

category

(6’)whose

the category

(A,

and

morphisms

g:

whose B

—>

B’ for which

the

and

A

6

objects from

A

0b(%), are

L»

then

the

category

comma

those

g-morphisms

B

A

to

—f'—> B' are

that those

of A

have

over

domain

/B—T’B’XI commutes.

in (A, ‘6) is defined

according

to

the

A,

‘6’-morphisms

triangle A

Composition

‘6 is

composition

in ‘6’.

Sec.

New

4

From

Old

29

DEFINITION

4.19

lf ‘6’ is any the category

A, and B

g:

C alegories

(6?, A) whose

whose

A

0b(‘6),

e

objects

the

then

those

are

B—>A

from

morphisms

for which

3'

—v

and

category

the

which

(6-morphisms B'

to

A

—,

of @

category

comma

have

codomain

‘6’-morphisms

those

are

A is

over

triangle

B—>B’

commutes.

A) is defined according

in (‘6,

Composition

the

to

in ‘6.

composition

EXERCISES 4A.

Prove

(Example ‘6

Q'

or

IS

4B.

(6

Let

homdA, A) bob

hamaM, A) Q is

not

or

.4C. of Q

is

=

a

subcategory

a

{b}, subcategory

“For

any

4D.

Prove

Qoobject B,

.51

then

4F. then

4H. of the

that a

Prove

(in the

that

by:

of

a

a

is

~

of

category

b

by:

subcategory,

o

b

and

A and

b

=

of

set o

a

morphism

b. Determine

=

monoid

morphism aob

a;

the condition

that

with

B is

a

=

set

whether

identity

every

Q-morphism."

subcategory of Q, then Q

quotient category quotient category a ~

(full) subcategory

a

OM? and Q is of Q.

on Q and congruence .91 such that w/ 4: on

category

every

a

‘6’,and ‘6’ is

has

,

be

classes

a

a! ~

is d

quotient category

is

a

a

of 9,

then

quotient category full

a

‘6’.

=

subcategory

subcategory

which

is

a

of

Q is

in the

sense

of Definition

=

3.8.

(RIF)

0

(Inc)

a

of%’, of

Q,

Q/~.

quasi-ordered

a set-indexed family of small categories. Prove that the product 1'I( Mar ‘61),“ together with the composition operation defined

fl[(F° G) a

a

object

one

by: is

ac

=

a

of 35(6)).

“We

morphism

if

as

congruence

sense

Let

if .21 is

regarded

Show

4G. class

that

induces

~

A

is

submonoid

be replaced by:

can

W, and g is

of

subcategory

be

only

IS a

that:

Show can

if Q

object

identity in ‘6 associated

the

(b) if Q is a (full) subcategory (full) subcategory of 9. 4E.

is defined

composition

in the definition

that

object (i. e., if Q

one

of e.

identity of g (4.1(4))

a

exactly

of ‘6 if and

a

where

Show an

(a) if Q is

be

with

category

a

with category exactly one {(1,b}, where composition is defined with exactly Q also be a category

=

b. Let

=

if ‘6 is

that

3 50)» then Q is the empty category.

30

Categories 41.

categories 95’,and (6’2,show

For any

011“, that if Al,

and

B,

and

011%,)

e

«3)

x

11011thng[(A,, A3), (3;, 32)] Show

4].

if

that

is

d,

a

0N)”.

(full) subcategory 4K.

Prove

categoriesd 4L.

x

Let

that

if

hamzlmh 3,)

=

of

x

32). [1011192012,

for each

g,

x

l, 2,

=

.

.

.

then

n,

.

‘6’".

x

d, 3, a! ’, and 3’ are non-empty categories, then the product x 3’ are .2!’ and .98 3'. equal if and only if d

Q and .5” be

($0.6,

i

xdzx-uxfln

‘62

x

then

0b(%2),

e

(full) subcategory

a

at, is

that

0b(‘6".) x 01w.)

=

A2. B;

C Imp. Ill

a

=

family of categories.

that

Prove

=

the

union

disjoint

of the

morphism classes

U (Mar (6",), is!

with

together the

the

Show

4M.

that

operation

composition

same

of Definition

sense

that

as

given in 4.10,

is

in

mtegory

a

3.8.

if

.91, is

a

(full) subcategory ol' ‘6, for each 1.]

#11151;

i

=

l, 2,

.

.

.

,

then

n,

Us!"

subcategory of

isa(full)

‘6’,[1‘62 Lin-11‘6". 4N.

Show

that

for each

40.

Form

the duals

category

of the

@, I!0m:6(A,B)

following

are

self-dual.

(a) (b) (c) (d) (e) (f) (g) (h)

%’ is connected. V is a quasi-ordered class (in the sense ‘6’ is a partially-ordered class. ‘6 is a totally-ordered class. of 35(7)). ‘6’ is a monoid (in the sense if is a group.

fe

Mar

[6

Mar

4?.

@ and there is some 9 ‘6’and for ally, It 6 Mar

Show

that

each

condition

e

statements

of them

of 35(6)).

Mor‘6 ‘6’ such of

hamg.,(13,A). and determinewhich

=

such

that

that/‘0 the

g

second

g

o

andfc

f

is

an

hare

definition

identity.

definedJo of

g

category

=

fo

It.

(3.8) is

self-dual.

4Q. If S is if and

Establish

the

following

consequence

of the

duality principle:

then S holds for all categories a categoriml statement, only if S” holds for all categories satisfying P”.

satisfying

property

P

‘6 can be considered to be a full subcategory 4R. Show that every category of full W2 can be considered to be a of and for A 6 each $1, ‘6”, subcategory 0b“), to be subcategories of (‘32. (A, W) and (‘6’,A) can be considered

Sec. 4

48.

New

Show

Categories

From

Old

that the following pairs of categories of “essential sameness" will be defined more (a) TopBun and the arrow category Topz;

[This sort

31

“essentially the same". precisely later (cl‘. 14.1)]. are

and the comma (b) 'I‘opBun,9 category (Top, B) (for any topological space B); (c) p'I'op and (P, Top) (for any singleton space F); (d) pSet and (P, Set) (for any singleton set P); (e) The category of bi-pointed sets and (A. Set) (for any set A consisting of exactly two elements).

IV

Special Morphisms and Special Objects

of categorical

Perhaps the purpose trivially trivial.

show

is to

algebra

that

which

is trivial

is

————P. FREYDt

In the

roles; these

certain

of sets,

category

special types

of functions

play distinguished

are:

identity functions

injective functions surjective functions bijective functions functions

constant

the identity morphisms, which in arbitrary Chapter III, we encountered in Set. For the categories are the obvious analogue of the identity functions of in of elements are terms which of functions. other classes usually defined In

domains

their

and

codomains.

will

we

now

furnish

“element-free“

characteriza-

investigate the corresponding categorical concepts. distinguished categorical analogues for some Set; namely, the empty set and the singleton sets.

we

will

and

tions

find suitable

will

§5

SECTIONS,

RETRACI'IONS,

AND

Likewise,

objects in

ISOMORPHISMS

Sections 5.1

MOTIVATING

If f following 1' From

:

A are

——>

PROPOSITION

B is

a

function front

a

non-empty

equivalent:

Proceedings of the Conference

mt

Categorical 32

Algebra

set

A

to

the

set

8. then

the

Sec.

5

Sections.

and

Retractions,

33

lsomorphisms

(l) f is injective (i.e., one-to-one). (2) There

exists

A such function g: B to function composition).

with respect

inverse"

of

1A (i.e., f has

=

“left

a

[:1

DEFINITION

5.2

A

A

morphism

L

‘6-section) provided

of

g

that g

—¢

some

that

1A (i.e.,fhas

=

B in

exists

there

be

to

in ‘6 (or

section

a

‘oo-morphism3-9—,A

some

“left inverse“

a

‘6 is said

category

a

such

a

that

in %).

EXAMPLES

5.3

(l) A morphism in Set is from

function

empty

(2) A morphism is

-f-

A

the

is

non-empty

a

section

of direct

embeddings Y in

Top is

if and

words,

is not

the

injective

and

and

injective only

if it is

in R-Mod

sections

the

(up

are

summands.

section

embeddings of

the

is

set.

if and

of Y.i‘ In other

retract

a

a

only if it

a

of B. In other

homeomorphism)

to

to

B in R-Mod

(3) A morphism X L embedding and f [X] is (up

set

empty

summand

direct

f [A] to isomorphism) a

the

if and

section

a

only

if

is

f

topological in Top are

a

words, the sections

retracts.

(resp. topological spaces) and if a 6 Y then the function X x Y defined by f (x) (x, a) is a section in Set (resp. Top). (What f: X is its “natural” left inverse?) [Note that the image off is a “slice" or “crossof the product, which is in one-to-one section" correspondence (resp. homeo(4) If X and

Y

sets

are

—.

=

morphic)

motivates

X. This

to

our

use

in Definition

“section“

of the word

5.2.]

of the following situation: in (4) are just special cases (5) The sections described the Let f: X Y be a morphism in Set (resp. Top, Grp, R-Mod). Consider of the as a subset product (resp. subspace, subgroup, submodule) graph off X x Y. Then the embedding of X into X x Y defined by x l-> (x,f(x)) is a in question. section in the category _.

5.4

PROPOSITION

A

If

L»

B and

B

L)

C

sections

are

in

a

'6, then

category

A

u)

C is

a

section.

k

c

Proof: Let B L» g 18. Then

A and

—"—> B be morphisms

that

such

h

of

IA

=

and

=

(hok)c(gof) Thus 5.5

C

(g 0f) has

a

left

=

hcx(kc‘(l):.f=

inverse.

he

lflof=

hof

=1A'

E]

PROPOSITION

If f

and

g

’6' and

is

a

section,

of Y if and Y is called .1 retract space A that leaves each point of A fixed. Such a

function,

morphisms of

are

a

category

g

c

f

then

f

is

a

section.

TA subspace A of function continuous topological retraction.

topological

a

r:

Y

—>

only if there r,

is some is called

a

and

Special Morphisms

34

Special Objects

Chap.

I V

Proof: By the definition of section we know that there is a morphism h such that h (g of) is an identity. Thus, by the associativity of composition, [:1 (h g) f is an identity; hence, f is a section. o

o

o

Reflections PROPOSITION

MOTIVATING

5.6

If f is

(1) f

:

A

—»

B is

function,

a

then

the

following

equivalent:

are

surjective.

(2) There inverse"

A such that fog function g: B with respect to function composition). is

—r

some

Proof: By

the

of Choice

Axiom

there

is

a

l, (i.e., f has

=

that

function

"right

a

assigns

to

each

'

b

B

e

some

of the set

member

1[{b}]. C]

f

DEFINITION

5.7

in B is said to be a retraction (6-morphism A —’—» ‘6’-morphism 83—» A provided that there exists some in ‘6). (i.e., f has a “right inverse"

Q

A

(or

‘K-retractiou)

a

that

such

f

og

1,,

=

EXAMPLES

5.8

A

(l)

in Set is

morphism

a

only if it is surjective.

if and

retraction

if and only if there exists a is a retraction B in R-Mod (2) A morphism f: A S of A and an isomorphism h: S —o B projection’rp of A onto a submodule in R-Mod are such that f (up to hop. In other words, the retractions isomorphism) exactly the projections of modules onto their direct summandsxl’f if and only if there is a continuous (3) A morphism f in Top is a retraction h such that f h r. In other words, the r and a homeomorphism retraction in Top are (up to homeomorphism) exactly the topological retractions. retractions —>

=

o

=

[This motivates Section

and retraction

Proof:

Let

SW) be the

and

there

Mar“)

6

566°") is the

Then

f

in Definition

5.7.]

exists

Mor(‘6°') and there

e

This

is

f If S is A

a

precisely a submodule S that leaves

dual nations.

are

statement: some

9

Mor(‘6) such that

e

g °g

f

is

K-identit

a

y.

statement:

Translating this into f e Mor(‘€) and there

p:

“retraction”

of the word

use

PROPOSITION

5.9

f

our

exists a

some

statement

exists

the statement

some

Morfif”) such that

e

g

about g

that

9?,

we

f

is

0.5 g is

a

i6’-identity.

a

obtain:

e

Mor(‘€) such that f

f

is

a

’6°?idcntity.

g 0.5...

retraction

in ‘6.

S is a of A, then a projection of A onto each point of S fixed. Such a homomorphism

E] surjective exists

homomorphism

if and only if S is

of A. summand Bis a “retract" of A if and only if it is a direct summand H Thus. an R-module of A (53(2)). have seen that Bis a “sect” of A it‘ and only ifit is a direct summand an in R-Mod object B is a “retract" of the object A if and only if it is a “sect" is true for every category. (Why?) sponding statement

direct

of A. Also

we

Consequently, of A. A

corre-

a

Sec. 5

Sections.

5.10

L»

A

B and B

Proof: If f and ‘6’°'-sections

i.e.,

f

g 0.3

From

now

while

5.1]

‘K-retraclion.

a

‘6-retractions, then according to Proposition 5.9 they their composition f ago, 9 in ‘6’” is a (WP-section (5.4); (fl-retraction. C]

a

will

‘6-retractions, then A air»C is

are

always indicate the proofs of dual propositions or an application of the duality principle (4.15). After sometimes not even for theorems, provide the dual statements as an exercise for the reader. implied

on

we

they

will

this

leaving

C

thus g is

or“,

since we

L are

g

and

f

=

theorems, a

35

lsomorphisms

PROPOSITION

If are

and

Rerracrions.

task

not

all

are

PROPOSITION

and g

If f

‘g-morphisms and

are

Dualize

Proof:

g

5.5.

Proposition

is

f

o

a

then 9 is

retraction,

a

retraction.

I]

Isomorphisms 5.12

MOTIVATING

If f (1) f

is

:

PROPOSITION

B is

then

function,

a

thefollowing

are

equivalent:

bijective.

(2) There 5.13

A

—>

existssome

function

g:

B

—’

A such that g

of

l Aand f

=

o

g

=

l5.

E]

DEFINITION

‘6-morphism is said to be an isomorphism in provided that it is both a “(f-section and a ’6’-retraction inverse" and a “right inverse” in ‘6’).

‘6

A

5.14

(or a ‘6’-isomorphism) (i.e., it has both a “left

EXAMPLES

(I)

In any

(2)

A

category,

morphism

identity is

every

in Set

is

an

isomorphism.

isomorphism

an

if and

only

if it is

if and

only if it

bijective.

(3)

A

(4)

A

morphism

(5)

A

morphism in BanSp, is an isomorphism ifand only ifit is a homeomorphic isomorphism, and a morphism in Bansz is an isomorphism if and only an isometric linear isomorphism.

morphism in Grp isomorphism.

linear if it is

(6) A monoid of its 5.15

(5.9).

is

in

a

Top

group

morphisms is

an

is

is

isomorphism

an

an

isomorphism

if and

only

if and

only if. considered isomorphism.

as

if it is

a

is

a

a

homeomorphism

category

groupatheoretic

(15(7)), each

PROPOSITION

lsomorphism

is

Proof: The

notions

['1

a

self-dual of

notion.

section

and

retraction

are

duals

of each

other

Special Morphisms and Special Objects

36

5.16

PROPOSITION

In any

the composition

category,

Proof: Immediate under composition (5.4 5.”

Chap. IV

fact

from

the

and

5.10).

is

of isomorphism sections

that

isomorphism.

an

and

retractions

closed

are

[:I

PROPOSITION

is

If f (l) f

is

a

‘6’-morphism,then the following

a

equivalent:

are

W-isomorphism.

(2) f has exactly

right inverse, h,

one

and

exactly

Proof: Clearly (2) implies (I). To show that f has some right inverse h and k. Clearly show that h

definition

that

(1) implies (2),

we

k.

=

know

k. We need

left inverse

some

and h

left inverse, k,

one

by only

=

=ko(foh)=(k°f)0h=

k=kol of the above

proposition, denoted byf". usually

Because

f.

It is

5.18

is

then

isomorphism,

an

speak

of

of the inverse

isomorphism

an

f

"

')’ 1. E]

'

is

an

isomorphism and f

(f

=

DEFINITION

An

A of

object

‘6’ provided 5.20

may

E]

PROPOSITION

If f 5.19

we

loh=h.

a

category

there

that

exists

‘6’-isomorpl1iewith B. Qf-isomorphism f : A

is said to be

g

B of

object

an

—.

some

PROPOSITION

For

‘6’, “is isomorphic with"

category

any

yields

equivalence

an

relation

on

Ohm). since Reflexivity holds an from the fact that iffis follows transitivity holds since isomorphisms

identities

Proof:

5.2]

under

is

also, and

one

E]

composition.

DEFINITION

Let

93 be

(I) .9 is said there

closed

f"

then

isomorphism, are

Symmetry

isomorphisms.

are

is

subcategory of

a

be

to

fi-object

some

(2) 9? is said

be

to

every

‘6~objectthat

5.22

EXAMPLES

them

that

of %’

B is

provided @isomorphic

with

isomorphism-closed subcategory {Iii-object is isomorphic with some

an

numbers

and

for

that

functions

each

%-object C.

C.

of itself

between

’6 a

'1

provided -object.

them

is

a

that

dense

of Set. of all

(2) The category between

is

subcategory

B such

of all cardinal

(l) The category

subcategory

dense

a

’6.

is

a

dense

subgroups

of

permutation subcategory of Grp.

groups

and

homomorphisms

Sec.

5

Set-lions,

(3) If Fis

field. then

a

(4) If F is

field. then

a

subcategory of (5) If

X is

Isomarphisms

full

the

37

all finite powers

subcategory of

of all finite dimensional

F" ofF

vector

'

F

is

a

dense

F.

over

spaces

of all powers

subcategory

of F is

dense

a

F-Mod.

topological space with three points subcategory of all subspaces of powers

and

a

then the full of

the full

of the category

subcategory

and

Retractions.

X

’

three

exactly of X is

a

dense

sets,

open

subcategory

Top.

(6) BanSp2 is both

dense

a

(7) A full subcategory in ‘6 ifand only ifd?

and

.3 of

a

subcategory of BanSpl.

isomorphism-closed

an

‘6 is both

category

dense

and

isomorphism-closed

’6.

=

EXERCISES Show

SA.

g"

c

f"

iffand

(i.e.. when

SB.

Show

have

several

may

that

5C.

Let

elements.

either

that

general

right

inverses.

a

morphism

fbe

Show

that

is defined, then

side

in

isomorphisms

are

g

section

a

may

have

in the

category

are

equivalent:

the following

so

‘6. then category is the other and they in

a

left inverses

several

of all

which

sets

(fa g)" are equal).

and

have

a

=

retraction

least

at

two

(a) fis an isomorphism. (b) f has exactly one right inverse. (c) f has exactly one left inverse. Do

these

least

two

members?

SD.

Let/and

section

and

SE.

‘6-morphisms. Show that but not conversely. retraction.

(35(7)), category only isomorphism. 5F.

if the monoid

that

a

Let :69 be

then

a

of natural

is the

zero

of all

category

g be

g is a

Show

in the

hold

equivalence:

same

ifg

fis

an

under

numbers

only section,

c

topological

the

with

spaces

isomorphism,

thenfis

is considered

addition

only retraction.

at

and

thus

a

as

the

of ’6.

subcategory

that fi-section is a %’-section any (resp. .fi-retraction, fi-isomorphism) W—retraction, (resp. g-isomorphism). (b) If 38 is a full subcategory of ‘6. show that every .‘VZ-morphismthat is a (6-section is necessarily a fi-section (resp. ‘6-retraction. (ti-isomorphism) (resp. £~retraction. fl-isomorphism). that 39 is full, (b) above that without the requirement is false. (c) Show

(a) Prove

50.

Prove

‘6-retraction.

that

if

‘3

is

a

quotient then

Z-isomorphism).

category

the

for ‘6 and

equivalence

class

iffis

f

is

a a

’6‘-section

(resp.

’Z-section (resp.

‘Z-retraction.‘z-isomorphism). 5H.

only the

if fr

A g

(if-morphism cf f. Prove

quasi-inverse

=

of

some

g

is said that

to

every

‘6-morphism.

be

a quasi-inverse ’6-morphism that

for has

the a

(if-morphism] quasi-inverse

if and is itself

Chap.

Special Morphisms and Special Objects

38

§6

AND

EPIMORPHISMS,

MONOMORPHISMS,

1V

BIMORPHISMS

Monomorphisms PROPOSITION

MOTIVATING

6.1

If f is

(1) f

:

A

B is

-r

a

function

then

sets,

on

the

following

equivalent:

are

injective.

f k, it follows that h (2) For all functions It and k such that f h f is “lcft-cancellablc"with respect to function composition). o

Proof: Clearly (1) implies (2). If f satisfies (2) and from a singleton set into A, f ((2), consider functions of which has image {b}. [:1 {a} and the other

a, b e A such

that

f (a) has image

of which

one

k (i.e.,

=

c

=

=

DEFINITION

6.2

@morphism morphism) provided follows

h

that

that

for all

said

be

in ‘6 (or f o h

monomorphism %’-morphismsh and k such to

a

(i.e., f is “left-cancellable“

k

=

—I—> B is

A

A

with

that

to

respect

‘6’-mono-

a

f composition

k, it in ‘6).

=

o

function

on

EXAMPLES

6.3

that

category (1) Every morphism in a concrete underlying sets is a monomorphism.

is

injective

an

the

(2) ln Set, Grp, SGrp, Ab, R—Mod, Rug, POS, Top, Top,, CompTz, LinTop, BanSpl, and Bansz, the monomorphisms are precisely the morphisms which are injective on the underlying sets. Notice that in Set, Grp, SGrp, Ab, R-Mod, Rng. Comp'l‘z, and BanSp,, the monomorphisms are ”essentially" the emthere but in POS, Top, Top;, LinTop, and Bansz, beddings of substructures, are monomorphisms which are not embeddings. A satisfactory categorical later (see 340). concept for “embeddings" will be discussed there

of divisible

s!

In the category

(3)

abelian

which

groups not

monomorphisms Q/Z, [Consider the natural quotient Q and Z (= the integers) are each considered are

are

—>

(4) In the category preserving functions.

underlying base point as

the

sets.

0 and

space

morphism function.

of

t See

E. H.

connected

pointed

there

are

group the

injective where

on

Q (= the

abelian

as

spaces

monomorphisms

under

groups

and that

homomorphisms, underlying sets. rational numbers)

addition]

continuous are

base-pointinjective on the

not

with [Consider the pointed space (R, 0) of the real numbers the pointed space (X, l) where X is the circle 5' represented

of all

complex

with

numbers

modulus

I. Then

x

H

e" defines

a

(X, I) that is a ‘6-monomorphism but not an injective a covering projection and the “unique lifting property" each that covering projections‘l‘ is equivalent to the statement

p: (R, 0) Notice that

covering projection in

’6’ of

and

‘6 is

Spanier,

a

—>

p is

’6’~monomorphism.]

Algebraic

flipology.

New

York:

McGraw-Hill,

I966,

p. 67.

Sec. 6

(5) There

L

X

is

Y of

f is

not

disc

into

the

In the

(6) (in the

the

the

class

homotopy

h'l‘op (the homotopy category embedding of the bounding circle

of

in

monomorphism

a

that

such

Top

usual

of

Field,’r and

of

3.5(6)),

in any

is

morphism

every

which

category

is

quasi-ordered

a

class

monomorphism.

a

PROPOSITION

6.4

If A L) morphism. is

Y in

39

disc.]

category

sense

L

X

monomorphism

a

topological spaces). [Consider a

Epimarphisms. and Bimorphisms

Monomorphisms,

B and

L» C

B

Proof: If (gof)oh monomorphism/o

a

then

A

g»

C is

then (gof)ok, go(foh) go(fok). a and sincefis fo k, monomorphism h

‘6-mono-

a

Since

=

=

h

g-monomorphisms,

are

=

k.

=

g

E]

PROPOSITION

6.5

If f and g monomorphism.

‘é-morphisms

are

=oh=

Pr00f: for: k. [I

6.6

PROPOSITION

=rok

Every ‘g-secrion

and

Proof: If gisaleft

then

monomorphism,

a

=g°tf°k)

=~gou-m

is also

is

of

g

=(gsflsll

=

f

is

a

(gnnek

‘6-monomorphism.

a

for f, then

inverse

foh=fok=go(foh)=go(fok) =(go/>oh=

(go/)ok lsk=h=k.

=lch= The

of the

converse

Top, the but not

embedding

does

proposition interval

open

into

hold

not

closed

a

since, for example, in

interval

is

a

monomorphism

section.

a

In any

(1) f

is

an

(2) f

is

a

category,

An

6.6 it is both

a

following and

isomorphism

fog=

and I

a

retraction.

is

a

retraction

=(fog)°f=

section

and

and

that

and

let g be

lof=f° o

in each

field 0 #

l.

a

retraction.

D

right

a

I 1.

=

that

by Proposition Thus, (I) implies (2). Let f be a

retraction.

f is a monomorphism, g f isomorphism. Thus, (2) implies (l).

1 Recall

since

equivalent:

are

a

monomorphism

monomorphism

that

the

isomorphism. monomorphism

Proof:

an

an

PROPOSITION

6.7

so

above of

E]

so

inverse

=fc(g°f)

Hence, f is

off.

=f°l a

section,

and

therefore

40

Special Morphisms

and

Special Objects

Chap. I V

Epimorphisms 6.8

MOTIVATING

If f f

( l)

:

is

A

PROPOSITION

B is

a

function

a

on

following

are

equivalent:

surjective.

(2) ftmctt'ons h and k such f is “right-cancellable"with respect For all

that to

If

Proof: Clearly (1) implies (2). {1, 2} by:

h, k: B

then the

sets,

h

k of; it follows that of function composition). A

f:

h

=

Bis not

—>

surjective,

=

k (i.e.,

define functions

-r

"[3]

I‘ll/[4]]

{l}.

=

{1}.

=

and

W

6.9

buth

kc],

Then,hof=

9': k.

m1]

—

=

{2}.

[:l

DEFINITION

‘6’-morphism A L) B is said to be an epimorphism in ‘6 (or a ‘E-epik of, it morphism) provided that for all ‘6’-morphismsh and k such that h f follows that h k (i.e., f is “right~cancellable” with respect to the composition in fi). A

o

=

=

EXAMPLES

6.10

(I) Every morphism in a concrete category which is a surjective function on the sets is an underlying epimorphism. (2) In Set, Grp, Ab, R-Mod, POS, Top, and CompTz, the epimorphisms are precisely the morphisms which are surjective on the underlying sets. [The proof B is an epimorphism in Ab for Grp is not immediate (see Exercise 6H); if A -—{—> or R-Mod, let [1, k: B Blf [A] be the induced quotient map and the zero map, —r

respectively.] (3) X

There

.L,

Y of

is

an

f

is

X

epimorphism not

of the real

line onto

In

the

an

L»

Y in

Top

such

that

the

homotopy class

in

hTop. [Consider the covering projection by: x H e".] are functions with precisely the continuous functions A B for which the closure of f:

epimorphism

the circle, defined

epimorphisms images, i.e., the continuous f [A] equals B. [If A 1—»B is an epimorphism, let C be the disjoint topological of union of two “copies" of B where the corresponding points of the closure and let h and k be the two natural from B have been identified, maps f [A] the epimorphisms are and Bansz to C.] Likewise, in BanSp. precisely the dense with images. morphisms abelian a morphism A i) B is an groups, (5) In the category of torsion-free (4)

Top2

dense

—>

if the

epimorphism

if and

this category,

epimorphisms

only

factor

need

group B/f[A] is not be surjective.

a

torsion

group.

Thus, in

Sec.

6

Monomorphisms. Epimorphisms.

(6) In Rng and the

Sgp

and

there the

rationals, integers Q morphism in Rug and in SGrp. [If lief: kofandifn/meQ,then

h(n/m)

then

h(n)'h(l/m)-h(l)

=

that

epimorphisms

are

k(n) h(l/m) -

'

k(n)‘h(l)-k(|/m)

usual

It and

k

Bimorpliisms

41

surjective; e.g., if Z is embedding Z 1-) Q is an epiare homomorphisms such that not

are

k(n)-h(l/m)'k(l)

=

k(m)

-

the

and

k(l/m)

=

k(n) h(l/m) -

=

k(n)-k(l)‘k(l/m)

=

h(m) k(l/m)

-

-

k(n/m).]

of finite

(7) In the category

scmigroups, there are epimorphisms that are surjective. [Consider semigroups A {0, a“, a”, a“, an} and A defined by: {an}, each with binary operation the

=

'”"—

P“

an

this

A and

operation, epimorphism (Howie

6.11

B

and

and

566°") interpreted

Proof: Dualize o

g

is

f

and

semigroups

the inclusion

B

—+

A is

1967).] dual notions.

are

all

11, k

e

all II, k

e

=

fa

k

II

=

=

k.

’6’ is:

about

statement

a

Mor(’6), fo [1

Mor(’6), I1 nf is

of g-epiniorphisms 6.4.

[1

a

‘6-epimorphism,then

g is

Dualize

Proof:

=

k

of:

It

=

k.

E]

Proposition

r6’-epimorphism.

a

Proposition

6.5.

a

‘é-cpimorpliirm.

[1

PROPOSITION

Every

'6-reiraclion

Proof: Dualize

is

a

(ti-epimorpliism.

Proposition

6.6.

[3

PROPOSITION

In any

( l) f is

an

(2) f

an

is

the

category,

following

isomorphism. epimorpliism and

Proof: Dualize thus

(1%":

PROPOSITION

If

Even

a”

ifq=m

PROPOSITION

The composition

6.15

if

statement:

as

fe Mor(’€), andfor

6.14

lsbell,

Mor(‘6), and for

fe

6.13

0

finite

are

epimorphism

S(‘6’) be the

Let

Proof:

6.12

=

PROPOSITION

Monomorplu'sm

Then

B

-

—

With

not

though always

the

a

notions

of

equivalent:

section.

Proposition

be handled

are

6.7.

[3

cpimorphism symmetrically. in

and

monomorphism are categories

well-known

dual their

and

can

behavior

42

and

Special Morphisms

to be far from symmetric. appears categories that we have considered, the

often

Special Objects instance

For

Chap.

in most

I V

of the concrete

monomorphisms are precisely those morphisms monomorphisms (i.e., injective functions) on the underlying sets. However, it is quite usual for epimorphisms in concrete categories not to be epimorphisms (i.e., surjective functions) on the underlying sets (e.g., in SGrp, Mon. Rug, Topz, BanSpl. and Bansz). Actually, there is a good reason for this, which will be explained later (see §30 and Proposition 24.5). that

are

Blmorpbisms DEFINITION

6.16

‘6-morphism is said to be a monomorphism

A

that

a

it is both

bimorphism in W (or a %’-bimorphism)provided an epimorphism.

and

EXAMPLES

6.]7

(I) For every category ‘6’,each ’6‘-isomorphismis a ’6’-bimorphism. (2) For the categories Set, Grp, Ab, R-Mod, POS, and Top, the bimorphisms

precisely those morphisms that are bijective on the underlying sets. Note that in Top and POS they need not be isomorphisms. as a category (3) In each quasi-ordered class considered (35(6)), every morphism is a bimorphism.

are

(4) A monoid

is cancellative

each

of its

(“8

DEFINITION

morphisms

A category an

is said

is

to

if and

only if, bimorphism.

a

be balanced

considered

provided

that

as

each

a

category

of its

(35(7)),

bimorphisms

is

isomorphism.

6.l9

EXAMPLES

Comp'l‘z are balanced. (2) Rng, Sgp, 'I‘opz, Top, LinTop, and POS are not three, epimorphisms need not be surjective functions; morphisms need not be embeddings.) (I) Set, Grp, Ab, R-Mod,

(3) A partially ordered

and

class

considered

as

a

category

balanced. for the

last

is balanced

(For the first four, monoif and

only if

it is discrete. 6.20

PROPOSITION

The

composition

Proof: (6.4 and 6.2]

Monomorphisms 6J2). [:1

and

a

(if-bimorphism.

epimorphisms

are

closed

under

composition

PROPOSITION

If but

of ‘6-bimorpltismsis

not

g

‘6.biniorp/Iism, (lien f conversely. [:1 o

f

is

a

is

a

monomorphism and

g is

an

epimorphism,

Sec.

6

Subobjects and Quotient 6.12

Epimorphisms, and Bimarphismx

Monamarphisms.

43

Objects

DEFINITION

A

subobjcct or monomorphism. lff called

a

object Be 012m also happens to be

L

B is

pair (A, n

where

A

then

(A,f)

is sometimes

is

a

a

section,

a

of B.

sect

DUAL

an

NOTION:

quotient object; retract. [l.e., (f. A)T _I_, A is an epimorphism, and (f. A) is

provided that B thatfis a retraction]

is a

quotient object of B retract of B provided a

DEFINITION

6.23

(I) If (A, f) and (C, g) are subobjccts of B, then (A,f) is said than (C, g)—denoted by (A, f) s (C, g)——ifand only if there A

morphism

—"—> C such

that

the

be smaller

to

exists

some

triangle A

l I

h

l

i v

C

X

B

/

commutes.

(2) If (A, f) S (C, g) and (C, 9) s (A,f), then (A, f) and to be isomorphic subobjccts of B; denoted by (A, f) z (C, 9).

such

said

quotient object (f, A) is larger than the quotient object by (f, A) 2 (g, C)—ifand only if there exists some morphism

(g, C)-denoted

1—»C

are

(l)* The

DUALLY:

A

(C. 9)

that

the

triangle A

B

V X

: I

:

¢ C

commutes.

(f, A) and (g, C) are isomorphic quotient objects—denoted by (f, A) z (9. C)—ifand only if(j§ A) 2 (g, C) and (g. C) 2 (f, A). Notice that even formally take a subobjcct ofa subobjcct though one cannot (since a subobjcct is a pair rather than an object), it is clear that if (8, f) is a subobjcct of A and (C, y) is a subobjcct of B, then (C,fe g) is a subobjcct of A. of a subobject is a subobject. a subobject Hence, in this sense

(2)‘

t For

quotient

aid in recalling

objects. that

we

write the codomain

A is the

pair as (f. A) rather than (A. f) as a mnemonic off. off rather than the domain

device

to

Special Morphisms and Special Objects

44

C Imp. I V

PROPOSITION

6.24

Subobjects (A, f) and (C, g) of .

.

there

only If

exrsts

a

.

that

are

and

(A, f)

B

isomorphic subobjects of -——> C such that g h f.

If

and

In

Isomorphtsm

unique

Proof: Suppose

B .

A

=

o

(C. g)

isomorphic

are

o

Since

subobjects.

(C, g), there is a morphism h such that g h f. Sincefis (A,f) morphism, h must be also (6.5). (C. y) s (A, f) implies that there is a k k such thatfc g. Now s

a

=

mono-

morphism

=

go(hok)=(go/1)ok=fok=g=golc, and a monomorphism, h k lc. Hence, h is a retraction is an it monomorphism, isomorphism (6.7). Uniqueness of [1 follows from the ll fact that g IS a monomorphism. Conversely, if A —» C IS an Isomorphism such that g h f, then clearly (A.f) S (C, y). Similarly,fo h" 9 shows that (C, g) s (A, f). Thus, the subobjects are isomorphic. [3

since

Thus

g is

a

o

=

so

.

.

a

.

.

.

.

=

=

COROLLARY

6.25

is

a:

on

the

class

of

that

the

class

of all

relation

equivalence

an

all

subobjects of

‘6-objcct

any

B.l___| of 6.25,

Because

know

subobjects of an object B partitioned equivalence isomorphic subobjects. Thus, via the Axiom of Choice (l.2(4)) for every ‘K-object B there exists a system of reprez sentatives for the equivalence relation on the class of all subobjects of B. Such a system of representatives will be called a representative class of subobjects

6.26

is

we

into

classes

of

of B. DEFINITION

6.27

‘6 is said

A category

class

representative

to

co-(wcll-powcred). [l.c.. quotient objects which is a set.] MOTION:

DUAL

class

of

6.28

EXAMPLES

(I) The categories and

be

well-powered provided that each subobjects that is a set.

of

object

every

Set, Grp, Top, Topz, BanSp,.

and

Bansz

has

a

representative

are

well-powered

a

co-(well-powcrcd).

class of all ordinal numbers (2) The partially-ordered (35(6)) is well-powered but not co-(well-powered). Notice for

has

(cf-object

that

each

to

say

that

’6-object B,

‘f-objects so

because

of

objects.

such

that

in any

a

category

there

for

each

category

can

be

ithere

‘6’,there

considered

as

is

a

category

well-powered is equivalent to saying that only a set (X,), of pairwise non-isomorphic is some B. This is monomorphismfiz X, is only a set of morphisms between any pair —»

Sec.

6

aml

Monomorp/tisms. Epimorpltisms,

45

Bimarplu'sms

EXERCISES

6A. is =

in the category of commutative cancellative if and only if for all I) e B there exist a., a2

that

Show

epimorphism [(az). [Hi/II:

an

There

group.

68.

the

commutative

each

epimorphisms

For

any

i:

U

be defined

be defined

Embed

Z—morphism A L) B,

by:f(g)

X

f0

g;

and

[: U {hom(B. by:[(g) g

X)

|

=

Xe

—.

DMZ);

if and

is

an

—.

U

b

abelian

out?»

{IIom(X. B) I Xe

U

0b(‘6’)}

{Itom(A, X) | Xe

only if f is

injective

injective function

an

and

that

f is

a

function.

on.

(full) subcategory

a

+

f(a,) an

°f-

=

Let Q be

in

let

Ohm:

e

that

6C.

semigroup

that

B

let

f is a Z-monomorphism Z—epimorphism if and only if f Prove

A such

6

L»

surjections.]

are

{hom(X. A) |

cancellative

A

semigroups,

fiebimorphism) is not (resp. fi-epimorphism, Q-mommorphism necessarily a ‘g-monomorphism (resp. Z’epimorphism. Z-bimorphism). that is a ftf-monomorphism (resp. ‘g-epimorphism, (b) Prove that every 3-morphism 93-bia is 9-monomorphism (resp. Q-epimorphism, necessarily Z’-bimorphism) morphism). (c) Compare these facts with those of Exercise 5F. (a) Show

that

a

‘3 be a quotient category 0N6 Z’-monomorphism (resp. Z’epimorphism, Z-bimorphism) Z"-bimorphism)? ‘é-monomorphism (resp Z’- epimorph_ism, 6D.

Let

(a) Iffis

a

lffis

(b)

Z b-imorphism) (resp. Z’epimorphism. (fiemonomorphism

a

then

then

must

f

be

a

must

I be

a

‘g-monomorphism (resp. ‘6—epimorphism,Z-bimorphism)? Prove

6E.

that

(f. g) is

a

in the

monomorphism

category

arrow

Z1,

then f is

a

in Z’.

monomorphism 6F.

if

Prove

that

iff

Form

the

dual

is

Z’-epimorphism

a

and

g

of

is

a

‘E-section. then

9

is

a

‘6-section. 66. 6H.

(a) Show G

group

of 6F.

Group Morphisms of the (finite) if K is a subgroup and group homomorphisms f.. [1: H —.

K

by adjoining p:

X

leaves

the

Consider

[Hinu —>

H, then group G such that

that

a

set

single

X be the

permutation

all other

elements H

—>

1?. Let

element

which

=

the

set

G be the

by:

ft(h)(s)= fol)

=

a

(finite)

f2(h)}-

of X fixed. G

exists

{hK I It 6 H} of all left K-cosets of H, of X, and let permutation group the elements eK and K. and (= K) interchanges

from

X obtained

new

Define/“[2:

{/16 ”MW

=

there

MK

if

s=mq

K

if

S:

(MAUI)C I’"~]

K

J

(b) Use part (a) homomorphisms;

”0

g:

in Grp

epimorphisms of finite

Algebras algebra is a triple (A. 0» f4) where that the following holds:

such

function

A

the

in the category

likewise

precisely the surjective

are

groups.

Induction

61.

induction

An

that

show

to

IV

Chap.

Special Morphisms and Special Objects

46

AandOAeDandeD]

c

D,thenD

c

homomorphism (A. 0.4.1:.) 9— (B. 03. In) A B such that g(0A) 03 and g(f4(a)) —>

in

that

a

induction

IndAlg of induction that are monomorphisms

the

them. there are underlying sets. (b) Show that IndAlg is a quasi-ordered

class

a

—.

A is

a

that

so

is

a

function

A.

e

homomorphisms on the injective functions

not

(15(6)).

A

and

algebras

category

between

fA:

algebras

for each

f5(g(a))

=

A and

0,4 e

set.

A.

=

between

=

(a) Show

A is

is

of its morphisms

each

a

bimorphism. Show

6].

then

(35(7)).

category

is

morphism

every

numbers

of natural

if the monoid

that

a

addition

under

but

bimorphism,

is considered is the

zero

only

as

a

iso-

morphism. 6K.

if ‘6’ is

that

Prove

in

that

Show

\6M,

isomorphic subobjects 6N. ‘60.

the dual

Prove

that

6K.

‘6’, it is possible for (X, f) and (Y. 9) to be nonobject Z. even though X and Y are @—isomorphicobjects.

category

a

an

of

statements

and

TERMINAL,

INITIAL,

relations

Corollary 6.25.

and

6.24

Proposition

of sets

the category

(15(2))

AND

ZERO

every

set

Q

B.

is balanced.

OBJECTS

PROPOSITION

MOTIVATING

Q has the property function from g to B. set

Proof: It is

the

empty

that

function

for from

to

B, there

exists

and

one

only

E]

DEFINITION

object X in object) provided that An

7.3

‘g-retraction.

a

Objects

The

7.2

of

Form

§7 Initial

of Exercise

the dual

Form

6L.

one

‘6’-epimorphism is

such that every

category

‘6 is balanced.

then

'I.I

a

a

if is called an initial object for $ (or category all for (cf-objects B, Itomgu’, B) has exactly one

a

‘6-initial member.

EXAMPLES

SGrp, and Top has a unique semigroup and the empty space).

(1) Each empty

of Set.

(2) Mon, Grp, Ab, and groups

and

R-Mod

each

have

initial

initial

objects (the

R-modules).

(3) The ring Z of integers is

an

initial

object

object (the empty

in Rng.

trivial

set,

the

monoids,

Sec. 7

Initial.

(4) BooAlg has initial

(5)

Field

has

(6) A quasi-ordered if it has

only 7.4

class

smallest

a

two-element

considered

as

a

Y,

are

Objects boolean

algebras).

has

category

47

initial

an

object

if and

member.

‘g-inilial

two

Let

Proof: definition.

Terminal

L)

X

X and

objects,

Y L

Y and

isomorphic.

X be

the

morphisms

guaranteed

by

the

By uniqueness

XflLX=X‘—X.X

and

and

retractions.

Thus, f and g

sections

are

Yfly so

vi.

=

isomorphisms.

Y.

['3

Objects

DEFINITION

object X in a category % is object) provided that for all objects An

7.6

and Zero

PROPOSITION

Any

7.5

objects (the object.

initial

no

Terminal.

called

a

B in i6.

terminal

object

for ‘6 (or

a

hom.‘(B.

X) has

exactly

one

‘g-teminal member.

EXAMPLES

SGrp, Mon, Grp, Ab. R-Mod, Rng. Top, Lin'I‘op, and BooAIgT objects (the “‘singletons"). h as no terminal (2) Field'l‘fl‘ objects. as a category has a terminal considered A class (3) object if and quasi-ordered only if it has a largest member. (l) Each

of Set,

has terminal

7.7

PROPOSITION

object and terminal

Initial 7.8

are

dual

E]

concepls.

PROPOSITION

Any

Zero

’6'-rerminal

two

Proof:

7.9

object

Dualize

objects

Proposition

are

isomorphic.

7.4.

C

Objects DEFINITION

A

’fl-object is called ’6-initial object

both

a

7.I0

EXAMPLES

(I) Grp,

pTop

Mon,

have

zero

Ab,

a

object

zero

and

a

for Q? (or

’6'-terminal

a

object) provided

BanSpl,

it is

Bansz,

pSet. and

objects.

(2) Set, Top. SGrp, Rug, R-Alg, BooAlg, POS, and Lat do ? I-‘or boolean H' Recall that

that

object.

'I‘opGrp. LinTop.

R-Mod,

’6‘-zero

algebras, for each

we

do

not

require that

field. 0 aé l.

0 5% l.

not

have

zero

objects.

and

Special Morpliisms

48

Chap. I V

Special Objects

PROPOSITION

7."

Any

'K-zcro

two

objects

E]

isomorphic.

are

EXERCISES

Determine

7A.

categories given 73.

in

initial. terminal Examples 2.2 and 3.5. if X is

that

Prove

and

the

(resp. ‘6-terminal, ’6-zero) object. then

‘6-initial

a

homgu’. X) 7C.

(a) Prove (b) Prove (c) Show

that

if Q? is connected

that

(b) is false if the condition

are

‘6’ has

a

X and

terminal

a

if X is

that

and

X is

is

a

initial

an

monomorphism. object. then f is

’6’ is connected

that

’6-initiai

a

{Ix}.

=

@-morphism. object, then f

if X is

Prove

following

A be a

that

7D.

(a) (b) (c) (d)

XL»

Let

objects (when they exist) of the

zero

object

and

Y is

a

a

monomorphism.

is deleted.

r6’-te1'minal object, then

the

equivalent: object. are isomorphic.

zero

Y

’10m%(Y.X)

9*

Q.

%’ is connected. ‘6’ be

with

initial

that

(f, g) is a mono‘62 if and only if both I and g are monomorphisms morphism in the arrow category in TopBun if and only if both in ‘6’. (Thus. for example, (f. g) is a monomorphism are fand g injective.) 7E.

Let

a

category

§8 CONSTANT

If f: following is

(l) f

(2)

POINTED

ZERO

MORPHISMS,

CATEGORIES

a

a

function,

i.e., f

[A]

is

the

to

set

B, then

the

that

for

singleton.

a

andfor allflmclions ‘facrored through" a singleton C

A

set

non-empty

r, s:

C

—»

set.

fo

A,

r

=

fo

s.

D

DEFINITION

‘6-morphism

A

(I) A each

(2)

f

function from equivalent:

sets

be

can

B is

constant

all

For

(3) f 8.2

a

A are

Prove

PROPOSITION

MOTIVATING —.

object.

MORPHISMS,

AND

8.1

an

is

constant

C

6

01266)

A coconstant a

constant

notions).

A

L)

morphism and

B is said

for all

morphism

morphism

‘6 (or

in r,

s

e

a

if-constant

hom.‘(C, A), f

in Q? (or in “6"”

be

to

(i.e.,

a

o

morphism) provided r f s. =

‘6’-coconstant “constant"

o

morphism) provided that and

“coconstant”

are

dual

Sec.

8

(3)

A

zero

‘6’—constant

morphism

in ‘6’

morphism

and

(or

a

Morphisms. and Painted

Zero

Morphisms,

Constant

a

‘g-zero

‘g-coconstant

49

Categories

morphism) provided

that

it is both

a

morphism.

EXAMPLES

8.3

g or morphism if and only if A (I) In Set or Top A L) B is a constant with in these are functions coconstants is a The categories f [A] singleton. only empty domain; hence, these are the only zero morphisms. =

B is a constant Grp, R-Mod, Mon, LinTop, BanSp,, or Bansz, A —f-> morphism, zero morphism) if and only if f [A] is morphism (resp. coconstant of B. the identity element {X, Y}, homg(X, X) {l x}, (3) Let X and Y be distinct infinite sets, 0b(‘€) Y". Then every X) Q, and hode, Y) homg(Y, Y) {1,}, hode, %’-morphismfrom X to Y is simultaneously a bimorphism and a zero morphism.

(2)

In

=

=

=

=

=

PROPOSITION

8.4

is

If f

@-constant

a

the

composition is ‘6~zero)morphism.

(resp. g-coconstant, defined, h f og is also o

g-zero) morphism, then whenever a ‘g-constant (resp. ‘é-coconstant,

If r and s Proof: By duality we need only to give the proof for constants. with common domain such that r and s are defined, g-morphisms g g then iffisconstant,fo(gor) Thus, (hofog)or =fo(gos). (hofog)os; so that h f g is a constant. 1:] o

are

o

=

o

e

PROPOSITION

8.5

i)

‘6-morphism, and T be a @terminal object. Then (1) then and 96 implies (2). If; furthermore, hom.6(T, A) Q, (1) (2) are equivalent. Let

(1) f (2) f

A

a

be

can

is

B be

a

factored through constant morphism.

T.

A L T L B. If r, s: C Proof: Suppose that A L» B A, then since there is only one s. Hence, h go r morphism from C to T, gor g h g so r that 3. is a constant s, Thus, f f f morphism. Let f be a constant morphism and g e hom.6(T, A). Since T is a terminal object, there is a morphism u: A T. Because f is a constant, we have —»

=

=

o

o

c

=

o

0

=

o

-»

f=f° Thus, f 8.6

be factored

can

1A =f°(y°u)

through

T.

=

(f°g)°u.

[:1

PROPOSITION

If f is equivalent .(1) f (2) f

is

a

is

a

(3) f (4) f

is

a

can

a

%’-morphism and

morphism. morphism. coconstant morphism. be factored through X. zero

constant

X is

a

zero

then object for “ts”,

the

following

are

Special Morphisms and Special Objects

50

Chap. IV

Proof: By definition, (1) is equivalent to ((2) and (3)). Also, since ‘6 has a zero Hence, since X is a terminal object, @ is connected. object, (2) is equivalent to (4) (8.5). Likewise, since X is an initial object, (3) is equivalent to (4) (dual of 8.5). [:1 8.7

LEMMA

B, where f is If f, g: A morphism and hom.‘(B, A) aé a, —»

Let h: B

Proof: morphisms

f

=

morphism,

w-coconstant

a

g.

Then, by the definitions

(f°h)°g

=

is

g

of constant

la°g

=

and

=

coconstant

E]

9-

THEOREM

In any

(1)

then

[A =f°(h°9)

f=f° 8.8

A.

-’

‘B-constant

a

(6, the following

category,

For all A, B

(2) For all A,

B

6

012(6), hom.6(A, B) 0b“), hom.‘(A, B)

6

equivalent:

are

contains

contains

a

contains

(3)

For

all A, B

e

0b(‘6’), homgM, 8)

(4)

For all A, B

e

(5)

For

0b(‘6), homg(A, B) contains 0b(‘6), homg(A, B) contains

and

at

(6)

There

morphism. constant morphism. coconstant morphism. one constant morphism

exactly one exactly one exactly one

all A, B e least one coconstant exists

morphism.

zero

least

at

zero

morphism. function” selecting exactly one element out of each that the composition (front the left or the right) of a selected morphism is again a selected morphism (if the composition

“choice

a

hom.‘(A,B) such morphism with any is defined).

set

Proof: We will show (I) => (2) => (6) => (5) => (3) => (1). Since (1) is self-dual and (3) is dual to (4), this will imply that all of the conditions are equivalent. (1) (2)

=>

from

Immediate

(2). (6).

=»

“selected”

the

Let

8.4, the

(6) Then

f

a

and

r

f

of selection, f f is a coconstant

(5) and and

(3)

Let

(3).

a

is

there g =>

=

o

a

i»

A

Let

(5).

=

o

B

of

the

"selected"

are

s

be

=

f,

g

e

unique morphism with

zero

morphisms

in

is

a

hom.‘(A, B) be

constant

any

and

h: A

_.

aé

z.

morphism. By morphism is a

zero

let

C

r,s:

A.

—+

hom¢(C, A). By the uniqueness morphism. By a dual argument, Q is connected

morphisms. By (5),

constant

morphism

coconstant

the

morphism,

s.

o

a

by (l), Ito:n,,(B, A)

“selected“

Hence, f f morphism.

r

be

morphism

composition

Proposition zero morphism.

since

the lemma,

B.

by the

Hence.

lemma, f

=

h

h.

(1).

Let

f:

A

_.

‘6-morphisms, then r f consequently are identical. phism. [j o

B be

and

s

morphism.

constant

a o

f

Hence,

are

f

constant

is

a

If

Bi;

morphisms

coconstant

and

C is

from so

is

a

A to a

zero

pair of C and mor-

Sec.

8

8.9

DEFINITION

@ is said

A category

pointed provided

51

Categories

it satisfies

that

of the

one

8.8.

PROPOSITION

which

(1) Every category

has

a

(2) Every full subcategory of As it turns

out, with

categories 8.11

be

to

of Theorem

equivalent conditions 8.10

Morphisms, and Pointed

Zero

Morphisms.

Constant

a

pointed. pointed category is pointed. is

object

zero

pointed categories are “essentially” zero object (see Exercise l2F).

full

the

the a

D of

subcategories

EXAMPLES

(l) Grp, R-Mod,

and

LinTop, pSet, pTop,

Mon,

the

of infinite groups

category

pointed.

are

POS,

(2) Set, Top, SGrp,

and

Lat,

the

of

category

bipointed

sets

are

not

pointed.

EXERCISES Show

8A.

function

on

the

need

be

true.

not

88. is

a

concrete

a

underlying that

Prove

boolean

in

that

A

i.)

with

algebra

sets

is

B is

a

a

constant

morphism

coconstant

constant converse

BooAlg if and only if f [A]

in

members.

two

or

one

’6. every morphism that is a morphism in ’6. but that the

category

Suppose that A L; B 1+ C are ‘d-morphisms. and g =fis a constant morphism, then fis a (a) Show that ifg is a monomorphism constant morphism. and g f is a zero morphism. then (b) Show that if’6’ is pointed, g is a monomorphism, is a zero f morphism. (c) Form the duals of (a) and (b). 8C.

a

8D.

that

Prove exists

(a) There (c)

a

8E.

morphisms 8F. then

there

86. then

there

morphisms 8H.

terminal Let?

domain

Show exists

that that

f

at y.

the a

then

L;

if X

unique

a

Establish

constant

fact

that

with

a

following domain

equivalent:

are

A.

monomorphism.

constant

the dual

I.»

Z

[0 1'1

Y is

a

X

[:1 Y.

is

a

and

Y

iffandg

8.5.

in

-"—)Y such

connected

the collection

of Proposition

that

morphism

constant

morphism if’d

Prove

are

g-constant

fit.

7- g

correspondence

one-to-one

hOMg("’. Y)

Form

A is

and

be connected

exists

the

category

monomorphism

object.

such

in

connected

a

with

(b) Every morphism A is

in

constant

a

category

between

org-constant

the

a

that and

connected Ii

2

f

category,

f.

=

W, X, Y

collection

morphisms

e

011%”),

of ‘6—constant in

humid/Y. Y).

Chap.

Special Marphisms and Special Objects

52

81.

determined 8].

Prove

and If ‘6 is

that

in

that

it

a

pointed category, selects exactly the a

pointed

category,

prove

the zero

that

“choice

function"

morphisms. the

following

(a) A is a zero object for g, (b) hamgm, A) {IA}. =

8K.

Prove

that

“‘6’ is

pointed” is

a

of 8.8(6) is

self-dual

statement.

are

equivalent:

1V

uniquely

V Natural

and

Functors

Transformations

first

be observed

It should

auxiliary

basic

our

one;

transformation.

.

.

that

the

concepts

are

whole

of

first notions

our

is

category

a

a

essentially

and of

functor

a

an

natural

.

——S.

One

of

concept

essentially those of

of

a

category

was

EILENBERG

that

of

a

AND

class

S. MAC LANET

of “structured

sets”

(called objects) together with a class of “structure-preservingfunctions” (called morphisms) between them. In Chapters III and IV we have seen that generally as it is the morphisms it is not so much the objects and how they are constructed. when one’s attention that is the focal of and how they are point composed, wider investigating categories. In this chapter we step back and take a somewhat classes and as structured looking at view—considering categories themselves them. Later we the “structure-preservingfunctions" (called functors) between will see that, analogously with the earlier situation, much of the importance of of the categories themselves, but the theory of categories lies not in the structure them and how they are composed. Actually, we in the functors between rather further can one (and do) go by defining and investigating “morphisms” step natural transformations. As expected, the between functors. These are called of and how are of natural transformations they composed is the essence study “functor theory“. §9 9.1

FUNCTORS

DEFINITION

Let

‘6 and

whereFis isms 1’ From

a

of 9

be

9

from

function

(i.e.,

Transactions

F:

triple (‘6, F, 9) class of morphisms of ‘6 to the class of morphMor(£2)) satisfying the following conditions:

categories. the

Mor(‘6)

of the American

A

functor

from

‘6 to

Q

—r

Mathematical 53

Society

58 (I945).

is

a

Natural

FUIICIOI'S and

54

Transformations

V

Chap.

then F(e) is a Q-identity. (1) F preserves identities; i.e., if e is a ‘6—identity, dom(f) (2) F preserves composition; F(fo g) PU) F(g); i.e., whenever cod( g), then dom(F(f)) cod(F(g)) and the above equality holds. =

=

o

=

Instead

tively,

of ‘6’

“F:

—v

writing “((6, F, 9) 9”, or %’ 1—»9,

is

functor",

a

“Fis

or

from

functor

a

usually write,

we

%’ to 9".

and write F instead functions, we usually abuse the notation ‘6’ and codomain F has domain for that we write, example, domain is a small category is called a small functor.

for any objects and

Because

there

category

is

identities

—»

F“) An immediate

F[hom.‘(A,8)] F

Obviously, then, any functor “object-function”F: 0b(‘€)

2

Q?

of

this,

denoted

(2)

If ‘6’ is

a

function, then (3) If i7 is

—°+i

1' If X’ function

(6&4

c

is

a

that

functor,

X,

l”

1:: A”

-.

Q is

quotient

function

canonical @

by lg. subcategory a

c

‘6-objects A

9

—>

ham

can

all

and

and

B,

functors

by

of

means

of £2 and a

of

category

called

each

to

the canonical

Y and f: X the Y’ for which _.

Y is

a

called

f unctor,

called

f unctor,

assigns

a

E:

the

identity

Mor(%’)L>M0119) the inclusion

the

restriction

offto

functor

is the from

functor

object

on

‘6’

inclusion ‘6’ to 9.

Mor(‘Z’)is the morphism f its equivalence class f, then or natural functor from %’ to (Z’. ’6, and

Q: (Worm?)

such

function

that

[[X']

—»

c

square

X_f, is called

their

restrictions.

x'---9--->y'

commutes.

its

from

easily be recovered of the restrictionsT

FA.FB

describe

%, (%’, lmm, ‘6’)is

( I) For any category and

that

FUNCI'ORS

OF

EXAMPLES

9.2

“hom-set"

their

and

functions

often

shall

we

such

). lhomiAJ)

F Because

Q-objects,

Itomg(F(A), F(B)).

c

0b(9)

-’

of

class

1,“).

=

for all

is that

consequence

whose

Q. A functor

(A H 1,.) functors identities, preserve denoted (also “by abuse of

(3.2) and because F: g a unique function 9 induces each functor of from the class notation" ‘6'-objects to the by F) for each ’6-object A between

Also, as with of ( B) is the homomorphism

unique morphism

I'm

,

for which

the

=

from

A/A’ A/A'

to

B/B' induced

square

A

4L)A / A’

fl

EHO)

‘9

h

B—>B/B’

commutes

functor.

(9) There

(where called is

the

horizontal

the abelianization a

functor

[i from

arrows

are

natural

projections).

Then

H

is

a

functor. the

category

CRegT,

of

completely

regular

56

and

Farmers

Hausdorff

Natural

Transformations

assigns

to the category CompTz of compact Hausdorff spaces X its Stone-Cech compactification [3X and to each space

tinuous

function

i»

X

Y the

unique

continuous

V

Chap.

function

that

spaces to each

con-

[3(f) which makes

the

square

,8 is

commute.

For

called

Stone-(Zed: functor.

the

pointed topological space X, let I'l,(X) be the fundamental of X and for each morphism X L) Y in pTop, let H1(f ) be the function group from l'l,(X) to HAY) that assigns to each equivalence class of closed paths [p] the equivalence class [fa p]. For each such f, IT,(f) is a group homomorphism. I'Il is a functor from pTop to Grp, called the fundamental group functor. Ab that assigns to each space X the free (11) There is a functor ho: Top abelian of X. If X 1—»Y, then generated by the set of components group is determined how: 110(X) 110(Y) by: (10)

each

-+

—)

Ito(/)(C) (where C is

a

the

for each

then

of

component

‘6 is

(12) If

X). [to is called of

category

integer

n

there

the induced

=

the

contains

0th

f [C]

homology functor.

chain

is

a

complexes of abelian homology-functor H": ‘6

groups Ab defined

——>

H..((G.-, don)

Hn((fi)‘ez)

of Y that

the component

=

(35(5)), by:

Ker(d..)/lm(d..+1)

=

homomorphism:

in: K0r(d..)/1m(dn+1)

-’

Ker(d.2)/lnl(d.§+ 1)

Actually different “homology theories” of algebraic topology can essentially be obtained from the category of topological by defining appropriate functors to the spaces category of chain complexes and looking at the “compositions” of each such functor with the homology functor H... 9.3

PROPOSITION

L,

lfd 9.4

Q

i.

‘6

are

functors,

then

64"

a!

'6 is

functor.

a

C]

DEFINITION

functor

The F and The

reader Q.

G

o

F of

the

above

proposition

is called

the

composition

of

G.

rately) to

36 and

A

have

may

both more

as

a

noticed

function

precise

from

statement

that

in

Proposition Modal) to Morwt) of the proposition

9.3

regard

we

and

would

as

a

be

F

(inaccu-

functor

fromsl

the

following:

If F

(d, F, 38) and (which is denoted

=

functor Because

it is

when

(Q, G, 9?) G by F). =

(.11, (NEW)

then

functors,

are

is

a

o

for funetors

confusion

no

G

and

descriptive

more

“arrow-notation”

in

instructive,

have

we

9.3.

Proposition

We

used

less

the

continue

will

precise

to

use

it

likely.

seems

DEFINITION

9.5

is called

A

if

57

szctars

S cc. 9

triple M, F, 9) 66”, F, 9) is a functor

Notice ‘6 to

that 9.

Nevertheless,

“covariant functors.

functor

from g

f unctors" Observe

will

we

functor

contravariant

from

if and ‘6’ to

(‘6, F, 9”) is

if

only .02 is

usually

the

notation

use

For this reason,

‘6 to .02 if and

from

functor

occasionally

to 9.

when

functors

not

a

a

only functor). from

functor

F: ‘6

for

9

—>

from

a

called

sometimes

are

wishes to distinguish them functor of a contravariant

one

the notion

that

contravariant

(or, equivalently,

contravariant

a

a

contravariant

is for the most

part

we for its use is the fact that in many instances principal reason over their For have a built-in preference for some categories opposites. example, Set” and Top” as we do about Set and Top. But we know just as much about when working with the latter categories. we are psychologically more comfortable

superfluous.

The

9.6

EXAMPLES

(1)

For

OF

CONTRAVARIANT

functor

any

F

FUNCl'ORS

(Q?, F, 9)

=

there

contravariant

associated

two

are

functors, F

(2)

The

*

(W’, F, 9)

=

9°: Set”

functor

subsets of A and defined by 9(f)(C) all

called

—>

the contrarariant

Set, which

each

function

f "[C],

is

to =

*F

and

a

assigns A

L)

(‘6, F, 9°”).

=

each

to

B the

contravariant

set

9’01) of 901) 9(8) y—UL from Set to Set,

A the set

function functor

power-set functor.

Top“? —> BooAlg that assigns to each topological space function the boolean algebra of its clopen subsets and to each continuous —’ F defined Y, the boolean homomorphism F(f): F(Y) (X ) by F(f)(A) f :X functor from a to is contravariant Top BooAlg. f "[A], functor

(3) The

F:

-—>

(4) If

f

then the

is the

functor(‘):

linear

functionals

linear

transformation

functor

from

of finite dimensional

category 57"" F

over

.97 that

->

and

f: W

.9" to

:7, called

assigns to each

to

the

duality

each

linear

spaces

vector

space transformation

by fly)

I7 defined

—)

vector

=

=

go

f, is

a

contravariant

functor.

that assigns to each Top” topological is real-valued functions the vector lattice C(X, R) of all continuous variant functor from Top to VecLat.

(5)

The

(6)

The

X

the

functions

functor

functor

C:

—»

C“: Top"p

VeeLat

—>

C*-algebra C*(X, C) is

a

contravariant

field

the

F, V,the space Vof f: V a W, the

over

C*-Alg‘that of

functor

space a

X

contra-

assigns to each topological space continuous all complex valued bounded from Top to C*—Alg.

58

and

Functors

(7) Each functor

Natural

Transformations

Chap.

of the

of algebraic topology cohomology functors Top to Ab; i.e., a functor from Top” to Ab.

from

is

a

contravariant

DEFINITION

9.7

If the domain is sometimes functors 9.8

of

functor

a

called

a

is the

product of two categories, Similarly, one may define

bifunctor.

OF

(l) The cartesian

trifunctors

BIFUNCTORS

product functor

(_

(_

Set

_):

x

X_.)(A,B)=A

U

X

g)(a, b)

a

Set, defined by:

xB—viD

(f (0). 9(1)».

=

product functor

tensor

Set

x

B

x

(_x_)(j;g)=fxg:A

(2) The

the functor

of n-variables.

EXAMPLES

Where

then

® _):

(_

(—®—)(A,B)

Ab

x

Ab, defined by:

—»

A 63 B

=

(_®_)(f,g)

Ab

=f®ng

® 3-»

C®D

where

(f® (3)

The

disjoint

® bi)

9X2 a,-

union

functor

(_

d3_)(A,

tb

(_ B)

Set

__):

®

Set

x

901))Set defined

—»

e‘) B

A

=

Z (f(ai)

=

(._d3_)(fig)=fd9g:A

08—»CGD

where

9.9

f(x)

if

i=1

if

i=2.

[g(x)

.

U‘WM')‘

_

THEOREM

If

F: 42!

(1) For each

w

x

A

e

—>

‘6 is

bifunctor,

a

0b(.nf), there

is

an

then

assaciatedfunctor

F(A,_):

(denoted by)

«1;,

e

—.

=

F(A, B)

defined by:

F(A,_)(B) and

F(A,—)(h)

=

F04. h)

and

(2) for

each

B

e

GHQ), (here

is

an

F(_,

associated

B):

.ss’

funcror (denoted by) —>

@,

defined by: F(—, BXA)

=

F04. B)

by:

and

V

Sec. 9

Functars

59

land F(—! 3X9)

Proof: FHA, 13) is morphisms

(l). (2) follows

We will prove

in ’6’ since

identity

an

I!)

o

F(14

=

=

a

1‘, g 0/1)

o

and

identities

Thus, FM, _) preserves

F((IA, g)

=

F(1A,!I)° F(1.4.h)

=

a

(1,4,11))

F(A,—)(g) °F(A.—)(h)-

=

compositions.

[:1

DEFINITION

9.10

The

functor 9.9 is called the right associated F(A, _) of Theorem F and A, and the functor F(_, B) is called the left associated

functor

respect to with respect

important

most

B.

to F and

functor The

analogously. Clearly, F(A, _)(l a) B” are functor. Now, if 8—5 B’—”—)

in Q, then

FM, _)(g

with

Fis

F(g’ '11)

=

bifunctor

its associated

and

functors

will be studied

section.

ately in the next

EXERCISES 9A. Verify that the following (a) The (covariant) power set functor

9M)

an where

9’(f)(C)

9;

(b) The squaring functor

warm/1)

=

)2:

(

Set

of all subsets -»

9(3)

( =

Set. defined by:

a

)2(A)

(

=

A2

PM L» B) =F:A1

—.

B2

(f(al),f(az))-

(e) The ith projection functor

for

product category

a

nli’é’, defined

Set. defined

->

f [C l-

=

wheref 2(01.dz)

Set

the collection

=

i» B)

functors:

are

x

(6’2 x

x

(gn fig)“

by:

nl(fl7f2u---sfl|) =1;(d) The ith injection functor

for

a

sum

category

[Ifig‘ fig; defined

11%: H"'

by:

.“l(f)

=

(f. “-

”gm

by: of A

separ-

necessarily

if F: ’6

that

Show

93.

is

Q

-.

then

functor

a

o———)e

to

that

Show

a

preserves

compositions.

essential.

[0“. Exercise

9D. the

is not

from

\l

1categories, which i.e., “identity preservation“ is

necessarily

a

of two

classes

the morphism

between

function

F is not

under

I——>o

o———>o

9C.

image of%

the

V

Chap.

Transformations

functor,

a

[Consider

of 9.

subcategory

a

Natural

and

Humor:

60

functor;

48.]

Show

that

functors

Prove

that

if each

coincide

which

on

objects (identities)

necessarily

not

are

same.

9E.

F: ‘6

then

(35(7)).

In

9F. funetor

a

_.

of g and

is

9

which

Ab

assigns

just

one

monoid

a

object (i.e., a monoid homomorphism.

given in Example 9.2(8). define a torsion subgroup abelian A its torsion to each group subgroup A‘,

F: Ab

functor

torsion-free

only if it is

if and

that

to

with

{I is acategory

functor

a

similar

manner

Ab

T:

and define

a

_.

Ab which

..

assigns

to

each abelian

group

A the

A/A‘.

group

Define

9G.

a

functor

the category

from

of

quasi-ordered X, the partially-ordered 005

to the

sets

category

is obtained assigns to any quasi-ordered set by identifying those members a and I) of X for which a s b and b s a. that assigns to each 9H. Let F: 0b(NLinSp) OMCompTz) be the function on X linear space X the closed unit sphere in the space of all linear functionals normed of some contravariant (furnished with the weak‘-topology). is F the object function from NLinSp to CompTz? functor a Define 91. Top whose value at any (bi)l‘unctor (_ x _): Top x Top pair of spaces is their topological product. with one 9.1. Show that if (G, +) is an abelian (considered as a category group x x and G G G G is a product category (D: object), then the direct product G

[’05, which

that

set

—>

_.

—>

defined

by GU, g)

9K.

Prove

=

that

f

g is

+

if F

:

.d

a

bifunctor.

.9! and

—»

Gzttv'

Fx

defined

G: g

~

It,

x‘6-m1o)’

then x

there

exists

functor

a

.9,

by: (F

(nth.

x

k)

(Ft/I), C(10),

=

product of two categories at, x tel: together with the pro. x that 512 .912 has the property .911 and nzzd. jection functors nlmfl x d; then there ’6’ are and G: and F: %’ functors, if '6’ is any category any d; d, P: ‘6 exists a unique functor :1, x .1312such that the diagram 9L.

Prove

that

the

a

—>

—>

—>

—>

(

.t/,

V Y. P

.c/2

/'2 ”RV? 1/1 .C/g X

commutes.

category .53" LI .31; together with the [15.51, —> .9], LI 5:12.1': l, 2 has the property that if?! is any category and K2531; —> 9, then there exists a unique functor Q: 51/, U .212 9M.

Prove

that

the

sum

—~

diagram

injection functors and 9

#9

sz.

such

that

the

Sec.

10

Hom-Functors

61

SQ

V‘N :Q

«5/1

.3

i9 '

Proof: Clearly

Itomdlm

lB)(x)

In

=

thus, ImmgUA, In) is the identity function

Itom.6((f, g)

o

(It, k))(x)

[10111301 of, o

g

the

k)(x)

o

g

=

Immg(f, g) and

=

x;

/l0m%(/i, 8).

set =

ItamgUt, k)(x) of

=

identities

Hence, hamc preserves 10.2

=

on

IA

oxo

=

a

g

k

0

x

o

It

Also

of

I10"1(‘( f, g)(hom(6(h, k)(x))

llama/1, k)(x).

o

composition.

[:1

DEFINITION

110mg: Q?” x ’6 morphism functor) for the category to home and A, i.e., homg(A, _): The

ol' ‘6’ with

functor

respect

to

A; and

the

—>

Set

95’. The ‘6’

the contravariant

T’P

set-valued

associated

is called

left associated

hom-l'unctor

the

right

Set,

—’

liom.‘(_, A): is called

is called

the covariant

functor, —,

of ’6 with

hom-functor

functor

i.e.,

Set, respect

to A.

with

(or

respect hom-functor

and Natural

Functors

62

‘6’

Occasionally, for simplicity, the subscript Notice pressed when denoting the ham-functor. hom(A, _)(f)

=

hom(A,f)

=

Itom(f, A)

=

is sup-

naming the category for a morphism f

that

i.e., hom(A,f)(x)

fo_;

V

Chap.

Transformations

=

[0

x,

=

x

of.

whereas

A)(f)

hom(_,

=

cf; i.e., hom(f, A)(x)

_

PROPOSITION

10.3

For any

‘6-object A,

'6’ and any

category

Itom.6(_, A)

of W"

with respect

Since

the

A.

to

Set

-#

[:1

sequel

will

we

’6’

of

ham-functors

contravariant of W”, in the

functors

with

——>

with

A is identical

to

respect

._).

homwpw,

=

Set of %’ ham-functor hom(_, A): W” the covariant ham-functor hom(A, _..)2 ’6’”

words, the contravariant

In other

have

We

exactly

are

need

formally

covariant

the

investigate only

to

hom-

covariant

ham-functors. Hom-Functors

IntemnI

In

with be

than

Set.

regarded

Since, case

cases,

Hanna:

(6”

ham

U

(l)

be

to

be considered

similar

I0.4

as

moreover, out

turns

=

will

‘6

the

use

and

9

->

Ab"‘p

notation

“Ham”

if

—»

U: 9

Set

instance

Ab to Al). In this

x

rather

is

“ham”.

than

the

Note

and

in

that

if

then

functor,

forgetful

EXAMPLES

Ham:

Ab”

Ab

x

R-Mod”

(3) Ham:

Top”

x

Ah.

—.

R-Mod

x

Top

topology). (4) Ham: NLinSp” defined by

x

—>

R-Mod

—>

Top

NLinSp ”fl

(where —’

(where R is commutative). HomM.

NLinSp

B)

(where

sup{llf(X)ll I IIXH

=

=

the

has

compact-open

B) has

Hom(A,

the

norm

|})-

DEFINITION

If a? is

contravariant

concrete

a

functor

category as!

L,

with

Q is called

forgetful functor a ham-type functor

Oboe!) such that ltom(_, be denoted by Hom(._, A). usually

exists

be

can

Ham.

(2) Ham:

10.5

from

functor

a

as

we x

o

homgM, B)

set

be considered

structure

a

other

supplied in a natural way as an so that it itself can object of a category For example, if A and B are objects in Ab, then hom(A, B) can an abelian (if we define (f + g)(a) to be f (a) + 9(a)). group for all morphisms f and g of Ah, the function lme', g) can in this an abelian homomorphism, the ham-functor group the

instances,

some

some

A

e

A)

=

U0 F.

U: .93

—‘

Set, then

provided In

this

case,

a

there

that F

will

Sec.

10

63

Hom-Fwtctors

Often

ham-type functors sidered simultaneously as an theories duality hinge upon 10.6

EXAMPLES

I.e.,

contravariant

OF

arise

when

is

there

that

object A the categories an

object of each of these ham-type functors.

HOM-TYPE

functors

be

can

.31 and

a.

Q,

where

con-

Many

FUNCI'ORS

of

the

form

Hom(_,

A):.a!

—.

the

triangle .n/

Ham (_.A)

"23’

1U

ham(_,A)

Set commutes.

d R-Mod

(1)

3

A

Mod-R

R

Hom(X, A) is called: The dual linear

(2)

LinTop

C-Mod

C

module

adjoint

module

NLinSp

C

BanSp1

(4)

LCAb

LCAbT Ab

(5)

CompAbfi

linear

Ab

CompAb Top

BooAlg

R/Z

21'???

BooAlg

ZTTTT

BooSpTTT

The

Top

R

Rug

Comp'l‘;

C

locally

of

the

com-

abelian

group

dual of

the

compact

algebra of X or the clopen subsets of X

Stone

The

of

X

group

or

space

algebra ring of

functions

(10) CBanAlg

of

space

X

of the

group

character

boolean

(9)

functionals

X

The The

linear

space

X pact abelian group The character group

character

algebra (8)

linear

The

abelian

(7)

of all

R/Z

R/Z

of all

dual, or conjugate) topological space X

conjugate (or dual) Banach normed

group

(6)

X

the module

X)

The the

on

(or

of the

(or the module (3)

or

functionals

The

on

of X

on

dual

boolean of

space

the

X

real-valued

continuous

X

The carrier

space (or maximal of the commutative Banach

ideal

space)

algebra

X

of locally compact T LCAb is the category abelian and continuous groups homomorphisms. of compact abelian and continuous 1T CompAb is the category groups homomorphisms. of boolean fit 8005;) is the category Hausdorll‘ (i.e., totally disconnected spaces compact

spaces) and continuous ’{1’T1'2 is considered

algebra.

functions. as

either

the

two-element

discrete

space

or

the

twoelemcnt

boolean

and Natural

Functors

64

Transformations

Chap.

V

EXERCISES 10A.

Show

that

there

exist

(tri)l‘unetors

natural

two

F, G: Set”

Set"P

x

Set

x

Set

—.

where

F(A, B, C)

hom(A

=

x

B, C),

and

That

is, describe

how

=

functors

act

these

F: %’ -v

Every functor

108.

hom(A, hom(B, C)).

GM, B, C)

on

9 has

bifunctors:

set-valued

associated

two

_):

hom(F_,

morphisms.

W”

x

9

-r

Set

9"”

x

g

—o

Set,

and

F_):

ham(.., where

_)(C,

D)

hom(__, F_)(D,

C)

ltom(F_,

=

ham(F(C),

D)

and

how

Describe

functors

these

§ll

act

on

=

ltom(D, F(C)).

morphisms.

CATEGORIES

OF

CATEGORIES

the role of “morphisms between already seen that functors assume categories”,i.e., the composition of functors is a functor (9.3), and, since their composition is the usual composition of functions, composition is associative with respect to the composition. Bebehave like identities and identity functors cause of this, one is tempted to form the “category of all categories". However, two technical difficulties arise. First, the “category of all categories” would have objects such as Set, Grp, and Top, which are not sets, so that the conglomerate of all objects in the category would not be a class (l.2(l)). This violates part (i) of the definition of a category (3.1). Secondly, given any two categories g and .62, it is not generally true that the conglomerate of all functors from ‘6 to 9 forms a set. This violates However, if we part (4) of the definition of a category. to that are to small restrict our attention sets, i.e., categories, then the categories We

above IL]

have

problems

are

eliminated.

PROPOSITION

There

exists

.l/

a

(0, .ll, (tom, cad, 0) where 0 is the class of all small of [humans between small categories, dam and cod are and codamain, respectively, and each F e -l/ its domain

category

is the class

all

categories, functions that assign to is the usual composition of functors

Proof:

Every

functor

Thus, the required classes

between and

o

in .11.

categories is

small

functions

can

be formed.

a

set

Since

(Exercise the

11A). composition

Sec.

11

Categories of Categories

of functors

is

composition, the “matching", “associativity”,and are easily verified (3.1). Thus, we need only show morphism class” condition; i.e., if

really function

“identity existence” the “smallness

65

conditions

of the

g

=

9

=

(0%, "It?! dOMg, COdg,°%’)

and

small

are

then

categories,

{Fl is

I g is

{g

Fis

Now, since % and 9

set.

a

(09, «([9, dome, coda, 09)

function

a

from

functor

a

from

be sets, sets, Jig and J19 must to .119} is a set (l.l(3)). Hence, {‘6} x

Jig

of functor

(l.l(3)). By {(6} x .97 x {.02}.Thus, by 1.1(1),

to

{Fl is

a

F is

(9.1),

functor

a

from

each

functor

‘6 to

from

.9"

that

so

9"

x

g to 9

=

{9} is belongs

9}

E]

set.

11.2

9}

are

the definition

a set

‘6’ to

DEFINITION

The

the

previous proposition is called the category of small categories and is denoted by Cat. Cat is actually quite large; for example, each of the categories Set, POS, in it (see Exercise 12F). It is nevertheless Mon, Grp, and Ab can be fully embedded unfortunate that we cannot form the “category of all categories”. Also, as we shall see lead to entities that would be categories later, other constructions were it not for the two “smallness” conditions for required categories, namely: given by

category

(1) 0b(‘6') and Mar“) must be classes, and (2) For each pair (A, B) of (cf-objects, lmm.6(A, B) must be For this reason, we consider the following more general

set.

a

notion

of

a

quasi-

category. ll.3

DEFINITION

A

(i)

quasicategory is

(9 and

all

are

(ii) dam and cod

(iii)

0

is

a

such

quintuple

g

(6', .ll, dam, cod, 0) where

=

conglomerates, functions

are

function

from

J!

to

0; and

from D

into

a

=

{(f, g)

5

all

x

-I/

| dom(f)

=

cod(g)}

all ;

that:

(l) If (fog) is defined €0d(f°g) cod(f);

(i.e., if (f, g)

e

D),

then

dom(fo g)

=

(2)1ffogand

h

ofare

defined,

thenh

o(fog)

=

(hof)

og;

=

dom(g)

and

66

(3)

and

Functors

For each A

(a) f (b) e (Compare 11.4

e

a

o

e

0, there exists

f

whenever

=

g whenever

=

g

this

f

with

some e

o

e

Definition

Natural

o

Transformations .1! such

e e

that

V

Chap.

dom(e)

A

=

cod(e) and

=

is defined, and g is defined.

3.1.)

PROPOSITION

is

Every category

a

D

quasicategory.

PROPOSITION

11.5

There

(6,3, dom, cod, 0) where (C is the conglomerate of all categories, 3' is the conglomerate of all functors between categories, dam and cod are functions that assign to each fitnctor its domain and cadomain, respectively ,' and is the composition of functors. D exists

a

quasicategory

o

DEFINITION

11.6

quasicategory described of all categories and is

The

category One that

of

entities

of the a

primary uses category) is that it

such

as

@119”.

in the above denoted

of the allows

It should

the

quasi-

by @4192

notion

of

be noted

a

talk

to

one

with

is called

proposition

that

quasicategory (as opposed to about “naturally” occurring most'of

the notions

associated

terminal

categories, e.g., monomorphisms, objects, functors, etc., can be for quasicategories as well. (An exception, of course, is the fact that in lack ham-functors into general Set.) Actually, our main use quasicategories for quasicategories will be as a device that allows more convenient expression. since are not the main of our Thus, quasicategories object study, we will not often be concerned with their “internal workings”. Also, we will never have a need to like the of all consider “quasicategory something quasicategories”. This is fortunate have to be revised to handle since, if we did, our foundationstwould in another it—we would find ourselves Russell-like paradox (see Exercise 11B). close to the motivating examples, i.e., Set, Grp, and Top, To keep our attention in that which follows, we will only consider categories (in the restricted sense) in instances where those except using quasicategories materially simplifies or defined

clarifies

matters.

EXERCISES 11A.

Prove

that

every

118.

Show

that

one

functor may

not

between form

small the

categories

is

a

set.

“quasicategory of all quasicategories"

by obtaining a Russell-like paradox from the assumption of all quasicategories that are not full sub-quasicategory

that

one

can.

[Consider

the

objects of themselves]

Special Alorphisms in (6’54? in “6.21.? are the monomorphisms whose precisely those functors functions between classes are underlying morphism injective. (b) Show that each functor whose underlying function on morphism classes is surjective is an epimorphism in (641.7. llC.

(a) Show

that

12

Sec.

Properties

of

Functors

67

it ., let N be the monoid of natural numbers considered (c) Let 2 be the category a category, and let F: 2 N be the functor such that F(f) 1. Show that F is an in (egg; it even is not on the cpimorphism though surjective underlying morphism .

—.

as

=

classes. the constant

(d) Determine 11D.

that

Prove

Cat

is

§12 this

In

section,

well

as

but

well-powered.

PROPERTIES

will

consider

@411?!" is

that

OF

not.

FUNCTORS

certain

properties enjoyed by properties possessed by special functors. we

other

as

in (62:51.97

morphisms

all

functors

DEFINITION

12.1

‘6’ 1-) Q is said

(l) A functor the

that

image

P has property

property g

L»

92 is said

to

F of each

under

categorical property 1’ provided morphism (or object, or diagram) in ‘6 with preserve

the

P in 9.

reflect

the

categorical property P provided that whenever image morphism (or object, or diagram) in g has property P in 9, then the morphism (or object, or diagram) must have property P in ‘6. (2)

under

the

to

F of

a

PROPOSITION

12.2

identities,

Every funetor preserves triangles.

isomorphisms, sections,

and

retractions

commutative

definition

the

thatfo

g

and commutative

identities

That

Proof:

of functor

F(g)

(9.1). Suppose that

9

preserved follows from and j; g e Mor(‘6’) such

a

Fan)

=

fi-section

F(f°g)

=

and

and

retractions.

functors

preserve

F( f )

F(f)°F(g).

=

must

be

a

Consequently,

woretraction. F

must

Hence, F iso-

preserve

[:1

morphisms. The

be

must

sections

preserves

’6’ I»

are

IA. Then,

=

Inn) Thus,

triangles

fact

that

isomorphisms implies that if X and Y are topological spaces such that for some n, the homology groups H,,(X) and HH(Y) Y are not homeomorphic. This fact provides a are not isomorphic, then X and relatively simple way of showing, for example, that the torus, Sl x S‘, is not homeomorphic techniques was 12.3

is

a

with one

2-sphere, 8’. Being able to for the emergence major reasons

the

of the

solve

problems by such of algebraic topology.

PROPOSITION

If g1 and functor,

9

are

then

connected the

categories,

following

(l)

F preserves

constant

(2)

F preserves

the terminal

are

morphisms.

object.

%’ has

equivalent:

a

terminal

object X, and

F: ‘6

—»

9

Natural

aml

Functors

68

Transformations

V

Chap.

Proof: h

(1)

that

Suppose

(2).

=>

F(X). Since

D 3

3i»

X

X is

morphism

constant

a

9

(8D), its image F(lx) 'rm must be constant. h so that 9. Thus, F(X) is a terminal object.

Hence,

=

oh

1,")

lr(x)og;

=

=

(2)

Suppose that C 3—»C’ through X (8.5); i.e.,

factored

CLtC' But this

F(X) is

As

into

statement

forming

the dual

of the

functor of

and

W”

F”:

97""

—*

9""

are

is

F: ‘3

F'":

‘6"

[:1

l2.3.

9, then in the dualization (Notice in particular that the

—>

9"".

—v

We

reversed.)

Proposition connected

illustrate

now

each

Replacing

F” preserves

the terminal

Proposition

12.3:

this

by

category

process by its opposite,

the

following

F preserves

coconstant

F preserves

the initial that

from

certain

property,

then

the

sections,

functors

preserve

retractions

functor

between

their

‘6’ and

an

initial

9,

we

have

object X, and

duality principle, it follows

all have

categorical

object X,

and

the

F: ‘€

dual

—»

of

9 is

a

morphisms.

preserve a

terminal

object.

functors whenever

a

equivalent:

are

about

statement

a

If? and Q are connected categories, ‘6 has functor, then the following are equivalent:

Notice

has

morphisms. object.

into

back

’6”

categories,

then

afunctor, constant

this

one

(8.5).

or property has a categorical statement its and category by opposite translating involving the original categories. If such a

functor

a

is not

F'” preserves

Translating

(1) (2)

be constant

must

have:

If ‘6'” (I) (2)

each

the functor

with

begin

we

every

property

or

involves

property

process, direction

we

XLC'.

object, FU)

by replacing

a statement

or

be

can

F(C‘) "—"lF(X) L“).F(C’).

=

earlier,

out

is obtained

this back

f

FUNCI'ORS

pointed

was

that

dual

terminal

a

WITH

DUALITY

Then

morphism.

that

implies

since

Thus,

constant

a

CL.

=

F(C) 5’! F(C')

12.4

is

(1).

=>

the dual could

we

them,

have

concluded

(cf. Proposition

statement

other

opposites one categories by about the original categories.

For

property.

that

if all functors

have

a

example, knowing that all using duality alone that all

12.2). It should involves

be mentioned

that

categories and “dual" statements may be formed by replacing at a time and translating back into statements or

Since

property

these

statements

just

two

involve

contravariant

See.

12

Properties of Flmctors

69

duals” of the original statement. For functors, they are called “contravariant duals of Proposition l2.3 are the following: example, the two contravariant If %’ and 9 are connected, X15 (1 ‘6-mtltal object and ‘6’ —> 9 IS a contravarlam functor from ’6 to 9. then the following are equivalent: ‘6’-coconstants

F carries

(l)

is

(2) F(X)

Q-terminal

a

object. .

If ‘6 and 9 are connected. functor from ’6’ to 9, then (1) (2)

F

As

before, the formation

is

(X )

9-initial

a

exercise

implicit

X

statement

a

following

r

.

object and ’6 equivalent:

(cf-terminal are

.

9

—»

.

ts

a

contrat'anant

Q-coconstants.

to

object. of duals

for the reader.

will, for the

In the

sequel,

involving functors,

property

or

.

ts

the

%’-constants

F carries

Q-eonstants.

to

we

most

when

will

be left

part, we

speak the

mean

as

standard

a

of “the"

dual

“covariant

of

a

dual".

DEFINITION

12.5

I"

A functors!

(1) full provided

.

.

J?

a

.

said

IS

each

that

be:

to

ham-set

restriction hom(F.-I,FA')

F

of F is

IhomtA.A')

surjective.

(2) faithful

of F is injective. provided that each ham-set restriction (3) (an) embedding provided that F: Mama!) More?) is an injective A (4) dense’r provided that for each Be abut?) there exists some such that F(A) is isomorphic to B. —»

One or

is

Notice

only

preserve I 2.6

0b(.sa!)

of the

identities).

on

that

if F” some

to

each

objects" (i.e., and

e

distinguish between embeddings and faithful functors from Grp, Mon, Top, example, forgetful into Set is faithful, but none is an embedding. It is easily seen that a an that it is both faithful and “one-to-one on embedding provided

For

R-Mod

functor

be careful

should

functors.

function.

functor

a

has the

F is full

(resp. faithful, (an) embedding, dense) if property. Thus, for example, if all faithful functors

same

they

property,

must

also

from

a

the dual

preserve

property.

EXAMPLES

functor

(1) Every canonical full and

f unctor

(2) Every inclusion

(3)

A category

each

functor

T A functor

category

to

quotient

a

category

is both

dense.

with

‘6’ is

with

a

of

subcategory into quasi-ordered class (in the

domain

this property

a

a

category

sense

(6" is faithful. is sometimes

called

"representative".

is

an

of 35(6))

embedding. only

if and

if

and Natural

Funcrors

70

field

no

six-element

a

from

from

f unctor

from

The

forgetful

The

inclusion

(8)

The abelianization

Field

Set is not

to

full and

[There is

dense.

not

set.]

functors

(6) (7)

V

is faithful.

(4) Every forgetful functor (5) The forgetful functor on

Chap.

Transformations

functor

full.

but not

Grp to Set and Ab to Set are dense, Top2 to Top is full, but not dense. from Grp to Ab (92(8)) is dense, but

full

not

or

faithful. We have natural

a

way a

will

we

a

[2.8

faithful

a

U: M

U: .m' two

pair (51, U)

—»

Set is

_.

concepts

are

category" in

CONCRETE

OF

is

category

functor

“concrete

the term

use

DEFINITION

SECOND

A concrete

is

a

%’,there

category

and

category Thus, the category.

concrete

the future, 12.7

a! is

where

pair (42!, U) rise to

with any concrete and a faithful categoryd that

seen

can

be associated

in

Set.

Conversely,

every

faithful

functor, gives In essentially the same.

a

the

following

CATEGORY

where

d

is

a

category

and

r

Set

—»

.

.

h

F(A) M F(A')

that

(I) Suppose

phisms

foh F(h)

left-cancellable,

is

=

F k

I1. Therefore,

=

(2) If F(A) 53’. F(A’)

is

a

F(fo r)

all-constant

F(fo 5).

=

0

=

fo

5;

$3

Since

FU)oF(k).

monomorphisms. and

HI)

HS),

=

0

if A"

:3

A, then

Since

i.e.,fis

A

)

morphism

Hr)

F

injective,fc r (3) If the triangle

“nu

ho

=

If A”

human»

F reflects

F(f) that

aQ-monomorphism.

=fo k, then F(f)oF(h) F(k). Thus, since

that

such

injective,

.

reflects monomorplusms, eprmorphtsms, coconstam morphisms, zero morphisms, and .48

Proof:

is

U: .21

PROPOSITION .

so

sense:

functor.

Every faithful functor d —9 morphisms, bimorphisms, constant commutative triangles.

is

a

Lo

FA'.FA’

’

Infill-.4»

constant.

Thus,

F reflects

constants.

F(A)—F—U)>F(A')

M 1N9) F

(A')

are

mor-

F0)

is

Sec. 12

Properties then

commutes,

F(h)

F(g)

=

o

F(f) F

is

The a

h

injective,

o

g

M

r

-

.F

Imfihfrf’

F reflects

of the

proposition

commutative

follow

triangles.

from

the fact that

faithfulness

is

D

concept.

PROPOSITION r

42/

Every functor and

71

F(g of). Since

=

f. Thus,

remaining pans

self-dual

12.9

=

Functors

of

.49 that

_.

.

15

.

and

full

.

faithful reflects sections,

.

retractwns,

isomorphisms.

Proof: If F(A) 51: F(A') is a 3-section, then there is some fl-morphism L, such that F(A’) F(A) hoF(f) If“). Since F is full, there is some safA such that h. Now morphism g: A’ Hg) F(g of) F(g)o F(f) since F is and hoF(f) Thus, faithful, F“). go]: 1,4 sofis an If“) =

—.

=

Hence, F reflects

sl-section. the fact that 12.10

=

=

fullness

sections.

faithfulness

and

F reflects

That are

both

follows

from

I]

concepts.

THEOREM

dense

Proof: By the above two propositions and duality, preserves monomorphisms and constant morphisms. the following:

Since

retractions

self-dual

Every functor .2! L, 9 that is full, faithful, and monomorphisms, epimorphisms, bimorphisms, constant morphisms, zero morphisms, sections, retractions, mutative triangles.

Let

=

=

A'

L) F is

A

be

ail-morphism

an

dense, there

Since F is full, there h o s.

are

is

and

sat-object si-morphisms q an

A"

qll, A,

I

A” and and

morphisms, isomorphisms,

we

To

8::3,F(A')

let

r

need

only

do this,

be

any

we

reflects

coconstanl

and

show start

cam-

F

that

with

3-morphisms.

£—isomorphismF(A”) 3—»B. that F(q) s and F (r) g

a

such

=

F(A')

is B

9i lit F(A’)

la!) A

and

preserves

F(A)

o

=

and Natural

Functors

72

Monomorpht'sm Preservation If f is a monomorphism,

RD”!

=

F(f)ogos

=

Fm°F(q) F(f°q)

Constant

If

f

is

a

F(f)ohos

=>

F(f)°F(r)

*4

is

=

FO’)

=

=

h

=

o

='

(since

r

(since f

h

=

g

Thus, F( f) is

s

o

=>

F(f°r)

fo

r

1’01)

=

=

=

q

q _=

g

V

then

F(f)°g

f0

Chap.

Transformations

s

(since

s

faithful)

=>

monomorphism)

a

is

F is

an

=

epimorphism)

monomorphism.

a

Morphism Preservation constant morphism, then fo

q

fo

=

FLD°F(II)

Hence, F( f ) is

F(D°F(r)

=

chgos F( f)

=.

r

=

o

a

F(f)ohos=>

F(f)

=

g

constant

=

oh

(since

s

morphism.

is

an

epimorphism).

E]

PROPOSITION

12.11

composition of full (resp. faith/ill, embedding, dense) functars (resp. faithful, embedding, dense). [j The

”.12

is

full

PROPOSITION

subcategory of a category is ajitll (resp. dense) subcategory if and only if inclusion flmctor is full (resp. dense). E] A

the

hom-Functors

Properties of 12.13

PROPOSITION

Each

couariant

Proof: If

B

horn-jimctor, hom(A, _),

—I»C

(howl/1 —)(D)(x)

is

a

=

.

Thus, hom(A,._)( f) is an (See also Corollary 29.4.)

monomorphism,

preserves

monomorphisms.

then

f

since

(howl/l. ——)(f))(J’)=>f°

injective function;

i.e.,

is left-cancellable,

f

°

=

x

=

x

=

a

Set-monomorphism.

y

y-

E]

Sec. 12

12.14

of Functors

Properties

73

DEFINITION

‘6-objcct P

A

’6’-projectiveif and only if

is called

‘6

honnAP, _):

epimorphisms. Dually, Q

preserves

12.15

PROPOSITION

An

each

only if

Set

OF

PROJECTIVE

OBJECTS)

‘6’-projectiveprovided that for each ’tf-epimorphism B —I»C

.

and

->

and

10.3).

(CHARACTERIZATION

P is

object

Set

—r

‘6”

epimorphisms (see Proposition

preserves

functor

‘Z-injective if

is called

homq,(_, Q):

the

P

morphism

L

.

C, there

h

.

exzsts

morplusm

a

P

B such

—s

that

the

triangle

D

commutes.

Notice

12.16

that

the

above

gives a completely projective objects Using duality, it is then acterization of injective objects; namely: in ‘6.

object Q is ’6-injectiveprovided each morphism C L Q, there exms

An

.

that

.

“internal easy

to

characterization” form

of

internal

an

char-

B and for each ‘6’-monomorphismC —f—> h morphtsm B Q such that the triangle .

.

—.

a

commutes.

These

mining

internal

thc

categorical

concepts

hom-functor A

L,

12.17

and

that

characterizations

B is

(if-object

characterizations

projectives

a

are

injectives have

been and

monomorphism if

easier

to

work

particular categories. defined “internally“ also

hence

and

often

in

could

have

been

with Notice have

defined

only if hom(C, _)( f)

is

when

that

detersome

“external"

externally;

e.g.,

injective for each

C.

EXAMPLES

is categorically (R-Mod)—projectiveif and only if it is a (l) A left R-module R-module and is (R-Mod)-injectivc if and only if it is an injective projective R-module. for our tcrminology.] [This provides the motivation

and

Funetors

74

V

Chap.

Transformations

principal ideal domain, then A is (R-Mod)—projectiveif and only free and is (R-Mod)-injective ifand only if it is divisible. Grp, BooAlg, and R-Mod, the projective objects are precisely the retracts free Boolean algebras and free modules objects, i.e., of free groups,

(2) If

R is

if it is

(3)

Natural

1n

of free

a

(cf. 3|.lO).

BooAlg-injective if and only if it is complete. is BooSp-projective if and only if it is extremally dis(5) A boolean space of a Cantor is and connected BooSp-injective if and only if it is a retract spacefr if and if is it discrete and is Toponly (6) A topological space is Top-projective is and if it indiscrete if and only non-empty. injective A boolean

(4)

is

algebra

DEFINITION

12.18

‘6’-objectS

A

is called

for ‘6 if and

separator

a

’6

hom(S. _):

only

if the functor

Set

->

is faithful.

OF

(CHARACTERIZATION

PROPOSITION

12.19

for ‘6’.

coseparator

NOTION:

DUAL

SEPARATORS)

.

‘K-objectS

A

is

for

separator

a

‘6’ if and

I

.

only if

whenever

A 2;

B

distinct

are

9

(é-nzorphisms,there

exists

a

fi-morphism S _",

A such

that

S—xaA—IsBaéSLALB.

The

“internal” A

functions

these

are

such

that

A

é;

B

distinct, then hom(S, f) ¢

are

characterization

above

description

‘6’-objeetC

is

a

with domain

hom(S, A). Hence, there hom(S, f)(x) 96 hom(S, g)(x); i.e., fox aé g by reversing each of these implications. [:1

be obtained

can

12.20

and

9

hom(S, 9). But x e hom(S, A) converse

is faithful

If hom(S, _)

Proof:

of separators.

motivates

Dualizing it, ‘0”if and

terminology

our we

obtain

the

and

is o

some

x.

gives following:

The

an

I

only if whenever

are

distinct

precisely the non—empty sets. Top, Top2 or Comp'I‘z are precisely the non-empty

spaces

for

coseparator

A :3

B

9

there ‘6’-nwrphisms,

exists

a

g-morphism

B

L» C such that

ALB—LegeA—‘sB—LC EXAMPLES

12.21

(l)

The

separators

for

(2)

The

separators

for

in these

(3)

The

Set

categories. of integers group

are

Z under

addition

'l' A space is extremnlly disconnected provided that if and only if it is homeomorphic Cantor space space.

is

a

separator

the closure with some

of every power

for open

of the

Grp

and

set

is open

twmpoint

for Ab. and is discrete

a

Sec.

12

Properties of Funclors

(4) The monoid (5) For

every

(6) The

sets

of natural

ring R, with

The spaces

(7)

R is

least

at

numbers a

N under

elements

with non-trivial

addition

is

a

for Mon.

separator

for R-Mod.

separator

two

75

for Set. precisely the coseparators subspaces are precisely the coseparators

are

indiscrete

for Top.

(8) The two-element for the category The

(9)

of

discrete

To~spaces. R/Z group

(10)

The circle

group,

it is

a

coseparator

(1 1) The two-element

(12)

The

(13)

The

closed

is not

discrete

and

indiscrete

not

is

a

coseparator

To-spaces.

two-element

dimensional

that

space

is

a

a

interval

is

algebra a

the

a

for

coseparator for

coseparator

CompT2

complex numbers, C, regarded BanSp, and for Bansz (Hahn-Banach theorem). (l4) None of the categories Topz, Rng, or SGrp has as

Banach

a

a

zero-

compact groups.

BooAlg. for

or

is

space

a

of

category

for Ab, and considered as of locally compact abelian

coseparator

is

for

coseparator

for the category

boolean

unit

is

space

a

CRegTz. for

coseparator

coseparator.T

EXERCISES 12A.

Give

lZB.

Prove

example of

an

equivalent:

(a) (b) (c) (d)

F preserves F preserves

constant

F preserves F preserves

zero

which

12D. on

12E.

Let

that

the

zero

does

not

monomorphisms.

preserve

objects and d

.i) Q,

then

the

following

morphisms. morphisms. morphisms. objects.

example of a surjection

an

is

not

Prove

(a) F is full. (b) F is one-to-one Prove

zero

image ofG;

the

which

coconstant

Give

l2C. functor

if .1! and .676have

that

are

functor

a

that

when

the

and when

F: .9!

on

_.

:3 be

full

a

functor

which

composition F faithful. then

it is a

is not

a

surjection and

a

dense

objects.

on

functor

that

has

G of functors

a

is full, then

F is full

G is faithful. one

of the

following properties:

objects.

“image of .2!" under

F is

a

subcategory

of 3

(cf. Exercise

98).

(a) Prove that the categories Set, POS, Mon, Grp, and Ab can be fully embedded in Cat (i.e.. for each of these categories. 5!, there is a full embedding functor F: s! Cat). [Consider. e.g., the subcategory of all small discrete categories and the subcategory of all small one-object categories] 12F. -.

1 To prove this. use the following facts: (I) for each Trspace X there is a T,-space map from Y to X is constant. There exist arbitrarily large fields and

Y with

more

than

one

point such

tinuous

(2)

arbitrarily

large simple

groups.

that

every

con-

(b) Prove that a mtegory with a zero object. 120. as

and Natural

Erector:

76

a

is

pointed if and only if it

be

an

fully embedded

that f: A —> Bis an epimorphism in Men between one-object categories A and B) is an

12H.

Prove

reflects initial

that

objects,

in

V

category

a

if and only if f (considered epimorphism in Cat.

Prove

functor

Chap.

Transformations

and full, faithful, and dense preserves objects, projective objects, injective objects, separators,

functor

every

terminal

that

is

and

dense

and coseparators. Show

12].

that

but that

identities,

Show

12J.

full, faithful,

a

that

a

functor

does

reflect

necessarily

not

embedding does reflect identities.

every

A is

@object

for ‘0”if and

coseparator

a

only if hom(_,

A) is

faithful.

and

coseparators

of

projective and

injective objects

and

separators

quasieategories.

to

Categories and Subcategories of Set provided that there is U, (.21, U) is a concrete

Concretizable

12L.

notions

the

Generalize

12K.

A category .9! is said to be concretizable U: .2! —» Set (i.e., provided that for some

faithful

some

category).

functor Prove

the

power

set

following: (a) (b) (c) (d) (e) (f)

category is concretizable. Every subcategory of a concretizable with a separator is concretizable. Every mtcgory Cat is concretizable. The category of sets If the category (g is

relations

and

then

concretizable,

mtegory % is concretizable. functorg: Set” -, Set is faithful.] (g) Every small category is concretizable.

and

iffe

is concretizable.

is the

so

then

If the

UM)

them

between

[The

[If ‘6‘ is small,

A) | a U{Iwm.,(13,

=

category ‘62. contravariant

arrow

is %’°”.

so

Mor(‘6’), let 00‘) be defined by U(j')(g)

for each

A

e

01:66),

let

012m}

5

fo 9.]

=

is concretizable. category [Let 0 be the class of all ordinal numbers, sets and let A be a set disjoint from for each a e 0 let B, and C, be disjoint non-empty each B, and each C,. Let

(h) Not

every

01166)

{A}U {Czlaew}.

(B, | :16ka

=

Let

Itamg(X, X) 110%(A, 3,)

Q

=

Itamg(B,, 3,) Immewm A)

home“. Cr.)

for each

{Ix}

=

=

Q

=

=

{have} (It:

Immth

=

Ca)

Itomg(C,, B”)

=

{"3 °fzv

for

all

for all

¢

:1

=

a

{le °f=}

if

(9, fl

0.

e

a

¢

0;

e

3;

0.);

UL) (It: °f=

9,

{93°f¢} g

0b(%’);

Iromg(C,, Cl!)if

{fa};

=

6

hontg(C,, A)

=

=

Immewm C3)

X

6

¢ 9; a

¢

°f.);

[3;

Sec.

13

Natural

Verify that U: 93’ —>

this

Transformations and Natural

determines

a

‘6

category

and

77

lsomorpltisms

that

is

there

faithful

no

functor

Set.]

(i) A category ‘6’ is concretizable only if there is some embedding

§l3

NATURAL

if and

only if it E: ‘6

functor

be embedded

can

in Set; i.e., if and

Set.

—.

TRANSFORMATIONS

AND

NATURAL

ISOMORPHISMS

Until

now

and

another"

to

have

we

We

are

one

functor

functors

able

now

morphisms

seen

define

to

a

as

natural

a

as

words, functors

Proposition

13.7). already “natural way" to

formations is

a

has

dimensional same

to

study

functor.

these

in

given

no

“natural

Historically,

the

to

way"

i.e.,

it

shown

the

on

go from

theory adequately

trans-

that

category

dual

in

precise

of natural

was

was

to

transformations

concept

second to

category

situations;

“getting from regarded as morphisms

(as will be made the

Chapter I, where

field F

a

is

for

l)

another".

of

way

and natural

identity functor

the

over

similar

and

motivation

a

go from

spaces but that there

first dual

the

been

vector

category,

to

that

a

object

one

to

category

be

can

categories (as has been made precise in §l be regarded as morphisms between functors Recall

one

transformations

between can

“getting from

of

way

“getting from

of

way

In other

to another".

as

functor

there

of finite on

the

the

identity functor developed in order deal

with

natural

transformations. DEFINITION

13.1

Let F: as!

(l) (F,

A

natural

G)

I],

G: .52!

.4? and

—>

tramformation

where

0b(.d)

n:

—>

98 be functors.

->

(or functor morphism) from F to G is a triple Mor(.€&) is a function satisfying the following

conditions:

(i) (ii)

For

each

'1‘:

F01)

For

each

taxi-object A, 11(A)(usually —>

denoted

by m) is

a

:B-morphism

C(A).

L, A',

A

d-morphism

the

diagram

"A

F(A)—>G(A)

A

if F(A’)-;l>G(A')

N!)

GU)

A’

commutes.

(2)

A natural

that

for each

(3)

F and

if there

G

exists

transformation

A, '14 is

d-object are

a

said

natural

(F,

to

be

a

n,

G) is called

a

natural

isomorphism provided

w-isomorphism.

naturally isomorphic (denoted

isomorphism

from

F to

G.

by

F a

G) if and only

and

Funcmrs

78

If

(F,

domain

the

functors,

called then F (resp. G) is sometimes G) is a natural transformation, with of to the situation (F, n, G). Analogously (resp. codomain) the notations (F, n, G), 11:!" —> G, F 1-» G and (’1‘): F -> G are used r],

the

:1 and

notation

by abuse of triple (F, n, G).

and

interchangeably between

V

Chap.

NaturalsTransformalions

does

usually

one

distinguish

not

EXAMPLES

13.2

1; from any functor F to itself, which (l) The identity natural transformation assigns to each object A the identity morphism on F (A ), is a natural isomorphism. transformation 11 ('14) from the (2) For each field F, there is a natural identity functor on F-Mod to the second dual functor, which assigns to each vector ",4 defined by: space A the linear transformation =

(nA(X))(g) the second

that

Note

dual

is

functor

9(x).

=

precisely

Hont(—, F)

HOM(—, P)".

c

where

F): (F-Mod)"

Hom(_, In

exactly

identity

functor

functor

and

abelian

(3) n

the

2

variant

be

each

A, there

set

a

set.

Then

there

is

a

ham-functor

product by A” (_ x A): Set

functor

(A

is x

Set, defined

->

a

(5) There Grp

is the

is

inclusion

to the

the

H

:

value

=

from

Grp

—o

Ab

functor. at

each

=

Set

on

to

the contra-

(b. a). the

subgroup functor

commutator

lam

A’ is the

Foot.

be the

abelianization

There

is

whose

value

at each

A

group

group

A is the

commutator

A.

functor

natural

a

A

(where

2)

homomorphism

inclusion

whose

locally compact isomorphism. natural isomorphism

=

—»

identity functor A’

(6) Let

of

natural

('13) from the “left isomorphism n Set to the “right product by A" functor

natural

transformation

natural

a

Grp (92(8))

—>

the dual

g"[{0}].

=

_): Set by:

hom(_,

second

by:

na((a. bl) F:

spaces is

the

to

category latter

My) (4) For

the

on

from

transformation

The

9, defined

f unctor

set

linear

functor.

two-element

a

contravariant

the

power

dual

the second

to

from (11A)

=

category

natural

a

of normed

identity functor

the

{0, 1}

=

define

can

one

way,

the

from

groups

Let

same

on

F-Mod.

—.

transformation

canonical

L

subgroup

(92(8))

of

from

homomorphism

A/A' A).

and

K: Ab

16,,

to

-*

K

Grp o

H,

Transformations and Natural

Natural

13

Sec.

79

lsomorphisms

from the nth homology functor Hfl to (7) There is a natural transformation the (n 1)st homology functor H“- ,, which assigns to A the connecting homo1(A). morphism 64: H,(A) ',(X) to each homomorphism 1r,,(X) (8) The assignment of the Hurewicz the nth from a transformation natural X is homotopy functor topological space Grp. Grp to the nth homology functor H": Top 1:": Top —

—’

,,_

—>

—->

-.

(9)

If Q? is any

category

and

L»

B

C is

%’-morphism,then

a

there

is

a

natural

transformation :1:

defined

_)

hode,

by

M9) (10)

homgw, _)

—)

U be the

Let

from

functor

Set

=

from

functor

underlying to Grp. There

‘’f.

g

Grp

natural

exist

Set

to

and

let F be the

free

transformations

'1

=

('14): lse.-*

s

=

(ea):Fo

WI:

and U»

law

U(F(A)) is the insertion of the generators and as: F(U(B)) —>B unique group homomorphism induced by the identity function on U(B). (11) Let [3: CRegT2 CompT2 be the Stone-éech compactification functor and let T: CompT2L> CRegTz be the inclusion functor. There exist natural where is the

114: A

—’

->

transformations

(fix): lCReuTz 7°13 -’

=

'l

and 3

(5r): 5°

=

T

1Com“;

"

for each space X, "X is the usual embedding of X into and for each Y, e, is the unique homeomorphism induced

where

(12)

For

each

set

A, there 0

=

is

(no): (—

x

by

compactification, identity on Y.

the

transformation

natural

a

its

A)°hom5..(A._)

-»

ls“

where "a:

is defined

B"

(13)

For

—>

B

by "BU. a)

q is called

A

x

the evaluation each

set

A, there

homo((_

natural is x

a

=

f“)-

transformation. natural

isomorphism

A)"x ls“):

Set”

x

n from

Set

—»

Set

the

bifunctor

to

V

Chap.

Transformations

and Natural

Functors

80

bifunctor

the

(lsflop

homo

Set‘"D

hom(A,_)):

x

Set

x

—>

Set

where 11361 CB“ is defined

by

(nuc(f)(b))(a) (14)

abelian

each

For

(0‘)},

—’

is

A, there

group

f (b, a).

=

natural

a

isomorphism

n from

the

bifunctor Ham to

0

H(A) G

F(/)

H(f)1

(D

F(B)i>a(a)i>nw) for

each

A

composition

of

B in

the

isomorphisms

is

H, and

of F, G, and

domain an

from

the

fact

the

that

{:1

isomorphism.

PROPOSITION

13.5

composition of natural

The are

L;

transformations such (n e) are defined and are

natural

and 6

o

o

transformations that

6

equal.

o

n and

E]

is associative n

o

s

are

,'

i.e., if 6, n, and

defined, then (5

o

n)

s o

t:

Sec.

13

Natural

A natural

there

exists

5

°

(I) If

G is a natural transformation 1]: F isomorphism if —> F some natural 6: G such that 6 on transfitrmatian —)

Assume

n: F

G is

—>

that

—F—; a ands!

d

natural

a

Clearly, 6: (2) Iféon

G

—>

1m, 13.7

PROPOSITION

There

natural

a

o

=

exists

all

define

o

by

6(A) 6

transformation,

lam.

=

the function

o

e

Hence, each

6A

=

n

ifi'.

=

l,-

=

and

n

6

o

15.

=

0b(s¢),6A 01),, (6011),, '14 is an isomorphism. [:1 =

=

91, dom, cod, 0) where 8 is the conglomerate quasicategory (“5-, conglomerate of all natural transformations, dam and cat! assign to each natural transformationits domain and codomain, is the composition of natural transformations defined above

a

91 is the

jitnctors, functions that respectively, and (13.3). [I are

13.8

o

DEFINITION

The

Analogously functors

them,

described

in the above

and

‘6. by 97011.11" with (daily, when

quasicategory

category of all functors

Star

only if 1,- and

=

.93.

la,thenforeachA

=

and

More?)

—>

Irandn 06 6,, (n 6),,

=

11,,

F is

1»

isomorphism,

6: Obese)

to

and

Ia.

=

Proof:

of

81

Isomorphisms

THEOREM

13.6

'1

and Natural

Transformations

obtain

we

to

the

between a

proposition is

called

the

quasi-

is denoted

situation

small

categories denoted by

category,

and

natural

Func

(see

restrict

we

attention

our

transformations Exercise

between

13H).

Products

We next 13.9

investigate

another

of

way

natural

composing

transformations.

PROPOSITION I"

Let sy’ natural

g;

ll

93 and

Q

‘6’ befimctors and

? Then

transformations.

for each

A

let :1: F

G amid:

H

-v

K be

the square

0b(.s4),

e

—>

(HoF)(A)H(l;(H°G)(/0 56A

5m

(K°F)(A)

5090)“)

K (‘74)

commutes.

i/‘u: 01709!) Mor(‘6') is the above i.e., diagonal of square,

Furthermore, A to the

-»

NA

then

(H

o

F, n, K

o

G)

is

a

=

66A ”01.4)

natural

°

=

the

function that sends each d-object

KUIA) 554,

transformation.

°

82

and

Functors

The

Proof: 6 is

and

To Since

that

see

p is

natural

a

Transformations

because

commutes square natural transformation

a

I] is

Natural

'14: H to K.

from

HA)

natural

transformation, let the square transformation, a

Chap.

G(A)

—>

is

—f—> A' be

A

V

fi-morphism

a

Jed-morphism.

an

".4

F(A)——>G(A)

A

G(D

H!)

f

F(A’) —')G(A')

A'

"A!

commutes.

H to

Applying

this,

we

that

see

the left square

of the

diagram

"A

(Horxmmmoaxmfiuxooxm (H°F)(D

(HoG)(f)

(KOGXD

°G)(A')

G)(A')—>(K °F)(A’);,7—>(H°

(H

“A" commutes.

Since

00'):

G(A)

G(A') is

—»

transformation

from

H to

transformation

from

H

13.10

is called

the star

a:

o

6 is a is

Thus,

a a

natural natural

transformation

H

product of 6 and

11+ K

F

o

n and

o

G constructed

is denoted

by 5

in a:

Pr0position

13.9

'1-

PROPOSITION

The

(6

K

since

DEFINITION

The natural

13.11

K, the F to

o

£3-morphism, and right square commutes. G. [:] a

11)*

s

Proof: Either

i.e., whenever

is associative;

product and 5 a: (n

star

a)

*

side

defined and

are

is defined

i

are

we

cl

r

>0.A’

the

((5

*

definition

'1) * 5)A

=

=

of

G)(€A) (5

we

have

a

It

a are

defined,then

situation

such

as:

”4,92

l6

G

K

product several

star

n and n

l"

>(' 6)

T

Applying

only if F

S

/

t

equal.

are

if and

6

>

times,

we

9 obtain

K(G(5A)) (KOISA) 66°34) K(G(3A) USA) 6(FoS)A (6 “ ('1 * 3)),4- Cl

(K

°

°

°

°

*

")SA

°

=

=

°

Sec.

Natural

13

Isomorpln’sms

and Natural

Transformations

LA\V) (INTERCHANGE natural v, p, q, and transformations any

83

THEOREM

13.12

For

(t'°u)*('t°8)

(“10°01”):

=

i.e., when IIu-leflsitle is thfilu'tl, then

so

is the

For the left side to be defined,

Proof:

a

right

side and have

must

we

they a

are

equal. such

situation

as:

.m—f—va—Kva

in

al

.91_G)Q_L_)(/;

"l is defined, then so is the of natural transformations;

Thus, if thclcftsside and

composition (0'

°

But since

l1) * ('1 3)),1 °

it is

a

1"

natural

(V ll)!“ K((u °

=

°

transformation

°

and

right. By

3»)

=

the definition

"MA

°

#114

GM) "—3H(A)

°

is

Kala) a

of star

°

product

K934)-

morphism

in a,

the square

mam—filmwmn in

F‘

GM)

Hm

WLU‘KA»

L(G(A)) commutes.

Thus, the

middle

two

of the above

terms

expression

can

be

replaced

to

obtain

vmfi) But

by

the definition

of star

For

typographical

transformation

on

*

°

F,

0

’1) (11t 8)),4- Cl °

we

of

often

writing I r for the symbol

use

example, (’1* F)A

=

firm)

and

(F

K024).

this is

instead

reasons.

the functor

Lola) "5(4)

product, ((v

13.13

°

*

8)‘.

=

Fm)-

the

identity natural

“F” itself.

Thus, for

84

Functors

Given

the

Natural

(GODEMENT’S

COROLLARY

13.14

and

“FIVE"

and natural

functors

Transformations

Chap. V

RULES)

transformations, U ———>

is K>’€

L349

sf

4—)

Vtfl

in

H

w

—>

the

——>

1“6—09”;

hold:

following

(I) (GOF)*§ (2) 5*(K°L)

0*(F*€) (5*K)*L

=

=

(3)1tv*K=1wK (4) Fr lu =1F=u (5) F*(€*K)= (F*€)*K (6) F*('t°€)*K= (F*'I*K)°(F*~E*K) (7) The square

review ”*U

utV

H°U——>H°V [hi

[:1

commutes.

PROPOSITION

13.15

There

exists

a

((5, 9t, dam, coa’, *) where (E is tlte conglomerate the conglomerate of all natural transformations, dam is the

quasicategory

of all categories, ‘Jt is function that assigns to each natural of its domain, cat! is the function codomain

of

The

13.16

its

and

above

that

assigns

*

is the star

proposition

together

domain, ‘Jt of

all

the category each natural

transformation

natural

to

product of with

transformation transformations.

natural

13.7 shows

Proposition

transformations

that is the domain

be

that

of

the

[:1 the the

conglomerate thought i.e., with the “usual ways; morphisms of a quasicategory in two different composition” or with the “star product” *. Also, these two types of composition are intimately linked by the “interchange law” (13.12). Thus, the triple called the “double quasicatcgory”, ./V .919" of natural (91, o, as) is sometimes can

as

c

,

transformations.

EXERCISES in general that Show (6), (7), (8), (9). (10), (11),

13A.

132(5),

none

or

of

(12) is

the a

natural natural

transformations

isomorphism.

of

Example

Sec.

Natural

13

138. BC.

Prove

that

Show

that

then

phism),

[(F"", t], G”)

the trifunctors

of Exercise

2

{0, 1} be

=

the

between

two~element

a

covariant

10A

morphism functor( )2 (see Exercise

isomorphic.

naturally

are

Prove

set.

ham-functor

85

Isomorphisttts

if (G, It. F) is a natural transformation is a natural transformation (F°", u, G”) (resp. called the opposite of (G, t], F ).] is sometimes

Let

13D.

and Natural

Transformations

that

hom(2, _)

on

isomor-

natural

(mp. natural

isomorphism).

there

is

Set

and

the

natural

a

natural

a

iso-

squaring

9A(b)). F

13E.

Show

formation.

then for Q

wtegory 13F.

be

r] can

:53a

regarded

functors

are as

functor

a

and from

:1: F

J!

to

—.

G is

trans-

32 (where 92 is the

arrow-

(4.16)). that

Show

group” functor 130.

if .9!

that

there

F: Set

—’

are

at

Grp

to

least

two

natural

“free abelian

the

the “free

transformations

from

group” functor

C: Set

-*

Grp.

Let

Homnws,(_, 2): 3005p”

BooAlg

-»

and

2): BooAlg

Hammm(_, (see Examples 10.60) and (8)). Show

BooSp”

-o

that

2) HomeooMg(—. 2) E ”Oilluoosn(——. °

'

locum

and

2)

HontnooMg(_,2) Homms,(_, o

Let Func

13H.

In

131.

subquasicategory of .97021.A/‘(6’whose categories. Prove that Func is a category.

In

l3J.

‘

U)

F)°(F‘

the

la

=

it)

'1“-

=

that

show

Example 1120]),

(T ”9°07"

T)

mam:

1])

1r

=

that

(8' Show

l3K.

exists

are

that

(6

and

objects

that

Example 13.2(10), show

(U‘8)°('l' and

1.1005,".

be the full

small

between

functors

2

a

natural

13]...

.. if F. Gm! transformation :1: F

that

Prove

as if, considered functor. identity

Q

that

a

a

functor

are

a

functors

and

a

is

pointed,

then

there

G.

endomorphism

group on

—»

t,,.

=

is

an

one-object category,

inner it is

automorphism if and only isomorphic to the

naturally

and

Functars

86

§l4

Natural

AND

ISOMORPHISMS

Chap.

Transformations

V

OF

EQUIVALENCES

CATEGORIES

for problem of determining what it means two categories to be “essentially the same”. We begin by introducing the notion of “isomorphism” of categories, which seems appropriate at first glance, but which is too strong. The weaker concept of “equivalence” of categories is In this

shown

consider

we

F: .21

A functor it is

that exists

is

there

that

is said

a

—>

in the

isomorphism

an

functor

a

0: fl

(2) Categories

Q

such

s!

->

and 59

said

are

isomorphism

an

isomorphism from a! to 9 provided ; i.e., provided that there quasicategory (6:49 that G F 13. 1d and F0 G to be isomorphic (denoted by a! g 9) provided be

to

an

o

=

=

them.

between

EXAMPLES

14.2

(1) Every identity functor (2)

sameness”.

for “essential

notion

be the proper

to

the

DEFINITION

14.1

(l)

section,

There

is

is

isomorphism

an

an

isomorphism.

from

Rng

itself

to

sends

which

each

ring

R to its

opposite ring R”. (3)

For

every

ring R,

(4) The category Rng ring of integers).

R-Mod is

is

with

isomorphic

isomorphic

with

the

Mod-R*.

category

Z-Alg (where

(5) Ab and Z-Mod are isomorphic. (6) The category BooAlg is isomorphic with the category together with ring homomorphisms. (7) For any category if, (g x g)” g ‘6’” x ‘6”.

Z is the

of boolean

ringsT

category (8) Let A be a one-element (resp. two-element) set. Then the comma sets the of is with the category of pointed (resp. (A, Set) isomorphic category bi-pointed sets) (see Exercise 48). (9) TopBun g Topz, and for any topological space B, TopBunB '5 (Top, B) (see Exercise 45).

(10)

A

of Set

subcategory 14.3

(see Exercise

PROPOSITION

If

F: M

F is

—»

Q is

t

only

if it is

with

isomorphic

some

12L).

OF ISOMORPHISMS) (CHARACTERIZATION the then following are equivalent: aflmctor,

isomorphism. (2) The function F: Mama!) (3) F is full and faithful and is a bijecu’on.

(1)

if and

is concretizable

category

an

A boolean

ring is

a

ring in which

—>

Mor(93)

the associated

each

element

is

a

bijection.

object function is idempotent

with

F:

respect

0b(.9!) to

—>

Ohm?)

multiplication.

Sec.

14

Isomorphisms and Equivalences of Categories

87

The

equivalence of (l) and (2) follows from the fact that for any on ‘6’, the identity function category Mor(‘6’) is the identity functor on g, The equivalence of (2) and (3) follows from the one-to-one correspondence between and identities in objects any category. 1:]

Proof:

The category F-Mod of all vector spaces %’ consisting of all objects of the form F the

obviously not view, they are

same

nor

even

“essentially

characteristics.

The

main

the

field F and its full

subcategory (i.e., all powers of the field F) are isomorphic. Yet, from a categorical point of same", i.e., they have the same categorical

difference

over

a

'

between

the

lies in the

two

whereas

in ‘5’

fact that

isomorphic objects many objects isomorphic to any given object in F-Mod. Below we will define “equivalence of categories" in such a way that two categoriesaal and Q will be equivalent provided that the only difierence between them lies in the fact that in one of them some times than in the other—in other objects might be “counted” more words, provided that the categories obtained from .51 and Q by “counting" each object just once, are isomorphic. any

two

14.4

are

identical,

exist

there

different

DEFINITION

‘6 is called

(l) A category

skeletal

provided

that

‘6’-objects are

isomorphic

identical.

(2)

A skeleton

14.5

of

‘6’ is

category

a

a

maximal

full skeletal

subcategory

of %.

EXAMPLES

for Set. (I) The full subcategory of all cardinal numbers is a skeleton F' is a skeleton (2) For any field F, the full subcategory of all powers

for

F-Mod.

(3) For

field F, the full subcategory of all finite powers of all finite-dimensional vector F. over spaces

any

the category

F

"

is

a

skeleton

for

PROPOSITION

14.6

‘6 has

Every category Proof: If

a

let A g

B

skeleton. that

there

is

@isomorphism from A to B, 0b(‘6). Hence, by the Axiom of Choice (12(4)), it has a system of representatives 3’. Let a be the full subcategory of ‘6’ that is generated by J. Clearly, Q is skeletal and is contained in no other full skeletal of ‘6’. subcategory I] is

we

then

a

14.7

PROPOSITION

Any 14.8

skeletons

two

relation

equivalence

an

of

mean

a

category

a

on

are

isomorphic.

[:I

DEFINITION

A category

if and

only

d

is said

if .91 and

to

.48 have

be

equivalent to a category isomorphic skeletons.

.9

(denoted by

.95!

~

30)

14.9

Chap.

Transformations

V

PROPOSITION

“is

The relation

of

and Natural

Functors

88

all categories.

equivalent to"

is

relation

equivalence

an

on

the

conglomerate

I]

PROPOSITION

14.10

Skeletal

categories

equivalent if and only if they

are

isomorphic.

are

[:1

Categories .9! and Q have been defined to be isomorphic provided that Q. Likewise, we will define special functors is an isomorphism F: d called “equivalences”in such a way that sat and Q will be equivalent provided that there is an equivalence F: a! —» Q.

there

—>

THEOREM

14.11

de

If

Q

—t

is

the

following

equivalent:

are

dense.

full, faithful, and (2) There is aflmctor G: (3) There is a functor G: ezFOG—> lasuchthatFtn (1)

then

functor,

a

F is

Q

.9! such

—»

Q

—>

that F0 G

and natural

a!

(8:17)"1

=

la, and

2

G

o

F g

isomorphisms 11:1,,

anthe

id. G

-r

o

F and

(th)-l.

=

Proof: Clearly (3) implies (2). We will show that (2) implies (l) and (1) implies (3). (2)-4(1). Suppose that G: Q —» s! such that Go F g 1,, and Fe G a 13. Let n:

'4

G

—>

(i) If

A

F be

o

fi

natural

a

A'

isomorphism. such

.sl-morphisms

are

FU‘)

that

=

F(g),

and

if

9

A

A’ denotes

----->

either

f

A—Lm

9, then

or

the square

um)

16°th)=G°F(9) A’——;'-—>G i

i

o

AI

commutes.

Thus, '14- f Thus, F is faithful.

f g. (ii) Since =

=

o

F

o

G a

la,

11A. og

know

that

0(a)

g

that

(since

"A. is

a

monomorphism)

G is faithful.

for each

13(3)

B

0b(Q),

e

3.

=

F is dense.

(iii) Suppose from

so

Likewise,

we

Fe

Thus,

F(A')

that

A to A' such

F(A) i» F(A’). that

Then

f

=

n?

o

G(g)

the square

A——fl4——->G°F(A)

fl A’

1607) or

—>

G ’14!

°F(A’)

GoFU’)

o

'14 is

a

morphism

Sec.

lsomorphisms and Equivalence: of Categories

14

G aF(f )°'1.4 so that G(g)onA G is faithful, G 1’0"). Since G(g)

Thus,

commutes.

=

89

(since

1],,

is

an

g F(f). Concpimorphism) sequently, F is full. => (3). Let 4 be a system of representatives for the equivalence relation 2 (1) on 0b(.;s’). [Such exists by the Axiom of Choice (l.2(4)).] Since F is dense and and reflect isomorphisms, for each functors since full and faithful preserve is a unique member of d (which we will denote Be 0M9?) there by G(B)) such that F(G(B)) ; B. Thus, G : 0b(Q) 0b(.sz¢)is well-defined. Again using 8 6 0b(.@) choose one the Axiom of Choice, for each isomorphism —> B. F(G(B)) as: We will now define G on the morphisms of Q. If B L. B’ is a 3-morphism, =

=

0

-.

then

the

square

A»

new»

6.8!,g o

0

CR

9

new» F is full and

Since

commutes.

B

5'

T there

faithful,

f: G(B) such

that

F(f)

well-defined

=

cg.‘eg

function

0

G:

M0493)

F(G(ln)) So

=

that, since F is faithful, [in

i.

B’

L. B",

F(G(h N)”

C(g)

831013083 C(13)

0

in this

[and

=

For

any

B

6

establish

manner

a

0b(.%),

IF(G(B)) Fawn)=

=

Hence

lam.

=

It 0g

F(G(h))

=

C(B’)

-.

Model).

-.

unique sci-morphism

a

G preserves

identities.

then

5;)

=

let

53. We

is

Thus, since F is faithful, G(/:

c

Consequently, G is a functor. natural isomorphism from F

a

£8

(CE)

=

F(G(g))

°

g)

C

=

GUI)

=

a

h

o

F(G(h)

G( g),

Clearly, by G to I”.

0

the

so

88') (5;.1 cg o

°

as)

c

0(9))-

that

G preserves definition of G,

composition. (as) is a

s

=

1,, to G F, note that for each is a A e 0b(.n(), a“: Fe G F(A) FM) Q-isomorphism. Thus, since Fis full that Go F(A) such and faithful, there is a unique d-isomorphism 11‘: A PM) 8;) (l2.9). Since r.“ is natural, for any A _’_.A' the square To

establish

a

natural c

isomorphism

from

c

—’

—.

=

'1

8FA=F("A) FM) -———>F°G°F(A)

F071 F(A')

F"G°FU)

-.—l————>F°G°F(A') am.=FlnA,l

refiect

G

f

A’

Hence, :1

definition

the

Clearly, by

natural

a

°

F l f)

°F(A’)

G

T

(:14) is

=

the square

(12.8),

commutativity

—:A—>G 0F (A)

A

commutes.

from

isomorphism

of n, F

*

(e

=

n

F)".

t

1,, to Also by

G

F.

o

definition

the

F0166”) 501(5).Hence, Fl’la—(in)spam. By the definition Thus F0 C(83) 55' 3,, 656(3) F(t15(},)) F(G(sn)), Erma)since Fis faithful, (n G)"(B) (G c)(B). [:1 of

of

=

=

11,

e

=

=

=

0

V

Chap.

Transformations

functors

faithful

Thus, since

commutes.

Natural

and

Functors

90

G, that

It

=

a:

SO

DEFINITION

14.12

F is called

A functor

(l)

of the above

conditions

(2) (F, G,

n,

and

functors

thatFtn

is called

a)

l

n:

G

—v

Gulf)"

=

equivalence provided that it fulfills the equivalent theorem, i.e., provided that it is full, faithful, and dense. an provided that F and G are equivalence situation l are natural a: F G F and isomorphisms such an

—r

o

o

and

6*8

(11*0)".

=

PROPOSITION

14.13

is

(l) The composition of equivalences then

equivalence.

an

is F’”.

(2) If F is an equivalence, (3) If (F, G, n, e) is an equivalence situation, so

then

so

(G, F, a", :1").

is

and (2) follow from the fact that each of the properties—fullness, under and density—is closed (3) composition and is self-dual.

Proof: ( l) faithfulness, from

follows

the fact

for

that

(f‘ ')‘l

isomorphism],

any

=

f.

E]

PROPOSITION

14.14

If J

is

then

there

that

P

o

skeleton

a

exists

E

of functor

a

‘6

P:

E is

Id (i.e.,

=

‘6’ and

category

a

J

—>

section

a

J 9+ ‘6 is the inclusion

E:

functor,

(called the projection of ‘6 onto J) such in ‘1?st ). Furthermore, both E and P are

equivalences. [3 THEOREM

14.15

Categories F: .sl

—>

Proof: respectively, P: .2!

-

exists Eo J

an o

If .n/

and

equivalent if

are

9,

then

there

that

.97

J?.

~

such

a

a? and the inclusion J:.c7 isomorphism

P: .9!

—>

Conversely, d

and :28

s!

and

there

only if

exists

equivalence

an

Q.

J9,

J? is if F

fling!

By

skeletons

.94 is

—*

are

the

—»

J? is

an

inclusion

a? and

J? of .2! and

proposition, projection there definition, equivalences. By Thus must be an equivalence).

equivalence, .r? and

:23,

the

above

the

J?C-> J?

J? (which equivalence (14.13).

an :

E:

exist

52

L

J?

and is

I}? are the

skeletons

projection,

for then

Sec.

Q

14

Fo

o

.97

K:

a? is

equivalence. Thus, Q F .57 and Q are skeletal, the associated object bijection. Hence, it is an isomorphism (14.3), E]

since be

Isomorplu'sms and Equivalences of Categories

a

alent.

—»

EXAMPLES

14.16

an

OF

o

of

(2) Every equivalence (14.14).

K is full

a

function that

so

and

for

faithful, and

Q

.31 and

F

o

33

0

K must

equiv-

are

EQUIVALENCES

(1) Every isomorphism is inclusion

91

equivalence.

an

skeleton

a

and

projection

every

W is naturally isomorphic (3) If F: 9? conversely (see Exercise 14D). —>

1%, then

to

F is

onto

an

a

skeleton

equivalence,

is

but not

field F, the category of finite dimensional vector spaces over the of F-matrices a category (15(4)). [Given basis, consider equivalent matrix associated with each linear transformation] For

(4)

F is

any

the

to

any field F. Then the

For

(5) over

is

its dual

to

F, let a! functor

be the

Hom(_,

equivalence,

an

but

of finite

category a!”

F): not

an

adjoint (equipped isomorphism. (7)

the

strong

sends

vector

each

spaces

vector

space

isomorphism.

(6) Let .‘l’ be the category of reflexive locally convex The functor Hom(_, C): I!” —> .S!’ that spaces. with

dimensional

.al that

—.

an

Hausdorff sends is

topology),

each

linear such

equivalence,

an

topological space to but not

its an

Let 9? be the category of reflexive Banach spaces and norm-decreasing linear The functor Hom(_, C): 9?” —v 9? that sends each reflexive

transformations. Banach

(8)

space

LCAb

Let

functor

its

to

be

Hom(_,

abelian

group

conjugate the

space

category

R/Z): LCAb"" its

to

of —»

is

equivalence, but not locally compact abelian an

LCAb

of characters

group

that

sends

is

equivalence,

an

each

an

isomorphism. Then

groups.

the

locally compact but

not

an

iso-

morphism. Then the functor (9) Let CompAb be the category of compact abelian groups. -* that sends each abelian Ab, Hom(_, R/Z): CompAb” compact group to its is an equivalence, but not an isomorphism. (discrete) group of characters Honi(_, 2): 8005p” (10) The functor BooAlg that sends each boolean a

space

to

the boolean

isomorphism. (l l) The functor A to

is

an

its carrier

algebra

Hom(_,

of its

clopen

C): C"-Alg"”

—>

is

CompTz

C) considered

i.e., Hom(A, space; equivalence, but not an isomorphism.

The

subsets

as

equivalence,

an

that a

sends

each

subspace of

the

but not

an

C*—algebra space

C",

preceding examples indicate quite strongly that the concept of equivmore important categorically than the concept of isomorphism of categories. Moreover, since equivalences preserve (and reflect) all “essential” Theorem 12.10 and Exercise l2H), one might even categorical properties (see define a property of categories to be “categorical" provided that it is preserved by equivalences. alence

is much

last

The

of

few

the

Natural

and

Functors

92

preceding

Transformations

examples

Chap.

motivate

also

the

V

following

definition. DEFINITION

14.17

3’8 are

(1) Categories .al and

dually equivalent if and only if a!” and 33

called

are

equivalent. .31 is called

(2) A category

self-dual

that

provided

dually equivalent

to

itself.

EXAMPLES

l4.l8

‘6”

‘6’,g and

(I) For any category

dually equivalent.

are

of R-matrices

ring R, the category (2) For any commutative [Consider the functor that sends each matrix (3) The category

of finite dimensional

(4) The category

of reflexive

(5)

it is

The

category

vector

Banach

spaces

of locally compact

abelian

(6) The category

of compact

abelian

(7) The category

of boolean

spaces

(8) Comp'I‘z is dually equivalent (9) Set is dually equivalent (see Exercise 14H).

is

its

transpose] over

spaces

any

field is self-dual.

is self-dual.

is self-dual.

groups is

Ab.

dually equivalent to dually equivalent to BooAlg.

groups

C”-Alg.

to

of

the category

to

to

is self-dual.

(35(4))

complete

atomic

boolean

algebras

EXERCISES Show

MA. if there

is

148.

an

that

F: d

embedding

each

that

Show

of

each

to

isomorphic

with

a

subcategory of .673if and only

93.

—»

the

of each of the other

subcategory (or even equivalent)

is

categoryd

a

Set, Grp, and

categories

two,

but that

two

no

of these

Top is isomorphic to a categories are isomorphic

other.

Finilely Generated

Spaces be finitely generated provided that topological of a disjoint topological union of finite spaces. quotient (a) Prove that for any space X, the following are equivalent: 14C.

A

is said

space

to

(i) X is finitely generated. of any (ii) The intersection c X, then (iii) "A

family of A”

sets

open

U

=

it is

a

topologieal

in X is open.

{0}"

«16A

(where

“"’

denotes

closure).

of Top consisting of all finitely generated spaces (b) Prove that the' full subcategory of quasi008 (resp. finitely generated Tyspaces) is isomorphic with the category of partially-ordered sets). sets (resp. POS ordered MD.

naturally

Prove

that

isomorphic

an

to

isomorphism 1.4- [See Exercise

F :5!

—’

l3L.]

d

on

a

category

is not

necessarily

15

Sec.

Funcror

14E. in fact, if

gories d

that

and

are

Determine or

whether

it is

not

Show

The

14H.

(a) Prove (b) Prove

algebras

Set is

that

Set”

set

of atoms

(c)

Prove

of

Show

is

equivalent to

that

if X is to

(4!, U) and (Q, V) be following concepts:

in Set and

initial

an

object

an

isomorphism

(2) There

exists

an

equivalence

(3) There

exist

isomorphisms

(4) There

exist

equivalenoes sz

—~

(5) There

exists

equivalence

H: d

an

H: .2! H: d

H: d

exists U

=

an

isomorphism

Set

be embedded

can

e, then the

category

a

Discuss

.4? such

Qand

Qand

the

that

U

that

K:

U

Set

Set

K:

Q and

a

H: d

of the

in Set".

comma

category

V

=

=

V

o

o

H.

that

Set such

—~

that

K

U

c

Ko

U

isomorphism K: Set

an

the

H.

Set such

—o

between

relationships

V

=

=

V

H.

o

o

H.

—>

Set

such

—.

Q and

equivalence K: Set

an

—.

Set such

VOH. H: d

exists

an

isomorphism

(8) There

exists

an

equivalence H: d

categories which

Q such

_.

—»

3

satisfy (8) above

(515 FUNCI‘OR

such

that that

U a

called

are

U a

V V

o

equivalent

o

H.

H. concrete

categories.

CATEGORIES

DEFINITION

If .9! and

categories, then objects precisely the functors by 9" and is called the (quasi)category (quasi)category Let, Q].

Q

are

are

the

full

from

subquasicategory

w

of functors

to

of

$41.14” W

Q is denoted

by [51, $3] or

from

or

.9! to Q

the functor

algebra is called atomic if and only if each of its elements x is the suprea with a s x. (a is called an atom of all of the atoms provided that it is an immediate ofO; i.e., 0 95 aand il'O < y S a, then y = a.)

T A complete boolean successor

that

.9 such

—’

—»

—.

(7) There

mum

all subsets

VoH.

(6) There

whose

of

categories.

concrete

exists

15.]

atomic? boolean complete atomic

Categories

(1) There

Concrete

complete,

(6’.

Let

thatKo

of

category

homomorphisms. [Hint Each the complete boolean algebra of

be embedded

can

Equivalence of Concrete

thatK°U=

the

to

A.]

isomorphic

l4].

initial, terminal, monomorphisms, and pointedness.

self-dual.

not

complete A is isomorphic Set”

that

141.

equivalences between categories preserve and coseparators, injective objects, separators morphisms. zero morphisms, connectedness,

boolean

algebra

“categorical property“. i.e.,

a

Category Set”

that and

is

that

projective, epimorphisms, constant zero,

“smallness"

not

or

preserved by equivalenoes.

and

(X, ‘6’) is

small

=

MG.

boolean

cardinality of

93

categories is not preserved by equivalenoes; infinite cardinal B any numbers, then there are equivalent cateand .9 such that card (Mor(.d)) a and card (Mor(Qi))= [3. a

MP. whether

Prove

Categories

and Natural

Functors

94

V

Chap.

EXAMPLES

15.2

(1)

Transformations

by 1, then for any just one morphism is denoted @, [1, ‘3] is isomorphic to ‘6. -—->is denoted by 2, then for any category ‘6, the functor category ‘62 (4.16). [2, ‘6] is isomorphic to the arrow-category

If the

category If the

(2)

with

category

category

0—).

(3)

If the

the functor

we

Ham(F,

have

now

$.21?"

Ham:

G

=

Ho

0

where

H0m(d,

category

any

triangle-category

Ham-functor"

(6.915",

—»

the

for

then

by 3,

to

“internal

an

(6.91.7

x

G)(I-I)

is denoted

[3, $0] is isomorphic

category

that

Notice

\l,

category

‘6,

%3 (4.17).

for

W421?

fl)

=

namely,

,

[$1, 93]

and

F.

PROPOSITION

15.3

If 5/ morphic to

small, then for

is

A functor

from

function

[471,fl]

is

iso-

category.

a

Proof:

the quasicategary

3,

category

any

F: a!

6(17‘)of F. Since

.9! is

Mar

actually

a

set,

a

is Mar

so

triple

with

.93. Associate

.2! to Mar

Mar

.48 is

—>

each

.9!

x

(.91, 17‘, .93)where F is a functor F, the graph F[Mor .21] (1.1(3) and (4)).

F

=

such

Hence, since

G(F) be

it must between

a

set

Similarly,

(l.l(l)). of

objects

Mar

c

to

[.21,Q]

also

dam:

.l/

—»

be

a

(D and

‘6

Let

set.

=

natural

Obd

((9, all, dam, cod, 0),

{canny—we}

.1!

=

{G(I1)|t1:F—>H

dom(G(n))

—»

0

are

C(dom n)

=

1;

=

(F, 17,H)

1'][0bd],

x

=

all

transformation

graph C(17)of 77.Since

the

0

cod:

F[M0r .91],

x

each

associate

C(17) c it must

.2!

where

defined

where

EH60}

by:

and

cod(G(n))

G(cod n),

=

and e:

is

{(f, g)

6

.II

x

J!

l dom(f)

cod(g)}

=

—>

all

defined by

Since

all members

of (9 and

001)

°

.1!

are

0(6)

=

sets,

GUI 8)°

0 and

.11 must

be classes

(1.2). It is

easily verified that ‘6 satisfies the matching, associativity, and identity existence that conditions (3.1). To show that each morphism class is a set, suppose F, H z a! —) Q. Let S

=

H(A)) U{homQ(F(A),

|

A

6

0b

.21}.

Sec.

15

Functor

Since

S is

a

?(0b

.11

x

union

S)

of

is

a

set

a

Itomg(G(F), G(H)) that

so

it must

be

and

indexed

(3)).

{C(n) |

=

(l.l(4)(i)).

set

a

of sets

family (1.1(2)

95

Categories

by

a

set, it is

(l.l(4)). Thus,

set

a

But n: F-r

H}

9(01;

c

at

S),

x

[:1

PROPOSITION

15.4

F

If is

a!

.98 —G_‘:

Next

will

we

that

properties itself

from

often

that

see

are

n: F

—»

G is

53" if and only if it is

Immediate

Proof:

3"

in

isomorphism

an

and

ftmctors

are

in

general to

common

inherits

Theorem

the

natural

a

13.6. a

natural

a

transformation,

then

i;

isomorphism.

[3 n in Q” inherits

morphism

all of the m,

A

the

012091); whereas

6

categorical

the category

of E.

categorical properties

PROPOSITION

15.5

morphism n in [$1, 3?] has one of the properties: “isomorphism",“monomorphism", “epimorphism”,“bimorphism”.“constant morphism”, “coconstant morphism", or “zero morphism”,iffor each A e 0b(.ss’),M has the corresponding A

property. from Proposition 15.4. Proof: The proof for isomorphisms is immediate If each '14 is a monomorphism and g”and 6 are morphisms in 3" such that then for each A e 0b(.al), 11‘ g' 6, n n g" '14 6‘. Hence, for each A, 5,. 6A; i.e., g“ 6. The proofs for other cases are left as an exercise. [:1 o

=

o

o

=

15.6

=

o

=

PROPOSITION

If the category 9'6 has one of the properties: “has terminal object”, “has a zero object", or “is pointed". has the corresponding property. [.91,13]

an

then

object", “has

initial

for

any

.21,

category

functor F: s! Proof: ".4? has an initial object X, consider the constant for all Then for 66 by F(f) l, fe Mor(.a’). 0mg"), n any “the unique morphism from X to G(A)", ('14): F—v 0 defined by: m since for each A —I—> A’ the square clearly a natural transformation —.

defined

=

=

F(A)=

xi»

a

9 =

is

C(A)

F”)=ll‘l JGU) F(/II)=

must

(There

commute.

reason,

it is clear

93" has

an

initial

that

is

only

n is the

object.

one

only

X

TG(AI)

morphism natural

from

X to

transformation

G(A').) For from

F to

the

same

G. Thus,

and Natural

Harder:

96

proof for terminal unique zero morphism

the

B to B'

from

let

Cm

=

(Cm-1‘:F

G be defined

-+

For

(8.8).

F, G: d

by (Cm),

Crumw-

=

15.7

square

lam

G(A')

W

commutes, by the uniqueness of the zero each (Epcl‘ is a zero morphism, so is fire

evaluation

The

G(A)

F031

morphism from F(A) to G(A'). Since (15.5). Thus, Q” is pointed. [I

categories naturally give rise

Functor

any

denote

9?,

->

Foo—M» F(A’)

pointed, let C33. pair of functors

If .6? is

is similar.

objects

The

Chap. V

Trarwformatlaus

to

a

type of bifunctor

new

called

the

functor, defined below.

PROPOSITION

For any

categories

at

and w, there E:

93”

is

(bi)fuuctor

a

.sl

x

—>

9

definedby: A)

EU? ,

andfor each

n: 17—. G

A

andf: HOLD

functor

called the evaluation

for

E is weltdefined

Proof:

=

—>

FA(

=

)

,

A’

003°

=

".4

"11'

PU):

°

9".

because

6(1)

0

"A

=

"A:

0

F0),

since

I] is

a

i.e., since the square

transformation,

F(A)

—flA—> G(A)

16(1)

mnl F(A’)

—;—>

G(A’)

A!

commutes.

Consider

the

Ear, Thus, E preserves

identity (I F, 1,.) on (F, A). 1,4)

=

FHA)

°

15(4)

=

]F(A)° INA)

identities.

(H, A”). Suppose (F, A) 9-1). (G, A’) “493»

=

INA)

=

l£(F.A)'

natural

15

Sec.

Functor

E((§.g)°

E(é °mg°D

01.17)

Hence, E preserves If E: 3"

15.8

the

left

relative

a!

associated to A.

mom

and

functor

H(g)°(6m°

=

functor.

a

[1 for each

bifunctor, then

A): 3"

E(_,

.48 is called

—>

C(f))oru °E(n.f)-

E(ég)

=

is therefore

.93 is the evaluation

—.

(6 “0.4

°

ém)°(GU)°nA)

compositions

x

”(gof)

=

H(g)°(H(f)° (H(g)°

97

Categories

A

0b(d),

e

evaluation

the

functor

Clearly, E(_,

A)(F)

E(_,

A)(n)

=

F(A)

=

"A

for each

F251

Q,

—*

and

that

Note

for each

the functor

Also frmctor

for

F: s!

that

note

C: $3

—>

a“

functor

E(F, _):

a!

is a.“constant

for any two categories 9? and at, there defined by: the constant

Q-object, B, C(B) is d-object is B and whose

each

B

value

f

=

functor from d~morphism

each

at

—f-> B, C( f)

(C(f))A

—’

3

is

is the natural

for

each

then

the

a! is

to

functor”

value at

.9 whose

13,

transformation from C(B)

to

A.

Ji-object

THEORENI

15.9

g“

Mor(.43").

e

F.

fi-morphism, C(B) defined by:

for

n

.99, the right associated

—»

each

each

for each

If at, Q, and ‘6" are are isomorphic.

categories,

functor (quasi)categories 66“)“ and

"a

Proof: Define F:

[5:1,[5%,fl]

[.31 x 93, ‘6]

-»

by 3)

(I'(F))(A. and

if

(a, [3)! (A, B)

(F(F))(a. fl) And

ifn:

F

-’

=

Q

to

(A'. 3') =

F(°¢)s'° F0003)-

G

By straightforward x

-t

F(A’)(/3)° F(or)s

r('1)(A. B) .2!

(F(A))(B)

=

Q and

it

arguments, that

1'01) is

a

can

=

be

('10:;shown

that

transformation

natural

1'(F) is from

Clearly, I'(l,-)(A, 3)

=

((IF)A)8

=

“Fouls

=

Inna),

a

functor

l'(F)

to

from

HG).

so

and

Funclors

98

that

position

If F L

identities.

F preserves of natural

Natural

transformations

°

I‘(g’ n)

Hence,

°

not/1,

Hi)

=

o

3)

=

1‘01),so

0

A, then have

we

V

Chap. definition

the

by that

of

com-

for each

33),

x

(EA ’14))! (EDD ("108

=

((6

i,

0b(.;/

6

"11):;

=

°

G'

(13.3),

(A, B)

1-0: "XII: 3)

Transformations

°

°

=

WIN/1, 3)

(“5)

=

F preserves

that

FM)“.

°

3)and

compositions

is thus

a

functor. Now

define A:

[d

the

right

529.a]

x

[51, [38, €]]

—.

by A(H)(A) and

if

a:

A

and

if

a:

H

=

associated

functor

H(A, _)2 a?

A’

_.

”(a, In)

(A(H)(°‘))n K

-o

(A(°’)A)B

“(may

=

Again, straightforward

be used

from

transformation

d

to

can arguments ’6‘”and that A(a) is a natural

(Aunhh

(MT

(luhuz)

=

If H i»

identities.

Thus, A preserves °

0'),4)a

A preserves [53, ‘6’]],then

«A

=

(1’

°

=

°

r(’1))4)a

=

that

from

A(H) is a functor A(H) to A(K).Now

Immune)-

L» L, then °

704.8)

and

is

so

(A(r('i))4)n

=

show

to

Imam)

=

K

0')(A.B)

compositions.

Hence,

[51,

‘6’

—»

“(mm

=

((13? Aa)4)a°

functor.

a

If n is

a

morphism

F(")(A.B) ("4)8'

=

=

Hence, A so

°

1‘01)

=

n.

that A

Likewise,

ifa

is

(r

°

a

morphism

13(0))(A’ 3)

in =

F

°

[d

=

x

lld'la’rcn. W], then

9,

I'(A(o))(A, 3)

=

Hence, F so

0

Mo)

=

a,

that 1‘0 A

Consequently,

F is

an

isomorphism.

=

Immacu{j

(A(a)4)a

=

“(min-

in

Sec.

thctor

[5

99

Categories

EXERCISES

ISA.

Prove

that

for any

categories .51

and

9, [51. {B}and Ltd”,W’]

are

dually

equivalent. 1513. In the proof of Theorem 15.9, show that: (a) For each functor F: d —» ‘6‘“,1‘(F) is a functor from as!

x

93

to

g.

transformation (b) For each morphism F .1, G in [.21. [$541], 1'01) is a natural from 1"(F) to F (G ). x to 4?". fromd .49 —. ‘6’,A(G) is a functor (c) For each functor 6:5!

(d) For each morphism from A(H) to MK). 15C. functor"

evaluation

H

1—»K in [d

x

3, ‘6], 11(0) is

a

natural

transformation

that .r! and w,the“constant Prove for any twonon-emptycatcgories. C: 93 —v w” is a section in (6.213“,and for each d-object functor A, the —» in (gaff. functor relative to A. E(_. is a retraction a?” A): 96,

VI Limits

in

theorems

Old

Categories

die; they

never

turn

into

definitions. E. Hewm

is

Category theory and smallness

—namely

essentially

conditions.

limits

involved

In this

colimits—and

and

with

chapter begin

concepts—general constructions some general constructions investigate the role that smallness

two

consider

we

to

relationship between them. In Chapter I we have seen that of cartesian the notion categorically as products in Set is essentially the same the notion of direct products in Grp or topological products in Top. The obvious of these concepts within their respective categories naturally leads to usefulness the categorical concept of “product". This is one particular type of a categorical in well-known limit. Other “general constructions” categories naturally lead to varieties of limits other (and colimits)—such as equalizers, kernels, interdirect limits. Later we shall see that the knowledge and sections, inverse limits, tells much about that exist in a given category of which limits and colimits reflected which limits or colimits are or of and knowledge preserved category, that functor. tells much about a functor by given

conditions

play

in the

§16

EQUALIZERS

AND

COEQUALIZERS

Equalizers 16.1

PROPOSITION

MOTIVATING f

If

A :1

B is

a

pair of functions from the

9

e

of

the set 100

set

A to the set

B, then the embedding

E

(I)

A

{06 has the following properties:

A

into

and

Equalizers

16

Sec.

E

e:

(2)f°e

=

(3) For

9(0)}

=

ace:

e':

function

any

function

lf(0)

afunction,‘

A is

a

=

101

Coequah'zcrs

'

é: E

E such

—.

E’

A such

—»

that

the

that

f

o

e’

=

e', there

o

g

exists

unique

a

triangle n1

m

such

A

—v

E such

that

the

that

f triangle

e’

o

=

g

o

e’, there

exists

a

5

B

-n1 —-

NI

(2) If

‘6’ is

one

of the

categories

Set

or

Top (resp. Grp

or

R-Mod) and if

A

as

a

the

is

:f:; B 9

are

@morphisms,

on

B that

contains

let

Q be the smallest equivalence relation (resp. congruence) all pairs (f(a), 9(a)) for a e A, let C be B/Q with the

Limits

102

appropriate

induced

Then

is

(c, C)

Let ‘6’ be the category

(3)

of

B

c:

locally connected

(E, e) is

then

is

If (E, e)

r

is

there a

o

e

=

Then

s.

that

show

must

f

(e

o

r)

o

é),B,

A

equalizer of

an

We

Proof: e

quotient

map.

is

g(x)}

=

inclusion

the

off and

continuous

functions. with

supplied E

e:

the

X is

—»

con-

g.

PROPOSITION

16.4

o

which

in %’

equalizer

an

induced

and

spaces

{x l f (x)

=

are

tinuous,

C be the

—>

and g.

é: Y ‘6’-morphisms,and E coarsegst locally connected topology for If X

C hop. VI

Categories

let

and

structure,

coequalizer off

a

in

=

g

such

is

t

o

o

e

=

But each

r.

of

and

s

r

is such

[:1

monomorphism.

a

OF

(UNIQUENESS

PROPOSITION

16.5

(6.22). Suppose that definition of equalizer,

a

e

A.

subobject of

a

o

that a

=

is

(E, e)

monomorphism (e r) so that by the

e c

unique morphism r. Thus morphism; hence s t

a

is

then

EQUALIZERS)

f

Any

equalizers of

two

definition é

e

=

each

q. Thus

o

(E, e)

The

16.6

above

(E, é)

S

of

equalizers

(loosely) of

a

of

pair

then

g,

that

e

é

o

p

(E, é)

(E, e); i.e., (E, e) and

s

the

by =

is

essential

no

difl‘erence between .

of this,

B. Because

often

we

speak

5

the

and

equalizer off

z

Coeq( f, g); (c, C)

DUALLY:

there A :,’

morphisms

and

q such

f

Equ( f, g) will be used to equalizer of f and 9. [Sometimes Equ(f, 9).] (E, e)

p and

A.

E]

.

.

two

(E, é)

and

that

shows

proposition

equalizer of f

an

unique morphisms

of A (6.23).

isomorphic subobjects

are

isomorphic subobjects of

are

agnd(E, e‘) is

of (E, e) of equalizer there exist

Proof: If and

B

A :3

[denoted by Equ(f, g)].

g

A is

is abbreviated

this

Coeq( f, g)

2

subobject (E, e) of (inaccurately)

the

that

mean

notation

The

or

c

to

an

e

z

Coeq(f, g).

:e

PROPOSITION

16.7

If (E, e) (I) f

=

z

Equ( f, 9), then the following

equivalent:

are

g-

( 2)

e

is

art

isomorphism.

(3)

e

is

an

epimorphism.

Proof: (1) e

o

(2).

=>

s

=

phism; (2) => (3). =>

=

1. Hence

hence

(3)

lff

(1).

an

g,

is

e

thenfe

l

retraction

a

isomorphism

=

g

o

and

1

so

that

is

there

(since (E, e) is

a

morphism subobject) a a

.r

such

that

monomor-

(6.7).

Trivial.

Since

e

is

an

epimorphism,fc

e

=

g

o

e

implies thatf

=

g.

[:1

Sec.

16

Equalizers and Coequalizers

Up of

pairs

this

point nothing has morphisms with common

stressed

to

that

said about

domain

in general,

cannot,

one

been

and

103

the existence

of

common

codomain.

that

exist

assume

they

equalizers

of

It should

be

within

a

given

category. 16.8

DEFINITION

‘6’ has

A category

domain

common

DUAL

16.9

equalizers provided

and has

NOTION:

codomain

common

that

pair of Qflmorphisms with equalizer.

every

has

an

coequalizers.

EXAMPLES

(1) Each

of

the

categories

Set, Grp, R-Mod,

and

Top has

equalizers

and

coequalizers. (2) The category of all sets, with at least two them, has neither equalizers nor coequalizers. 16.10

members

and

functions

between

DEFINITION

indexed (1) If ([1,),6, is a non-empty family of morphisms contained homg(A, B), then (E, e) is said to be a multiple equalizer of (h,),, denoted (E, e) z Equ((h,-),), provided that: .4; (i) e: E For all i,je (ii) l, [1,0 e hjo e; -v If e’: E’ A such that (iii) h, e’ hi e' for all i,j e I, then there

in

by

—v

=

o

unique morphism A category ‘6’ has of family morphisms

(2) has

é such

that

e

=

o

E

=

o

e

have

a

domain

common

a

.

multiple equalizers provided that that

is

each

and

a

non-empty

indexed

codomain,

common

multiple equalizer.

a

DUAL

16.11

multiple eoequalizer;

NOTIONS:

has

multiple coequalizers.

PROPOSITION

Each 16.12

multiple equalizer

PROPOSITION

An y two

subobjects.

is

a

subobject.

(UNIQUENESS

multiple equalizers of D

OF

the

1:]

MULTIPLE

some

EQUALIZERS)

family of morphisms

are

isomorphic

Regular Monomorphisms 16.13

DEFINITION

(1) If E L» and

[and

e

is called g such

A is a

that

‘6-morphism, then (E, e) is called a regular subobject’r of A regular monomorphismfi' if and only if there are ‘6’-morphisms (E, e) a: Equ(f, g). a

T Sometimes only the object E is (inaccurately) called a regular subobject of A if there is some e such that (E. e) is a regular subobject of A. for not calling these special morphisms if The reason “equalizers“ lies in the fact that this would lead to undue confusion when we define what it means for a functor to preserve equalizers. It will turn out that a functor may preserve regular monomorphisms without preserving

equalizers (see Exercise

24B).

Limits

104

is called

A category

(2)

representative

a

of

set

Chap.

Categories

regular well-powered provided that regular subobjects.

quotient object;

regular

NOTIONS:

DUAL

in

‘6’-object has

each

epimorphism;

regular

VI

regular

co-(well-powered). EXAMPLE-‘5

16.14

Grp, R-Mod, and CompT2 the regular monomorphisms are precisely are and the regular epimorphisms the monomorphisms precisely the epimorphisms. [To see that monomorphisms in Grp are regular, see Exercise In Set,

(1)

6H(a)-] Top the regular monomorphisms are the embeddings (i.e., homeomorphisms into) and the regular epimorphisms are the topological quotient maps). maps (i.e., surjective identification (3) In Top; the regular monomorphisms are precisely the closed embeddings. [If X 1—,Y is a closed embedding (i.e., a homeomorphism onto a closed subset), then let Y1 Y, let Yl Ll Y2 denote the disjoint topological union of Y1 Y, and Y2, let Y, LI Y; [1,: Y,In

(2)

=

=

—t

for i

l, 2 be the corresponding

=

q: be the

quotient Then

sz(x)).

map that identifies and Z is Hausdorff and

(4) In Mon, SGrp, monomorphisms; e.g.,

Rug

there

inclusion

the

and

embeddings, Y1 Ll Y2 (X, f) are

Z

—»

for each

let

x

e

X the o

q

and

points u,(f(x))

two

Equ(q #1: monomorphisms z:

o

111).]

that

are

not

regular

ZC» Q.

PROPOSITION

16.15

In any

(1)

every

(2)

every

‘6’.'

category

regular monomorphism in ‘6 ; and regular monomorphism in ‘6 is a W-ntonomorphism. “6-section

is

a

from

Proof: (2) is immediate L» B is a section. Then

there

Proposition 16.4. To show (1), assume is a morphism B 1—) A such thatg of

that

1,4. =fogofl Clearly lsof=fo IA (A,f) Equ(ls,fog). r =fo(gor). then such that Also if r is a morphism Igor (fog)or, is unique since f (being a section) is a Thus r factors through f. The factorization a is [:1 regular monomorphism. monomorphism (6.6). Hence, f A

We

claim

that

=

z

=

sections and monomorphisms lie strictly between a from in function since, Top any bijective example, is a monomorphism that is not regular, to a non-discrete discrete space space and the embedding of the unit circle into the unit disc is a regular monomorphism fail to satisfy certain In general, regular monomorphisms that is not a section. for conditions whose analogues have already been established of the convenient In

general, monomorphisms

regular

for

Sec.

Equalizers

16

and

105

Coequalizcrs

monomorphisms and sections; for example (even in the “respectable"category SGrp) the composition of regular monomorphisms is not necessarily a regular h is a regular monomorphism, h is not necessarily g monomorphism and if f 16] and 34K). Nonetheless, we do have a regular monomorphism (see Exercises of isomorphisms in terms of regular characterization the following convenient o

=

morphisms. PROPOSITION

16.16

For any

(1) f

is

an

(2) f (3) f

is

a

is

a

morphism f,

the

following

are

equivalent:

isomorphism. regular monomorphism and an epimorphism. regular epimorphism and a monomorphism.

Proof: Since every isomorphism is a section, and each section is a regular consequence monomorphism, (1) implies (2). That (2) implies (l) is an immediate of Proposition l6.7. Clearly (3) is the dual of (2), and (l) is self-dual. E] Kernels

An

special

important

R-Mod

of

case

equalizers

and

quite early—the concepts

observed

has been

coequalizers of kernels

in

and

Grp

and

cokernels.

DEFINITION

16.17

Let ‘6’ be

L.

pointed category.

a

fi’f-morphismand if 04”,is the unique zero morphism from A to B, then (if it exists) Equ( f, 0,43) is called the kernel of f2 Notation: Ker(f). (2) ‘6’ is said to have kernels provided that KerU) exists for each f e Mor(‘(t’). A is a (cf-morphism, then (K, k) is called a normal subobject of A, (3) If K 35—. and k is called a normal monomorphism provided that there is some morphism at such that fin (K, k) a: Ker(f). (1) If

A

DUAL

B is

the cokernel

NOTIONSZ

object; normal

16.18

a

B is

i is the

a

normal

Ab, R-Mod,

pSet,

or

pTop, and

if 0 is the

of B, then

i)

inclusion have

in

morphism

element

categories

z

and

Ker(f) and

kernels

h is the

and

natural

(II, By'[A]) map

from

z

Cok(f), B to

the

quotient. Thus,

cokcrncls.

monomorphisms in Grp are (up to isomorphism) the embeddings subgroups. Hence in Grp a regular monomorphism need not be

(2) The normal of

quotient

epimorphism.

(f"[{0}], these

normal

EXAMPLES

(1) If A L» distinguished

where

Cok( f ); has cokernels;

off;

106

Limits

'

normal.

Also

in

Chap. VI

of normal

the

Grp,

composition the alternating

[If/14 is

sarily normal.

in C aregories

the four

on

group

is not

monomorphisms elements

neces-

1, 2, 3, and 4,

then

V4 is

a

normal

{(1), (12)(34). (l3)(24), (l4)(23)}

=

of A4 and

subgroup

Z: is

a

normal

{(1), (12)(34)}

=

of V4, yet 22 is not

subgroup

normal

in

.44.]

EXERCISES

11‘A

16A. g(a) f (b) (A. L.)

%; B

‘6—morphisms,show that the following

are

equivalent:

are

=

z

Equ(f. g)-

Suppose that Show that the following (a) (E. e) z EMU. g).(b) (E, e) z Equ(m 0f, 168.

l6C.

Show

—"—) A, (C, It)

C

(c) For each isomorphism

is

m

that

of and

m

o

m

defined.

g are

g).

o

it' ‘6 has

that

g).

and

equivalent:

are

m

Eun;

z

monomorphism

a

and

equalizers

I is the

5.1.13 then

category

there

,

II

exists

a

F: W'

functor

pair (F(D), en) is 160.

16E.

and

that

usual

and

BanSpl

Show

that

the

and

transformation

have

Bansz

of torsion

category

formed

are

Show

natural

e

to l, E(_, l): ‘6’" ‘6 such equalizer of D(m) and 001).

an

that

coequalizer

a

that

->

Show

they

l6F.

g

relative

functor

evaluation

—.

by factoring

out

(e9)

=

from

for each

D: l

F to a

the

‘6 the

equalizers. free

the

abelian

torsion

groups

has

subgroup

after

coequalizers forming the

in Ab. that

any

non-trivial

group

(considered

as

a

category) does

not

have

equalizers. 166. Suppose that g mo g only if whenever I

has

equalizers.

and

=

m

is

a

Prove

regular

that f is a ‘g-epimorphism if and then m is an isomonomorphism,

morphism. Determine

I6H. 161. then

f

is

161.

Prove an

that

the if g

regular

monomorphisms

of is

regular

a

in POS. and

monomorphism

isomorphism. (a) Show

that

.

:\-\

=

g,

then

there

('3

(.___-_.__

\

(Mi—w;

[:1

commutes.

DEFINITION

17.2

If B is

‘é-object and (A ,, m,),

a

(D, d) is called

(l) d:

D

Bis

—>

(2) for each

d, (3) ifg:

m,

o

exists

a

a

in 9? of

intersection

an

family of subobjects of B, then the pair (A 3, ml), provided that

is

%’-morphism; is a @~morphism at:

a

1 there

is

D

A

—»

,

with

the

property

that

y, then

there

d;

=

C

-t

for each

Band

i

Lg‘:

e

unique g-morphism f:

C

—>

C

—v

D such

A, such that the

that

m; og,

=

triangle

6 \

To

show

Equ(r, s). By

that the

e

is

definition

an

epimorphism, let of equalizer (16.2)

there

(D, m),

of intersection,

there

o

=

roe

intersection

=

see,

is

a

and

let (E, morphism It: A

k) —»

z

E

110

Limits

such

that

e

koh.

=

Since

morphisms are closed (0,, m,),; i.e., for some]

k

is

I, (E,

m

o

that

since

is

m

monomorphism). It often

k)

monomorphism, Consequently, k is that

since

k

the

to

family

=mol

l; i.e., k is

(I).

c

=

isomorphism,

an

mono-

(DI. mi). Thus,

=

mjcdj=m

a

V]

Chap.

a

"Ickcdj= so

Categories

monomorphism (16.4), and composition, (E, mok) belongs

under 6

in

a

that

so

retraction r

(and a (16.7). [:1

s

=

morphism f can be written as a composition of an epimorphism e followed by a monomorphism m in several essentially different m e constructed above is charl7E). The factorization ways (cf. Exercise f acterized the fact that the e has an additional by epimorphism important property occurs

a

=

defined

the

below.

DEFINITION

17.9

(l)

o

morphism e is called an following two conditions: (i) e is an epimorphism.

A

(ii) (Extremal be

must

m

condition):

A.

(3) A category

is

has

DUAL

at

If

e

of.

m

=

where

is

m

it satisfies

that

epimorphism provided

then

monomorphism,

a

isomorphism.

an

B is (2) If A —'—t quotient object of

@-object

extremal

an

most

extremal

called a

set

(e, B) is called

extremal

an

extremally co-(well-powered) provided that of non-isomorphic extremal quotient objects.

extremal

NOTIONS:

then

epimorphism,

extremal

monomorphism;

each

subobject; extremally

well-powered. 17.10

EXAMPLES

categories Set, Grp, and epimorphisms and extremal morphisms. (I)

same

In

the

R-Mod,

extremal

monomorphisms

as

epimorphisms are

the

same

(2) In the category and

extremal

the

are as

Top extremal epimorphisms are topological quotient are (up to homeomorphism) embeddings. monomorphisms

mono-

maps

Top2 extremal epimorphisms are topological quotient maps are monomorphisms (up to homeomorphism) closed embeddings.

(3) In the category and

extremal

(4) In SGrp and and

there

are

Rug extremal

monomorphisms

epimorphisms that

not

are

are

surjective

homomorphisms

extremal.

(5) In BanSp, a morphism X 1—»Y is an extremal epimorphism if and it is a surjective bounded linear transformation and it is an extremal

morphism provided

that

there

is

some

mllxll

m

s

>

0 such

IIf(x)lI.

that

for all

x

e

X

only if mono-

Sec.

17

17.11

Intersections

111

PROPOSITION

Every regular epimorphism

Proof Now

is

and Factorization:

Clearly

.'

extremal

an

epimorphism.

regular epimorphism is an epimorphism (16.15 dual). L» B is a regular epimorphism and e m f, where m We know that there exist morphisms r and s such that

every

that

suppose

is

A

=

o

monomorphism. (e, B) z Coeq(r, 3). Thus a

mofor=eof=eos=mofos since

that

so

is

m

there

coequalizer,

monomorphism, exists a morphism

fo

a

r

fo

=

h such

that

Hence

s.

h

o

e

definition

the

by

of

Thus

f.

=

moltoe=maf=e=1oe since

that

so

hypothesis)

is

1. epimorphism, m h monomorphism. Consequently, m

e

a

an

o

=

Thus is

is

m

an

a

retraction

isomorphism.

and

(by

E]

PROPOSITION

17.12

For

an

morphism /,

y

tlte

following

(1) f

is

an

isomorphism.

(2) f

is

an

extremal

epimorphism

(3) f

is

an

extremal

monomorphism

and

equivalent:

are

monomorphism.

a

and

epimorphism.

an

Proof: (1)

This

follows

immediately from the facts that regular epimorphism and a monomorphism (16.16) and epimorphism is an extremal epimorphism (17.11). is

each that

a

(2)

(1).

=9

Then

f

(2).

=>

f

be

must

(l)

Suppose f

=

self-dual 17.13

that

f

f is

a

is

extremal

epimorphism monomorphism, so that by an

and

a

monomorphism.

the extremal

condition,

isomorphism.

an

(3).

o

l, where

o

isomorphism each regular

This

and

is immediate

(3)

is the dual

from of (2).

the fact

that

(l) is equivalent

to

(2), (1) is

E]

PROPOSITION

For

‘6’,the following

category

any

equivalent:

are

(1) ’6’ is balanced. (2) Each ’6’-epimorphismis an extremal epimorphism. (3) Each ‘6-monomorphism is an extremal monomorphism.

Proof: (1)

(2).

=

then

m

If

is also

e

is an

an

epimorphism and e epimorphism (6.13),

=

so

m

of, where

that

since

m

is

‘6’ is

a

isomorphism. (2)

=>

(1).

Immediate

from

the

preceding proposition

monomorphism,

balanced,

(17.12).

m

is

an

Limits

112

(3). and (3) is (1)

Immediate

c:

the dual

is

(l)

VI

Chap.

Categories

the fact that

from

of

in

equivalent

(2), (l) is self-dual

to

E]

(2).

PROPOSITION

17.14

If ‘6’ has equalizers and e (ii) of Definition 17.9(1), then

is

fi-morphism tlmt satisfies the extremal be an extremal epimorp/u'sm.

a

e

condition

must

Proof: We need only show that e is an epimorphism. Suppose that r and s e. Let (K, k) z Equ(r, s). Then k is a s are ‘6’-morphismssuch that r e monomorphism (16.4) and by the definition of equalizer there is a morphism I: =

o

that

such an

e

k

=

c

isomorphism.

It. Hence, Thus r =

since s

e

o

satisfies

the

extremal

condition,

category

%1

k must

be

E]

(l6.7).

DEFINITION

17.15

Let 6‘ and

be classes

.ll

(l) A pair (e, m) is called

an

of

morphisms of

a

of

(6, .ll)-factorization

a

%’-morphismf provided

that:

(i) f (ii) e (iii) m

m

=

o

e

6

e

6

J!

abbreviated

This is sometimes

by saying

that

f

m

=

o

e

is

an

(6”,sin-factorization

off. (6’,JI)-factorizable category provided that each W—morphism (6‘, .ll)-l‘actorization. ‘6 is called a uniquely (6“,JI)-factorizable category if and only if it is (6‘, VII)(3) m «E of the m e factorizable and for any two (6’, JI)-factorizations f the an It such that there exists same diagram isomorphism %’-m0rphismf,

(2)

‘6’ is called

has

an

an

=

=

o

o

/,.\ x:

a

commutes.

(4) If ./1 is composed of monomorphisms, QS’is called

and

provided that f 6 .II has a representative

(A, f) is called

then

JI-well-powered set of .ll-subobjects. if 6’ is composed of epimorphisms, DUALLY: quotient object and

B

l

.0

CT” and

if e morphism

commutes

exists

a

is

regular epimorphism

a

h: B

C such

—»

that

the

and

is

m

a

monomorphism,

then there

diagram

A—e>B III

",1"

/

9

I” K

C——>D "I

commutes.

Proof: such

that

Since

(e, B)

e z

is

,ne(for) that

since

is

=

go(cor)

=gc(eos)

monomorphism, f r there is a coequalizer, morphism h such that f m oh. it also follows that g [:1 so

m

there

regular epimorphism, Coeq(r, 3). Thus a

o

a

f

=

=

o

=

s.

h

o

are

r

and

Since

by the definition e

is

an

PROPOSITION

If

a

category

‘6 is

(regular epi, monoyfactorizable,

then

(I) ‘6’ is uniquely (regular epi, mono)-factorizable.

(2) The regular epimorphisms

in ‘6’ are

precisely

the extremal

of

epimorphism,

=

17.18

s

"10(fos),

Hence, e.

morphisms

epimorphisms.

114

Limits

in

Categories

VI

Chap.

Proof: If each

(1).

m e and f off by proposition (17.17)

then

of];

=

o

=

Tn

E is

o

there

is

a

(regular epi, mono)-factorization morphism Ii such that the diagram a

\ X R 3

Being

commutes.

first factor

the

of the

E is

since

morphism (6.5). However, epimorphism (17.1 1). Thus by (2). Immediate. [:1 A further

monomorphism m, I: regular epimorphism,

a

the extremal

of extremal

treatment

o

condition, I: must

be

must

it is

be

a

mono-

extremal

an

isomorphism.

an

morphisms and factorizations

appears

in

let (D, d) be

a

IX.

Chapter

EXERCISES ”A.

(X;. on), be

Let

of X such

subobject

a

family of subobjects

(a) for each i, (D. d) s (X,, m.) and (b) if (E, e) is a subobject of X such (D, d). Show

by (X1, mi)!the extremal

condition Prove

17C.

X and

object

that

example

an

that

that

for each

(D, d) is

i, (E, e) not

(X,, m,),

s

the

necessarily

then

(E, e)

intersection

s

of

if g is

(cpi, mono)-factorizable and f is a morphism that satisfies (ii) of Definition 1730), then f is an extremal epimorphism.

I: y, then 9 f is an extremal monomorphism and f 5.5 and 6.5 and Exercise (cf. monomorphism Propositions 16]).

extremal

an

that

Prove

178.

be

of

means

of the

that

if

=

o

must

17D.

An

f

o

Splitting of ltlempotems idempotent in a category W is a ‘K-morphism f: A a A with the property f 12 An idempotent f is said to split provided that there is a factorization A

where

that

=

r

a

Show

s

=

1.,

A

=

A

L)

B

3..

A

In.

that

the

following morphism A L)

statements

are

equivalent

for

any

category

‘6’ and

any

A in 6’.

idempotent (a) f splits in ‘6’. (b) I has a (retraction, section)-l‘actorization in w. (c) The morphisms f and l A have an equalizer in ‘6. (d) The morphisms f and l A have a coequalizer in g. 1713. two

Show

essentially

that

in the categories Top and Rug there different (epi, mono)-factorizations; i.e..

f

that

morphisms f

are =

mac

and

f

=

have ficé‘

Sec.

18

Products

where

c and E are epimorphisms, m isomorphism I: such that the diagram

and

and

115

Coproducts :7:

are

and

monomorphisms,

there

is

no

./ X

_....._..-——.. ,,

NA commutes.

§18 18.1

MOTIVATING

The

PRODUCTS

COPRODUCI‘S

AND

PROPOSITION

cartesian

product of a pair (A, B) of sets is a set P together with two A and 1:3: P B with the property that if C is (projection) ftmetions 75,: P -> —> B are functions, then there exists a unique function A, g: C any set and f: C h: C —> P such that the diagram ->

I]

commutes.

18.2

->

DEFINITION

A

V-product of

‘6’-objectand

a

19: P with the prOperty that

‘6’-morphisms, then (f, g): C —> P such

pair (A, B) of (ti-objects is a triple (P, 1:4, 1:3)where P is a B are A, an: P ‘6—morphisms(called projections) if C is any “(f-objectand f : C B are arbitrary A, g: C there exists a unique fi-morphism (usually denoted by) that the diagram —>

—>

-+

—>

A

/i" (f9)

commutes.

Quite A

x

B is

triple (P, 1:4, 1:3) is denoted by A x B. Usually the symbol (inaccurately) used to stand for just the object P, rather than the entire often

the

116

Limits

product. It should just an object.

be

kept

in

Categories

Chap. VI

in mind, however, that

a

product is really

a

triple—

not

The notion

of course,

be

translating importance 18.3

is dual

that

obtained

readily

this

back

and

to

to

into

make

a

the

product is coproduct, the definition of which can, by forming the definition of product in ‘6" and statement in ‘6’ (4.15). However because of its notion explicit, we state the definition.

DEFINITION

‘d-coproductof g-object and “A: A A

(e-objects is a triple 01A,’15, K) where K is a K are V-morphisms (called injections) with K, [13: B the property that if C is any tof-object and f: A C 9: B C are arbitrary then a there exists @morphisms, unique (of-morphism (usually denoted by) C such that the diagram U, g]: K a

pair (A, B)

—»

of

—v

-»

_.

->

A

“‘1 K K

[fig]

-----

ac

”1/ commutes.

The 18.4

triple (um

as,

K

),

and

often

just K, is usually denoted

by

A Li B.

EXAMPLES

A

Category

Grp Top pTop

commutative

(9)

a

Cat

or

R-Alg

(6.31.7

that

direct

product

product category

single partially ordered set (15(6))

It is clear

disjoint union free product direct

sum

topological (disjoint) sum topological sum with base points identified

fibre product (Whitney sum)

TopBunB

(7) (8)

A [I B

product direct product direct product topological product topological product

R-Mod

(6)

B

canesian

Set

(I) (2) (3) (4) (5)

x

infimum

=

A

tensor

(9L) A

B

sum

A

product =

A

easily generalize the notion of products objects to the notion of finite products. However, as in set theory, it consider products for arbitrary families of objects, to wit— one

can

B

(9M)

category

supremum

®n v

B

of

pairs

is useful

of to

Sec.

18.5

and

Products

18

117

C oproducls

DEFINITION

(cf-product of a family (21,),-e, of ‘6-objects is a pair (usually denoted (I'M/mm. (ROI-6,)satisfying the following properties:

by)

A

(l) “(A)”,

is

(2) for each j

"(’11)ch

‘0

e

‘K-object.

a

1, nj: ”(14,)”,

_.

A j is

‘6’-morphism(called

a

the

projection

Afl-

A i) pair (C, (fl),-E,), (where C is a‘g-object and for each j e 1.1}:C C denoted a "(44‘)”, by) (1“,): unique (co-morphism (usually for each j e I, the triangle —»

(3) for each there exists such

from

that

-»

(fr) C

+U(Aa).-.z

""""""

commutes. 18.6

DEFINITION

family (A i)“ , of V—objectsis u(A,),c,) satisfying the following properties:

‘G-coproduct of

A

(0mm,

(I) 1.101,)“,is each

(2) for

j

a

a

@-object.

e

I, 11,-: A, ->Ll (A I),-E, is

a

a

pair (usually

‘6-morphism (called

A, 11010;“). (3) for each pair (09“,, C), (where C is a ‘6-object and C) there exists a unique ‘6’-morphism(usually denoted f,-: A, from

the

by)

injection

to

for

[ft]: Home: that

for each j

e

each

jeI,

by)

_.

such

denoted

C

"

l, the triangle

”(Ai)i:I

)

""""""

C

commutes. 18.7

For

simplicity

one

often

writes

(n,)) (I'll/1i,

(inaccurately) 11A,-alone when denoting (11,-,11/15), and HA.) Also, ((11,), LII/15), is called

of B and

the Ith power

phism ((13),):

B

—>

B'

is called

a

product

when

(”A of the

for each

3,

n.)

or

sometimes

even

family (A i)“ ,. (Dually: I, A; B, then HA;

is

=

by B'. In this case the unique diagonal morphism and is denoted

is denoted the

or

mor-

by

118

Limits

A,,

A.

simply

or

Notice

that

when

(Dually: Ith I {1, 2},

in

Categories

'B; codiagonal morphism;

of B;

copower

Chap. VI V a;

V.)

of the notion

of

=

(tn-D (CI/1i, is

“essentially the

same"

as

(’41 so

that

this

“product 18.8

of

justifies pairs".

A2, “Al! 7:14;),

X

definition

the above

for

the

generalization

EXAMPLES

X is

‘g-terminal

object if and only if (X, E) is a product of an empty indexed family of ‘6’-objects. (2) If A is a ‘6~object,then (P, f) is a product of the self-indexed set {A} if and A is a ‘6-isomorphism. only iff: P and coproducts of arbitrary families of objects in the categories (3) Products and have the same names as those given in 18.4. Set, Grp, R~Mod, Top (4) The categorical product in the category of abelian torsion groups is not the group-theoretic product but is the torsion subgroup of the group-theoretic product. The coproduct, however, is the direct sum. is not (5) The categorical product in the category of locally connected spaces the topological product but is the cartesian of the product underlying sets locally connected supplied with the coarsest topology that is finer than the usual in this The however, is the (disjoint) product topology. coproduct category, (l)

a

->

topological sum. (6) The coproduct not the topological sum. The product 18.9

Hausdorff is Comp'l‘2 of compact spaces sum but is the Stone-Cech compactification of the topological in this category, however, is the topological product. in the

PROPOSITION

category

(SIMULTANEOUS

CANCELLATION) it

If (l'lA

that

property in

,-.

unison,

product of (A ,-),-E , and if

a

o

For

each

in the definition

of for

Returning the

is

for each i e 1, n,- h monomorphically”.

act

Proof:

m)

i, let f,-

a

n,

h

product,

h

=

(fl) the

o

k, then

1:; =

o

k.

category

11A

?: h

morphisms

1 are

with the

k; i.e., “projections,acting

=

k; then

by

the

uniqueness condition

[:| of sets,

consider

an

infinite

set

A,

set

the

x

each

yet these

A

=

{(a,b)|a,beA},

set

A2 Now

7:,-

o

=

to

moment

A

and

=

=

C

of A sets

are

x

A and

quite

=

{f: {0, l}

A2 is commonly

different.

At first

—+

A}.

called

glance

the one

product of A with itself, might think that they are

Sec.

18

Products

and

Coproducts

119

A x A “essentially the same” because there is a bijection g: A2 (f (0), f (l))]. However, since A is infinite, there is also a [defined by g( f) from A2 to A, and clearly A has no claim to the title “the cartesian bijection product of A with itself”. What has been forgotten is the fact that each of A2 and A x A has projections associated with it: til: A2 A defined by A defined x A A A defined fo(0); fizz A2 by foU); 11,: by A defined by (a, b) H b. The reason (a, b) H a; and n: A x A that the triples (A2, 5%,,R2) and (A x A, 7n, n2) are thought of as “essentially the same” is because the bijection g: A2 A x A “respects” these projections; i.e., for l, 2 the triangle

regarded

as

—>

=

-»

—o

—.

—>

——>

=

A°—>AXA

commutes.

A similar

“essential

holds

for any

categorical product,

as

the

shows:

following proposition 18.10

uniqueness”

PROPOSITION

OF

(UNIQUENESS

PRODUCTS)

A

each

If

of (l'lA,, 1:1)and (11A,, 1%,)is

product of (Ame,

a

/\

a

unique isomorphism

3:

HA

“A,

-’

‘

such that

for each j

e

then

there

exists

I, the triangle

UAi—‘flfii

x /;,. A,

commutes.

Proof: By the definition such

for each j

that

e

I the

of

there

product diagram

exist

unique morphisms

s

and

nni—’>1fii—'—>nA,

Hence

commutes.

for

each]

6

l.

tr,o(tos) Thus that 18.11

the cancellation

by s

c

t

=

=

(18.9),

property

1;“. Consequently,

rt,-

njc

lm‘.

In“ Similarly [:1 isomorphism.

s

is

a

category

an

=

t

o

s

=

it

can

be shown

COROLLARY

Any

two

terminal

objects of

are

isomorphic (cf. 7.8).

[:1

t

Limits

120

Because

of the

essential

product of a family for coproducts.

of

in

of

uniqueness than

rather

objects

C hop. Vl

Categories

products, product of

a

one

the

often

speaks of the family and similarly

categories, the projection morphisms injection morphisms are injective. This is not always the surjective N is the empty function, case. For example in Set the projection N x Q which is not surjective; and in the category of commutative rings the coproduct 0, so that the injection Q Q u Z; is not injective. Q LI 2; Q ®z 22 However, we do have the following: It is well-known

that

for many

concrete

the

and

are

-i

=

—’

=

PROPOSITION

18.12

’6 is

If

a

connected

(and dually,

retraction

then

projection morphism injection morphism is a section).

category,

every

every

’6’ is

in

a

Proof: Suppose that :rj: “A, Aj is a projection morphism. For each be i is], any morphism and letfj: A,#jletj}: Aj be [41.Then A,- A, ”A, such that nj (1}) bythe definition ofproduct there exists (fl): Aj IA]. Hence has a right inverse. E} rt, —>

—>

—>

—»

PROPOSITION

18.13

(K‘LE,

OF

(ITERATION

=

c

PRODUCTS)

disjoint family of sets. Suppose that for each (P, (1:0,,aK‘) product of the family (X 9,, 6K‘ of ‘é-objects and that (P, 7n) is the product of the family (P‘),e,. Then (P, (n, 1m,“ MK.)is the product of (Xk)ke u, it. Let

i

e

l

be

a

IS

the

pairwise

0

Xk. Suppose that C is Proof: Clearly for each i e [and k e K,, 7r}; m: P for each is I and k e K,, f,f: C a ‘6’-object and XK. By the definition of P‘ such that i e I there exists a for each unique morphism (fl): C product, for each k e K ,, a; (/2) fif. Again by the definition of product, there exists P such that for each i e I, rt, a unique morphism ((11)): C «fl» (1:). —»

0

—>

-*

o

=

-’

Hence

for each

i

e

I and

k

e

K

0

,.

the

diagram

«It» ”

x Since

established. 18.14

Hence

each

(P‘, (7:1))is

(P, xi

0

n.) is

1'1’

it, 2-:(Zt).) Pi

C

commutes.

=

a

the

if; X],

product, product

uniqueness of «ft» of the ATS. E]

the

is

readily

PROPOSITION

If (ll/1,, 1n) and (”33, p,) are products of the families (A,), and (B,),, respectively; and if for each i e I there is a morphism A , 1'» B,, tlten there exists a unique morphism (usually denoted by) TU, that makes each square

18

Sec.

Products

and

“f;

“A;

HB‘

)

"""

121

[:]

commute.

18.15

DEFINITION

The

morphism Hf, of the morphisms (f,),. If] {1, 2,...

above

DUAL

L] j} is the

NOTIONS:

is called

proposition n}, Hj} is sometimes

=

,

the

writtenfl

product of the x 1;. f2 x '

x

coproduct of the morphisms (1}), ; f

,

11f;

Ll

-

-

-

-

'

LI 1;.

PROPOSITION

18.16

In any

the

category

(l)

retractions

(2)

sections

is

is

section.

a

isomorphism. is a monomorphism. morphisms is a constant morphism.

constant

For

Proof: (A ,),

product of

retraction.

a

(3) isomorphisms is (4) monomorphisms (5)

Coproducts

and

an

each

([13,, pi)

commutative

let

is]

[,2 A, 3,, product of (Bi),.

be the

—.

let

(HA5, 7n)

Then

be

for each j

the e

I

product we

have

of the

square

"fa

HB‘

—)

"A;

Aj —f_)Bj i

(I).

If for each

together

fj,

commutative

is

squares

(nft)

°

P}

there

°

gj such

some we

(“9.)

see

j}

=

for each

that

°gj

thatfj

°

P1

=

j

P1

Hence

cancellation by the simultaneous property is a retraction. i.e., (Fifi) ([19,) Hf,lnafi to (2). Analogous (l). o

(l) and (2).

(4).

If h and

k

jel,p,o(l'lf,)oh Since each f] is (5).

=

I

P}

for

°

lan

the

pj’s,

we

have

that

=

(3).

cancellable

e

la’, then by pasting

=

cg,-

morphisms

are =

a

that

[:1

(11190]:

=

(Fifi)

o

k, then

for each

pjo(l'lf,-) ok,sothatforeachjeI,fjon}oh =flonjok. and since the nj’s in conjunction are left-

monomorphism

(18.9), it follows

Exercise.

such

that

h

=

k.

122

Limits

in

(ES, e1)

:5

EquLfb g1)

z

Equ(nf.-. not).

C ltap. V]

Categories

PROPOSITION

18.17

If for

each

i

e

1,

exist, if Heb Hf, and l‘lg,~

and

then

(“Eb Hes) (l.e., the product of equalizers

Proof: For

each

j

e

is

equalizer of

an

the

I, consider

the

product.)

diagram “f,

He,

HE,- —>rtA,- —’_,nB,. Hgi

l

i

e,

l

f:

E, ——>A,--:;B, I

morphisms. For each j, fjoejonj a). Fig‘ He), so that by the cancellagj that tion 11g,- I'Ie,. Now suppose He; C—"—> Ug,oli. Then for each j, I'M, is a morphism such that rifle/1 gjo (pie It). Thus since (5,, e) z Equ(f},g,), for each j, there fjo (p1 ch) exists a unique morphism C L B] such that ej k] p] It. By the definition of product, there is a unqiue morphism (k,): C "E, such that for each j, k,- 7:, (In). n}, p}, and

where

projection that l1], ”e; ai ‘31 It, implies for projections, I'lf, property

o

are

oj

o

o

c

=

c

=

0

=

c

o

=

=

=

o

o

—+

o

=

l

a;

l»)

p ,- It, so that by the cancellation property Hence, for each j, p ,- He, (k9 with to this because Also is 1:. unique respect property (ki) He,- (IQ) and is thus a monomorphism (18.16). [:1 He, is the product of monomorphisms 0

set

DEFINITION

A category (resp. finite DUAL

18.19

o

=

=

o

18.18

0

% has

set)

NOTIONS:

products (resp. has finite products) provided that for every family of ‘K-objects indexed by I has a W-product.

I, each has

coproducts;

has

finite

coproducts.

PROPOSITION

In any

category

%’, the following

are

equivalent:

and

Products

[8

Sec.

[23

Coproducts

(l) ‘6’ has finite products. ’6’ has

(2)

object and

terminal

a

each pair

product for

a

of objects

in ‘6’.

C]

PROPOSITION

18.20

In any

that

morphisms Proof:

a

has

the

products, regular monomorpltism.

category is

See

18.17.

Proposition

product of

a

family of regular

mono-

D

EXAMPLES

18.21

have

(l) The

following categories R-Mod, Top.

(2) The

category

Exercise

l8L).

Field

products

finite

products

nor

finite

products

but

neither

has

(3) NLinSp and BunSp| each have products. (See Exercise 18M.) (4) A partially-ordered if it has

if and

coproducts following:) 18.22

THEOREM

For

any

(1)

‘6 has

(2)

‘6’ has

(3)

g

is

(considered only if it is

set

and

both

as

complete

a

finite

Grp,

coproducts.

(See

(arbitrary)

not

has

category)

a

Set,

coproducts:

lattice.

products if (Indeed,

we

and have

only the

(FREYD)

small

(6’, the following

category

products. coproducts. equivalent to

a

equivalent:

are

lattice.

complete

Proof: If ‘6’ is a complete lattice family of %’-objects,it is clear that

considered

“At l

as

a

and

category,

(A,), is

a

inf (A i)

=

t

and

17"“ =

Since

SUM/1i)I

complete lattice is a skeletal category, categories are equivalent provided isomorphic skeletons, and products are unique only up to commuting isomorphisms, we have that (3) implies (l) and (3) implies (2). If a small category ‘6’ has products, then 93’ is a quasi~ordered set, considered as a category. For a

they have

to

suppose

cardinal i

e

the

number

contrary of

that

:3

Mor(‘€), and

B

are

for

‘6-morphisms. Let

distinct

each

function

f:

l

_.

{0. l}

I be the

and

each

I, let ,. ’I-A

By the definition some

A

je

l, f(j)

of product,

¢f(j)

so

”be

_,

for that

y

if

f(i)=0

h

if

f(i)=

eachf, since

1

3’. Now

sué ll, Rio

(r{)

iff 95 njo

¢

f, then for (r{). Thus

124

Limits

¢

(r{)

B’,

(r{). Consequently

there

in

Categories

2' different

least

at

are

C hop. VI

from

morphisms

A

impossible. quasi-ordered set, any skeleton, .9’, for since ‘6 has and, set, partially-ordered products, .9’ has infima. Thus it is a complete lattice. Hence (i) implies (3). Analogously (2) implies (3). E] to

it is

which

is

Since

V is

a

a

18.23

PROPOSITION

I, D, 1» D0

-

that

Suppose

each

for

is

.

I

.

the

ts

:",A,,

equalizer of Do

mi",

and or.» is the product of the family (A..), and f= (19:1), g (9,): Do l'IA, are the unique morphisms induced by the product. Then is the (C, d) equalizer off and y if and only if it is the intersection of the family (Db 91)!-

an,

—.

-v

=

If (C, d)

there that

z

o

—>

for each

Then

that

by

and

such

d

that

p

=

o

g, e,

d

implies that

1,. Now

o

for each

i, e,

suppose o

h,

=

k.

nl°f°k’

f k. Hence since projections, g k C there exists a unique morphism p: K of the family (0,, e,),. the intersection o

=

o

—»

9,

(C, d) is

k. Thus

=

for

property

(C, d) is the equalizer off =

=ft°€t°ht

=gi°ei°hi

the cancellation

o

=

i,

“t°g°k so

d

Eun, 9), then for each i, f, exists a unique morphism 1,: C D, such that d (K —"‘—s 0,) and K L» D0 are morphisms such that

Proof:

_:_,nA flDgiei D,

”I

k

K

Conversely, if (C, d) is there is a morphism 1,:

A,

—o

D, such that d

suppose f, k o

the

the cancellation

k is

that =

g,

o

k

definition

so

a

property

morphism

such

that

exist

there

of intersection,

there

for

18.24

(C, d)

z

Equ(f, g).

we

have

f

o

k. This

d

=

g

that

o

d. Now

for each

i, f implies that k. But such by e,- h, morphisms h, exists a unique morphism I) such that o

=

g

o

o

do 1') Thus

k

i

“fly”!-

=

projections,

that

each

o

=

=gi°el°ll

“t°f°d=ft°ei°lt Again by

family (D,-, e9” then for I,. Hence, for each i, e,

of the

intersection

the C

y].

=

=

k.

[:1

COROLLARY

If W has products, then 1:] regular subobject.

every

intersection

of regular subobjects

in ‘6’ is

a

Sec.

and

Products

18

125

C aproducts

COROLLARY

18.25

If

products (resp. finite products) and equalizers, (resp. finite intersections) of regular subobjects. [Z]

sections

‘6 has

then

% has

inter-

EXERCISES If the

18A.

_:Q

x

_

‘6 has

category

finite

show

products,

there

that

is

bifunctor

a

@fi‘gdefinedby:

x

_(A,B)=A

x

_

B,

x

if

and

sz—sA’ x

_

where

f

x

g is the

g:B—>B’

and

_(f,g)=fxg:A

unique morphism

x

which

B—-)A’

makes

the

B’,

x

diagram

A—[———>A'

TRARAT B—--€§-g--->A’

A

B'

x

x

”i B

is. ._—_)

B

I

g

Dually, if ‘6 has finite coproducts, show that there is a bifunctor V g ‘6, which assigns to each pair of @-objects (A, B) the coproduct (Cf. 9.8 and 91.) commute.

18B. the

functor

an

is

induced

a

if Q has

that

Show

has

F': ‘6’

functor

finite

—>

to

products

A

e

A

L»

18D.

from

if F: ‘6’ —»

and

Q,

x

_

then

Bz‘fi

x

_

B to

‘6' is

—»

in 18A.

described

_:

A [I B.

Show

1.3.

for each

set

I, there

53, defined by =

(F(A))'.

0b(‘€).

F’(f) for each

and

OMV),

B) of the functor

transformation

natural

PM) for each

Be

products,

(with respect

family (19),. 60,“,

18C. is

g

that

Suppose

the left associated that

Ll

_

—v

x

B

U Fm:

=

(F(A))'

_.

(FtBn'.

Mor(‘€).

e

If ('40::

,

is

a

family of ‘g-objects,

set-indexed

prove

that

the

following

are

'

equivalent: (a) (P, (n.)) is (b) For

each

3

a e

g-product of (A‘). 011%"), the function

homgtB, P)

-.

H h0m«(B, A,),

defined

by:

I

Bi» is

bijective (where I] hom(B,

A

1»...

(n.°f)m

,) is the cartesian

product

in Set

and

(n, of)“, is the

1

unique element that categorical

of

the

products

cartesian can

product

be defined

whose

in terms

ith

coordinate

of cartesian

is n,

products

of). of

sets

Conclude

(cf. 68).

126

Limits

l8E. B

f:

for each

that

f

in ‘6 of

product

f is

if and

morphism

constant

a

if there

that

is

the

only if

of is

n;

that

n,

is

f

o

a‘6’-object, Kand

A is

that

Suppose

i such

some

a

monomorphism

L

are

such

sets

AA and A"

are

powers

of A" and

subobject

and

that

constant

morphism then

(resp. section),

a

¢

K

L and

c

that A“ can be considered as being simultaneously object of A" by exhibiting a section 3 and a retraction

of A. Show

a

quotient

r

that

A“ is the

186.

Show

that

18H.

Prove

that

of epimorphisms

product

the

A“ =1“.

ALL.

product

of

is not

necessarily

multiple equalizers is

the

epimorphism.

an

of

multiple equalizer

product (cf. 18.17). 18!.

(object)

X

’6’-tem1ina| object.

a

A exists

x

that

Show

and

is

isomorphic

X is

a

terminal

For to

each

’6-object

that

A prove

the

product

A.

object in

Set

if and

only if each

Set-object

is

a

of X.

copower

Show

l8K. finite

X be

Let

l8J.

that

non-trivial

any

group

considered

as

a

does

category

not

have

products. Let ‘6’ be

18L.

(a) every

([3)

a

thatQ

.

the

family (A,)

(section).

monomorphism 18F.

such

VI

Chap.

i.

(b) Show a

(P, ('71)) is the

Categories

P.

_.

(a) Prove

is

that

Suppose

in

there

morphism exist

two

a

category

is

a

distinct

such

that:

and monomorphism; morphisms with the

(a) Prove that ‘6’ does not have finite products. (b) Prove that ’6 does not have finite coproducts. finite products that Field has neither (c) Conclude l8M.

A and

Let

(a) If P is the cartesian

sup{lia_. "bi”, and Bansz.

show

B be Banach

product that

domain

same

nor

and

the

same

codomain.

finite coproducts.

spaces.

of A and

(P. it,h nu) is

Bsupplied a product

with

the sup-norm of A and B in

(i.e., 11(1). 12)]! NLinSp, BanSpl, =

(i.e.. Ma, 1))“ product of A and Bsupplied with the sum-norm at + b). show that (Q. :rA, rt”) is a product of A and Bin NLinSp and in BanSp, but is not a product of A and Bin Bansz. of the categories (c) Show that neither NLinSp and BanSp. has products.

(b) IfQ is the cartesian

=

§l9

AND

SOURCES

SINKS

family (Xi), of objects in a category is FIX; is an object and (in), is a family of pair (ll/Y), (7:0,); domain (such as l'lX,, satisfying certain conditions morphisms with common comwith Such families of “in concert" monomorphically). morphisms acting mon domain codomain) (or dually with common appear frequently and thus As

defined

we

to

have

be

a

seen.

the

product

of

where

a

Sec.

deserve details very

that

technical

rather

will look

section

In this

special attention.

127

and Sinks

Sources

19

we

first

at

provide some of the sight, but that will turn

will

necessary out

be

to

useful. and

Mono-Sources

Epi-Sinks

DEFINITION

19.1

in ‘6 is

X0, pair (X, (f,),), where X is a (if-object and (f;: X X. In this case X is called the is a family of (6-morphisms each with domain of the source. and the family (X ‘), is called the codomain domain of the source (X, (f,),) is often denoted by (X, f,).] [To simplify notation a source provided that the f‘ can be simul(X, f.) is called a mono-source (2) A source (l)

A

source

taneously cancelled

-+

a

from

the

for

left; i.e., provided that

any

pair

Y

X of _—'_t J

s. 1} s for each i, it follows that r morphisms such that f.- r (X, f,) is called an (extremal mono)-source provided that (3) A source it is a and mono-source, (i) (2, g.) and each epimorphism (ii) (Extremal condition): for each source such that for each i, the triangle =

o

=

o

e

x——"—>x,.

\ fl Z

commutes, DUAL

e

NOTIONS:

must

be

sink

an

isomorphism.

in ‘6’; (fh

X); codomain

of

a

sink; domain

of

a

sink;

epi-sink; (extremal epi)-sink. 19.2

EXAMPLES

(l) (X, Z) is

a

mono-source

if and

only if for each object

Y there

is at most

one

morphism from Y to X. (Hence, in case the category is connected, this is equivalent to the condition that X is a terminal object.) if and only if f is a monomorphism. It is an (extremal (2) (X, f) is a mono-source if and if mono)-source only f is an extremal monomorphism. (l8.9). In fact each product is an (3) Each product (I'IX‘, at) is a mono.source (extremal

mono)-source

(Exercise

l9D).

Then (4) Let (1}, X) be a sink in one of the categories Grp, SGrp, or R-Mod. if the union of the set-theoretic if and is an only (extremal epi)-sink (L, X) images of the homomorphisms f,- generates X in the usual algebraic sense. in Top. If (X, f.) is an (extremal mono)-source, then X (5) Let (X, f.) be a source functions has the weak (i.e., coarse) topology with respect to the continuous 1}. then it is an (extremal mono)-source if X Conversely if (X, f.) is a mono-source,

has the weak

topology with respect

to

the functions

f,.

128

Limits

in

Chap. VI

Categories

(1}, X) be a sink in Top. If (f,, X) is an (extremal cpi)-sink, then X has functions strong (i.e., fine) topology with respect to the continuous fl. Conversely if (f;, X) is an epi-sink, then X has the strong topology with respect to the functions f,- only if (fl, X) is an (extremal epi)-sink. (6)

Let

the

PROPOSITION

19.3

(X, ff) be

Let

and

source,

X

f:

—>

a

(I'IXE, in)

source,

“X,

be the

product of the codomain (X 1)of the morphism for which all triangles

be the unique induced

f=(f.-> ——>HX'.

l“

f.-

X commute.

Then

(1) (X, f.) (2) (X, f.) morphism.

is

that To

is

a

is

a

if and only if f is a monomorphism. (extremal mono)-source if and only if f is an

mono.source an

extremal

mono-

It is also clear since (l'lXi, n.) is a mono-source. Proof: (1) is inunediate then is an extremal if (X, fi) is an (extremal mono)-source, f monomorphism. that is an extremal show the converse, (Y, g,) f monomorphism, suppose such that for each i the and e is an epimorphism source triangle x

Lu,

x A Y

By the definition

commutes.

such

for each

that

implies

that

the

i, g,

of

=

there

product, o

it,

9.

is

since

Now

unique morphism g: Y (RX, 1:.) is a mono-source, a

a

FIX, this

diagram 11X,-

X

X;

g

x. A Y

Thus

commutes.

morphism. 19.4

since

f

is

an

extremal

monomorphism,

e

must

be

an

iso-

[:1

PROPOSITION

If condition

coequalizers and (X, f.) is a source (ii) of Definition 19.1(3), then (X, fi)

Proof: Analogous

to

the

proof

of

satisfies the extremal (extremal mono)-source.

in %’ which

‘6 has

must

Proposition

be

an

17.14

dual.

E]

Sec.

19

Sources

and Sinks

129

Separators and Coseparators 19.5

PROPOSITION

For

(1) (2)

C is

a

‘K-objectC, the following g-object X, the

(X, hom(X, C))

source

(1) C

‘6’ has is

For

a

arbitrary

powers

of

the

a

C]

mono-source.

object C, then the following

equivalent:

are

coseparator.

each

‘g-object X,

the unique

morphism induced

X a

is

PROPOSITION

If

is

equivalent:

are

coseparator.

For each

19.6

(2)

any

by the product,

cwm

—»

monomorphism.

is (3) Each @—object

subobject of

a

some

power

C’ of C.

then Proof: If C is a coseparator, by the above proposition (19.5) is a mono-source, so that the induced C“°’"‘x'c’ (X, hom(X, C)) morphism X must be a monomorphism (19.3). Hence (1) implies (2). Clearly (2) implies (3). To show that (3) implies (1), let X be any ‘g-object. Then by hypothesis, there -»

monomorphism X i» C'. But f (n, f), so that by is a mono-source. But each 1:, f belongs to Proposition [9.3, (X, it) of ) so that Iwm(X, C). Hence (X, hom(X, C)) is a mono-source, by the above C is a for g. proposition, (l9.5), coseparator [3 is

index

an

I and

set

a

=

o

o

DEFINITION

19.7

Let Q be

which

has

products and let J1 be a class of monomorphisms in g. A ‘K-objectC is called an .ll-coseparator of ‘6 provided that each ‘tf-object is an Jl-subobject of a suitable C’ of C. In particular: (extremal mono~ power and (regular monomorphism)ocoseparators are called extremal coseparators are called morphism)-coseparators (Cf. 19.6.) regular coseparators. DUAL

a

(if ‘6 has coproducts and 6‘ is extremal separator; regular separator.

NOTIONS:

6-separator;

a

class

of

epimorphisms

in fi’):

EXAMPLES

19.8

(l) The closed for

coseparator

(2)

category

The

extremal

unit

interval

is both

an

extremal

separator

and

an

extremal

CompTz.

two-element

coseparator

discrete for

the

space

is

both

an

extremal

of zero-dimensional

category

and separator Hausdorff compact

an

spaces.

(3) CRegT; has (4)

For

same

as

the

no

extremal

coseparator.

categories Grp, R-Mod,

the extremal

coseparators.

and

Set, the coseparators

are

precisely the

Limits

130

in

Categories

Chap.

VI

Conditions

Stronger Smallness DEFINITION

19.9

‘6 is called

A category

strongly well-powered provided that

family (X ,), of ’6’-objects,there is fi-objects X with the property that there indexed

is

a

a set

mono-source

of

each

set-

pairwise non-isomorphic from X to (X 9,.

strongly co-(weIl-powered).

NOTION:

DUAL

at most

for

EXAMPLES

19.10

Set, Top, and

Grp

both

are

strongly well-powered

and

strongly co-(well-

powered). PROPOSITION

19.11

Every strongly well-powered category one-element

the

Proof: Consider

is

well-powered.

families.

E?

the properties of being well-powered and strongly wellcategory powered are, in general, different (see Exercise 19F dual). However, as the next that the category has proposition shows, under the often satisfied condition

For

a

the two

products,

properties

are

equivalent.

PROPOSITION

19.12

If a category ‘6’ has products, strongly well-powered.

then

‘6 is

well-powered if

and

only if

it is

family of Proof: Suppose that “’6is well-powered and (X i), is a set-indexed X be the of the Then the definition product family (X,),. by ‘6’-objects.Let ([1 i, m) of product, for each mono-source (A, A A, X r) with codomain (Xi),, there exists a unique morphism f: A FIX; such that for each is I, f,niof. is a for Thus each mono-source Also, by Proposition 19.3, f monomorphism. (X i),, there corresponds a subobject (A, f) of 11X p (A, ([9) with codomain than a set of pairwise non-isomorphic Since ‘6’ is well-powered, there is no more subobjects of “X 1. Hence there is no more than a set of pairwise non-isomorphic from A to (X i ,. 1:] ‘fi-objects A with the property that there is a mono-source —>

of Sources

Factorizations

=

and Sinks

DEFINITION

19.13

(Xi L» X, X) be

Let

of

monoMaetorization (1) (gi, Y) is Y

an

X is

sink

(f,, X)

in %. Then

if and

epi-sink, a monomorphism,

(2)

m:

(3)

foreachi,x,.£.x=

—.

a

only

Xi L

Y 1»

Xis

called

an

(cpl-sink,

if:

and

Xig—HYLX.

Analogously, one has [(extremal epi)-sink, mono]-factorizations mono)-factorizations.

and

(cpl-sink,

extremal

DUAL

NOTIONS:

source]-factorizations;

[epi, (extremal (cpi, mono-source)-factorization; (extremal epi, mono-source}factorizations.

mono)-

PROPOSITION

19.14

% has

and equalizers, well-powered and has intersections [(extremal epi)-sink, mono];factorization.

‘6 is

If

an

The

Proof: (1}, X)

Let

131

and Sinks

Sources

19

Sec.

that

be

a

of

part

are

is

proof

in ‘6’ and

sink

to the

analogous

proofs

of

"0).! be the

let (D j,

then

sink

every

in

Propositions 17.8 and 17.16. family of all subobjects of X

factorization

some

xi‘i‘l’.Djflur X‘u—‘)>X= (1}, X). By Proposition 17.7, the family (D j, mi)J has an intersection for each i, (D, m) that is a subobject of X. By the definition of intersection, each the D that for X such there is a unique morphism e,-: j diagram ,. of the sink

->

commutes.

We need

sufficient Definition m is

to

only show that (ei, D)

the

(extremal epi)-sink, and

an

dual

of the

to

such

for each

that

do this it is

(ii) of

condition

extremal

19.1(3) (see Proposition 19.4 dual). Suppose that (93, Z) is

monomorphism

a

it satisfies

that

show

is

a

sink

and

i, the triangle

X——>D

\/ fit) belongs

family (DJ, ”11),;i.e., there is some m m 1. Since m is a monomorphism, we have j 6 J such that m fit d; ti: (and a monomorphism); hence an isomord} 1. Thus fit is a retraction is an (extremal epi)-sink. C] phism. Consequently (e,-, D) Then

commutes.

(2,

m

0

2

=

0

=

to

the

o

=

o

EXERCISES 19A.

is

Suppose epi-sink if and

an

l9B.

Suppose

(f,, X) is a sink and only if f is an epimorphism. that

that

one

has

the

source

(A,f,).

Prove

the

each

i.

f,-

=

f.

Prove

that

(f,, X)

factorization

AflBi= of the

for

ALCFL'KB

following:

(resp. (extremal (a) If (A,f,) is a mono-source monomorphism). morphism (resp. extremal

mono)-source),

then

Ii is

a

mono-

I32

in C ategories

Limits

(b) If h is Obtain

as

a

results

earlier

19C.

(a)

and

monomorphism

a

some

Interpret functor

from

natural

transformation

then

mono~souroe,

monomorphisms

(A. f,) is

a

V]

mono-source.

corollaries.

as

two

ways:

of the form

category

a

a

in ‘6’ in the following

source

a

(C. 51,) is

about

Chap.

%.

into

(b)

as

a

between

l9D.

Prove

that

each

I9E.

Prove

that

if 9,”has

71,) is

product (le,.

and

products

from

functors

two

(extremal

an

C is

discrete

a

mono)-source.

g-object.

a

into @.

category

then

the

following

are

equivalent: coseparator. (a) C ts an extremal (b) For each object X (X, hom(X C )) is an (extremal (c) For each ‘g-object X, the unique morphism induced is

19F.

a,

sé

the

Consider

ordinal

each

let a.

a

51:13]:

and

following subcategory ‘6’ of Set. Let a be a, be unequal sets such that for any two

A, each

—'

C"°"'(""C’

and

set

a

for

ordinals

a

and

ordinal

a.

A," for each the

are

where

for

ham-sets

following

a.

ham(A,. A,) hom({a}, 21,)

=

all four

=

both

front

functions

from

functions

A, to 21,. {a} to A,.

specified by the above are empty. epimorphisms are precisely the surjections. and conclude (a) Prove is a c0o(well~powered) category. (b) Prove that ‘6 is not strongly co-(well-powered). All ham-sets

not

is either

homomorphisms) In

19H.

a

co-(well-powered) has

there

is

a

equalizers, factorization

a

regular

monomorphism.

complete lattices

strongly

or

show of

that

(and complete co-(welI-powered). a

sink

(1}. A)

Obtain

I91.

Separating

Let g be any

Proposition

category.

Sets

17.16

an

cpi-sink

(f,. A)

must

m

be

isomorphism.

an

Obtain

corollary. l9].

is

=X.£‘_.BL.A

AA m

of

the category

not

or

that

category

only if whenever

if and

whether

Determine

l9G.

that ‘6

in ‘6’ the

that

with

3,

1.2.

objects of ‘6’ are the following sets: {a} and there {a,, a2}. Besides the identity functions.

The =

X

by the product

monomorphism.

extremal

an

mono)-source.

as

a

corollary

to

Proposition

l9.|4.

l6G

as

a

Sec.

and Colimits

Limits

20

that

(a) Prove

for each

.9’ of fi-objects, the following conditions

set 1'

(i) For

X

pair

any

133

‘6-morphisms,

Y of distinct

:1

there

equivalent:

are

exists

S

some

.5” and

e

a

hzs—t

Xsuch ~75goh. thatfoh (ii) For each @—objectX, the pair (

U

hom(S, X), X) is

epi-sink.

an

56V

.9’ is called

(b) Prove

separating

a

if g

that

is

5” is

‘g-objects, then

for ‘6

set

connected

a

separating

a

provided

set

it satisfies

that

has

that category for ‘6”if and

only

(i) and (ii). and

coproducts if

LI{S|

S

e

5’ is

3’} is

a

set

of

separator

a

for g.

(c) Show

Set

that

x

Set has

but that

separator,

no

{(E. (QB. ({Q}, m} is

a

separating 19K.

for

Set

that

the

set

Prove

x

Set.

functor

§20 In this

section

x

_

LIMITS

AND

20.1

the notion

Set

x

—»

Set

(18A) is

not

faithful.

COLIMITS

functor, which is a generalization object”, “equalizer”, “intersection", and “product”. We have seen that using the categorical notion of product, one can simultaneously prove many theorems about particular products in various categories. Generalizing one step further, we will see that with the of limit one can simultaneously prove theorems about particular limits concept and products). For example, the fact that (such as equalizers, intersections, limits are “essentiallyunique" (20.6) will tell us immediately that terminal objects, and are intersections, equalizers, products “essentially unique”. Later, other special types of limits, such as inverse limits of directed systems, inverse images, and pullbacks will be introduced and will provide us with additional useful tools for working within the realm of category theory. we

of each

introduce

Set

_:

of the

of the limit

of

a

“terminal

notions

DEFINITION

If I and

‘6’ are

and

D:

1

D(i);

at

each

—>

D(i) L

V I;

Dim)

DU)

commutes.

In

object

other

words,

is L and

whose

if L: I

value

—>

at

@ is the

each

functor

constant

morphism

is

whose

value

1L, and if (L, (It)ie0b(!))is

a

134

Limits

Categories

in (K,then (L, (ldteoun) is a natural natural transformation from L to D.

source

is

in

a

A natural

DUALLY:

sink for D is

transformation

natural

from

To

simplify notation, (L, ((0,) or even (L, 1,) to

we

often

functor

write D, rather

the natural

K: I

than

only

where

((k,),co,,m,K)

D to the constant

denote

for D if and

source

sink

a

Chap. VI

—>

if

(macho) is

$9.60,",

a

(6‘.

D(i) and usually write

(L, Omega”).

source

DEFINITION

20.2

If D: I

—>

‘6’ is

then

functor,

a

natural

a

provided that if (E, 7,) is any L such that for unique morphism It: I: of D

limit

source

natural each

—»

source

j

e

(L, 1,) for D is called for D, then there is the

0b(I),

a a

triangle

h)

("r—-is ——-—>D

15 commutes

Exercise

natural

“terminal”

a

sink

A natural

DUALLY:

(L, 1,) is

that

[i.e., provided 20A)]. sink for D factors

(k,, K) is called a uniquely through it.

colimit

natural

of D

(see

source

provided

that

every

EXAMPLES

20.3

Let I be the category

(1)

lm2 .—}. ,

_., fl

(L, (1):”) is a limit of D if and only if (L, 1,) is an D(m) 11 D(n) 1,. ((k,),=.,'2, K) is a D(n) and I2 equalizer of D(m) colimit of D if and only if (k2, K) is a coequalizer of D(m) and D(n) and k2 D(n). k1 k; D(m) a (2) Let I be a category that is just a sink (A, it 110,110), and let D: 1 be a functor such that for each i, D(f,) is a monomorphism. Then (L, (1,), lo) of the family (D(A,), D(f,» is a limit of D if and only if (L, 10) is an intersection for each i e l. of subobjccts of D(Ao) and IO D(j}) l,fi’, then (L, (l,)) is a limit of D (3) Let I be any discrete category, and let D: I if and only if it is a product of the family (D(i)),-Eo,,,,,,and ((k,), K) is a colimit of D if and only if it is a coproduct of the family (D(n)),eoun. ‘8, then (L, (l,)) is a limit of D if and (4) lf 1 is the empty category and D: I only if L is a terminal object for ‘6 and (1,) Q ; likewise ((k,), K) is a colimit of D if and only if K is an initial object for {6’ and (k,) Q}. g is the identity functor, then (L, ((4)) is a limit of D if and only (5) If D: ‘6’ if L is an initial object of Z’ and for each A e 011%), 1,, is the unique morphism let D: I

and

—»

%. Then and

=

=

0

=

o

=

o

o

-»

=

o

—>

—>

=

=

—»

and

Limits

Sec. 20

Colimits

135

((kA), K) is a colimit of D if and only if and for each A e 0b(fi), k], is the unique morphism in this example the families ((4),,EON‘O’) and (kA)Asown from

L to A.

K is

terminal

a

from

object

of ‘6

[Notice that necessarily sets.]

A to K. not

are

PROPOSITION

20.4

Any limit (L, (l,-))of If

Proof:

Q

':;

L

a

D: I

functor

‘3, is

—>

‘6’-morphismssuch

are

(extremal mono)-.source.

an

that

for each

i

e

S

I, (Q, (l,- r),eo,,(,)) is anatural

then

is

unique morphism Q L»

a

of

r

and

To show

source.

s

is such

an

h

o

s,

D. Thus for

that

l,-

=

11. Hence

it is extremal,

that

o

for

L such

l, But each

l,-

=

r

source

o

there

c

r

o

the definition

by

each

i

of limit,

0b(l),

e

r.

Consequently (L, (1‘))is

3.

=

0b(l),

that

suppose

a

mono-

it has factorization

L‘—‘>D(i)=LL>R—fi->D(i), where natural g:

R

is

e

epimorphism.

an

source

for

L such

that

-»

Hence

D.

for each

Since

is

epimorphism, (R, (f,)) is clearly a by the definition of limit, there is a morphism i,f,l,- g. Hence, for each i e

an

o

=

[l°g°e=.fl'°e=’i=[l'°l so

,that since

(L, (l,-)) is hence

epimorphism);

e

a

is

mono an

source,

g

e

a

1. Thus

=

is

e

a

section

(and

an

[:1

isomorphism.

COROLLARY

20.5

Each

product 20.6

regular

is

PROPOSITION

exists

a

an

extremal

(cf'. 17.“ OF

(UNIQUENESS

of (L, (7.)) L unique isomorphism h: L

each

If

is

monomorphism (extremal mono)-source

an

(L, (19) and

is

a

—>

dual

LIMITS) limit of

that

such

monomorphism, and each and Exercise [90). {:1

thefunctor for each i

D: I

—>

0b(1)

e

‘6, then there the

triangle

l~

M

L

such

that

for

of limit all

1',

there

Lo]:

are =

I,

unique morphisms h: L —> L lick 7,. Consequently,

and

=

for all i.

=l,-:h=

lioksh so

that

hence

since h is

an

(L, (1,.)) is

isomorphism.

a

mono-source,

El

l,-=l,-51L keh

=

1L. Similarly

hok

=

ll;

136

Limits

20.7

in

Categories

Chap. V!

COROLLARY

Terminal

are

objects, equalizers, multiple equalizers, intersections, “essentiallyunique”(cf. 7.8, 16.5, 16.12, 17.4, 18.10). D Because

20.8

of the essential

uniqueness

and

products

of limits,

by an abuse of the language we often speak of the limit of a f unctor D: I g (when one exists) and denote it by Lim D. Dually we speak of the colimit of D (denoted by Colim D). Thus in general we write —v

D

z

(L, (l,))

Colim D

z

((k,), K).

Lim

and

sometimes

However,

we

Lim

L. For

D

z

(inaccurately) call the object example, when I is discrete, Lim

D

Colim

D

("0(5),

z

L the limit

of D and

write

7ft)

and

(u,, LID(i));

2

M

whenlts

—*_,.,

.

fl

Lim

D

a:

Colim

D

z

Equ(D(m), D(n))

and

and

when

I is the empty

the initial

Coeq(D(m), D00) D is the

Lint

category,

D is

object and Colim

terminal

object.

EXERCISES The

20A.

Suppose the

that

natural

of Natural

Category

D:

I

—>

‘6’ is

for

sources

L such

—>

object is of terminal 208. 20C.

(a)

Prove

a

limit.

Thus

Let

the

D where

morphisms f: L morphisms is that induced (a) Show that (6,, is indeed (b) Prove that (L, 1,) is a as immediate (c) Obtain essentially unique. (By the above, every limit those

Sources

functor.

a

from

@Dbe for

each

corollaries is

facts

the

if and

that

only if it is

limits

are

a

limit

the

sense

of D. and

mono-sources

object. We have already seen that theory of limits is equivalent

terminal

a

some

=

‘6.

terminal

in

from

morphisms

that

quasicategory. object of $0

a

quasicategory whose objects are (L, 1,) to (L, l,) are precisely i, I, l,o f; and composition of

the

terminal

each to

are

the

theory

objects.) Describe

multiple equalizers

Suppose

that

that

the

(i) (L, (I,)) is (ii) ((1,), L) is

D:

following a

a

natural

natural

I

—>

are

‘6’ is

limits

and

functor.

a

equivalent: for

source

sink

as

for

D.

D”:

I""

-»

‘6”.

multiple coequalizers

as

colimits.

137

and Colimits

Limits

Sec. 20

(b) and that the following are equivalent: (i) (L, (l,)) is a limit of D. (ii) ((1,). L) is a colimit of D”. General

20D.

It‘d whose

and

Comma

Categories

g and .98 3—)g —F—)

objects

f: F(A)

—>

are

those

the

then

form

triples

of

whose

morphisms

and

6(8);

functors,

are

the

(0. b): (14,}; B) where

A

a:

A’ and

->

b: B

—»

A

e

0b(.g¢), B

6

012(3),

3')

(A'J'.

that the square

B", such F

(A, f, B) where are those pairs -*

(F, G) is the category

category

comma

(A) —L’G(B)

16 F(A')—f—’>G(B') (b)

F (a)

is defined

of morphisms

Composition

commutes.

(a,b)o(é,

6)

by:

(aofi,bol}).

=

(a) Verify that (F, G) is indeed a category. whose value at ‘6”is the functor and G : l ‘6 is the identity functor (b) If F: ‘6 the comma the single object is A, show that (F, G) is isomorphic with category (‘6, A) and that (G, F) is isomorphic with the comma category (A, %) (4.18 and 4.19). on @, show that (F, G) is isomorphic (c) If each of F and G is the identity functor ‘62 with the arrow (4.16). category ‘6 whose value at the single object is A and G: l (d) If F: l —t ‘6’ is the functor whose value at the single object is B, show that (F, G) is isomorphic is the functor with the discrete 6.3., the set) hams“, B). category ‘6’ be functors defined GI 9. f and P2(f, g) by P10; 9) (e) Let P1, P2: Define “projection functors” 5%, Q2: (F, G) 3, and H:(F, G) —' 9‘ Q1: (F, G) such that the diagram —v

—>

a

—»

=

=

—>

—>

mfi—(F.G) —Q'—>.@

commutes. (f) Let I be the category C

0

I

AJ/ \,.fl r

and

let

and

D(s)

and

Q2

D:

I =

‘64:“?

be

P2. Show

that

-’

is the limit

of D.

the

functor

s

defined

(F, G) together with

by D(m) the

=

functors

F, D(n)

Q,, Pl

G, D(r) H, H, P;

P1,

=

=

0

o

H,

138

Limits

§21 In this section

notions Definitions 21.1

we

of limit and and

in C aregaries

PULLBACKS

investigate

AND

Chap. VI

PUSHOUTS

important special

some

cases

of the

general

colimit.

Examples

DEFINITION

(l) The

in %

square

PI

P —)D1

D2 ——>Do f:

is called D: I

—»

a

(P, (p;),=o',‘z) is

square provided that I is the category

pullback

‘6 where

D(m) fl, D(n) fl, is a pullback square if =

=

and

p0

=

fl

it commutes

=

op,

and

f; for

0

p2.

a

limit of the functor

In other

words, the square

any

commutative

that

the

square

of the

in the

diagram

form:

5L0!

D: —)Do f: there

exists

commute.

a

unique morphism

II: P

—>

P such

triangles

Sec.

2]

Fullback:

and

Pushouls

139

If

(2)

FLA

ml in D2 _>Do 3

pullback square, then p2 is said to be a pullback off1 along f2. In the case f I is a monomorphism, p; is commonly called an inverse image of 1} along f2. (3) We say that ‘6’ has pullbacks provided that each functor from is

a

that

to

Q? has

a

limit;

figure

each

i.e., provided that

o

1,.

——)

o

f, be extended

in (K can

that

to

become

pullback

a

square.

‘6 has inverse

images provided

figure

each

if:

0—)-

f, in ‘6, with

fl

a

monomorphism,

has

along f2;

be extended

to

become

a

pullback

pushout of f1 along f2; direct square; has direct images; in particular, the square

Pushout

NOTIONS:

DUAL

can

pushouts;

square.

image

of f1

004.,“

fxl lpl D: ——)P P:

is called

a

pushout

l is the

where

square

provided

that

((p,),=o‘1_2,P)

category

n\b.2 Don)

=

I]. D0!)

=

f: andpo

=

P1°f1

=

p:

°fz-

is

a

colimit

ofD:

l

—>

V,

Limits

140

and

limits

Since

colimits

and

pushouts the pushout) of (resp.

pullbacks

in

Chap.

Categories

VI

essentially unique, it immediately follows that Thus when they exist, we often speak of the pullback codomain (resp. pair of morphisms with common are

are. a

domain). EXAMPLES

21.2

(I) If

B

A and

of the

subsets

are

set

AL» C and

inclusions

C, with

BC» C,

then

AflBc——->A

[

l

BL———>C is

a

and

in

Top

(2) If f:

B

a

C is

a

A and

Similarly in Grp when are subspaces of C.

square in Set. when A and B

pullback

function

and

sets

on

A

B

are

subgroups of

C

C, then

c:

f“[AlL—>B A

l,

f

f"m

AL—aC is

pullback (3) If X i» a

square B and

in Set.

—g—> B are

Y

E has the

[This

motivates

morphisms

{(x,y)|f(X)

=

subspace topology,

the

=

terminology

90)}

C

X

image”.]

if

Top and

in

“inverse

X

Y

then

Y—>B

is

a

to

pullback

square

in

Top, where

p

x

and py

are

the usual

projections

restricted

£1

(4) The construction Rng, and BanSp,.

in

(3)

“works”

in many

categories;

e.g.,

Set, Grp, R-Mod,

to be topological bundles if in this example (X, f, B) and ( Y, g, B) are considered of (X, f, B) and space B, then (E, fo 1),, B) is the fibre product (= Whitney sum) of those two is the and with the categorical product and morphisms p,r p,,) (together (Y, g, B) Exercise in 21C). (see TopBunB objects

T Notice with

base

that

Sec.

Fullback:

21

to Other

of Pullbacks

Relationship

and Pushouts

141

Limits

last

example above indicates that in many categories pullbacks can be equalizers of products. Next we see that this is in fact always provided that the corresponding products and equalizers exist.

The

constructed true, 21.3

as

THEOREM

OF

CONSTRUCTION

(CANONICAL

PULLBACKS)

Let

A\/)C 3/3”

codomain. If (A x B, HA, 1: a) pair of ‘g-morphismswith common of (A, B) and if (E, e) z Equ( f 12‘, g 1:3), then the outer square be

is

a

a

product

0

o

TA°9

5—6/4 e

In,

X/lf wheel/\Af 847C is

a

pullback

square.

it commutes. If 443 Q A and so that Proof: The square is constructed B such that f of product, there qA g q,, then by the definition 433 Q A x B such that 71A oh is a unique morphism h: Q q3. qA and 113 h Thus (fo nA)oh (g as) h, so that since (E, e) is an equalizer offo 1:4 E such that e k h. Hence and g an, there is a unique morphism k: Q q3. Also k is unique with respect to this (HA e) k qA and (n30 e) k [:1 property since products and equalizers are mono-sources. —>

—>

a

o

=

—>

o

=

=

—>

c

o

21.4

o

o

=

finite products

‘6”has

and

equalizers,

T is

a

terminal

object, then the following

(1)

a

then % has

pal/backs.

PROPOSITION

If

is

o

=

=

COROLLARY

If 21.5

o

o

pullback

(2) (P,

pA.

square.

pH) is

a

product of

A and

B.

are

equivalent:

[:1

=

in

Limits

142

Chap.

Categories

Vl

Proof: Suppose that f: C

(1) => (2). object, the

A and g:

-»

C

—>

B; then

since

T is

a

terminal

square I C—>A

fill B——>T

that

by thatf=

so commutes, [11C -» Psuch

(2) is

a

PA and 21.6

The square

(1). product,

=>

pair

any

the

definition

of

pthandg

pullback, pBoh.

=

commutes,

since

morphisms

to A and

of

T is

a

B

there

terminal can

be

is

a

unique morphism

object. Since (P, pd, 1),) uniquely factored through

D

Pa-

COROLLARY

lf‘g has pullbacks

and

a

‘6 has

terminal

object,

then ‘6 has

finite products.

I]

be deleted in the above object cannot corollary, since every non-trivial group (considered as a category) has pullbacks (213) but no such group has finite products (18K). Also Field has pullbacks (21A) but does not have finite products (18L).

The condition

21.7

that

a

terminal

PROPOSITION I

If

A :3

B

are

‘6’-morphisms, if (A

B, 1:,“ 7:”)exists, and if

x

9

P

—p‘—>A

lulu/I) pal A—)(1A.9) A

is

a

(1)

pullback P1

=

square,

B

x

then

P2-

(2) (Pm)

z

Equ(,

(3) (Rm)

z

EqUU. a)-

)°k k

since

and

the

is

product

a

°

k

the square oh. [3 pl

thatk

=

square.

k. Then

a

is

pullback

a

°k'

("A °°k

=

there

square,

is

a

unique

h: K

—.

P such

COROLLARY

21.8

finite products and finite intersections,

‘6 has

If a category equalizers. [:1

then

‘6' has

THEOREM

21.9

For

(1) (2)

g

pullback

a

(“9° (la°g))°k

we

mono-source,

since

that

=

=

°k so

is

k,

=

(“4° (lAvf>) Since

143

category

any

(6’ has a has

‘6, the following

equivalent:

are

equalizers and finite products. pullback: and a terminal object.

Proof: (1)

of

is the

(2)

immediate

(2).

=>

=9

‘6’ has

from

Corollary family.

21.4

and

the

fact

that

a

terminal

object

empty product if Q? has (1). pullbacks and a terminal object, then by Corollary 21.6, finite products; hence by Proposition 21.7, ‘6 also has equalizers. E] an

of Pullbacks

Relationship

to

Special Morphisms

PROPOSITION

21.10

Suppose

that

the

diagram

.;i l

,4

commutes.

(i) If the (2) If outer

"outer

the “inner square

is

a

square"

is

a

pullback

square,

square" pullback

is

a

pullback

square

square.

[:1

then

so

and h is

is the a

“inner

square”.

monomorphism,

then

the

in

Limits

144

VI

Chap.

Categories

PROPOSITION

21.11

If

pi»:

”l l; A—>B

pullback square, Coeq(p1. 112)-

is

a

Proof: If f is

morphisms Q h:

Q —> ifgw. definition Hence

‘2' A.

a

regular epimorplu'sm if and only if (f, B)

regular epimorphism, By the definition

of

oh and

q;

then

U; B)

pullback

square,

a:

Coeq(q,, ‘12)for there exists

a

2

some

morphism

l:

P such =

a

then f is

that

=

q,

p,

oh

thengom

g°pz.

=

coequalizer, there is (f. B) z Coeq(p1.pz)« III of

h.

Clearlyfop, so on °P2°hs thatg°r11= k such a unique morphism =

p;

o

12,. Now

=fo

By

g°qz-

that

g

=

k

the

012

PROPOSITION

21.12

In any

is

A

category

Bis

a

monomorplzism if and only if IA

A ———>A

l

A——-->B

is

a

pullback

l;

[:1

square.

PROPOSITION

21.13

Every pullback of:

manomarpltism is a monomorplzism (thus in particular monomorphisms). (2) a regular monomorphism is a regular monomorplxism. (l)

a

(3)

a

retraction

is

a

Proof: Suppose

is

a

pullback

square.

retraction.

that

all inverse

images

are

Let

(I).

be

f

f°(r°h) that

so

since

is

f

h, k: Q

If

monomorphism.

a

g°(S°h)

=

P such

-»

r

oh

r

=

o

that

s

o

h

s

=

o

k, then

=f°(r°k).

=go

monomorphism,

a

145

and Pushouts

Fullback:

21

Sec.

k. Since

pullbacks (being limits)

k. simultaneously, so that h (q cg) us. Now if I: Q -> B (2). Let (A,f) z Equ(p, q). Then (pog)os such that (p g) at (q g) t, then by the definition of equalizer, there is t. Hence u A such that f some u: Q by the definition of pullback, g I: is unique with respect to‘ h. Moreover, s P such that t h: Q there is some this property since by (l), s is a monomorphism. Thus (P, s) z Equ(p g, q g).

cancel

can

we

mono-sources,

are

and

r

=

s

=

o

o

=

o

—v

o

=

o

—>

o

=

o

Exercise.

(3).

o

[:1

Congruences

f: A —» B is a group homomorphism, then mined by f is (in the elementary sense) the subset f (b). Obviously S can pairs (a, b) with f (a) =

A

A, and

x

projections,

if

SC»

m:

then

A

A

x

(according

to

is the

be

regarded and

embedding 21.3) the

Theorem

relation

the congruence S of A x A

If

A

consisting of all a subgroup of

as x

deter-

A

53A

the

are

31

square

71°"!

S——>A

M1 if A——>B

is

a

pullback This

square. motivates

our

next

definition.

DEFINITION

21.14

(1) If 0

—-fl

P 0

01 If '

—>

r

a congruence then the pair (p, q) is called pullback square, relation (2) A pair (p, q) of ‘6-morphisms is called a congruence there exists some @morphism f such that (p, q) is a congruence

is

a

21.15

relation

off.

provided relation

of

that

f.

PROPOSITION

Let

(p, q)

be

a

congruence

relation (1) (p, q) is a congruence the composition is defined).

relation

of

m

off. Then of, for each monomorphism

m

(for which

146

Limits

(2) if f (3) c z

=

It and It

o

g

a

h

=

p

Coeq(p, q) implies

in

Categories

q, then

o

that

(p, q)

(p, q)

is

a

is

C Imp. VI

a

relation

congruence

relation

congruence

of

of II.

c.

Proof: (1) and (2) follow immediately from Proposition 21.10. To notice that since c z Coeq( p, q), there exists a morphism g such that f Apply (2). El 21.16

any

(3) c.

o

g

square

following

(l) The

equivalent:

are

is both

square

pullback

a

and

square

(2) (p, q)

is

a

congruence

relation

of f

and

f

z

(3) (p, q)

is

a

congruence

relation

of f

and

f

is

(4) (p, q)

is

a

congruence

relation

and

f

Proof: implies (4) first that A

That

from

(p, q) is

C such

pushout

square

(i.e.,

a

pulation

that

a

pot

a

regular epimorphism.

(2) implies (3)

Proposition

21.1].

congruence = 1,. and

relation, there

that

suppose

square,

Coeq( p, q).

Coeq( p, q).

z

and

(I) implies (2)

follows

since

-*

pus/tout

a

41.6).

see

square,

t:

=

PROPOSITION

For

the

show

r

got

and

s

To

see

is immediate.

that

That

exists

(4) implies (1) a unique morphism

1,4-To show that the square morphisms such that rop

=

are

(3)

notice

=

is s

o

a

q.

Then

rolA

r=

Hence

ro

=

p

r

phism 21.17

PROPOSITION

(1) f

is

a

has

so

r

=

since

that

II of

=

s.

the following

category,

f z [:l

=S°]A

Coeq(p, q),

=3,

there

exists

a

unique

mor-

equivalent:

are

monomorpliism. is

(2) (l, l)

(3) f

q

that

It such

In any

o

=50q0f

rope]

=

a

a

relation

congruence

relation

congruence

off. of

the

form (p, 1)).

Proof: That (I) implies (2) follows from Proposition 21.12 and that (2) that r and s are morimplies (3) is trivial. To see that (3) implies (I), suppose definition r 5. Then the of that such by f f pullback, there is a phisms It and s s. It; hence r E] p p morphism It such that r o

o

=

=

o

=

o

=

147

and Pas/touts

Pnllbacks

21

Sec.

EXERCISES Show

21A.

it does

have

[Using

that

21B.

Show

2lC.

Generalize

category

Field does not have finite category the canonical construction form the pullback

though

even

pullbacks.

for any

that

that

(considered

group

every

the

the statement

as

products, in

Rng.]

category) has pullbacks.

a

in the footnote

to

Example

21.2(3) by showing

together

with p1

%’

pl

P—>A

17,1

1’

is

a

pullback of A

product

in '3’ if and

square

L)

C and

B

only if

3—) C

in the

P

m

C

category

comma

(W, C)

of @

over

and C

p,

is

a

(4.19).

and Prultouls Correspondence Between Fullback: of forming pullbacks and pushouts of commutative Show that the process squares of forming the that if P is the process yields a Galois correspondence in the sense pullback square 2] D.

Galois

L—>Ii

C—>D of the lower

of

corner

commutative

a

square

[Ifii C —-—->D

and

Q is the

process

of

forming the pushout A

square

—>l[

C —>K of the upper 2|E. Show

corner,

then

PQP

=

P and

Pasting and Cancelling Fullback

that:

(a) if the

smaller

squares

in the

figures

QPQ

=

Q.

Squares

148

are can

Limits

in

Categories

Chap. V!

pullback squares, then so is the large rectangle and be composed by “pasting their edges together“.

(b) if figure A

is

Fullback:

21 F.

Show, in any category specifically if

its

and

commutes

its left square

then

and

right

large square; are

square

i.c., pullbacks

pullback squares,

square.

and Products

‘6’,that

rectangle

outer

pullback

a

the

Commute

the

product of pullbacks is

a

pullback of the products;

fi——>2

P—>A

l J: iB—HC If 73—3’5 and

are

pullbacks and if A x A, B pullback (P, p1, p2) of

x

B, and

C

x

C‘exist, then P

x

15exists if and only

if the

2

A

XC mu? exists, and if they exist,

P

x

f’ and F

are

a

isomorphic;

moreover,

if I is the category

0 A

1

m

m

efit

.—A—")' n

and D: l then

P

figure

x

‘6’IS the functor

defined

by D(m)= f D(n): g, D(m)= f, and D(n) 9, f’ together with the six morphisms to A, B, C, A, B, and C‘indicated by the

->

=

Sec. 21

is

a

Fullback:

and Pas-hams

limit of D, and 1’ together with the six morphisms

cated

the

by

149

A",B,

A, B, C,

to

and

C‘ indi-

figure

\ko k)

3:.)

\ha

to;\\ is also

limit of D.

a

21G.

(j; g) is

Suppose that

a

Show

of X with

that

itself, then a

the

relation

congruence

21H.

(i) f is (ii)

v/ Q)

in)

constant

pair of morphisms (f, g) has if and only if it is a congruence

if f: X —> Y is a morphism the following are equivalent:

and

(X

coequalizer. Prove that of Comm 9).

a

relation x

X,

n1,

n2) is the product

morphism.

Xxxiwr

1 1f X—>Y

f is a pullback square; (iii) fo 2:, In :12.

i.e., (3,, n2) is

a

relation

congruence

off.

=

211.

has

a zero

(i) (K, k) (ii)

is

a

Show

that

if

object 0, then z

Ker(f).

pullback square.

f: X the

-»

Y and

following

k: K are

—>

X

are

equivalent:

morphisms

in

a

category

that

in

Limits

150

211.

L»

if X

that

Prove

Y is

morphism

a

Chap. VI

Categories and

Z is

object in

an

any

category,

then

XxZ—wY

7x

If

[“2 YXZ—H’ ”Y

is

a

pullback

square.

Multiple Pullbacks the multiple pullback of a sink ((f,),, A) is defined to be the limit (L, (1;), d) (if it exists) of the sink regarded as a functor D in the following way: the i is formed to be a discrete and a new category category indexing set I is considered I form: a terminal has the it from object I; i.e., by adjoining 21K.

In any

category

1°

m;

’"j

j.

m

and

d

the

D is defined

functor We

by D(m,)

o]

=

has category set-indexed every

that

say f, 01,.) pullbacks) provided that notion is that of multiple pushouts. =

a

,5, for each is 1. (Notice that for all i6 1, multiple pullbacks (resp. has finite multiple sink (resp. finite sink) has a limit. The dual

as multiple pullbacks. (a) Interpret intersections (b) Prove that a category has finite multiple pullbacks if and only if it has pullbacks. (c) State and prove analogues of 21.3, 21.4, 21.5, 21.6, 21.7, 21.8. 21.9, and 21C for multiple pullbacks.

21L.

Prove

if ’6’ has

that

pullbacks., then the following a regular epimorphism.

equivalent:

are

(i) Every epimorphism in Qi’ is (ii) If e is an epimorphism in ‘6" and

0—).

He o——)o

e

is

a

pullback Show

21M.

pullback ZlN.

(a) Show (b) Show is

of

a

then

square,

that

section

Pill/backs

even

is not

it is also

a

pushout

square.

is

though the pullback of a retraction necessarily a section (cf. 21.13).

a

retraction,

the

of Epimorpln‘sms

Top the pullback of an epimorphism is an epimorphism. that in Grp, R-Mod. Lat. Rug, and Mon the pullback ofa regular epimorphism that

in Set

and

regular epimorphism. that in general the pullback is not necessarily an epimorphism. a

(c)

Show

of

an

epimorphism

(resp.

a

regular epimorphism)

and Direct

Inverse

Sec. 22

15]

Limits

(Regular Epi, Mano)-Factorizations that if any of the which has pullbacks and coequalizers. Prove eategory Q? is then is satisfied, uniquely (regular epi, mono)-factorizable. following conditions 210.

Let ‘6’ be

a

(a) The pullback of each regular epimorphism in Q’ is an epimorphism. (b) The class of regular epimorphisms in ‘6’ is closed under composition. and reflects regular functor U: Q -» Set which preserves (c) There exists a faithful epimorphisms. of a g-morphism relation 1', let g z Coeq(p, q), and [Let (p, q) be the congruence is a monothat m mo let m be the unique morphism such that f g. To show consider: morphism, For (a): The diagram =

of suwessively constructed For (b): A factorization

g'

o

g

z

Coeq(p, q)

21?.

Show

to

that

is

Tom, for which I pullback square. in

pullbacks. =

m

investigate

now

of limit

and

every

commutative

dense

a

g‘ a regular

g with g' is an

the

fact

that

isomorphism] square

g is

embedding.

AND

INVERSE

additional

some

Use

epimorphism.

perfect. and k is

DIRECT

an

embedding.

must

be

a

LIMITS

important particular

cases

notions

of the

colimit.

DEFINITION

22.1

(l)

A downward-directed

that

each

(2) Any into

o

that

§22 We

h

show

a

pair

of elements

functor

from

category

is

class

a

‘6’ is called

has

a

a

lower

partially-ordered

inverse

the

with

property

bound.

downward-directed an

class

class

(considered

as

a

category)

system in ‘6.

D: I —. ‘6 is an inverse system in ’6, and (3) If I is downward-directed, called the inverse limit of D. is the limit of D, then (L, I.) is sometimes

(L, l,-)

Limits

152

%’ has

(4)

functor

inverse

D: l

are

VI

Chap. downward-directed

I, each

set

limit.

a

direct

in (6’; direct

system

limit; has

limits. is

an

apparent

defined

to

be

particular

for the “switch"

reason

Top, inverse

Definition ”free

and

22.l.

situation

defined

were

is similar

particular categories

are

in certain

historically,

limits

limits

to

in

“inverse"

when

terminology

and “direct”

limits

is that direct

This

in

inconsistency

a

colimits. such

The

Grp

as

consistent

manner

encountered

some

limits

with

earlier; e.g., the

product” of groups is really a categorical coproduct and the “Whitney of topological bundles is really a categorical product (18.4).

sum"

EXAMPLES

22.2

inverse

(l) Categorical notions

of inverse

limits

and

direct

(2) For

direct limits

(or projective)

categories Set, Top, Grp, and and

for each

that

provided

There

and

Categories

Upward-directed class;

NOTlONS:

DUAL

direct

limits ‘6 has

—»

in

R-Mod.

limits

and

Each

coincide

direct

with

(or inductive)

of these

categories

the

limits

has both

classical

in the inverse

limits.

a

given

functions

is

set

direct

a

into it is the direct

A, the family of all finite subsets of A together with inclusion system in Set, and A together with all of the inclusions of the system.

limit

(3) Similar to (2) above, each group is the direct limit of its finitely generated subgroups, each R-module is the direct limit of its finitely generated submodules and each partially-ordered set is the direct limit of its finite subsets. (4)

A Hausdorff

it is the direct

space is compactly generated (i.e., is limit in Top of its compact subspaces.

Hausdorff

(5) Every compact and every

spaces,

compact

of the concrete

In each

Field, direct an

each

limits

upward-directed 0b(I),

is

be

can

00')

set

is

an

inverse

an an

inverse

inverse

and

let D: I ifi

‘6’ be

a

:

W

Set

-.

be

metric

polyhedra. However, polyhedra. The reason composed (or iterated).

of

be

a

direct

=

and I be

in ‘6, where

system

for

Alfie/4r

appropriate forgetful functor coproduct (i.e., disjoint union) of (U(A,-)) in Set. Define on C by: U

compact

then

sj,

Dow) Let

of

only if

limit of limit

cannot

systems

limit

if and

categories Set, SGrp, Mon, Grp, R-Mod, Rng, in the following canonical constructed way: Let

A,- and

=

is

space

Hausdorfi~ space every compact for this lies in the fact that inverse not

(6)

is

space

metric

k-space)

a

the

and an

let

(,u,, C) be the equivalence relation

~

“if

x

if and

and

y

only

are

members

if there

is

some

of C with

k 2

U(A,-) and y e U(A,-), then x i,j such that U(fu,)(x) U(fj,,)(y)." x

e

=

~

y

22

Let

h: C

exists

C/~ be the corresponding natural unique “‘K-objectstructure” on Cl~ such

a

‘g-morphisms 1,: A,

U(Ai)

that

Cl~

—*

C. (1,, C) is the direct

—»

of the

limit

family. [Observe in particular that Field has direct limits but in essentially the (7) The direct limits in Top are constructed in (6) except that the topology on C/~ is taken that described such

one

all of the functions

that

11 o u,-

there

Then map. all of the functions

quotient

—»

h° fl‘: become

153

Limits

and Direct

Inverse

See.

direct

given

coproducts.]

not

manner

same

as

to be the finest

continuous.

are

PROPOSITION

223

Let I be

g is the restriction

—>

then the

following

(1) (L, ([9,) (2) For each of D.

[Thus

is

a

limit

limits

direct

limits

Dually,

of

an

“initial”

subclass

of I afunctor, L 1,: —» D(i),

Q? is

E.

J, there

—

inverse

I be

equivalent:

are

i in I

c

is some j in J such that j S i). If D: I -+ of D to J, and for each i e J there is some

i in I there

(i.e., for each E: J

class and let J

downward-directed

a

are are

is

1,:

some

L

-»

that

such

D(i)

(L, (l,-),) is

limit

a

by inverse limits of “initial” subsystems. by direct limits of “final” subsystems.]

determined

determined

Proof: (2). Let Let I, D(m)

(1)

=

is

natural

source

natural

h: R

->

If 22.5

J. Then

—

is

there

J is initial downward-directed, function a well-defined assignment yields

D.

(R, (r,),) the appropriate

This

that

L such

m:

j

-+

If

be

augmented existence

to

of

i.

(L, (l,),) (1,)1) is a

(R, (r,),) is also

E.

the

guarantees

triangles

J and

in I, and and (L,

I is

for

source

j in

some

a

a

provide unique

[Z]

commute.

COROLLARY

22.4

D:

I

source for Clearly (L, (l,),) is a natural for E, then, as above, the family (n), can

(1).

=

object of

an

11-.Since

of E, this of D.

limit

a

o

limit

a

(2)

i be

=

I

—>

I is

a

(t? has

downward-directed a

set

(L, l,-) with L

limit

with

a

smallest

D(io).

=

element

i0, then each functor

E]

PROPOSITION

Products

are

inverse

limits

of finite subproducts. Specifically,

that

suppose

(i) (X,), is a family of %’-objects; of (X l-),; ,-, 71,) is the product (ii) for each finite set J c I, (51X (iii) for all finite subsets J and X such that J unique morphism induced by the projections;

(iv) for each

i

e

I, h,:

—»

{11X j

X

,-

is the

c

K, pm:

is the HEX, SIX,—»

projection (iso)morphism;

and

154

Limits

in

Categories

for the inverse limit system (0) (L, (11)) is a natural source the finite products and the morphisms (pm) between them. Then the

following

1“,»is

o

D induced

by

equivalent:

are

(1) (L, (l_,))is the limit of

(2) (L, (h,

VI

Chop.

the

D.

product of (X;),.

Proof: Clearly h, 1",: L —> X‘. Suppose By the definition of product, for each finite set K such that for eachj e K, r“: R —> 12X,(1)

(2).

=>

o

°

7‘1 ’1: if]

Now

for

K, then

c:

eachj

nJ°pKJ°rK Hence

since

source

for the inverse

that

products

IJ

g

pgJ there is

so

r,. Thus, in

=

ht°hn°g and g is easily seen to be is the product of (X ,),.

(2)

(1).

=

definition each

i

e

unique

=

I, f,: R

X,. unique morphism -—>

fi-

=

71'1°"J-

=

Thus

r J.

=

rx

a

a —r

natural L such

hr°rm =fi this property.

to

Hence

(L, (h, 01(0))

for the inverse system D, then by the (R, (r,)) is a natural source is a unique morphism g: R —> L such that for of product, there

I

for each “W":

22.6

a

e

If

o

Thus

i

(R, (r 1)) is unique morphism g: R particular for each i e I, o

with respect

h, 1m Hence

"K

each

I, there is

c

hj°PJ(j)°pKJ°rK =1:i

-_-

mono-sources,

o

°

for

J

e

system D,

finite J,

for each

are

hi °me

=

that

since

finite set J =

products

c

I, if

ht°PJm°rJ are

i

h, rm.

o

g

o

6

J, then

=

hi°"m

=

mono-sources,

r,

=

=

ht°lm

°9

[J

E]

o

g.

=

751°IJ°9-

COROLLARY

If

% has

finite products and

inverse

limits, then

9 ‘has

products.

E]

EXERCISES 22A. of iunctor

Formulate or

general

definitions limits

and

for inverse colimits.

and direct

limits

without

using the notions

Sec.

23

C amp/ere C alegaries

228.

Prove

if and

spaces

22C.

that

only if it is

Show

of free abelian 22D.

that

abelian

an

is torsion

group

lt‘D:

l

there

is

and

Initial

if and

only if it is

Subcategories (sec correction p. 382 ) has pullbacks, and suppose that .I is e there is some 012(1), j 0b(J) and

any

‘6”is

4

a

I):

some

‘6’ is the restriction

functor, -v D(i), then show E: J

->

L

that

(1) (L, (l.),) is a limit of E. (2) For each i in 012(1) 0b(J). a

limit

(b) Obtain

sub-

direct

a

limit

of D to

following

is

1,:

some

J, and

full initial

some

m:

for each

i

sub-

j e

->

i).

0b(J),

equivalent:

are

L

a

D(i) such

-r

that

(L, ([0,)

of D. 2.3

Proposition

22E.

the

there

—

is

free

which category of I (i.e., for each is

category

its finite

topological space is the direct limit in Top of a finitely-generated space (see 14C).

groups.

Limits

(a) Let I be

a

155

as

immediate

an

corollary.

Objects, and Categories finitary (resp. strongly flnitary) provided that for each direct limit (1,, L) of a direct system D in ‘6, (F(l,). F(L)) is an epi-sink (resp. direct limit of F D) in 9. A ‘6’»objectA is called (strongly) finitary if and only if the functor —> Set is (strongly) hom(A,_):‘6’ finitary. A concrete (W. U) is called eategory that U is (strongly) finitary provided (strongly) finitary. Finitary Functars,

F: ‘6’ ->

A functor

9

is called

o

(a) Prove

for any

that

set

X the

following conditions

are

equivalent:

(i) X is finite. (ii) Xis a finitary object of Set. (iii) X is a strongly finitary object of Set. Prove that the following conditions (b) Let X be any R-module. (i) X is finitely presented (i.e., there exists an exact sequence

R’"—~

(ii) X is (iii) X is

a a

are

equivalent:

R"-X—>0).

finitary object of R-Mod. strongly finitary object of R-Mod. topological space. Prove that the following

(e) Let X be a (i) X is discrete and compact. (ii) X is a finitary object of Top. of Hausdorfl' (d) Show that the concrete category

spaces

conditions

finitary

is

are

but

equivalent:

strongly

not

finitary. which

(e) Determine

of the concrete

§23 Until

now

we

have

special categories. For objects); the category

categories given

COMPLETE

in 2.2 is

(strongly) finitary.

CATEGORIES

investigated limits of functors example, the empty category

.__——)l.

whose

domains

(which

yields

are

a few

terminal

Limits

156

in

Categories

(which yields equalizers); categories of

the

Chap.

V!

form

( (which yield multiple equalizers); discrete

the

categories (which yield products);

category

(which yield multiple inverse limits). We

also

have

pullbacks);

and

that

categories

seen

all of these

certain

downward

directed

such

Set, Grp, and Top have

as

of limits; i.e., if if is Set, Grp, or special small category I of one of the above types, each functor D: I this point, because of the myriad of possibilities for small

kinds

to be an

seem

impossible

task

to

verify that Set, Grp,

(which yield

sets

Top

or

Top, then for each —>

‘6’ has

a

categories, has

a

limit. it

At

might

limit for every

functor

from any small category. However, we shall see in this section that merely knowing that a category has only a few kinds of limits (e.g., products and equalizers) is enough to guarantee the existence of all limits of small functors. We

shall

also

“having all

of

surprisingly enough, limits” is equivalent to

that,

see

small

for

categories the property property of “having all small many

the

colimits". Definitions

and Preliminaries

DEFINITION

23.1

(1) If

I is

that

l-limits) provided

(2)

a is said

be

to

the category functor D: every

‘6’ is said

then

category,

a

complete provided

that

I

a

‘6’ is

g

has

be I-complete

to a

(or

to

have

limit.

I-complete

for each

small

category

I.

(3) is

‘6 is said

l-complete DUAL

to

be

finitely complete (or

for each

finite category

I°'-cocomplete

NOTIONS:

to

have

finite

limits) provided

(or

has

EXAMPLES

(1) All categories

are

l-complete,

(g

l°”-colimits); cocomplete; finitely

cocomplete. 23.2

that

I.

2-complete and 3-complete.

Sec. 23

Complete Categories

1 has

Q

_>3

a

has

-

:3,

terminal

has

.

.

all

categories

this form

-

:

;

has

.

has

has

of

pairs

pushouts

g is (”P-cocomplete for all categories I of this

for I of

that ‘6’:

means

object

coequalizers

coproducts

pullbacks

‘6 is l-complete

initial

an

has

equalizers

has

I

has

object

products of pairs

.

(3)

o

%’ is I""—cocomplete means that ‘6’:

g is I-completc means that ‘6:

(2)

.

157

form

has

multiple equalizers

means

that

‘6:

multiple coequallzers

Uw has

discrete

small

has

products

coproducts

0

§ -

;

-

ha

.

:

s m U

has multi P le D ushouts

It l P 1e P ullbacks

.

7 downward directed

has inverse

set

has

limits

direct

limits

.

23.3

PROPOSITION

category, % Ii(D)) be the limit

Let I be

let

(Ln,

a

an

of D. Then there

functor D: I unique flmctor Lim:W

and

Ilcomplete category, exists

a

such that:

(l) for

each

D, (oat-object

(2) for

each

n ‘6'-ntorphisrit

=

L;m(D) =

(m):

for

each

LD, and D

—>

E, the squares I D

—"—’>D (i)

LilmtD) Limb) L'

commute,

for each

is

0b(1).

n, '

E

llm(E)——IE> 0)

—» —»

‘6, ‘6

Limits

158

in

Proof: (L1,, :1, 15(D)) is easily seen to such exists a unique morphism Lilm(n) o

there

i. A

all

for

functor.

be

natural

a

that

shows

above

the

Hence

E.

commute

squares

defines

this

that

for

source

the

required

[:1

DEFINITION

23.4

If ’6 is

is called

there 23.5

DEFINITION

an

->

Analogously, if Q,”is I-cocomplete, ‘6" 6’. by Colim: —»

E) is said

functor

embedding provided that

in a

in the above

Q? constructed

‘6’ (with

category of l-limits

the formation

under

closed

a

%’, denoted

for

functor

subcategorysal of

A

for $1

functor

the I-limit

I-colimit

exists

Lint:’6'

the functor

then

I-complete,

proposition

(l)

verification

straightforward

Chap. VI

Categories

for each

be

to

functor

D:l-».nl, ‘6-limit

each

(2) A subcategory small

23.6

the formation

of l°”-colimits

formation

the

a

o

it‘d

is full in ‘6,

we

the

have

full complete subcategory of

It should

‘6

in ‘6;

(with embedding E),

necessarily following:

not

hold;

be

pointed

i.e., if .n/ is

a

that

out

a

complete category

the

of the

converse

complete category that s! is not necessarily

‘6, then

complete category Take, for example, at as the category (Example 219(8)) or s! as the category and ’6 as Top (Example 239(7)). Characterization 23.7

of I-limits,foreach

of ‘6’.

cocomplete

limit

a

D:

of D.

l

.91, and

—>

(Why not?)

PROPOSITION

A

not

under

closed

under

complete subcategory

a

complete subcategory of limit of E D, then (L, 1,.)is

is a

is closed

of ‘6.

subcategory lid

D lies in sf.

of ‘6’ that

a!

NOTIONSZ

DUAL

c

I, is called

category

(L. I.) is However,

of E

object

of

of abelian

is a

is

itself complete.

['3

above

proposition also a full subcategory complete subcategory

torsion

of locally connected

groups

and

does

of

’6’ as

topological

a

of %. Ab

spaces

Completeness

THEOREM

For

any

category,

‘6, the following

are

equivalent:

(l)

’6‘

(2)

V. has

(3)

‘6' has

(4)

’6' has

pullbacks and finite products finite products

(5)

‘6 has

finite products

(6)

((1 has

(7)

‘6' has

finite products and equalizers. finite products, equalizers, and finite

isfiniteb' complete. a

terminal

object.

and

pal/backs.

and

inverse

images.

and

finite

intersections.

intersections

of regular subobjects.

Sec. 23

Complete Categories

159

Proof: a (2). Pullbacks finite domains.

and

(2)

=

(3).

Corollary

21.6.

(3)

=

(4).

Inverse

images

(4)

=~

(5).

Finite

intersections

(5)

=>

(6).

Corollary

2L8.

(6)

=>

(7).

Corollary

l8.25.

(1)

terminal

objects

limits

are

of

are

inverse

particular

images.

(1).

=>

with

particular pullbacks.

are

Suppose that I is a finite category and D: (“0(1), 7:.) of the family (D(i)),eow, of ‘6-objects. For (7)

particular functors

l

‘6. Form

—»

the

product l-morphism ii» j,

each

let

(Em, em) 2.1 E‘l“(D(m)

o

1”).

”is

By hypothesis, the intersection (nEm, d) of the family (Em, em),,em,(,, exists. that (nEm, (12,0 d)) is a limit of D.

We claim

HD(i)

Clearly, for is also

natural

a

h: P

morphism natural

each

L, j, D(m)

i

—v

1100')

for D,

source

for

source

we

D.

such

n,- od nj ed. Now Then by the definition o

that

have

—"‘——>D(i)w

that

=

for

each

for each

D(m)otr,oh= Hence P

fm: is

i, 7ri i —'"—) j, o

=

product,

p‘. Since

=

such

In

definition

dog

—.

a

of

(P, (p9) there

(P, (p0)

is

a

is

a

njoh.

by the definition of equalizer, for each h. Thus by the E,,I such that e," of," P unique morphism g: nEm such that -»

h

that

suppose

=

there

is

of

morphism intersection, there

11.

Consequently, for

a

eachi

(“1°d)°9= and with 23.8

since

products

respect

to

this

are

nt°h=pi and

mono-sources,

property.

Consequently

d is

monomorphism, g (05,”, (n, d)) is a limit a

o

THEOREM

For

(l)

‘é is

(2)

‘6 has

any

category,

‘6’, the following

are

equivalent:

complete.

multiple pullback:

and

a

terminal

object.

is

unique C]

of D.

Limits

160

in

‘6‘ has

products and pullbacks. ‘6 has products and inverse images. ‘6 has products and finite intersections. (6) W has products and equalizers. (7) ’6’ has products, equalizers, and intersections (8) ‘6 is finitely complete and has inverse limits. (3) (4) (5)

C Imp. VI

Categories

of regular subobjects.

Proof: The equivalence of (1) through (7) can be shown with proofs analogous to those given in the finite case above. This is left as an exercise. Clearly (1) implies (8), since inverse limits are particular limits. That (8) implies (6) follows from the fact that products are inverse limits of finite subproducts (22.6). E]

expected, the above characterization it comes to establishing the completeness particular categories.

theorems

As

23.9

and

indispensable when cocompleteness properties of are

EXAMPLES

of finite sets

(l) The category

finitely complete complete. both

(2) The category

and

the category

finitely cocomplete,

and

of finite groups

is

of finite

are

but

co-

finitely complete,

topological spaces neither is complete or but

is

not

finitely cocom-

plete. (3) The categories Set, Grp, R-Mod, and Top are complete and cocomplete. sets is neither (4) The category of non-empty finitely complete nor finitely cocomplete. (5) The category Field is neither finitely complete nor finitely cocomplete. is finitely complete but not (6) Each of the categories BanSp, and Bansz complete. (7) The category of locally connected topological spaces is complete and cocomand equalizers are not the topological products and plete [however, products subspaces (18.8(5) and l6.3(3))]. is complete and torsion of abelian (8) The category groups cocomplete [products, however, are not the direct products (l8.8(4))]. of torsion free abelian is complete and cocomplete (9) The category groups [coequalizers, however, are not the quotient groups (l6E)]. (ID) A partially-ordered

finitely cocomplete largest member.

if and

(I I) If ‘6" is

category,

a

small

considered

set

(i) (6 is complete. (ii) %’ is cocomplete. (iii) %’ is equivalent to

only if it is then

a

the

complete

as

a

lattice

with

following

are

a

lattice

is

category a

finitely complete

smallest

equivalent:

(sec [8.22).

member

and

and

a

Sec. 23

Complete Categories Almost

Completeness Since

Means

reasonable

to

have

nothing

then

that

that

assume

do

to

Cocompleteneus and

completeness

161

cocompleteness

the presence

with

the

dual

are

of either

of the

condition other.

it

notions, in

might

would

category

a

seem

It is somewhat

surprising shows) for small categories the two conditions are equivalent. We have previously had another hint that completeness and cocompleteness are in some that a terminal related; way namely object in a is both the limit of the empty functor and the colimit of the identity category functor and we shall see that even in quite general (20.3(4) 203(5)). Presently, settings completeness often implies cocompleteness. That this is not always true is shown by the following three examples. In each example, the category ‘6’ is complete but not cocomplete. 23.10

presence

(as the last example above

EXAMPLES

(I) Let ‘6 be the opposite of the partially-ordered class of ordinals, % has no initial object; i.e., no empty coproduct. a category.

considered

as

of complete lattices and complete lattice homo(2) Let ‘6 be the category If X is the three-element no then there is of three lattice, morphisms. coproduct of X. copies [Hales, 1964.] (3) Let g be the category of complete boolean algebras and complete boolean homomorphisms. If X is the four-element complete boolean algebra, then there is no coproduct of countably many copies of X. [Gaifman, 1964; Hales, 1964.] The

show that under suitable smallness conditions, following theorems completeness in a category implies cocompleteness. This is one of the many instances that illuminate the fact that category theory is essentially a “twopronged” subject, consisting of : and

(1) general constructions; (2) smallness carried 23."

which

considerations; within

out

that

guarantee

“regions" (usually

certain

fixed

the

constructions

be

can

categories).

THEOREM

‘6 has

Every comp/ere, well-powered, extremally co-(welI-powered) category

coequalizers. I

Let A

Proof: quotient

.

9:;

(q, X) of

objects

co-(well-powered), there Since ‘6’ is complete, we guarantees

.

the existence

B

that

for each

(17.16), there

is

a

i, h,

for

which

qof

Let 6 be the class =

q cg.

Since

of all extremal

@ is

representative class (11,-,X,), of 6 that can form the product (l'lXi, 11,), the definition of a morphism is

a

h such

‘6-morphisms.

B be given

=

factorization

=

11,. oh.

(I1[)ZB-r Since

of h

‘6 is

extremally is

a

set.

of which

l'IX, (extremal

epi. mono)-factorizablc

in

Limits

162

Bivl‘lX, where

e

(e, C) is

since e

the

product

is

a

mono-source

e

o

=

a

monomorphism.

each

We claim

nich0g= is

n‘omoeog

monomorphism, morphism such that E f exists an extremal epimorphism m é. Since m is a monomorphism, isomorphic quotient objects, there E is

m

a

a

we

o

t

o

hj

have

l;

=

o

g.

é and

o

that

that

i, =

and

that

g. Now

f suppose Again by Proposition 17.16, there monomorphism m, such that li é g, so that by the definition of some I and isomorphism t such j e =

o

11X...

a

n‘ohof=hiof=h‘og

niomoeof= that

is

extremal

an

a

and

33—» cl.

=

epimorphism and m coequalizer of f and g. Since for

is

C hop. VI

Categories

é

is

a

of

=

some

é.

=

r/Cm

A—__,_’B————h—->ITX.-

D A, by: the

g-epimorphisms

are

the immediate . _

Aunt)

a‘

if

x

£12

if

.r

=

a2

a

if

x

=

at

a,

that

epi-sink

is

there

from

if

x

e {an 11. at}

x

—

#48) Show

of

successor =

the

a

pair (X. X)

§24

of

class

proper

supw).

=

‘K-objects

non-isomorphic

Y such

there

that

is

an

Y.]

to

WHICH PRESERVE FUNCI'ORS LIMITS REFLECI'

AND

preservation and reflection of of the important functors in mathAs it turns limits and colimits. out, many ematics functors) do preserve (including many of the forgetful and inclusion limits. In the next chapter we will see that the concept of limit preservation is of universal with the very important concepts maps and intimately connected situations. adjoint this

In

section

consider

we

notions

the

of

‘

24.1

DEFINITION

Let I be

(l) is

limit

a

limit

that

such

that

that

be

a

—.

(F(L). (F(li)))

provided

in .cl

source

functor.

a

whenever

D: l

is

of F0

a

limit

whenever

(F(L). (F(l,-)l) is

D: a

I

Fis .5! is

-.

—~

limit

said

D:

I

a

—->

.2! is

a

of Fe

to

functor

and

(L, (l,))

.9.

functor

D, then

and

(L, (1.)) (L, (l,)) is a

of D.

(3) preserve for every

(4) reflect for

l-limits

F: d

and

provided

of D, then

(2) reflect is

category

l-limits

preserve a

a

every DUAL

(resp. preserve finite limits) provided that (resp. finite category) I. category

limits

small limits small

(resp. category

NOTlONS:

colimits, reflect

reflect

finite

limits)

(resp. finite category)

preserve

provided

that

F

reflects

I-limits

l-limits

I.

I°”-colimits, reflect

(finite) colimits.

F preserves

1"".colimits,

preserve

(finite)

Sec. 24

Functors

It should

be

noticed

which

that

carries

with automatically from the general definition

the

Reflect Limits

and

Preserve

definition

above

is

167

general

a

it many translations to special one obtains the definitions:

F preserves and :t

equalizers provided

that

F preserves

I-Iimits, where

F presen‘es category I.

products provided

that

F preserves

l-limits

-

example,

l is the

category

.

Also from the and

of the characterization

proofs

completeness (23.7

24.2

and

23.8),

small

discrete

for finite

theorems

immediately

we

for each

obtain

completeness following theorems:

the

THEOREM

If s! equivalent

finitely complete

is

F preserves F preserves F preserves

F preserves

(6) F

preserves

24.3

THEOREM

If at equivalent:

is

(1)

F preserves

(2) (3)

F preserves F preserves

(4) F

preserves

(5) F

preserves

(6)

F preserves

(7)

F preserves

24.4

and

F: .24

-.

Q is

a

functor,

then

the

following

are

:

F preserves

(I) (2) (3) (4) (5)

finite limits. pullbacks and finite products finite products finite products finite products and

complete

terminal

objects. pullbacks.

and

and inverse

images.

finite intersections. equalizers. [:I

and and

F: to?

g

—o

is

then

functor,

a

the

following

are

Rug, BooAlg,

and

limits.

multiple pullbacks and terminal products and pullbacks. products and inverse images. products and finite intersections. products and equalizers. finite limits and inverse limits.

objects.

E]

EXAMPLES

(I)

The

Lat

to

or

For

cases.

which

one

forgetful functors Set preserve

reflects

(2) The does

not

and

from

reflect

Grp, R-Mod,

limits

and direct

SGrp, Mon, limits, but

of them

none

preserves

arbitrary colimits. functor

forgetful

from

Top

to

Set

limits

preserves

and

colimits

but

reflect either.

(3) The embedding functor

(4) The

forgetful

torsion)

groups

Exercise

24F).

functor to

Set

from from preserves

Ab to the

Grp

category finite

limits

preserves

limits,

of

finite but

not

but

abelian

not

colimits.

(resp.

arbitrary

abelian

limits

(see

Limits

168

hom-functors

All covariant

(5)

(6) If

(A

for each

_)2 s!

x

(8) Examples (6) and (7) above that

29.3).

d

—.

then

“constant

(9) Each

functor

functor"

si’

-’

if .r/

is

colimits.

.24 preserves

Cm!

and

limits

preserves

(sec 15.8)

both

preserves

colimits.

and

.2! be

(10) Let

—»

—*

equalizer functor

the

special cases of the general (see §25). Specifically if at is

commute

s!’ Collim:

the functor

l-cocomoplete,

then

-,

as

Limnsxl’.9!

functor

the

::

-

be derived

can

of limits

varieties

two

any

l-complete,

E reflects

both

provided

that

24.5

(see Theorem

A, the functor

d-object,

(7) 11‘s! has equalizers and l is the category E: sl’—> sat (see Exercise 16C) preserves limits.

limits

limits

VI

limits.

preserves

fact

C hap.

Categories

hom(A, _) preserve

(finite) products, then

has

s!

in

full

a

limits is

s!

and a

of :3 with

subcategory

and if a! is

colimits,

embedding functor E: 41‘» £3. complete, then E preserves limits

complete subcategory of

a.

PROPOSITION

[f

pal/backs, then

F preserves

that

Proof: Recall

is

f

a

F preserves

monomorphism

manomorphisms. if and

only if

1

is

a

24.6

and

pullback

E]

square.

PROPOSITION

If .2! has equalizers and F: .9! only if it reflects epimorphisms.

Proof: That previously (12.8). To show

reflect

functors

faithful

that

if

HI) epimorphisms, 24.7

=

e

then F is

equalizers,

has

epimorphisms

faithful if shown

been f

e) be the equalizer of

let (E,

the converse,

(NE), F(e)) so

.93 preserves

—¢

F(g), He) must

be

an

z

a

pair

9

1501400), Fly».

epimorphism (16.7). g (16.7). epimorphism; hence f be

must

X _'—’_, Y. Then

an

Since

F reflects

[:1

=

THEOREM

If

F :d

—>

(resp. I-colimits)

98 is

and

reflects isomorphisms them, then F reflects them.

and

faithful F preserves

Proof: Suppose that (F(Q), (F(q‘))) is a limit of

ad

Dz! F

o

D.

Then

(Q, (q,)) is (F(Q), (F(q,))) is

and

and

a a

d

has

I-limits

source

such

natural

source

that for

Sec. 24

F

D,

o

which Preserve

Functors

that

so

since

F is faithful

and

Reflect

Limits

reflects

(and hence

169

commutative

triangles)

for D. Since a! is l-complete, D has (Q, (q,-)) must be a natural source and there is a L such that for each is (L, (1‘)) morphism [1: Q q,1,0 h. Consequently, for each i the triangle

a

—>

limit

0b(I),

=

F(

F(Q)—q')>F(D.) .

Filthl F(I)

F(L)

that

since

both

(F(Q), (F(q,))) and (F(L), (F(l,))) are limits of morphism F ([1)must be an isomorphism. Since F reflects isomorphisms, h: Q L must be an isomorphism, so that (Q, (q,)) is a limit of D. The proof for colimits is similar. C] commutes, F o D, the

so

—>

24.8

THEOREM

If

limits complete and F: .524 Q preserves is faithful and reflects limits, monomorphisms,

.2! is

then F

F(r)

that

Let (E, e)

F(s).

=

show

To

Proof:

F

is faithful,

since

.

be

sat-morphisms

with

Equ(r, s); then

z

(F(E). F(e» that

and

nzr',

let

reflects isomorphisms, epimorphisms.

and

—»

Equ(F(r), F(Q),

z

F(s), F(e) must be an isomorphism (l6.7). Since F reflects s (16.7). isomorphisms, e must be an isomorphism. Thus r Since F is faithful, it must reflect limits (24.7), monomorphisms, and epimorphisms (l2.8). E] so

F(r)

=

=

In

§12

functors

that

continuation 24.9

have

we

investigated

full, faithful,

are

of this

the or

dense.

We

reflection

conclude

this

properties section

of

with

a

study.

PROPOSITION

Every full faithful functor reflects I-limits We

Proof: that

and

preservation

11)

at

(F(L), (F(I,))) for

source

give

L» 33, is

D.

Now

(F(Q), (F(q,))) h: F(Q) F(L)

a

such

—>

for limits.

proof

F is full

limit

a

is

a

and

of F0

D.

suppose

that

natural

source

that

and l-colimits

proof for colimits faithful, (L, (1.)) is a source Since F is faithful, (L, (I,)) The

(Q, ((11)) is also

for all i,

for

F

F(q,)

o

=

D,

HI.)

natural

a

there

so 0

a

Q

-----------

I.

I

>1.

mm

in 5!, and that be a natural

D,

for

D.

h

F(Q) ------->F(L)

17(4)

/'

Then

unique morphism ND.)

I, q,

Suppose

h.

F07.)

1.

must

source

is

category

is dual.

1),:1,

each

for

F(

1,) (F°D)(m)

\Fili) F(D,)

Limits

170

full, there is somef:Q

Since Fis

of F, f this property.

L such

—»

F(f) I, of and

=

Consequently

Every full and faithful functor F: at not A e 0b(.d)) only reflects limits,

one

F(A)

has at least

set-valued

about

in

[Z]

(where

F(A)

#

them

as

preserves

0b(.;l),

only

are

some

by the faithfulness only morphism with E for well.

at

least

Also,

if

F preserves and reflects of the rather surprising facts

then

will be treated

more

comprehensively

LEMMA

24.10

If

F and

G

then naturally isomorphic functors and l is a category, G does so. and [:] only if (resp. l-colimits) if

are

F

reflects I-limits

or

preserves

PROPOSITION

Each every

but

functors

These

Set

e

is the

of D.

_.

hand

=

VI

VIII.

Chapter

24."

functors.

A

some

24D). These

(see Exercise

colimits

for

elements

two

Chap.

that

for all i, q,(L, (1‘)) is a limit

that

has the property

in Categories

equivalence

and

preserves

reflects

both

Ninth:

and

I-colt'mits, for

1.

category

follows and l-colimits Proof: That each equivalence reflects I-limits immediately from the fact that it is full and faithful (24.9). We will show limit preservation. The proof for colimits is dual. 5? is an equivalence, D: I d, and (L, (1.)) is a Suppose that F: d s! such an there is an D. F is limit of Since equivalence G: 38 equivalence, that G F is naturally isomorphic to l, (NJ I). Hence by the lemma, —»

—>

->

o

((0 is

a

limit

of G

reflects limits

24.12

are

Some

figure. (F

Fe

D.

the

F)(L). (0

proposition categorical properties. of this

results

is assumed

be

to

F)(’.-))

a

functor

shows

section

are

with

domain

that

completeness in

summarized

and

the

.51.)

F preserves

monomorphisms

l "/5

completeand

Fprescrves

F reflects

llmllS

isomorphisms

L

{7reflects

F reflects

limits

epimorphtsms

/ F is faithful

so

it

,

above

of the

°

G, being an equivalence, is full and faithful; (F(L) (F(l,))) is a limit of Fe D. E]

Now

Hence

(24.9). that

Notice

pleteness

0

°

—-—-’

F reflects

monomorphisms

cocom-

following

Sec.

in Humor

Limits

25

171

Categories

EXERCISES 24A.

then

a

248.

same

U:

24C.

isomorphisms,

(a) If F(A) ¢ Q for some (b) If FM) has at least

d

has

initial

an

“F preserves

object P,

from

unique morphism

the property:

P to A.

coequalizers”

regular epimorphisms”.

preserves

extremal »

[Consider

the

->

Q preserves

limits

reflects

and

epimorphisms.

Set

is full and

.rl-object A, elements

two

F151

and

complete

F255!

Suppose that

24D.

is

F reflects

then

and

IA is the

Set.]

->

if a!

that

Prove

that

show

to

“F

property

Grp

limits

->

example

an

the

as

functor

forgetful

:13 preserves (F (P), (F ( [4»). where

limit; namely Construct

the

is not

if Fm!

that

Prove

F has

that

prove

for

faithful. limits.

Fpreserves

d—object A,

some

prove

that

F preserves

colimits.

Suppose that F, G, and H

24E.

limits

preserves 24F.

and

Prove

Let .n/ be

considering types

that

G

o

F

=

H. Show

that

if H

limits.

a

a separator full subcategory of 3 that contains E: s! Q .9 reflects regular epimorphisms.

of

for Q.

Prove

functor

§25

certain

such

F preserves

the

-»

the embedding

When

functors

then

finite limits

Set preserves whose image is

246.

limits,

are

of finite abelian forgetful functor from the category groups but not arbitrary limits. [Consider the limit of a functor Z/BZ -» Z/4Z -» Z/ZZ.]

that

into

that

G reflects

LIMITS

particular limits

can

IN

FUNCTOR

several instances limits, we have seen be “commuted”; for example, products

of

iterated

CATEGORIES

where can

of

be

(18.13), products equalizers equalizers products (18.17), can be composed by “pasting their edges together" (21B), pullback squares and products of pullbacks are pullbacks of products (21F). In this section, we will prove the general result that any two types of limits commute, obtaining in and also obtaining the consequence the process the above results as special cases w“ inherits the completeness and cocompleteness that each functor category .9. To do this we will ‘0”is a of begin by showing that ifD: I x J properties and ‘6 is I-complete and Jocomplete, then the limit of D can be confunctor first the limit of each of the associated structed “pointwise”, i.e., by finding right functors D(r', _), where i is an object of I, and then by finding the limit of the “induced" functor by these limits (or, alternatively, by first finding the limits of each of the functors D(_, j) and then the limit of the functor “induced" by these the technique with a special case that We first illustrate has been limits). are

~

considered 25.1

before:

EXAMPLE

Equalizers

and

products (of pairs)

Let I be the discrete

category

commute.

3

3

and

let J be the category

i

&

j.

172

Limits

Thenlx

in C aregories

Chap.

VI

Jis (tun)

(1.020.!)

(In)

"2"": (2 )j ') (2.")

(2 i) 9

Suppose a

functor.

‘6’ has

that

in

Then

'

products (of pairs) and equalizers, ‘6 we have the diagram bum

.

and

let D:

l

J

x

-

‘6 be

.

DU, 1):.

D(l,j)

D(U'l)

D(l.m)

D(2, 1) —*——» 00.1)“230

and D(2,_) are functors from J to ‘6’ whose limits are the D(l,_) and equalizers of the top and bottom pairs, respectively. Call them (E(l), el) a functor E: I defines This then W, and it is easy to verify that the (15(2), 2,). x limit of this functor, namely (E(l) E(2), “an, firm) is such that when the are with the other morphisms given, the result composed projection morphisms Now

—.

is

a

Lim

limit D

z

of D, i.e.,

(E(l)

x

E(2),

0

e,

n5“), D(l, m) oe‘

0

n5“),

IX}

(’2

”5(2), D(2, m)

3

E(l)

X

5(2)

17/ ,3.

e2

°

”E(2))'

D(l.m)

eI

,

5(1) ——"D(|.i)——->

E(l)

0

D(Lf)

D(l.n)

D(LM)

e,

m

——’__,

5(2) ——>D(2.i)

D(ZJ)

D(2.n)

Similarly, when we consider the functors D(_, i) see that they have limits which are products. Let F(i)

=

HI)

=

F(m)

F(n) Then

we

that

have

F is

(K, k) of F(m) and

phisms given, Lim [‘1

D

z

a

limit

a

functor

F(n)) is such

=

=

D(l, i)

x

D(2, i)

D(LJ’)

x

D(2,!)

D(l, m) D(l, n) from that

of D is obtained;

(K. 7:”

o

and

k, D(l, m)

o

x

x

I to

D(2, n).

k is

limit

composed

(namely with

the

the

ok, 1:2, ok, D(2, m)

a

1:2,

equalizer

other

i.e., 7!”

‘6, we

D(2, m)

J to ‘6’ whose

when

D(_, j) from

ok)

mor-

Sec.

25

Limits

in Functor

C aregories

I73

D(l,m)

D(Li)—>i

If" K

004')

00.7!) F(m)

If”

'

__>——->F(j)

—>F(i)

if”

F(n)

lfzj

D(Z. m)

D

(2.0 ——>D(2.j) D(an

Notice

that

in the first

and

in the second

the

equalizers of equalizer of

the 25.2

THEOREM

case

any

the

took

product of equalizers to obtain a limit of D, equalizer of products. Thus the product of pairs of morphisms in V is essentially no different from we

we

case

two

took

a

the

formed

product morphisms

(POINTWISE

from

and

for

is

each

a

unique functor F: I

morphism

m:

pairs.

EVALUATION)

Suppose that I, J, and ‘6 are categories, D: I x each object i in l, the right associated functor D(i, _): (Li, (lib)- The" (1) there

the two

i

‘6’ such

-r

i in I and

-s

J

g) is

—)

J

—>

afunctor, and for a limit, namely

‘6’ has

that

each

for each object i in I, F (i) object j in J, the diagram

=

L,

I,

F(i)

L,- ———>’ D(i, j)

=

F(In)

D(InJ) A

I

H?)

L:- —’—>D(i.j)

=

commutes.

(2) D has a limit if and only if F ifand only if(L, (1}0p9,“) is

has a

limit, and any

a

limit

source

(L, (p,),)

is

a

limit

of F

ofD.

Proof: I, let F(m) be the morphism from L, to L; induced by the fact that (L;, (1:9,)is a limit of D(i, _) and (L,, (D(m, j) I}),)is a natural a natural is F must source for 00‘, _). Since D(m, _) transformation, clearly hence it is a functor. the identities and compositions; By uniqueness preserve that F can be defined on morof limit, it is evident condition in the definition

(I) For each

m:

i —: iin

o

phisms in no (2) Suppose

is

a

natural

other

that

source

way.

(L, (p,),) is

a

limit

of R Then

(L.

(1}°p.-):u)

for D. Let

(R, (40):“)

174

also

j

in

Limits

be

L]

a

natural

for D. Then

source

Categories

for each

Chap. 1' in I and

object

each

VI

morphism

in J, ‘11“

Hence

is

there

a

r,: R

morphism

DU. )1) qu°

=

L‘ such that

—>

the

diagram

DUJ) 4-3

.

1’

r

--------‘-->L,-

R

no. u)

.

1:. 1

4;;

Dani)

commutes.

Now

for each

object j

lienm)". is

since

that

so a

(L3,

morphism ii)

in J and each

D(m.j)°l}°r:

=

D(m.j)°qu

=

(lib)is mono-source,F(m) a

limit of F, there

is

a

i in I

o

I]: R

unique morphism

=

=

r,

r;.

Hence

L such

_.

lion

=

qr.

since

that

(L, (p,),) diagram

the

q if

L‘ R

--)

——————

If1

D

(1.9.1.)

F0")

L

00".!)

I

I:

A

in). commutes.

‘

Since each

property is

a

limit

converse

exercises.

[:I

h is thus

mono-source,

j, q”

(11'p.)

=

c

o

h.

unique with respect to the Consequently, (L, (I; pan”) o

follows

readily,

the first part

does

as

of

(2). These

are

left

as

‘6’ is

a

COROLLARY

If 25.4

a

i and

of D.

The

25.3

,) is (1})

(Li,

for each

that

g

is

I-camplete

COROLLARY

Suppose funclor. Then

that

and

J-complele,

(COMMUTATION {6’ is I-camplete

Lim(Lim D(i',j)) I

1

z

then ‘6 is I

0F

and

Lim D “

in Functor

Limits

Sec. 25

175

Categories

EXAMPLES

25.5

The

result

above

l

yields

the

following previously

considered

iteration

discrete

.

.

_’

CM

products (18.13) equalizers of products

are

(18.17)

.M

.

pullback

-

if,»

,...---'>

of

products of equalizers

__~.

discrete

cases:

Result

J

discrete

special

be

can

squares

composed (ZIE)

°

N

discrete

In §15

properties

seen

That

of the

consequence

23.6

have

we

of a.

products of pullbacks

.

next

that

categories as” often completeness and

functor

inherit

the

categorical cocompleteness is a

for

is true

this

pullbacks of products

are

theorem.

THEOREM

Limits

in

functor categories that D: I

ically, suppose is the evaluation

functor

—>

be obtained

can

33" is afunctor and for each A

relative

evaluation.

by pointwise each A

to A

(15.8). If for has a limit (LA, (If),), then D has a limit (F, (q,),), where such that F (A) LAfor each A e 0b(.szl), and for each i F D(i). transformation (11546050,): =

0b(fl),

e

e

0b(s¢), F

e

:

a!

0b(l),

—o

Specif-

E ‘z 3" EA

D: I

o

.99 is

—v

.43

—9

Q

afunctor

q; is the natural

-»

Since

Proof: considered

as

from

I

x

at

Q"”

and a?

to

Evaluation

are

D

isomorphic,

(15.9). Hence, the result (25.2). C]

is

can

merely

be a

Theorem

COROLLARY

53, am! I

lfd, then

functor

(W)’

categories

of the Pointwise

restatement 25.7

a

the

so

is Q”.

are

categories,

then

if

a? is

I-complete (resp. l—cocomplete),

E]

EXERCISES

25A. and

(a) Use the pointwise evaluation

F and

G

are

if for each

A

e

objects in a”, 0b(.sz/). 11": FM)

theorem

then

r]: F

—»

C(A) is

a

show

that

if (It? has

pullbacks in 3" if and only monomorphism in 9 (cf. 15.5). monomorphism

G is

a

a

to

176

Limits

in

Categories

that if .43 has pullbacks and (b) Conclude for each balanced, mtegory d. 253.

Commutation

In this exercise

set

we

each

it

are

functor

D: I

x

J

when

we

is

in @ “direct

upward directed (6, the object parts of

-r

and

00', j)) Colimugm are

that

say

for each

that

means

and

pushouts

balanced,

of limits and colimits conoemed with only the “object parts” of certain

colimits; for example, here commute"

Chap.

set

I, each

limits

and

then

V]

9"

is

limits and limits

finite

finite category

J, and

DU, j)) L5m(Co'Iim

isomorphic.

(a) Show

that

and

coproducts

equalizers

in the

commute

mtegoriee

Set, Top,

and

Cat. in the categories Set, R-Mod, (b) Show that direct limits and finite limits commute and Lat. show that direct limits and Grp, BooAlg. [First pullbacks commute] in Set. [Let I be the (c) Show that direct limits and (arbitrary) limits do not commute natural

discrete

numbers

N with

category.

Define

the usual D:

I

x

order J

0031') and

if

f: (i, j)

->

and let J be natural Set

~9-

=

by:

{HENIII

2

(i, j)

DUN")

=

supil', n}.]

i)

numbers

N considered

as

a

VII Situations

Adjoint

The slogan is “Adjoint functors

arise

everywhere“. S. MAC

In

I we claimed that category theory allows one to make precise the of “universality”. In this chapter—the most important of the text—we show why this is so. Here we investigate adjoint situations—situations

Chapter

notion will

LANET

occurring so frequently are regarded as perhaps

and

in

§26 The

following well-known

Let

G:

set.

Then

Grp

—>

there

ua: B —> 00",) there exists a

Set

so

the most

many

diverse

useful

of all

UNIVERSAL

example

categorical

definition

our

of universal

forgetful (i.e., “grounding") functor, group FR (called the “free group on B")

be the

exists

a

such

that

unique

group

for

any

H

group

and

((71,) I I

G commutes.

Categories for the Working Mathematician. 177

function

any

homomorphism f: F3

j

that

they

notions.

MAPS

motivates

3 —>G

1' From

of mathematics

areas

—»

r;,, I

I

30(1)

if

i

t

(In

H

H

such

and

let B be

and

f: that

maps:

a

B

the

a

function —>

C(H),

triangle

'

Adjoim Situations

178

DEFINITION

26.]

Let A

01.9!

B

u:

G-universal

a

B

—»

map

C(A’) there

—»

be

3

—’

and

0b(.x:l)

e

(or

f:

VII

Chap.

functor

and

let

A

Be

0b(.93).

GM) is called a universal for B) provided that for

map each

for

unique .sxi-morphism f:

A

A' such

B——“—>G(A)

A

exists

a

a

A’

—>

with

pair (u, A)

B with

respect and

Ohm!)

6

that the

G

to

each

triangle

a

.

50(7) 5

f G

t

l,

(A')

A’

commutes.

If F:

DUALLY:

called

a

co-universal

provided provided f: F(A’)

that

(u,

that —>

u:

.531 —> .28 is

functor

a

and

B

0b(.93),

e

then

a

pair (.4, u)

is

map for B with respect to F (or an F-(co—universal)map for B) A) is a universal Le, map for B with respect to F ”P: .2!” 7» 578"”; —» B and for each A’ and each F(A) ail-object sly-morphism

B, there

exists

a

unique morphism]: A’

A'

—>

A such

that

the

give

three

triangle

FM’)

\l

kxeA

A=JupBX6A

is in general different

from

the

well-known

MacNeille

completion.

Sec. 26

Universal

Maps

181

by ‘ffacloring out” subobjects

(b) Qualiemsfarmed .2!

a

Ab

An

Grp

B/B’ (where

B’ is the

commutator

subgroup of

B) torsion-free

abelian

B/B (where B

Ab

torsion

groups

Ab“) (= abelian groups

reduced

G with

Ab nG

=

is the

subgroup of B)

B/nB

0) commutative

rings (no

B/r(B) (where r(B)

rings

nilpotent elements except 0)

{x l x is

e

x“

B such n e

some

0} is

=

=

that

there

N with the nilradical

of B)

(c) “Identifications" .m’

a

POS

As

quasi-ordered sets (and monotone funetions)

(where

B/~

equivalence

is the

~

relation

defined

on

B

b’ if and by b only if b s b’ and b' s b)

To topological

is the Bl~ (where equivalence relation on B defined by b b' if and only if b e {b'}‘ and

Top

spaces

~

~

~

b’ metric

pseudo-metric

spaces

e

{b}')

is the Bl~ (where equivalence relation on defined by b b’ if and if 0 only d(b, b’) ~

spaces

B

~

=

(d) “Modifications"of

structure

an

sets

599

a!

regular topological

locally convex topological

underlying

linear spaces

spaces

An

Top

the

regular modification

LinTop

the

locally

of B

convex

of B

modification

Adjoint Situations

182

(3) Universal For

the

.5! listed

categories

Hom~functor

with

internal

for

maps

respect

to

Hom-functors

universal

a

let

.9! .2! be the internal Hom(C,_): and let B e sat-object C, 0b(.d). Then (an, A3) to where A n and u, are as Hom(C, _), respect

below,

—v

the

map for B with defined in the table that follows. is

.324

“at B

A5 B

Set

VII

Chap.

C

X

(u3(b))(c)

=

(u,(b))(c)

=

H0m(C. AB)

—'

(b. 0)

R-Mod

®x

C

(where R is commutative)

B

Top (where C is locally Hausdorff)

B

x

C

8

A

C

compact

p'l‘op (where C is locally Hausdorff )1 .compact

(“3(b))(€)

or

B

x

C

where

each m:

b

natural

defined

1 If. for this of Band B

example. /\

one

C is called

(4) Co-universal

=

-.

u3(b’)

trans-

(m

,

by:

1c)

linear B

spaces

if

b’, u3(m): u5(b)

-*

us(m)c normed

(b, c), for

=

[1 e OMB); and

formation

convex

guished point, if (b, c) is in the wedge of Band C; (b, c), otherwise

=

(u,,(b))(c)

is the

locally

(b, C)

=

(smash product)

(6’51?

c

the distin-

(ua(b))(c)

Cat

b ®

® C

chooses the circle 5' for C. then the suspension of B.

where

Ham(C.

(1:5(b))(c) B) is called

=

b (8

the

c

loop space

maps

functor, B following examples E: d C» .4? is the inclusion B (where us: EM») fi-object, and (A8, us) is an E-(co-universal) map for is the “obvious" morphism). In each

of the

is —»

a

B

Universal

Sec. 26

abelian

torsion

groups

Ab") (see (2)b)

183

Maps

Ah

the torsion

Ah

the

subgroup of B

subgroup of B composed of

all elements

commutative

nilrings

x

of B with

nx

0

=

of B

the

nilradical

the

“locally connected

rings

locally connected

Top

spaces

refinement"

compactly

generated

“compactly generated

the

Top

spaces

of B

refinement"

finitely-generated

the

topological space whose underlying set is that of B and

Top

spaces

of B

whose

sets

open

trivially partially-ordered

POS

sets

the trivial with set

It is clear

and

colimits

that

G-universal G: d

—+

maps

partially-ordered underlying

B

as

in mathematics.

of functors

(resp. limits) (resp. G-(co-universal

261).

The

D:

I

—»

to

all of these

be

can

to

diverse

many

it

Moreover, d

maps)) relative

fact that

set

interpreted

suitable

a

be

can

as

functor

constructions

can

be

the significance essentially the same way once again underscores of categorical language. It also clearly classifies the notion of maps (and as we shall see later, adjoint situations) as an extremely

in

usefulness

universal

mathematical

important above

constructions

i.e., that

there

special Next

some

constructions

in B

sets

same

preceding examples (l)—(4) correspond

d’ (see Exercise

described and

the

classical

important

shown

as

that

the

the

are

of open

intersections

can

very

Later

(§28)

we

shall

also

that

see

actually be accomplished in essentially that yield most of the general theorems

most

the above

same

of the way;

examples

cases. we

earlier

thoroughly

are

concept.

introduce notions in later

the concept

(see examples sections.

of

“generation", which

26.4) and

which

will

is be

a

generalization of investigated more

184

Adjoinr

Chap.

V]!

DEFINITION

26.3

G: .nl

Let

—)

(l) A morphism A

Siluarions

r:

A’

.6? be B

g:

a

—v

functor.

G(A) is said

.nl-morphisms

are

such

that

A

G-generatc

to

G(r)

«2

g

0(3)

=

that

provided cg,

then

r

whenever s.

=

3

(2)

A

morphism

B

g:

-+

G(A) is said

extremally

to

G-generate

A

that

provided

A, and

it

(i) G-generates (ii) (Extremal condition): A and

codomain

a

whenever

lid-morphism)"

there

such

that

is

an

g

=

.ni-monomorphism G(m) f, then m

with

m

must

o

be

isomorphism. (3) Q-object B is said to G-generate (resp. extremally G-generate) the ail-object A provided that there is some G(A) that G-generates 38-morphism g: B A. (resp. extremally G-generates) an

A

—i

26.4

EXAMPLES

A G-generates A if and .9! is the identity functor, then g: B (I) If G: d only if g is an epimorphism; g extremally G-generates A if and only if g is an extremal epimorphism. .nl’ is the “constant functor" If I is a discrete functor (2) category and G: d (15.8), then (f,: A. A), G-generates A ifand only if(f,-, A) is an epi-sink ind; and ( f ,), extremally G-generates A if and only if (f,, A) is an (extremal epi)-sink in .d (19.1). categories Grp, R-Mod, Lat, BooAlg, (3) Let (5!, U) be any one of the concrete If B is a function from a set B into the underor U(A) Rng. g: SGrp. Mon, lying sct U(A) of an .d-object A. then g extremally generates A if and only if g[B] “generates A” in the algebraic sense; i.e., provided that A does not contain a proper subobject whose underlying set contains g[B]. In the case of groups and R-modules, whenever g: B U(A) generates A, it also extremally generates for semigroups, monoids, and rings. In these cases, A. This is not the case A of the subU(A) generates A if and only if the embedding e: C g: B is an the C of A, generated (in algebraic sense) by g[B], epimorphism. object our use of the term motivate “generates".] examples [These U(A) is a Top (resp. Topz). If g: B category (4) Let (:1, U) be the concrete Hausdorff set of a set B into the from a function space) underlying space (resp. in A); A if and only if it is surjective (resp. g[B] is dense A, then g generates A if and only if it is surjective and A is a discrete and it extremally generates -»

—>

-.

—»

—»

_.

—’

—»

-»

space. 26.5

PROPOSITION

a? preserves them, then equalizers and G: .9" A the it if satisfies G(A) extremally G-generates g: B extremal 17.14 and 19.4.) 26.3(2). (Cf. Definition

If

.5! has

—-

—.

Proof:

We

need

only

show

that

g

G-generates

A.

a

Eli-morphism

condition

Suppose that

A

(ii) of

E), A’ S

are

is

Universal

26

Sec.

a

d-morphisms such that C(r) monomorphism and since G

185

Maps

6(5) g. Let (K, k) equalizers, preserves

o

0

=

g

k

Then

Equ(r, s).

z

(G(K). C(16)) “‘4 15411400).0(5))Thus g

=

r=s.

by

60:) I]

o

of

h: B

morphism

a

k is

condition

by the extremal

11. Hence

is

there

equalizer,

G(K) such isomorphism, so

an

—-v

that that

PROPOSITION

26.6

If (u, A)

is

Proof: If

s

definition is

definition

the

6(5)

0

G-mziversal

a

and

of universal 11.

Hence

3:

there

maps s

=

is

u

extremally G-generates

such that

eel-morphisms

t are

then

for B,

map

a

G(s) u unique morphism

4>004)

6(5)

such

x

u, then

that

by the G(x) u o

0

A

013)]le i (I)

u

I

G(A’)

it

0

I.

=

B

Thus

6(1)

=

o

A.

I

A

G—gencratesA. the extremal

To show

that there is suppose codomain A such that u

condition,

d-monomorphism m definition of universality

with

an

there

is

morphism f

a

B—me}

such

that

B-morphism f and 6(3):) f. By the the diagram a

o

=

A

l

I

I

I _

_

5.6m if

1‘

4',

G(A’)

v

A’

commutes.

Hence

the

diagram "

B

\ commutes.

universal

maps

isomorphism.

E 1,. EmOfor

GLV 607107)

ll

also

A

0(A)

_

GL4!)

Consequently m

I]

of

=

IA. Thus

:

Y”C(A)

i xl

by the uniqueness condition m

is

a

retraction

and

a

in the

definition

monomorphism;

i.e.,

of an

C hop. Vll

Adjoim Situations

186

PROPOSITION

26.7

Universal

(u, A) and (u', isomorphism f:

-

essentially unique; i.e., if G: d B e GHQ), a G-uniuersal map for some A’ such that the triangle

A

a

are

maps A’) is -v

each

93 and

then there

is

of

unique

a

B——"-—>G(A) 4 i

l u:

GU)

4r

i 1r

G(A’)

A'

i

f

commutes.

f:

A

definition

the

Proof: By

A’ and g: A’

->

—»

of

A such

universal the

that

map

there

unique morphisms

are

diagram G(A)

u

/

GU)

B—“'>G(A')

W) But

commutes.

so

does

the

G(g°f)

N

GM)

triangle G(A) GOA)

B

G(A) Thus, since that

show 26.8

g 12 Similarly G-generates A (26.6), it follows that 1,, is an isomorphism. I] 14. f 9. Consequently f =

u

=

one

can

o

LEMMA

If

G: d

a

Q, 3,, B:

6

0b(.@),

and

l, 2; then for each morphism f: B, f: A, -—o A; such that the square i

o

(u,, A i) is B; there

a

—»

=

for B‘, unique morphism

G-tmiversal exists

a

map

B,—"‘—>G(A0 4: i

f

I

Eat?)

if

V

i

'

32—7,?)6041)A [I

commutes.

If universal

2

a

functor map,

G: d

then

the

—>

93’ has

preceding

the

property

lemma

will

that enable

each us

w-object to

define

has a

a

G-

functor

Sec. 26

121$

.331

—>

enable 26.9

Universal

Theorem

other

26.11). The things) to prove

187

next

somewhat

technical

that

F preserves

colimits.

lemma

will

LEMMA

Let

D: %

(ac, Ac) be (I)

(see

(among

us

Maps

Then

a

there

.49 and

—>

G-universal exists

a

G: .9!

.43 be

—)

flutctors, and for

each

C

0“?)

6

let

for D(C).

map

unique fimctor

F: %‘

—»

M such

that

(i) for each C e 0b(‘6’),F(C) Ac; and u is a natural (ii) (ac) transformation from D to G F. (2) Let ((kc), K) be a colimit of D. (i) If ((ké), K’) is a colimit of F, then there exists a unique (la-morphism 1": K G(K’) such that for each C E 0b(%’) the square =

=

o

—>

It

D(C)—”worm

Gut-g)

kc K

commutes.

F(C)

k'c K’

T606)

G—uniuersal map for K. G-universal map for K, then for each C

addition, (u’, K’)

In

is

a

(ii) Conversely, if (u', K ') is a there exists a unique k6: F(C) —> K’ In addition ((ké), K’) will be a colimit

such

that

the above

square

6

Oh“),

commutes.

of F.

Proof: (1).

If

there

is

D(C) M unique morphism f: Ac

CL) C', a

then

D(C'), so that by -> Ac. such that the

the

above

lemma

(26.8)

square

D(C)—u—G>G(Ac) 40 : .

'

DU)

:61?) i

{r

b

Ac. D(C’)TCI>G(AC.) commutes.

,

Define

f. Clearly, by the uniqueness, F : Mor(‘6) F(f) function, F(lc) I“, and if F is a functor it is the unique F (C) and each above commutes. Since the squares Ac square u Thus it (ac): D —t G F will be a natural transformation. =

-+

=

=

=

o

Marcel) one

for

is

a

which

do commute, to be

remains

such

that

o

D preserve

G and

Vll

C". Then compositions. Suppose that C—I»C’ —9—> compositions, each of 5 of and g f is a morphism x

F preserves

that

shown since

Chap.

Adjoim Situations

188

the square D

“C

Ac

(C)"_-)G(lAc)

i

I|

D(9°D

ix

EG(x)

4'

v

A cw

D(C')TC?G(AC') commutes.

__

since

Thus

°D(2). (i) By

G-gcneratcs Ac (26.6), g

no

f

o

=

go

F(g)o F(f)

f; i.e.,

F (g

the above

for each 9:

C

—.

=

compositions,

definition of F, and the fact that G preserves C' we have commutativity of the diagram “0

D(C)——>G(F(C))

We)

G(K’)

G(F(9))

0(9)

““5"

DtC’)T>G

GA(A )

I

A

commutes.

Thus, by uniqueness, fC for

F. Since

that

for each

((k'c), K’) C 6 0b“),

is

fci

=

colimit

a

the

o

F(g),

that

so

of F, there

((fc), A) is

a

is

natural

a

uniquef:

K’

—’

sink A such

triangle ké

———>K’

F(C)

if

fa

'v A commutes.

for the

Consequently,

diagram “c

D(C)

—>

G(F(C)) G(kc)

0%)

G(K')

/V

K—f—)G(A) we

have commutativity

right-hand triangle;

of the outer

from

which

fokc Since

((kc), K) IS a colimit, and uniqueness of f with respect

tion

and

Hence

the upper quadrilateral, and the for each C e 0b(‘€),

that

G(f)ou'okc.

=

The

the fact that (uc, F (u', K ') is a G-universal

square,

it follows

thus to

(C)) being map

epi--sink,

an

this

property a

universal

map,

f=

from

0( f ) its

o

u’.

construc-

G-generates F(C)

for K.

(ii) Suppose that (u’, K ’) is a G-universal map is a G-universal for D(C), we have for map of some K’ such that the square kg: F(C) —>

have

we

follows

for

each

K. Since C

e

each

(uc, F(C))

0b(‘6) the existence

Admin! Situations

190

C Imp. VII

Imiwmc»

no) I

l

l

gauge)

kc

l k; 4r

I

+

K—"—,->G(K')K' commutes.

We wish C

C'

—->

we

of the

triangle)

((1:3), K’)

that

show

since

of D and

colimit 9:

to

u:

D

G

—>

o

is

a

colimit

of F. Since

F is

a

natural

transformation,

for each

the

right-hand

of all

have

commutativity diagram

(except

most

at

((Irc), K)

is

a

ll

D(C)

——i———>G(F(C)) Dig)

G(th» u

a

D(C’)——C->G(F(C’)) G(kg.)

kc

G(ké")

kc' K

———.—’G(K’) ll

Since uc G-generates F(C), it follows that is a natural sink for F. To show that it is also

a

natural

natural that

sink

each

for F. [t is easy to for D, so that there is a sink

a

=

kg. F(g).

Thus

((kb), K') ((qc), Q) is see that ((G((1c) ac), G(Q)) is then a G(Q) such unique morphism h: K k};

0

colimit,

that

assume 0

—»

triangle K

"°/’“I I

l l

D(C)

i

G(q C )°uc

v

G(Q) commutes.

Since such

that

(11’,K’) is the

a

G-universal

for

map

K, there

is

a

unique lit

triangle

at?) El '9 commutes.

«0+-

K'

-§

Q

Universal

See. 26

h

of ‘1c and

But each

o

k}; is

191

Maps

morphism

a

such

x

that

the triangle

D(C)—"3mmF(IC) I

EGO!) 'g x

h°kc=G(qc)°uc

I

'v Q

G(Q) commutes.

since

Hence,

F(C), epi-sink and (u', K’) G-generates

uc

(kc, K) is an see that h is unique with respect of F. a colimit [:l Since

26.10

‘Ic

is

h

=

a

o

k}; for

G-universal

C

map,

it is easy

e

011%). to

Consequently (kg, K’)

this property.

to

each

is

COROLLARY

$. Then tlte fltll sub(fi-cocompletecategory and let G: .ss’ universal maps with respect to category a' consisting of all objects of a that have in Q; i.e., if D: g —> a has a G, is closed under the formation of g-coh'mits colimit ((kc), K) in Q and if each D(C) is in 9’, then K must be in .‘B’. I] Let

26.11

4% be

—>

a

THEOREM

G: d

Let G-universal

a

—>

be

a

fimctor

such

that

for each

B

6

ONE?)

there

exists

a

(:13, A n)-

map

unique functor F: a —r .2! such that A a: and (i) for each B e 0b(93), F (B) —r a natural F G is trandormation. (ii) 11 (nu): la each fi-colimits category ‘6’,and for (2) Moreover, F preserves 1‘, such that (3) There is a unique natural transformation a: F G Then

(1)

there

exists

a

=

o

=

——>

o

6':

0'1“» Geno—)0

=

050,

and

FflFoGoFflF: i.e.,for

each

A

F2917;

01201),

e

C(84) item °

and

for

each

B

e

=

law»

0b($), ams) °F (’13)

=

11-13)-

Proof:

(l).

This

the

identity

(2). K

6

follows functor

immediately from l3: Q a .43.

Suppose that D: W one), by hypothesis

—>

9?

there

is

is

the

a

a

preceding

functor

G-universal

with map

lemma

(26.9), where

D is

((kc), K). Since (nx, AK) for K. Thus by

colimit

192

Situations

Adiaim

Lemma

26.9,

have for each

we

((ké), AK)is

that

colimit

a

C

of F

0b(‘6’)a morphism kg:

e

D and

o

Chap.

each

F

0

0(0)

—>

VII

Ax such

square

D(C)—"”—‘°’->G(FoD(C)) room)

kui

(___--

0mg.)

kg,

AK=F(K) KTG(AK)=B(F(K))

commutes.

However, since 1]: 19 —> G F is a natural transformation, the morphisms Since "0(c) F(kc): FoD(C) —» F(K) also make the above squares commute. G-gencrates F D(C), it follows that F(kc) k'c, for each C, so that o

o

=

(04:). Ax) is

a

of F

colimit For

(3).

universal exists

D.

o

each

A

((F(kc)). F(K))

=

fi-colimits.

F preserves

Hence

011021),G(A)e 011(9),

6

G(A). Consequently, by map A unique morphism 3‘: F G(A)

a

that

so

("our F(G(A)))

the definition

for

such

—»

o

mommmloun

that

the

is

map,

a

G-

there

triangle

Page!)

I

sen—m-

l 605‘)

16M)

of universal

hh

5

v

G‘(A) commutes.

If

e

is

(34)

=

for each

triangle

natural

a

A

clearly

10 To show

that

s

is

°

But since

la

:1:

G

-?

F is

o

=

Gm

=

law)

a

°

"Gm

that

(s

t

F)

a o

(F

t

notice

=

°

°

(F

°

°

°

=

o

this

°

becomes:

°

have

we

(F° GXD.

natural

a

for each

’13

°

°

G)(A),

344' °

=

that °

A'. Then

->

transformation,

is

(Kama) F(lln»

A =

°

£4

F 06'

(SA): 1,, n) IF,

a

=

f:

°

°

G-generates

f° so

let

6054') (G ”(C(13) ’16“) C(BA' (F GXfD "6“)-

=

=

since

of the

GU) 16m C(84) 'lam GU) 60%) 'Icw) GU)-

°

natural

Glf° 5,1) ’lcm) Thus

commutativity

(see 13.13).

transformation,

GU° 84) "out

the

that

(G*e)o(ntG)

=

natural

a

then

transformation,

shows

=

transformation. B

6

To

012(9)

605nm) (G F013) "3)°

°

°

show

that

26

Sec.

Universal

1] is a natural

But since

°

C

"Gena;

for each

since

Consequently,

this is

transformation,

603nm)

193

Maps

GUI-13)) "8' °

=

"a

B, m, G-generates F(B),

F013)

°

Ema)

Ina)-

=

Hence

(€*F)°(F*tt)

1F- 1:]

=

PROPOSITION

26.12

Let

G: s!

Q be

ftmctor such that for each B e 0M3?) each of (:13,AH) and (fig,.43)is a G-tmiversal map for B. If F and F are the corresponding functors, the above A B and F (B) theorem, such that F (B) guaranteed by A}, then F is to F. naturally isomorphic —)

a

=

Proof: This follows immediately essentially unique (26.7). [3

from

=

fact

the

that

universal

are

maps

EXERCISES 26A.

Show 1.

Chapter from

B

[Note D.]

26C.

that

—>

1f

is the R—ng

of

in;

is the

Theorem

Rng has 26.11?

26E.

26F.

fields is

reflective

260.

State

Use

the covariant

has

a

Let

6:54

G-universal

26H.

(a) If B is prove

that

D of map

whether

the duals dual

an

to

Maps B

e

Lat

two —>

induced

is the

the contravariant

if G: s!

G preserves Initial

->

a? is

functor

show

such

and

that

each

guaranteed

i.e., the

and

B is

commutative

of Theorem

that

by

U-universal

functor

(12.4)]

“squaring

identities

functor

forgetful

duals a

the

26.11.

have

necessarily

generators”,

Set

the

to respect Theorem by

not the category of algebraically closed of the category of commutative fields.

that

and

do not

is the

with

U:

[including show

then

map,

3? and

the

or

with

guaranteed

What

lattice

B, where

subcategory

Universal ->

to

are familiar, determine 26.] l. by Theorem

you

(which forgets the identity),

map.

“free

the

Determine a

functor

forgetful

map

functor

of rings that

category

respect

universal

a

the induced

G-univcrsal

a

Determine

map (it, As) with two-element set.

has

set

every

Describe

(9A(b)).

26D.

object

example in (3) of 26.2 is implied by Theorem a bilinear morphism f: B -> Hom(C, D) induces

of the examples of 26.2 with which colimit-preserving functor F guarantwd

Show

functor”

Rng

second

each

For each

unique induced

:

the

that

C to

x

268.

G

how

every

26.11.

@-object

limits.

Objects

0b(.’3).

initial

object of 3d and following are equivalent:

(i) (u, A) is a G-universal map (ii) A is an initial object ofa’.

for

II

is the B.

unique Ill-morphism

from

B to

GM),

a

194

Adjoiu! Situations

Chap.

VII

whose value at the single object a is B, prove that the (b) If B: l -> 3 is the functor following are equivalent: (i) (u, A) is a G-universal map for B. (ii) (o. u, A) is an initial object of the comma category (3, G) (see 20D). an alternate are (c) Use part (b) to obtain proof that universal maps “essentially unique" (see 26.7).

Colimils

26]. Let .1! and

d

I be

Universal

as

51' (IS.8). let D: I

to

Maps

d

—>

be

a

an

(a)

u

.sfimorphism. Prove (14,): D -» GM) is

functor

-»

00')

u): be

.2!’ be the “constant functor" and for each ie functor, 0b(l) let

G :41

let

categories,

G(A)(i)

—v

from

A

=

that: a

=

natural

transformation

if and

only if ((14,), A) is

a

natural

sink for D.

(b) (u. A) is let

Now

a

G-universal

(m): G-generates u

=

D

if and

map

(c) (d) u extremally G-generates A if (e) Using Proposition 26.6 obtain (20.4 dual). 26.11 obtain (f) Using Theorem u

dual). (g) Dualize

each

of the above

that when

a

colimit

we

were

a

a

proof that

new

an

colimits

that:

(extremal epi)-sink. are (extremal epi)-sinks

proof of the commutation

new

of colimits

(25.4

(a)-(f).

ADJOINT

FUNCI'ORS

considering what it should

inverse". our first notion with However, since functors

of D.

Prove

only if ((u‘), A) is

and

results

§27 Recall

only if ((11,).A)is

transformation. C(A) be a natural A if and only if ((14,). A)is an epi-sink. 4

of

that

mean

for

a

functor

to

“have

between

isomorphism categories. “strong” type of inverse rarely occur, a more suitable notion was found in the conjunction of the notions full, faithful, and dense (i.e., equivalences between categories). Equivalences (although “weaker” than isomorphisms) have been seen to occur relatively frequently (14.16 and l4.l8) and to be of such “strength" that they preserve and reflect categorical properties. an

An

even

notion

weaker

of

functors, yet the notion is still In the last section we have seen

fi-object together with two

each

has

a

Gouniversal

natural

or

an

a

functor

such

a

left

a

of

that

has

such

that

a

functor

was

which

in

some

has

sense

there

is

an

inverse, is

right adjoint. Indeed, plethora of “strong" enough to be intensely interesting.

map, transformations

that

if G: d

then

n and

a

is

exists

a

-’

there s

(G‘E)°(’7"G)=

la

(ssF)O(F*q)=

1;.

and

such

that

a

a

functor

functor

such F: 3

that —»

d

27

Sec.

Adjoint FHIICIOI'S

This

section

27.1

DEFINITION

(I) If

a!

is devoted

and

w

(i) (ii)

such

Gm! :1:

further

a

study of such

G and

categories,

are

transformations

to

19,—»GoFandstoG—v G

02.

is called

adjunction

an

and

adjoint situation,

or

G: (M,

is denoted

by

Q)

briefly by

more

G

simply by

(2) If (n, a): right adjoint of the

F —{

F is said

G, then

of F, 11 is called

‘

to

G. be

the unit of the

left

adjoint of G, G is said to be a adjunction and a is called the counit a

adjunction.

(3) A functor exist F, q, and

sz a

such

is said

Q

—v

that

(n, a):

right adjoint provided that there other some

has words, a functor functor, and it has

a a

to

F —{ are

G,

have

G.

a

left

Similarly

n, and

a

such

left

that

adjoint provided

adjoint provided right adjoint provided

F: a

—.

that

there

a! is said to have

(n, s):

that

it is the

that

it is the

F ——l

G.

a

In

right adjoint adjoint of

left

functor.

some

The next situations 27.2

natural

are

a

Fir;

F —-1

of

n and

G, and

(n, s): F—l or

and

1d, =

(n, s): F—l or

functors

are

QandFuéa—nd,

—o

EmpocopfiiF= this

F

situations.

that:

6017ch (iii) G"'—6>

then

195

proposition (14.12).

follows

immediately

from the definition

of

equivalence

PROPOSITION

(1) If (F, G,

n,

a)

is

an

equivalence situation,

then

('1. c): F—I

G

and

(8".n"):G—l (2) If G: a! that

is

—>

.93 is

an

equivalence, left adjoint and

then

F. there

exists

functor F: a right adjoint of G. [3 of examples adjoint situations, a

93

—r

at

simultaneously other concrete we wish giving some to the between and universal clarify relationship adjoint situations maps. to Theorem 26.] whenever a functor G: .9! 33 has the I, According property that each Q-object has a G-universal map, this gives rise to an adjoint situation. The next theorem shows that each adjoint situation can in actually be obtained Before

a

—>

this

way.

Chap. VII

A djoiut Situations

196

THEOREM

27.3

G: s!

Let

3.

->

(1) if each B e 0b($) has a G-universal map (nu, A a), then there exists a unique (In) and for each B e 0b($), adjoint situation (:1, a): F —l G such that n A F(B) B. (11,a): F -——| G, then for each (2) Conversely, if we have an adioint situation B 6 017(9), ('le F (B)) is a G-tmt'versal map for B. =

=

first assertion

immediately from Theorem definition of adjoint situation (27.1). To show (2), suppose that a unique morphism f: F(B) to find B We wish f: C(A). triangle The

Proof:

26.”

follows

—»

-v

B

e

and

the

0b(fi)

and

that

the

A such

ma) 8-1:»(Goll7XB) l .

I

.—

i

6(f)

E

f

v

t

A commutes.

Let]

=

But since

5,.

1,,

:1:

0

Then

HI).

->

G

o

GU)

°

F is

a

=

'ln

natural

0(a)

°

(0

°

F)(f)

°

and

transformation

(G*E)°(n*G)

=

m;-

since

15.

this becomes

603,1) 'icm °f °

Heneef To

f

makes

the

show

uniqueness,

C(11)“),

=

triangle

Then

since

=

f-

that h: F(B) —» A is suppose transformation and is a natural

a

of the

commutativity

have

=

commute.

(8* F)°(F* we

loot) °f

n)

=

a

morphism

with

since

1p.

diagram Ins)

F(B) —————>F(B)

/~’

”gym (F°G°F)(B)

(Foam)

(FOG) (A) Hence 27.4

It

=

8,.

o

F0").

[:1

and

F is

it

——T———->A

COROLLARY ..

If each of F isomorphic to F.

a

left adjoin! of the functor G, then

F is

naturally

Adjofm FHHCI'OI‘S

Sec. 27

Proof: Proposition 27.5

This

is

26.12.

EXAMPLES

an

immediate

the

above

theorem

and

I] OF

ADJOINT

33

SITUATIONS

—-——l

G

(left atdjoint)

Grp

Set

free group

R-Mod

Set

free R-module

Top

Set

discrete

space

Rng

Mon

monoid

ring functor

Rug

Ab

tensor

complete

uniform

uniform

spaces

functor

CRegTz

Stone-Eech

uniform

of

consequence

F

d

197

(right adjoint)

functor

rcfcrencef

functor

2620):!

forgetful functor

26.2(l)a

forgetful functor

26.2(l)a

forgetful functor

26.2(l)b

forgetful functor

26.2(l)b

inclusion

functor

26.2(2)a

inclusion

functor

262(2)::

functor

inclusion

functor

262(2)::

quotients

inclusion

functor

26.2(2)a

inclusion

functor

262(2)::

inclusion

functor

26.2(2)b

forgetful

functor functor

ring functor completion

spaces

CompT;

compactification functor

BanSp.

NLinSp

completion

cancellative

Ab

group

abelian

of

functor

semigroups Field

integral

field of

domains

functor

quotients

(injective homo-

morphisms) Ab

T Note

Grp

that

c0ouniversal

either map

abelianization

functor

the existence of a universal map for each A e 0b(.v/) gives rise to

for each an

:uljoint

B e

01:02?)

situation.

or

the

existence

of

a

Chap. VII

Adioint Situations

198

G

F

93

.2!

torsion

functor

inclusion

abelian torsion

Ab

rcferencef

(right adjoint)

(left adjoint)

26.2(4)

subgroup

functor

groups

torsion

Ab

torsion

l'ree functor

inclusion

functor

26.2(2)b

functor

which

inclusion

functor

26.2(2)b

free

abelian groups

Ab“)

Ab

B to

BInB

Ab

Ab“)

inclusion

locally

LinTop

locally

functor

convex

modification

convex

takes

(subgroup of elements of order being a divisor of nyfunctor

26.2(4)

inclusion

26.2(2)d

functor

functor

linear

topological spaces

functor

inclusion

compactly generated

Top

compactly generated refinement

26.2(4)

functor

spaces

R-Mod

R-Mod

(where is

®R C

._

Ham(C,

_)

2620)

R

com-

mutative) pTop Grp

pTop

suspension functor

loop

pointed

fundamental

Eilenberg-Mac

arcwisc

functor

group

I'll

space

functor

space

26.2(3)

Lane

functor

connected

(with homotopy equivalence spaces

classes

of

maps) 1' Note that eo-universal

either map

of

the existence

for each

A

e

a

universal

01:01) gives rise

map to an

for each

adjoint

B

6

012(9)

situation.

or

the existence

of

a

Sec. 27

Adjoim Funclors

———|

F

a

.2! d

d

G

(right adjoint)

(left adjoint)

d'

(where

199

“constant

Colim ’

has I-

reference?

functor“

261

functor

colimits) 43’ (where a! has

d'

“constant

functor"

261

Lim '

functor

l-lim i ts)

T Note

the existence

either

that

comniversal

for each

map

THEOREM

27.6

Gm!

Let

following

—»

universal

a

.21, n: I,

—*

for each

Be 017(9) adjoint situation.

map to an

If

Proof:

we

G:

Thus, in

the

I,”

P"

3"”,

we

G

—.

that

see

Fis

o

a

—»

in

morphism

[9”, WW],

category o

0"".

Again, translating

0"“ [51, a]

inc”

we

a

in

morphism

n: 6""

o

F”

Imp.

—.

Similarly

the statement:

G=GLC

reversing aopn

Go'4——

F’"

P'oG”

viz.

the arrows; G"

=

'6”

G”

the

Clearly, applying

pro;

F”

=

1,»

F”

F”.

situation

adjoint

(1:, n): G” —l

27.7

[3,

—'

53], it is also

(see ISA).

have

we

0“"

r

F°Po

6970

rap."

the

have

we

the

have

F" Thus

G

[41", WW]means

to

G°’

:33 and

F: 9?

.21, then the following

->

natural

left adjoin! of G: i.e., there exist G. melt tltat (n, a): F —-l (2) TIM: associated set-valued biftmctors (IOB) (1) F

is

a

liom(F_,

53""

_):

.91

x

equivalent:

are

transformations

7) and

a

Set,

-’

and

(7...): 93"?

lmm(_,

.9!

x

Set

—>

naturally ismnorpltic.

are

Proof: (1)

=>

define

(2). If (n, c): F ———lG, then

‘1an) ll‘

018,.(f)

G-univcrsul

map

Consequently,

for 23,. is

C(A), lff: B morphism f: F(B) ->

—.

=

then

the property the triangle

again by

A such

that

imam

3

,

(a) I

i.e.. such

that

jcctive; hence, bijective. need only show that if

015,40)

Thus

also then

the

square

=

to

_)

so

that

since

for universal

of universal

G_) by

maps

(nu, F(B)) is maps,

f

there

is

=

a

g.

some

ma)

50(7)

if 'v A

Consequently a,” is 0th, =f. that a is a natural transformation,

C(f)

and

Itom(_,

lla-

i

establish

—r

I

G(A) commutes.

°

G(g)om,, G(f)ous B, by the uniqueness condition an injective function.

then

gum),

=

C(f)

=

ltom(F_,

a:

A—f—>A’

surwe

201

Functars

Adjoim

27

Sec.

(17(3), mfléhom

ham

(a, com 110mm. 0(1))

(9) .f)

hamlF

ham (F(B'),

A’)a—>Itom(B', G(A')) 3'11,

commutes.

Let

e

x

hom(F (B), A). Then

(“324’ IIOM(F(9),f))(X) °

aw(f°

=

G(f°

=

But since

19;3—» G

u:

G(f° X)

Thus

°

the square

avg

F is

o

G(f° X) (G

G(f)°(G(X)

=

(handy, G(f))

°

°

=

113'

“13)09

=

that

this

F)(9)

°

’13)

B 6

hom(F_, _) 013(9)), let

27.3,

6—)

ham(_,

is

a

natural

map for B. Suppose that f: B

ham

—>

for each

that

B

e

(:13, F(B)) is

0mg),

GM). Applying the commutativity

(F(B),

homunfitazhtf»)

f

Hence,

the

(F(B), A) The»:

In,” e Izom(F(B), F(B)),

the element

we

(B, G(A)) have

=

(D) “3,1015;

=

(llamas: 6(“1341 (f ) “mamxlrwfl

=

(“an hamllnup DEE} (f)))(1mn) °

°

=

6015410))

triangle

a—J’Lmopxa) {

f

commutes.

of the square

(B, G°F(B)) Han—"3591mm:

nomupwvaggm) ham

iso-

“mnmunml

=

only show

need

we

->

G—universal

to

'hr-

is

hamtg. G(f))(G(x)

=

°

513.006)-

°

a:

m:

Theorem

°

transformation,

natural

a

17(9))

°

commutes.

(2) = (1'). Suppose morphism. For each

By

x

17(9))

°

x

F(B) I

I

gamma» imam t

l

G(A)

A

°

’13-

a

202

To

Adjaim Silualians show

that g: F(B) —* A such that suppose 6(9) 0 m, f. of the above square, with a;}(f) replaced by y, we have

uniqueness, Again by commutativity “54(9)

This F: 59

left

—»

is

=

ammonium). mama)»

=

“Oman, 0(9))

=

Thus, since a“

Chap. VII

bijective,

g

=

°

“armxlrwfl

=

0(9)

°

’18

=

f-

«5"(f). [3

theorem, together with earlier ones, tells us that if G: .2! —. Q and .21, then there are at least four ways of describing the fact that F is a

adjoint

of G

:

(l) by

means

of

a

(2) by

means

of

a

(3) by

means

of natural

family (:15, F(B)) of universal maps; family (G(A), ad) of co-univcrsal maps; transformations and

nzlaaGoF

echG-vl,

suchthat

floopooflc=

(:2,

,

and

FflFoGeF£F=F£>R (4) by

means

of

natural

a

a

The

fourth

used second

as

the

way

is

isomorphism

(02“): hom(F_., _)

=

—v

hom(_,

perhaps the quickest and easiest to of adjoint situation. However.

definition

way is often

easier

to

establish

and closer

G_).

state

in

to one‘s

and

is thus

practice. intuition.

the

often first

or

Sec.

27

Adjoim Functors

203

EXERCISES 27A.

Let

and

H

K

be

considered

groups

f: H -v K be a group homomorphism following are equivalent:

(i.e.,

as

one-element

functor

a

from

categories. and let K). Show that the

H to

(a) f is an isomorphism. (b) I has a left adjoint. (c) f has a right adjoint. 278. be

is

(i.e.,

B be classes

functor

a

from

considered

discrete

as

B). Show

A to

that

eategories,

the

following

and are

let

B

f: A -> equivalent:

bijective function. is an equivalence. is an isomorphism. has a left adjoint. has a right adjoint.

f I f f f

(a) (b) (c) (d) (e)

Let A and

function

a

a

27C.

Let A and

B be sets

and

9M) and ?(B) are partially-ordered (15(6)). Show that: (a) the induced

let f

A

:

B be any

e

by inclusion,

they

function. can

Since

the power

be considered

as

sets

categories

functions

fl

Jim/1)»

9(3)

and

f"[

1:909)

—>

(h) f[ ] is a left adjoint off"[ ]. (c) Using the fact that right adjoints preserve

f“[r‘\C‘] for any

family (C,) of Let A be

27D.

totally-ordered be

901)

functors.

are

a

functor

(a) Show and

R

are

a

subsets

nf"lC:]

=

each

set

and let R be the real

of A and R

as

27E. »

if g has a left adjoint, then 9 is upper thought of as ordered topological spaces).

Let

‘6’ be

a

category

(considered and

let 9: A

as -»

a

R

function).

that

by: F(A) F

F is

that

Show

(f )

f

=

fl

U

adjoint of G.

left

a

A U A

=

that ‘6 is a category that has coproducts and A is a fixed object in G be the “copower functor” that assigns to each set I the object ’A of A) and to each function f : I -> J the unique induced morphism (i.e., the lth copower a ’A (for which fo II: f: ’A pm), for each ie I). Show that F is a left adjoint of 27F.

Suppose

%’. Let F: Set

->

=

‘6

-b

Set.

(Cf. 27E.)

Prove

that

every

hom(A, _): 27G.

has

that

functor

left

a

adjoint

preserves

monomorphisms.

Given (1], c): F—l G: (5%, .46). Show that: (a) G is faithful il’f all cA’s are epimorphisms. (b) G is full iff all cA‘s are sections. (c) If all ad’s are regular epimorphisms, then G reflects limits.

27H.

both

271.

Dualize

27].

For

the unit 27K.

naturally

of the

each

isomorphic

if

that

for each

F is

for each

B

g:

f: F(B)

_)

hom(F_,

Prove

that

if F: Q

a

for each

A

e

is

there

r:

H

G

thatM'

Suppose —>

id

are

isomorphism

flu

8A

th).

°

functors

are

unique adjoint situation

a

and

0—)

ham(..,

—>

F -|

(1), a):

G such

(lemurs) "l

“GA.A('6A)-

natural

—

-—G—) a and

g3

Ls!

transformations

such

GflGoHoGEm: (a) Show that (b) Show that (i) 33’ has (ii) .93 has

to

adjoint of G, and G is F.

G...)

hom(_,

G: .53! —» g

=

_

o

left

0b(.ss’) 5A

27N.

=

_)

'79

and

isomorphic

G(f)°

=

.9! and

hom(F._. then

isomorphism, for each B 6 0mg?)

that

a

C(A)

-»

natural

a

determine

familiar,

A,

-*

at:

is

are

A

—-)

09mg) 27M.

you

of G, F is

naturally

amU) and

which

()1,a) : F ——-| G, then there exists aunique natural at:

such

27H.

adjunction.

left adjoint

a

G, then

to

that

Prove

of the

if F is

that

of 27.5 with

examples

the counit

and

Prove

27L.

(a), (b), and (c) in Exercise

statements

G does if Q has

not

any

necessarily one

have

of the

equalizers, coequalizers, (iii) idempotents in Q split (17D), then C does have a left adjoint.

are

functors

that

630. a

left

properties:

adjoint.

and

6: la

-’

G

a

H and

Sec. 27

270.

only if

Show

there

is

27?.

a

for any

that

Funclar

Ilom(l, tl5.2) preserves

(b) State and Hom(_, l).

F—l

that

(c) Suppose “constant

the

corresponding

limit

(6.131.9-

—’

———1(00414121190.

for

contravariant

the

.46); l is a category; (l5.8). and

functors

(23.4).

internal

Cdzal

d’

->

[JI'mdnd’ d

-»

ham-functor

C”: 33

and

.9' Dim“:

and

%'

—>

.9

->

are

that

F—{

o

(digs) oLim'dz

G

that

(FC —) and

Imm-functor

functors

Cd and

adjoint of G if and (la. G)thnt

(Fo_)

assertion

01(51,

functor“

are

then

similar

a

prove

left

a

i.c.. if

at).

are,

F is

categories (F, id) and and Q (see 20D(e)).

d

to

$51.9-

-):

adjoint situations;

F——l

then

comma

Categories I. the internal

cutcgory

205

Lat,

x

and

an

Adjoinls am!

(a) Prove

1)

ifd

that

isomorphism betweenthe with the projection functors

commutes

Show

thcmrs

Adjoim

that

conclude

G

o

C9 —-i

=

and

le"

I-limits. adjoints preserve (d) Using (a) and (c), obtain

Limaw _) o

alternate

an

0—):(M'H‘E)

(G Lil'ma

=

about

(1) For all (2) For all

this

i.e., that

same;

proof of the commutation

27Q. Adjoims and Galois Correspondence: (A, - -> G :d Q and F: .43 .sl are order-reversing functions. statements

the

essentially

are

of limits

Consider

(cf. 25.4).

classes

quasi-ordered

=

right

the

and

following four

situation:

a

e

A and

a

e

A,

a

for all b

B,

e

a

F(G(a)) and for




following four conditions:

(1) F—l

(b) Show

a!

59); Le, Fis

of",

isomorphisms

equivalences. (2)

a

transformations

=>

(3)

9

(4)-

left

adjoint of e:

from

F0

G

G.

15.,

->

G to G

o

F

o

and G and

I]:

la

from

a

G

F to F

F.

o

o

G

c

F.

(c) Let ‘6’ be of

Chap.

Adjoim Situations

206

a

morphisms

that has pullbacks and pushouts, let A be the class category in 9? with common codomain, with the quasi-order relation

of all


—>

k

9

only if there

if and

is

commutes, and let B be the class

quashorder

relation

morphism

a

r

of all pairs of defined by:

5

1»




at

1 only if there

with

W

U

—>

if and

domain,

common

s

such

that

the diagram

commutes.

and 6: (A, (B. s) Using pullbacks and pushouts, define functions F: (B, s) —> (A, -

28.5

EXAMPLES

(l) “"6:

Bis an ail—morphism, then identity functor and]: A an of is the same as an (extremal G-generating, mono)-factorization (f, B) (extremal epi, mono)-factorization off(cf. l7.l5), 51’ is the “constant functor" (2) IfG: 41/ GM) functor(l5.8)andf= (fl): B is an .d’-morphism.then an (extremal of G-generating, mono)-factorization as an (f, A) is the same [(extremal epi)-sink, mono]-factorization of the sink (cf. 19.13). ((fl). A) .d

—>

.a/ is the

—>

—¢

(3) If then

U:

Grp

—>

—.

Set

is the

forgetful functor and f: G-generating. mono)-factorization

(extremal morphism) the factorization an

B—f» um)

=

3—9. 0(2) &

B

-.

of

U(A)

U(A) is a function. (f, A) is (up to iso-

210

Adjaint Situations Z is the

where inclusion 28.6

FACTORIZATION

.2!

If G: d

of

subgroup homomorphism,

is

the pair

and

g

intersections

and

equalizers, and limits, then for each (ii-morphism of the form (extremal G-generating, mono)-factorization.

an

(D... d‘), of

Let :1 be the class

Proof:

—>

U(m)

=

has

well-powered,

(f, A) has

A

generated by the set f[B], m: is the unique function with f

VII

A is the o

g.

LEMMA

.98 preserves

-r

A

Chap.

all

of A that

subobjects

the

i)

3

are

functar C(A),

part

of

factorization

some

L

B

C(A)

B‘_‘. G(D,) fl'flL GM)

=

of J (f, A), and let (A, m) be the intersection (17.7), where for each i, m limits (6(3), U(m)) is the intersection of the d,ok,-. Since G preserves a Thus there is B such that the C(A) family ((110,), G(d,)),. morphism g: diagram of

=

—§

l ——————>

B

C(21)

Ill

\\\

\\

0(1),.)

g\‘\

(Nd/v

A(:(k‘)

C(m)

\\J

C(71) for each

commutes

need

We

I.

e

show

that

g

extremally

it is sufficient

to

show

only

26.5

Proposition

i

B

I123

indeed so

m

A is

1+

0(2)

c

II

=

since

that

dj

monomorphism

the extremal

by

condition;

a

d}-for somej

is

and

e

=

1. Thus

([1. =

monomorphism,

a a

retraction.

hence

Incl]

an

then

be

an

(A, m

o

isomorphism. If It) belongs to 0/

have

we

l;

must

=

=

djokjoh

k,- h. Consequently El isomorphism. 0

h is both

a

COROLLARY

28.7

intersections has that Every well-powered category E] (extremal epi, mono)-factorizable (cf. l7.16).

and

equalizers

is

sink

in

COROLLARY

28.8

If sf

that

Notice

L. 6(2) 9‘1”6(2)

0

(II-0'2 so

B

=

monomorphism, then It CHI) g’, where h is a monomorphism. ...

=

g

that

9 satisfies

that

1.

if

i.e., that

where

G-generates

has

and equalizers, well-powered and has intersections [(extremal epi)-sink, mona]-factorization (cf. l9.l4).

51 is an

then

[:1

every

28

See.

Existence

2H

of Adjoims

THEOREM

28.9

If a! is complete and well-powered, Q-object extremal/y G-generates at most objects, then G has a left adjoint. B be

Let

Proof:

for the class

set

ofall

C(A‘), A i), be a representative C(A) extremally G-generates Lemma set (28.6). this is a G-solution Theorem (28.3). [3

which

A. Then

for B.

according to the Factorization Apply the First Adjoint Functor

28.10

SECOND

Let a! G: s!

be

FUNCTOR

—>

THEOREM

complete, well-powered and extrema/[y co-(well-powered). Then a left adjoint if and only if the following two conditions are

9? has

—.

limits, and each of pairwise non-isomorphic .9!-

set

a

.99 preserves

—’

let (8 fl» g: B

.‘B-objectand pairs (g, A) for a

ADJOINT

G: a!

satisfied: (I) G preserves

limits.

(2) Each B-objecl extremal/y G-generates

at

most

of pairwise non-isomorphic

a set

.nl-objects.

Proof: That adjoint has been necessary, need only

the

first recall

conditions

two

shown

in the that

sufficient

are

for

the

preceding theorem (28.9). right adjoint preserves

To

each

verify that (2) holds. a w-object and suppose

Let B be

objects ofd such that for each ally G-generates A. Since G ('15, A5) for B (27.3). Hence such that the triangle

A

thaw"? is

has

left

a

for each

class of

is

A e .9? there

show

limits

of

that

left

a

they

(27.7). Thus

are we

pairwise non-isomorphic C(A) which extremexists a uniyersal map a morphism L: A Afl

somefi: adjoint, there

.1? there

e

a

existence

B

is

—~

-¢

n

B_R’G(AR)

All?

I

|

36W) l

a.

{r

{r

C(A)

A

A

commutes.

We claim show

to

that

that

each

A

is

extremal

an

epimorphism.

T

0

show

this, it suffices

if

ABE”: =ABL>AL>A is

a

factorization

morphism have

the

off).

(l7.l4).

following

since

[A extremally

Consequently

for

each

is

m

f

=

A

factorization

32» But

where

If indeed

c

y where

m

is

a

then

must

m

monomorphism,

be

an

iso-

then

we

offA:

6(4)

=

B

3?,

M A. this

generates A e

monomorphism.

a

m

(L.

A) is

0(3) M implies an

that

extremal

C(A). m

is

an

quotient

isomorphism. object of AB.

2l2

Adjoint Situations since the

Thus

objects in co-(well-powered), 3? must

Q

are

be

a

C Imp. VI I

and

pairwise non-isomorphic [I

.91 is

extremally

set.

It is

interesting to observe that in general both of the smallness hypotheses of the Second Theorem are needed Adjoint Functor [for Well-powered, see Exercise and for see Exercise 28C; extremally co-(well-powered), 28D]. The next theorem a remarkable in the case where the provides simplification category .31 has a coseparator. Recall that we have earlier seen that this condition (in with is a smallness conjunction well-poweredness) actually quite strong condition

(23.14).

SPECIAL

28.11

ADJOINT

that

Suppose for each fimctor (1) (2)

G has

a

d

FUNCTOR

is

G: .2!

THEOREM

well-powered, complete, and has —» 3 the following are equivalent:

a

C.

coseparator

Then

left adioint.

G preserves

limits.

Proof: Clearly (1) implies (2) (27.7). To show that (2) implies (I) it is sufficient to show that (2) implies that each B—object,B, G—generatesat most a set of pairwise non-isomorphic art-objects (28.9). Suppose that 9? is a class of A and that for each a suitable 69?, pairwise non-isomorphic d-objects morphism B

G-generates

9—”)G(A)

A.

by the definition of product, for each A [1,.such that for each f e Itom(A, C), the diagram Now

A

L

e

.69 there

exists

morphism

a

Chom(A.C) ”’

r c

each

Since

each [1,. must coseparator, from 9,. G—generatesA, the function

Since

commutes.

C is

be

monomorphism (19.6). ltom(A, C) to [1001052,G(C)) for those objects A e 9? for which C”°"""-c’ C’"’"‘(”'G“")(18F), and

a

a

by f H G(f) g, is injective. Thus —» ltom(A, C) ¢ Q, there is a section s): (A, 3,4 11,.)is thus a subobject of C‘B'Gm’. If Izom(A, C) defined

a

0

is the terminal object T. Thus or of Chom‘B-G‘C” (A, [1,.) is a be a set. means that Q must

for each

subobject E]

A

Q, then

=

9?, either (A, 3,. of 7'. Since .d is e

11,.)is

C"°"’("'c’

subobject well-powered, this

o

a

implies, for example, that it's! is any F :9! of the categories Set, R-Mod, Top, or Comp'l‘z, then any functor a and has if and if it limits has a left adjoint right adjoint if only preserves Observe

that

only if it preserves

the above

colimits.

theorem

one -—r

9?

and

Sec. 28

Existence

The

following adjoint functor theorem V are forgetful functors from concrete

U and 28.12

213

of Adioints is often

applicable when the categories to the category

f unctors Set.

THEOREM

If

the

and

commutes,

and

diagram of categories

if

the

ftmctors

conditions

following

satisfied:

are

(1) .9! is complete, well-powered, and co-(well-powered), (2) G preserves limits, (3) U has a left-adioint, (4) V isfaithjitl; then G has

a

left adioint.

Proof: By Theorem 28.9, it suffices to show that each Q-object G-gcnerates a set of pairwise non-isomorphic .rl-objects. Let B e Obw) and suppose that .92 is a class of pairwise non-isomorphic d—objects such that for each is some A 599, 89—”)G(A) G-generates A. Since U has a left adjoint, there U-universal map (n, X) for V(B). Now for each morphism g,“ at most

V (9,4): so

that

that

since

the

(M, Z) is

WI?)

U-universal

a

V(G(A))

->

exists

there

map,

UM).

=

a

unique 574:1

—>

A such

triangle

V(B)—"> (1(2)

2

.

I

: '3 .a

._

W94)

V(gA)

1

‘5,

v

U(A)=(VoG)(A)

A

commutes.

We claim

that

5,. is

an

To

epimorphism. is

°

r

that

£74-

°

5

=

this, suppose

see

Then

UO‘)

(“(7,1)

U“)

V (9,.)

(V

°

ll

°

U (5A);

hence

(V Since

V is

faithful,

we

°

(7)0)

have

.9. Hence implies that r powered), this implies that =

°

C(r)

o

9’.

5,. is

each 9? is

=

a

set.

=

6(3) an

[:1

°

G)(S)

°

V (9,.)

But since 9,. G-generatcs A, this epimorphism. Since a! is co-(wello

9‘.

Chap. VII

Arljaim Simulioux

2l4

the

that

except

ever,

V reflects

that

condition

theorem

(2) in the above

condition

that

Notice

limits.

diagram

replaced by

all conditions

eliminates

then

This

of functors

be

can

the

G whatso-

on

commutes.

EXERCISES 28A.

that

ifs!

objects,

then

Prove

surjection

on

is

(12d

and

complete

the following

a? is

4

which

full functor

a

is also

a

equivalent:

are

(a) G has a left adjoint. limits. (b) G preserves the examples of 27.5 in which 51 is complete, use the various of the given of this section to obtain proofs of the existence theorems For

283.

functor

adjoint adjoint

situations. as a class of ordinal ‘6 be the partially-ordered numbers, considered —v Q let G: be and ‘6’” with one the be I. only morphism; category category; from ‘6" to .96. the unique functor

28C.

bet

let 3

that:

Show

(a) ‘6” is complete. (b) ‘6’" is cMwell-powered). (c) Each Q-object extremally

at

G-generates

most

set) of

(actually the empty

set

a

W'hobjects. (d) ’6’” has a coseparator. limits. (e) G preserves G has no left (f) adjoint. that

Conclude

A

e

on

A induces

morphism f:

each

A has

is

a

A

Adjoint

of

partially ordered

07,4 and

for

each

sets,

=

the closed

on

{xla

=

between

two

[f(a). m5] and

f,:

[11,inf{x|

Set

be the

a

e

A, the order

interval

s

x

5

MA}. is in

d-objects

Mares!)

if and

only if for

the restriction

x,f(x)

s

a

=

mall

U01). m5]

-*

bijection. U: .9!

Let

(a) d

a

is

complete. of the underlying two

B in [’05

—>

member

greatest

a

well-order

a

A,f[[a, null

e

a

Second

POS

of the category

subcategory

[a,m4] A

the

follows:

as

0b(.rs’) if and only if

relation

in

and

Let .9! be the

28D.

defined

“well-powered" hypothesis is necessary Theorem. in the Special Adjoint Functor

the

Theorem

Functor

indioes

i and

forgetful functor.

[For products, partially sets

by: ((1,)

s

Show

(1),) if and

that:

underlying set P of the product only if a, 5, b,- for each i and for any

order

the

k

0rd[a,. b,]

=

0rd[a.. b.]

or

0rd[a,. bl]




0rd[ak. b‘]

=

mi,

or

(where

0rd

X is the

ordinal

number

of

X).]

and

bk

=

m.

Existence

28

Sec.

limits. (b) U preserves and .s/ is well-powered (c) U has no left (d) adjoint.

that

Conclude set

of

215

of Adjoims

strongly co—(well-powered). each

.93-object extremally G-generates .9l-objects, in the Second Adjoint Functor that

the condition

pairwise non-isomorphic

at

most

a

Theorem,

be deleted.

cannot

28F..

following slightly modified

the

Prove

of the Second

form

Functor

Adjoint

Theorem: Let a!

if and

adjoint

only if the following

Each

C):

a!

Q-object G-generates

28F.

.3!

Let

be

a” be the

—»

conditions

two

G: s!

Then

co-(well-powered).

_.

a? has

left

a

satisfied:

are

limits.

(a) G preserves

([3)

and

well-powered.

be complete,

at

pairwise non-isomorphic

a

and

well-powered,

complete

“constant

of

most

set

functor

functor"

for

and

each

d-objects. let

I,

category

(15.8).

limits. (a) Show that each C , preserves Theorem proved above (285), (b) Using the form of the Second Adjoint Functor obtain a new proof of the fact that d is cocomplete if it is strongly co-(well-powered) (cf. 23.13). of discrete category functor. be the inclusion

Let 5!

286. let (Rd

9

->

(a) Show that G has no left adjoint. that the condition that (b) Conclude of Theorem 28.12. hypothesis 28H.

functor.

An

(Freyd) that

Suppose

d

is

Top, and

=

from

be deleted

cannot

the

Completeness Condition

Weak

With

Theorem

Film-tor

Adjoim

limits

G preserves

let fl

spaces,

—. Q is a in which idempotents split (170) and G: d category are conditions if and if the two G has a left adjoint only following

a

that

Prove

topological

be the

satisfied:

(a) Each Q-object has

(fl)

For

each

a

G~solution

small

category for G D, there exists C(A) such that for each

source -.

each

D: I

natural

a

o

h: B

set.

I and

s!

»

such

that for

(A, (k0,)

source

(8, (I0), is D

and

a

a

natural

morphism

I. the triangle

is

3 l

"

l

h

|

'v

GOO—GWGUL) commutes.

[To Q-object (5) [1,:

to

A

obtain

obtain

4» A for a

that

show

(at) and ([3) begin with

B. To do this a

singleton which

solution

601*) f o

=

sufficient,

are a

solution

set

construct

a

for B, considered

L,

as

a

map

functor.

for

each

and

take all d-morphisms set (E L» C(A), A). Then and use this as a functor (I?) again fl Consider

factorization B

universal

C(A)

=

B

L.

C(A). 6(2) “(—m’»

use

to

Adjaim‘ Situations

216

Then

is

a.

map

obtain

a

(retraction, for

8.]

morphism

u:

A

->

A

fl

X such

section) factorization

.2

=

of

that A me

m

o

Chap. Vll u

is

an

idempotent in d.

If

L» 2 3—»Z u,

then

(G(r) of; X) will be

a

universal

VIII Set-Valued

A harmless

Functors

cat.

necessary

——W.

SIIAKBPEARET

First

special interest for several reasons. of the motivating examples for category theory are actually i.e., pairs (53!, U) where .2! is a category and U is a faithful functors

Set-valued

with

domain

of

are

of the

Secondly, due to many functors are usually easier

.31.

Set, set-valued example, in

this

chapter

we

will

see

to

that

than

if G: a!

->

all, most categories,

concrete

functor set-valued nice properties of

well-known

handle

of

functors.

arbitrary Set

is

functor,

a

then

For and

if

G preserves

for at least one a G-universal set, map non-empty Also if a! is cocomplete and there is a G-universal map for a singleton that G has a left adjoint. set, then there is a G-universal map for each set—so is the fact that together functors A third reason for the importance of set-valued functors— a whole class of set-valued with any category 42!, there is associated

is

there

limits.

Set. We shall Izom(A, __): a! Q has a left adjoint provided that for G is naturally isomorphic Iwm(B, _) This then provides yet another approach

namely

the ham-functors

funetor

G: d

functor

set-valued with

—>

domain

.91.

—»

o

§29

see

that

arbitrary w-object B, the

each

an

hom-functor

to

some

to

adjoint

situations.

HOM-FUNCI‘ORS

usefulness

of ham-functors

is that

only in as we shall see but do all also, functors) (as preserve diagrams of “detect" concert this section, acting in diagrams. Moreover, they commutativity One

of the

for

reasons

the

commutative

1' From

77w Merchant

of

Venice.

217

they

not

Set- Valued

218

they

limits

preserve

thus

(and

Chap.

thctors

and—as

monomorphisms)

VIII

whole—“detect”

a

well.

them

as

29.]

PROPOSITION

Let C

é;

D be

a

Then

pair of d-morphisms.

the

following

equivalent:

are

9

(1) f (2) For =

g-

each

d—objectA, hom(A, _.)(f)

=

hom(A, __)(g). then

Proof: Clearly (1) implies (2). If(2) holds,

f=f° The

1c

above

llOIn(C,f)(1c)

=

proposition

can

=

be

IIOM(C.g)(1c

rephrased

as

1c

(1°

=

=

CI

9-

follows:

if and only if for triangle (or diagram) in .9! commutes image triangle (or diagram) under hom(A, _) commutes. A

d—objectA, the

each

COROLLARY

29.2

r

Let

.9!

@ be

$5

a

pair of fimctors and let

:1

(m: FM)

=

family of W-morphisms indexed by 0b(.e’). Then the following (1) (2)

n

=

For

G is a (114):F each ‘K-object C, ->

hom(C, _) is

natural

a

n

=

be

a

equivalent:

transformation.

(hom(C, 114)):hom(C, _)

transformation.

G(A))

o

F

—>

hom(C, _)

o

G

I]

THEOREM

29.3

Let D: a!

the

t

natural

are

—v

following

—>

are

‘6’ be

afunctor and let (L, equivalent:

(1) (L, (1,4))is a limit of D. (2) (hom(C, L), (hom(C, IA)» is

a

limit

L

'4, D(A)) be

of hom(C, _)

o

a

in W. Then

source

D, for each C

5

0b(%).

Proof: (l)

=

(2).

Since

functors

preserve

commutative

triangles, it is

0b(‘€), (hom(C, L), (hom(C, IA)))is a natural source for hom(C, _) source D, (Y, (5)) is also a natural fA(y) e hom(C, D(A)) and commutativity ol' the triangle C

e

o

hon:

Y

V

(C, D(A)) D(m)°_

IA. ham

(canon)

A

in

A’

for then

clear

that

hom(C, _) for

each

for each o D. If y

e

Y,

Sec. 29

‘

Hom-Fxmctors

219

for D, 0021) oj;(y) jj‘.(y). Thus (C, (jj.(y))) is a natural source so that there exists a unique morphism g,: C —v L such that for each A the triangle that

guarantees

=

1);A c

/ "no

'a c

D(A)

+-A h

Now

commutes.

the

for each

y e Y let

g(y)

9,. Then

=

g:

Y

—>

I:om(C, L),

and

triangle

ta

[mm (C,

that

of limit, there

for each

A the

is

a

triangle

[mm (B, 13)

ammo“)

: I

I

F

ham

i 4',

ham Hence

commutes. so

that

f

=

F09)

o

the

hom(B, IA)

(B, L)

(hom(B, 1,.) F)(l,,) makes

(B, D(A))

=

triangle B I I

64

I

f

I I

I

: V%

L

llanl(B, bA)(IB); i.e., [A F0») 0

=

b A,

Set- Valued

220

Funclors

Chap.

uniqueness, suppose that f’: B -v L also But then for each @—object C, each of liom(C, f) such that the triangle

To show

commute. commute.

function

x

makes and

VIII

the

triangle hom(C, f ') is a

ham (C, B) hamiC. bA)

1 l

x

i

ham (c,

D(A))

l

~lv ham

ham(C, l4)

(C, L)

in the definition of limit, by the uniqueness condition Izom(C,f’) for each C 6 0b(%). Consequently f f (29.1). [I

Thus

commutes.

'

Irom(C, f)

=

=

COROLLARY

29.4

(l) The following are equivalent: (i) f is a Z’qnonomorphism (ii) liom(C, f) is an injective function for (2) The following are equivalent: (i) T is a ‘6’-terminal object (ii) liom(C, T) is a singleton set for each

each

C

C

E

0b(i6’).

0b(‘6).

e

Proof: (1) f is

monomorphism if

a

and

only

if

_i_,

1—71!

1

is

pullback (21.12)

a

and

in Set

monomorphisms

are

the

precisely

injective

functions.

(2)

T is

object if and only objects are the singleton

terminal

a

Set terminal At this limits. 29.5

is

a

category

such

that

Since

S is

(L, (I,,)) a

yields

arises

partial

a

functor, and in

[:1

sets.

question naturally

is

in ‘6

retract-separator

as

to

when

reflect

lwm-functors

answer.

->

(6 be

a

functor

(liom(S, L), (Izom(S, IA))) retract-separator,

a a

natural

source

reflects

Ail-limits

for

it is

a

and let is

a

(L,

L

limit

of

so

that

separator,

L D(A)) be liom(S, _) liom(S, _) is o

in M

a source

D: d

faithful.

->

Set. Thus

for D.

Let (15',((1,0) also be a natural source for Izom(S, _)

natural

(19.7), then Iiom(S, _)

5%. Let D: d

Proof:

is

the

proposition

next

limit of the empty

a

PROPOSITION

If S each

point

The

if it is

for D. Then

source c

D,

so

that

there

(liom(S, B), (liom(S, gA))) h a unique function

exists

Sec. 30

such

Represcnrable Flmcrars

that

the

221

triangle ham (S, B)

i

with) 41,.)

I I

i

I:

ham (S, D(A))

I

i

ham (5', L) for each

commutes

i.e., for

A ;

each o

9,.

Now

since

morphisms

S is m

a

r

such

each

i

’s L

l)

such that for each

i, q

o

is

a;

B

l, and each A

e

o

m)=

org coproduct

by the definition

[Aolr(x).

=

there

(r

Hence

B

-»

some

(01,-),’S) of

copower

S and

that B

By the above, for

x

retract-separator,

and

S

x:

Mr

=

0

B

=

1'1)

0b(.e¢) a W m) e

o

o

unique morphism q: ’S —» L (01,), 1S)is an epi-sink, the diagram

exists

there

In). Since

a

B

q-mlEA—4 3—7

r

h(r°u-)

D(A) ———'I—-—>

L

0b(.e¢). unique morphism that q Consequently (L, (14)) must be a for each

commutes o

A

e

is the

m

Since

[I is

makes limit

unique

the outer

of D.

and

is faithful,

Irom(S,_)

for each

commute

square

A.

I]

COROLLARY

29.6

If

B is

limits.

a

then

set,

nan-empty

hom(B,_):

Set

—»

Set preserves

and

reflects

I] EXERCISE

29A. extremal

Prove

that

separator

in

its!

.e’,

then

§30 Since

is

category theory it is essentially the same, those functors that functors,

representable functors.

With

:1in

isomorphic

two

to consider, only natural are naturally isomorphic

them

we

can

an

FUNCI'ORS

REPRESENTABLE

in

and A is category and reflects limits.

complete, co-(well-powcred) Irom(A, _): .2! -> Set preserves a

obtain

entities

together to

yet another

regarded

are

them. way

with These of

the are

as

Immcalled

investigating

universal

link

The

situations.

adjoint

is

and

maps

Let G: d

G is said

(A, 6)

be

to

of G. In this

functor.

a

pair (A, 6) where a natural isomorphism. representable provided that there

A

_.

30.2

Lemma.

be

Set

—*

representation G is 6: hom(A, _) (2)

of

a

also

we

case

is

G

say

that

A

represents

A

is

d-object

an

exists

a

and

representation

0.

PROPOSITION

Representable functors Proof: Immediate naturally isomorphic (24.10). [:1 30.3

the

DEFINITION

30.1

(I)

VIII

representability and adjoints is through seemingly technical, yet nevertheless

between

provided by

extremely useful, Yoneda

Chap.

Functars

Sel- Valued

222

limits.

preserve

the fact that hom-functors

from

identical

have

functors

limits

preserve

limit-preservation

EXAMPLES

“usual"

a!

forgetful functor representable and is represented by:

Set

any

The

U: s!

—.

Set is

singleton

set —

EN— {0}, +)

SGrp

‘

‘_

"wi’im'

NT)”

MTO

-

7

(z.

Grp

+) 7

(Z i)

Ab v

--

R

R-Mod

Rng

2m

BooAlg

.

./'\.

.,

\ / 7

’1pr P08 77 W

E: ,

,,,7,,,

6 space any singleton set—x anysingleton partially-ordered lattice anysingleton *7

7

7

,,

,., l complete

°

lattices '0

a

A,“

(29.3) and properties

Sec.

30

30.4

OF UNIQUENESS If a functor G: d

then

A and

B must

223

REPRESENTATION

Set

is

represented by each of the objects isomorphic. More explicitly, if —>

be

(6c): hom(A, _) is

Functors

Representable

A and

B,

hom(B, _)

-—+

natural

isomorphism, then 6A(1A)e hom(B, A) is an isomorphism. assignment 6 H 6‘04) provides a bijective function from the set of all natural isomorphisms from hom(A, _) to hom(B, _) onto the set of all isomorphisms from B to A. This will be proved in that which follows. for the reader to prove these statements Nevertheless, it should be instructive directly before continuing on. a

Moreover

30.5

the

NOTATION

the

Throughout with

domain

common

of

conglomerate homwmai', G). YONEDA

30.6

G‘ : a!

If

remainder .2! and

natural

of this

section, whenever fl, then

codomain

common

transformations

from

F

to

.fl-object. then there

is

F and

G

[F, G]

will denote

functors

are

the

G; i.e., it will denote

LEMMA

Set

—>

and A is Y:

an

[hom(A, _), G]

a

bijective function

GM)

—>

definedby ‘5 whose

Y’: is

640,4)

'—’

inverse

C(A)

[hom(A, _), G]

—*

defined by H

x

5

(in).

=

where

63(f)

for

[Y

all

f

and

e

Y'

hom(A, B). are

called

the Yoneda

Proof: Clearly Y is function; i.e., that Y’(x) To

see

this. let f: B

(5c

°

function.

a

6 is

=

C and

—»

functions

hOM(A,f))(9)

this is true

for each

9

e

first wish

e

A.] show

to

that

transformation

natural

a

let g

for G and

We

for

Y’

each

€c(f° 9)

hom(A, B),

°

=

/

C

x

6

a

GM).

G(f° 500‘)

0(9))(X) we

have

=

(GU)

°

C'a)(g)-

commutativity

of the

e

3

is also

hom(A, B). Then

(00') Since

50706)

=

ham (A.

Bin—B—>G(B)

homer/)1J'Gm hom (A,

C)E—>G(C) C

diagram

4' is

Thus

a

Y', for each

of Yand

(Y chcc

Y

0

Y'

YE)

=

640,.)

=

Y'

Y is the

0

6 be Y'(6A(1A)).Then

is

of 6 this

naturality

let f be any

=

5,

is its inverse.

Y’

that

so

G. Then

——>

(GU)

o

611(14)-

=

from

morphism

(fir/WHULI

=

B. Then

A to

))(1,i)~

which

5.004);

by

is

the above

500,4).

3

is

(6n°/t0"t(r‘l.f))(l,t) 6

hom(A, _)

C(lnxénunn

=

(C(f)

Henee

=

5AM).

=

§n(f° L.)

=

of 6, this

By the naturality

the

the definition

by

Icon“)-

COAX-Y)

=

let 6:

identity,

.e‘l-objcctand

let B be any

€30")

By

G. Now

definition

by

M14) Now

to

icon.

that

)’(5) Let

hom(A, _)

GM)

6

x

Y')(-\')

°

=

To show

from

transformation

natural

Chap. VIII

thetors

Set- Valued

224

Y

a

53(f° L.)

=

=

630')Y is

Consequently l[lrom(A._).G]'

=

bijective

and

Y’

E]

COROLLARY

30.7

Let (A,

be

B)

7: homd(B, A) that

with each

associates

Then

pair of d-objects.

a

B

f:

Yoneda

the

[hom(A, _). hotn(B, _)]

_)-

traanormatt‘on 6

A, the natural

-—r

mapping

(4'5) defined

=

by: 63(9) is

a

natural

to

“f;

y

for each

that

show

isomorphism,

we

f

:

B

next

above,

A is

~

ofa

functor

E: .9!”

30.8

FULL

EMBEDDING

is

a

7U)

the functions

g

fit” embedding.

c

f

B

A .E

lignjofiu’hn

THEOREM

then

category,

=

if

[at Set].

—>

EM) E( Dd g)

only

the restrictions

are

.9! is any

and

[ham(A, __), Itom(B, _)]

—»

E

If

hom(A, C),

isomorphism if

an

that

prove

)7: Itomd(B, A) defined

9 e

E]

bijective function. In order

=

=

E: d”

hom(A, _)

-»

[.al, Set] defined by: for

for sat-morphisms

each B

si~object

—I—~ A

and

A A 1—» C,

is

a

Sec.

225

Functors

Representable

30

identities and Proof: By the way E is defined, it is clear that it preserves Since in any category, morphism sets are compositions. Hence it is a functor. the above on corollary (30.7) objects. By pairwise disjoint, E must be injective Thus E is a full embedding. be full and faithful. E must [:1 COROLLARY

30.9

Every category (resp. small category) can and cocomplete quasicategory (resp. category).

Proof: Since categorysl (25.7).

defined in 30.7.

them

and

complete

cocomplete,

is

so

in

complete

a

for

[.91, Set],

any

E]

)7: homd(B, A)

Let

a

is

embedded

fully

COROLLARY

30.10

is

Set

be

natural

Then

be the

[hom(A, _), hom(B, _)]

—>

eel-morphism f

an

is

sit-isomorphism if

an

function only if 7( f)

isomorphism.

Proof: Every functor (l2.9). [I

and

isomorphisms,

preserves

full

OF REPRESENTATIONS) (UNIQUENESS If each of (A, 6) and (B, é) is a representation of the then there exists an isomorphism f: A —» B with 6 0 Y(f)

30.11

Yoneda and

embeddings reflect

COROLLARY

=

functor G: 5. E}

.9!

-+

Set,

ail-object, and Y’: G(A) —> [hom(A, _), G] is the Yoneda that G an function, then by 30.l0 in the case liom(B,_), element of G(A) is Y' onto a natural hom(B, A) mapped by isomorphism if and only if it is an sl-isomorphism. The question naturally arises as to what If

G: .r/

A

Set,

—>

is

an

=

=

those elements of 0; Le, can we characterize happens in the case of other f unctors whose under Y’ values are natural The reader who is G(A) isomorphisms? interested in a general answer should see Exercise 30D (Universal Points). We now focus our attention on the case where G F, for some hom(B,_) —» F: .si Q and some a as either xiii-object, B. (Actually this can be considered =

specialization depending upon one’s point of view.) The of this problem will yield a fundamental relationship between reprefunctors and universal and maps; consequently between representable and adjoint situations.

generalization solution sentable functors

or

as

a

THEOREM

30.12

G: .9!

Let natural

—»

be the

9)}, B be

transformation, Y:

(l)

o

a

(2) (A, 6)

natural is

a

(3) ( Y(6), A)

6: Itom(A, _)

Yoneda

function.

Then

o

G] the

isomorphism. a

hom(B, _)

——>

and

representation is

Q-object.

[hom(A, _), hom(B, _)

corresponding

6 is

a

G-universal

of hom(B, _) map

for

B.

o

G.

—>

hom(B, G(A))

following

are

equivalent:

o

G

be

a

226

Set- Valued thclors

Chap. VIII.

Proof: Clearly, by the definition of representation, (l) implies (2). To show that (2) implies (3), suppose that (A, 6) is a representation of lwm(B, _) G. B If B then since is a Clearly Y(6) 64(14): G(A). f: G(A’), 6‘, bijective function, there exists a unique morphism ] : A —; A’ such that 64.0) 12 Also o

—>

=

->

=

since

6 is

a

natural

transformation, ham (A. lmm(A.

miumm

Applying

this

I

is the

(B, G(A’)) 5—>hom A! to the element

commutativity

from

unique morphism

A' for which

A to

G(A) l

Thus

(l),

(Y(6), A)

show

must

that

is

G-universal

a

We

have

the

triangle

4 :

iGU) if

,

we

1.4e ltom(A, A),

6.4-0) =f-

=

1’(5)=5A(|,1)

commutes.

6(7))

hom(B.

(GU) °6A)(1A) Hence

(a, G(A))

7)

ham (A, A’) commutes.

the square

V

V

G(A')

A'

map

for B. To show

that

(3) implies

Jal-object A’,

for each

6A.: hom(A, A')

lmm(B, G(A’))

»

bijective function. This is easily established since if f e hom(B, G(A')), then and only one because map for B, there exists one (Y(6), A) is a G-universal I e ham(A, A') such that the above triangle commutes; i.e., such that is

a

GU) 540..) °

But, again, since 6 is

a

natural

f-

transformation,

C(f)

°

6.40)

6A'(])' D

=

THEOREM

30.13

Let G: d

(1)

=

a

—»

ea be

.‘B-object B has

a

a

fimctor.

Then

G—univcrsal

map

if and only 1_'/‘lmm(B, _)

o

G is repre-

semable.

(2) is

G has

a

left adjoin! If and only iffor

each

Q-objcct B,

the

flmctor hom(B, _)

e

representable.

Proof: Immediate and adjoint situations

from

the

universal

preceding maps

theorem

(27.3).

[:1

and

the connection

between

G

See.

THEOREM

30.14

funclor, then an sit-object A represents G if and only if A G-universal objectfor a singleton set P ; i.e., provided that there is a singleton P and a morphism n: P map for P. G(A) such that (u, A) is a G-universal G: s!

If

is

a

set

227

Functors

Representable

30

Set

—>

is

a

—>

Proof: It is easily seen functors hom(P, _): Set bijective function

preserves

exists

Set, and

ls“:

[hom(A, _), G]

B:

that

there

that —>

reflects

and

a

G]

o

o

G]

hom(P, G(A))

——>

(A, 6) is a representation of G if of hom(P, _) G, and this is the case if Then

corresponding Yoneda bijection. only if (A, 3(6)) is a representation only if (Y(B(6)), A) is a G-universal

and

induces

Let

isomorphisms.

be the and

this

that

the

between

isomorphism

Set, and

—+

[hom(A, _), hom(P, _)

——>

[hom(A, _), hom(P, _)

Y:

natural

a

Set

o

for P

map

(30.12).

E]

COROLLARY

30.15

G is

A

set-valuedfunctor G-universal E] maps.

and

representable if

the

only if

singleton

have

sets

COROLLARY

30.16

If

a

preserves

This

G has

set-valuedfunctor limits. [3 corollary

be

can

G-um‘versal

a

map

considerably strengthened,

for as

singleton

a

the

set,

following

then

G

theorem

shows. THEOREM

30.17

If

a

non-empty

limits.

must

preserve

30.18

Theorem

for

map

at

least

one

limits.

30.13, hom(B,_)

B is non-empty,

limits.

G-unit‘ersal

a

o

G is

hom(B, _)

Hence

representable.

reflects

limits

(29.6).

it preHence G

E]

full subcategory of Set, containing at least one Set preserves limits. [:1 embedding functor E: s!

Let .2! be

Note

to

Since

Set has

COROLLARY

Then the

the

that

map,

an

In

discrete but

non-empty

a

set.

-+

set-valued.

not

—>

B, then G preserves

set,

Proof: By serves

G: s!

set-valuedfunctor

G does

analogue particular, not

if G: Set

that

on

space

30.17

of Theorem

preserve

set, limits.

then

—r

is not

valid

for

Top is the functor each

discrete

functors

that

space

has

sends a

that

are

each

set

G-universal

PROPOSITION

30.19

functor G : s! .si-object A for which A

—>

Set has

there

exist

a

left adjoint if and only if ’A. arbitrary copowers

it is

represented by

an

228

Set- Valued

Proof: G has (27.3). Since each follows

from

P has

a

left

set

I

Lemma

G-universal follows

a

ization

thus

Functors

Chap. VIII

adjoint if and only if each is isomorphic to the copower

26.9 map from

that

each

(u, A)

and

Theorem

I has

set

A has

G-universal

a

then

If .d is cocomplete, representable. [:1 The

G: .d

Set

—v

has

only

left adjoin! If and only if

a

(see corollary has a partial converse theorem ties several of following together the previous sections (for the case of set-valued

of

and

if

above

Exercise the

G is

31A).

results

of this

section

functors).

THEOREM

30.21

G

Let

(ii),

or

(l) G

s!

:

Set be

-9

a

functor. lfsal satisfies either of the following conditions (1) through (6) are equivalent:

then the conditions

complete, well-powered, and has a coseparator. complete, well-powered, extremally co-(well-powered), and each a set at most of pairwise non-isomorphic si-objects. y G—generates

.21 is

.51 is

extremal]

a

is

a

G-universal

map

(4) There

is

a

G-universal

map

(5) There

is

a

G-universal

map

left adjoint.

(6) G

is

30.22

EXAMPLES

of

(l) The category

at

obvious

functor

forgetful

complete

elements

two

for for for

each

set.

least

at

the

one

set.

non-empty

singleton

set.

[:1

representable.

than

set

limits.

preserves

(2) G has (3) There

more

if and

'A. The character-

copowers

The

(i) (ii)

map

map P, it

set

COROLLARY

30.20

(i)

G-universal

a

’P of the singleton

arbitrary E]

30.14.

I has

set

U: s!

has

lattices —>

is

complete and well-powered, and the representable. However, no set having

Set is

U-universal

a

(2) The category d of complete boolean extremally co-(wcll-powered), and the representable. However, no infinite set is a concrete (5!, category (3) There such powered, and co—(well-powered) not representable. (See Exercise 28D.)

map.

algebras is complete, well-powered and Set is forgetful functor U: M has a U-universal map (cf. Exercise 300).

obvious

-»

is

U) that that

U

:

sf

complete, cocomplete, -»

Set

preserves

limits,

wellbut

EXERCISES

30A.

each

For

(S: hom(A, _)

->

of

U (where

the

examples

in each

case

in

A is the

30.3,

find

a

natural

object specified).

isomorphism

is

Sec.

30

Show

308. lishes

and

a

Functors

Represenrable

natural

for each

that

isomorphism

229

lemma (30.6) actually .91, the Yoneda category between the evaluation functor (15.7) E:

Set”

x

.5!

—>

Set

N:

Set“

x

5.!

->

Set

estab-

the functor

defined by:

N(G. A)

[hom(A, _),

=

N('I,f)(6)(/n(9) Consider

30C.

the full

”A(6A(g°f))~

=

E: a!”

embedding

G]

[d, Set]

->

of Theorem

30.8.

(a) Show that if .92 is the full subcategory of [5%,Set] consisting of all representable functors fromd to Set, then [421,Set] is the “colimit hull” of .9? in the sense that for each

functor

Fe

there

0b[.:/, Set]

exists

((kc). F) such that that

(b) Prove

G: .a’

that

A is

a

morphism f:

A

~>

e

0b(9?)

—>

and

[5%,Set]

and

some

Coli‘m D.

((kc). F) general is not colimit preserving. consisting of all limit preserving limit

and

x

colimit

preserving and

functor.

ail-object,

an

e

Points

Set be

-»

C

limit

a

Universal

30D. Let

E is

for each

D: g

functor

some

Ob“), D(C) preserving functor, but in (c) Show that the full subcategory ‘5 of [531,Set] functors. is both complete and cocomplete. (d) Show that the embedding E: d” —v ‘3 is both that the “colimit hull“ of its image is all of (E. sink

e

a

X such

if 029’

A pair (a, A) is called a universal point of G provided C(A), and for any such pair (x, X) there is a unique .5!that G( f )(a) x. =

0b(.2!), and Y’: G(A) -> [hom(A, _), G ] is the Yoneda function, then the following are equivalent: (at) (a, A) is a universal point of G. of G. ([3) (A, Y’(a)) is a representation -» e Show that if Gad A Q, 0b(.si), Be GHQ), and u: B -> C(A), then the (b) following are equivalent: (a) (u, A) is a G-universal map for B. G. (B) (u, A) is a universal point of Iiom(B, _) -» -> let l Let G: d and F: Set be a Set functor whose value at the single object a: (c) is the singleton set {p}. Show that the following are equivalent: (at) (a, A) is a universal point of G. ([3) (t. f, A) is an initial object of the comma (F. G), where category

(a) Show

that

Set,

-»

A

e

o

f2 (P) defined

is the function Let

30E.

that

property Y c X, then such

that

set

[hom(l. _)

(b) Illustrate a

there

a

two-element

for o

a

02.x! set

a

Show

(see 20D

and

26H).

well-powered, if (u, A) is

that

U-universal

subobject

each

=

complete,

a

limits.

exists

Suppose that that

be

U preserves

(B, m) is

30F.

(a) Show

(M, U)

by f(p)

GM)

-*

map

(v, B) for

concrete a

Y, and

with

category

U-universal a

map

morphism

the

for X and m:

B

-»

A

of A. —>

Set

I, there

is

has a

a left adjoint bijective function

F.

from

the

set

G

a

F

(l )

to

the

G, G ].

the above set.

result

for the

cased

=

Grp. G is the forgetful

functor,

and

I is

thclars

Set- Valued

230

Complete Bo'alean

306.

(5!, U) be the

Let

Chap.

VI]!

Algebras

concrete

of

category

boolean

complete

algebras

and

complete

homomorphisms. is

thatsl

(a) Prove (b) Prove

complete, well-powered, and extremally co-(well-powered). exists

there

that

U-universal

a

finite set.

for each

map

topological space. A subset A of X is called regular open provided that (where “int" designates “interior" and “cl” designates “closure"). Show that the set R(X) of all regular open subsets of X is a complete boolean algebra with the to following operations: respect

(e) Let X be

im(clA)

a

A

=

vM AM A’

=

=

im(cl(uM)),

for

im(nM),

for

Q

for

A

im(X

=

A),

—

M

R(X)

c

M

#

c

R(X)

R(X).

e

Let X be the set of all ordinals (d) (Solovay) Let K be an infinite cardinal. as a discrete topological space. Let P dinality less than K, considered of X, with projections n": of copies countably many topological product is the boolean Prove that algebra 110’”) extremally U-generated bya complete set. [Show first that R(XN) is extremally U-generated by the family =

{Rn-'(OINEN. Secondly,

that

show

car-

countable

66 X}.

R(XN) is extremally U-generated by Mm." l m,

with

X" be the XN -r X.

Ir 5

the countable

set

N},

where

Arum

=

{x

XNI Hui-Y)S 7f..(-\')}-]

e

(e) (Gail'man-HaleSoSolovay) Prove that each infinite set extremally class of pairwise non-isomorphic d-objects. proper is not cocomplcte and that U has no lel‘t adjoint. (d) Prove theta! 30H.

Let

let D: d

and

its unit

(:1, U)

be the usual

that 301.

D is

representable, and

Dualities

(.91, U) and (a,

Let

a

category of (commutative) C‘-algebras, with each C‘-algebra X that associates

disc

D(X) Prove

concrete

Set be the “unit disc l'unctor"

-r

U-generates

contravariant

l'unctors

=

and

{xlxeX

lell

5

l}.

U is not.

but that

Represenrability

V) be

categories

concrete

G: d

—.

.46 and

F

2

93

that —’

dually equivalent;

are

42¢ such

F

a

G

z

1,,

G

o

F

2

I“.

i.e., there

are

that

and

Suppose that the JJ-object A. C(A). F(B) and E =

=

A represents Prove that:

U and

(a) U0?) z Vtfi). (b) V G 2 Iromd(_, A) and U F 2 Imma(_, 39 has products, then show that If, moreover,

the

.‘E-object

~

o

o

E).

B represents

V, and

let

See.

31

Free

for each

(c)

exists

X there

d-object

a

such

monomorphism C(X)

mx:

231

Objects

Bum

-»

V(mx) is the embedding of

that

wax»

homdflt’,A")

z

into

hamsfl(U(X). UM‘» Compare these results

with

hamsfl(U(X), m?»

z

the

examples in

§3I 31.1

10.6 and

FREE

(V(§))U‘X’ z HEW”).

=

14.18.

OBJECTS

DEFINITION

If G: .2! the

X, then

set

is called

is

functor

set-valued

a

A is called

G—free

a

object

and

(u, A) is

X and

over

the

G-universal

a

morphism

the insertion

provided 3|.2

Set

—»

of the generators X into A. We say that for each set X there exists a G-frce object.

that

u:

map X

for

C(A) objects

—>

at! has G-free

EXAMPLES

then the U-free objects of .se’ (5’, U) is the concrete category of groups, are Likewise, free R-modules, free rings, free monoids, exactly the free groups. free semigroups, free lattices, and free boolean algebras are exactly the U-free in the sense of the above for the corresponding concrete definition. objects, categories with forgetful functor U. lf

31.3

NOTATIONAL

Throughout is

and

functor,

a

31.4

REMARK

the remainder will

we

ofthis

simply

use

section, the

We

will

“free"

term

that

assume

rather

than

G: .5!

—~

Set

“G-free”.

PROPOSITION

(I) An JJ-ObjCCIis free

the empty

over

set

provided that

it is

initial

an

object of

.d.

d—objeet is free over a singleton set provided that 55/ has free objects if and only if G has a left (It/joint. set has a free object. then G (4) If at least one non-empty An

(2) (3)

In the then u:

X

u

preceding chapter extremally G-gencrates C(A) need

—»

shows:

Let a!

the empty

objects.

set.

but

full

be the and

for

be

not

let G:

is not an generators that it is rarely the

A

have

that

(26.6). However,

if(u. A) is an

any

insertion

limits.

E]

G-universal

map, of the generators

injective function as the following simple example subcategory of Set whose objects are the singletons and .n/ Set be the embedding functor. Then 5! has free —>

have

more

injective function. case

seen

preserves

G.

an

that

sets

we

it represents

that

insertions

than The

one

element,

the

following theorem

of generators

are

not

insertion

of

the

shows, however.

injective.

Fmtctors

VIII

Chap.

THEOREM

3L5

contains

lfs!

least

at

B, the

free object for function. injectit'e

two

for which GM) has

insertion

than

more

the generators

of

X

one

L

element,

C(B)

is

an

G(A), there is some u(x) u(y), then for all functions f: X A such thatf hence Since GU) :1; f(x) fly). G(A) has at least elements, this implies that x y. [:1 If

Proof:

B

A

object

one

each

then

I:

Valued

Set-

732

—»

=

-»

=

=

o

=

Every free R—module is not hold for arbitrary

projective. A corresponding theorem categories. For example, the discrete Hausdorff spaces are U-free, yet they are not projective in the concrete category A theorem will become if we true, however, modify (Topz, U). corresponding does

somewhat

definition

our

of

known

to

be

concrete

projectivity.

DEFINITION

31.6

6“ be

Let

the

that

provided

of d-morphisms. functor hom(P,_):sl

class

a

An

tel-object —>

Set

P is called

6-projective morphisms in d” to

sends

surjective functions. regular epimorphisms, then "6' is the class of all extremal epimorphisms, then If 6' is the class of all sat-morphisms f for which called sur-projective.

P is called

regular-projective. extremal-projective.

G( f ) is surjective, then

P is

EXAMPLE

31.7

If 6 is the class if it is

of all

in .21, then

epimorphisms

P is

6-projective

if and

only

projective (cf. l2.l4).

PROPOSITION

31.8

Each

C

..

exists

a

is

free object

stir-projective.

for X and let f: A —> B and (u, A) be a G-universal map B be d—morphisms such that 0(a) is surjective. Then, clearly, there h: X function G(C) such that the square Let

Proof: g:

P is called

of all

If a is the class

——>

C(C)Fg;>G(B) commutes.

(a, A) is

Since with

6‘03) 0

u

h.

=

that

31.9

o

h

for

°

N

=

0(9)

°

6(5)

°

=

N

f (26.6), i.e., hom(A, g)(h)

=

X, there

exists

a

unique 5:

A

—>

C

=

0(9)“

I!

f. Hence,

=

G(f)° A is

a,

sur-projective.

E]

PROPOSITION

If a

g

map

Consequently

GUI0 5) so

G—universal

a

.9! has

morphism

for each al-object A there exists a free object A and such that G (e) is a surjection. In other words, each si-object

free objects, e:

A

-+

A

then

Free

Sec. 3]

is

a

smjective image of

be chosen

so

as

to

be

233

Objects

If G

free object. Furthermore,

some

is

then

faith/ill,

e

can

epimorphism.

an

Proof: Let (u, A) be a G-universal map A such that the triangle morphism e: A

for the set

G(A).

Then

exists

there

a

—>

G(A)

—“——>G(2)

3:

:

ie

G(e)

E

'am

I}

*

A

G(A) Since

commutes.

G(e) is

a

faithful, it reflects epimorphisms, 31.10

that

so

be

must

e

a

surjectivc function. If an epimorphism. E]

G is

PROPOSITION

lfd has flee objects, equivalent: (l) (2)

be

it must

retraction,

A is

stir-projective.

A is

a

retraction

“retract” r:

A

—»

of

a

then

sl-object A, the following conditions

each

for

free object;

i. e.,

there

is

are

fl'ee object A and

some

some

A.

Proof. To see that (I) implies (2), recall that according to proposition (3! .9) there Is a free object A and a morphism e: A G(e) Is surjective. Consequently by (I) there exists a morphism f: that the triangle

—»

the

previous

A such

that

A

a

A such

A

—»

A be

A .

I

[I

f / 1’

IA

’/ A

g’

A———>

Hence

commutes.

e

is

a

A

retraction.

(2) implies (I), let A be a free object retraction. Then there is a morphism m with rem B be morphisms with G(g) surjective. Since A g: C there exists a morphism f : A —> Csuchthat g of f r. To

see

that

=

->

=

has the property

that A

Z/

\114

i\ C

g°f=g°f°m

=f°r°m

=f°LI

A

———>

=fl

if

B

E]

o

and

let

r:

1,4- Lct f:

A

a

B and

—»

is

sur-projectivc (31.8), f m Consequently I =

o

234

S

Now

is

set

non-empty

a

that

thetors

for any

concrete

First,

however.

separator.

Chap. VIII each

category, establish

We

free

object over following more

the

a

result.

general 31.11

will show

we

Valued

et-

PROPOSITION _

H

If

Z”

:

.‘Z is

—»

for 9, then C

separator

is

a

H-tmirersal

an

.

B be

A z;

a

for D, and

map

D is

a

‘6.

for

separator

f

Let

Proof:

is

faithful, (u, C)

.

of

pair

‘6—morphtsmssuch

that

for

‘6-

each

9

k1C

morphism

A,fo k

—>

=

”(1’) Since

(it, C) is

for each

H(f)

(”(10

H-universal

an

9-morphism H(g). Hence,

=

°

k. Then

c

g

c

u)

for each

”(9)

=

k:

C

(”00

°

°

since

I] is

faithful,f=

g.

I5

0

”(g)

=

o

for 9, it follows

h,

that

E]

COROLLARY

31.12

If (s! G ) is ford. C] Our

then each

concrete,

,

consideration

next

must

that

this

may

not

true,

as

free object

is the

free

have

sets

over

a

set

non-empty

is

a

separator

question of whether or not non-isomorphic objects. The example following 3L4 shows

non-isomorphic 3|D and 321 show that. Indeed, Exercises always the case. even be true for “algebraic" categories. However. in many cases it the following proposition shows: is not

it is

LEMMA

31.13

Suppose thatsi has an object A with] < card(G(A)) is the insertion into the free object A otter of the generators then card X card Y. 3 G—generatesA, Let card

Proof:

A, we

X

k. card

=

Y

=

and card

m,

C(A)

X

n.

=

No. If u: and if g:




l

—>

then .1! is

algebraic, U has

(32“)

D: a

is

Since

Proof: proposition complete.

7;: A

that

d

it also

be

U-universal

D(i) such

a

that

a

left

adjoint,

reflects

small map

U0)

complete and them.

functor, for L. Then 0

u

need

we

each

reflects limits. by the

above d

is

of U0 D, and

let

only show

let (L, (1,» be a limit for each i there is a

1;. Now for

=

limits, and

it preserves Thus

and

U preserves

that

unique morphism j, the l-morphism m: i —»

equality

U(lj)ou

=

I]

=

(U0 D)(m) 01,-

=

(U0 D)(m)

o

U(l,)o

u

=

U(D(m)ol,-) ou

the fact that

and

and

Algebraic Categories

Sec. 32

l,

that

U—gcneratesA (26.6) implies

u

241

thctors

Algebraic

0011) 07,.

=

DU)

(U °D) (i)

Consequently, (A, (7‘))is natural such that that

for

source

the

for each

U

U01)

i,

Hence

D.

o

natural

a

I, f.

o

U(A).

=

Let

(0, B)

F, s: B morfphisms

(q, Q)

z

that

so

limit, it is

a

a

is

a

L

—»

so

mono-source,

=

a

=

11° IL

surjection and, coequalizer

as

for

map

such, is

of

be the

U-universal

a

I,-

=

C. Then

regular epi-

a

pair of functions

some

exist

there

unique

that

U(f)

o

and

v

s

=

U6)

0

0.

Coeq(F, S). Then the equality

U(q)°r

=

=

implies that there U(q) is surjective,

U(T,or")ov

is I:

Consequently, for

U(q)°(U(F)°v) U(qocov

=

U(q°F)°v

U(q)o(U(§)ov)

=

=

Hones

I: f. unique function It: L —» U(Q) with U(q) must be surjective also. Furthermore, for each i, =

a

U(7,)or=

=

This, together with

71

be

A such

—>

r

=

a

(L, (I,)) is

Since

U03)“:

=

imply that f u 1L. Thus f is Therefore (f, L) must morphism.

Let

D,

equations Info"

Ci;

exists

there

o

=

(U(A), (00,») unique function f: U(A)

for

source

the

each

[infer

=

l,ofos=

U(7,)os

u

C

--—>U(B)

Jam \‘f‘a 5

M

—>

V

U(A)

—————>L

Since

U(7,c§)ov.

that v U-generates B, implies that i there exists a unique morphism (1,: Q —> fact

(11° ‘1-

L

=

o

7,0 F

=

00‘) such

i,

o

5.

that

242

Set-Valued

thctors

Chap.

VIII

Now

Ulqt)

°f

U(q.-)

=

U(q)

c

since

=

Ulqt eq)

00;)

=

=

Inf.

f is a surjection, U(q,) I: l,, for each i. This provides a factorization of the (extremal mono)-source (L, (0) (19.13), where h is an cpimorphism. Hence h is an isomorphism. Thus (U(Q), (U(q,.))) is a limit of U D. Since U reflects limits (32.11), (Q, (q,)) must be a limit of D. D so

that

oh

o

=

o

32.13

COROLLARY

Each and

algebraic

U preserves

Proof: 32.14

and

(.11, U) is uniquely (regular epi, mono)-factorizable,

category

reflects

these

from

Immediate

factorizations.

Theorem

32.3.

[:1

THEORENI

is

Every algebraic category

cocomplete.

U) is algebraic, then a! has coequalizers. Thus it remains to be shown that .a’ has coproducts (23.8). To do this, it is sufficient to show that functor“ for each small discrete category functor G : a! -’ d’ I, the “constant

Proof: lf(.d,

left

limits (28F(a)) so that adjoint (261). It is easily seen that G preserves show since a! is complete and well-powered, it suffices to that each .af’-object a set of at most pairwise non-isomorphic d~objects extremally G-generates family of .d-objccts (A ,), is the domain of at (28.9); i.e., that each I-indexed most a set of pairwise non-isomorphic (extremal epi)-sinks. has

a

coproduct in Set of the family (U(A,-)),, and let (u, A) be a U-universal map for LIU(A‘). We claim that if (Ag-1L)B, B) is an (A ,),, then B must be (the object part of) a (extremal epi)-sink with domain at most a set of such pairwise non-isomorphic is of A. (There regular quotient regular quotients, since .52! is regular co-(well~powered).) Since ((11,),UU(A, ) is a coproduct, there is a unique function g such that the top left triangle in the following diagram commutes: Let

((11,),LIU(A‘))

be the

0(9)

UM)"

0(3)

U(m)

”(0

y

”I

Uta)

U(e)

(it. A) is

Since

g: A

—~

(g, B)

is

B such a

a

U-universal that

map

the middle

for

L1U(A,), there triangle commutes.

exists

a

unique .d-morphism

It remains

to

be shown

that

of A.

regular quotient

Let _

A

be

a

above

L

a

=

A L»

C L

(regular epi, mono)-factorization of ('1'. Then diagram and Corollary 32.9 imply that for

B the

each

of

commutativity i there

exists

a

the

unique

243

Funclors

Algebraic Categories and Algebraic

32

Sec.

C with U(h,.) U(e) u 11,-. Hence since U is faithful, d—morphism 11,-:A,sink ((g‘)“ B). But since m is a monoof the m is a factorization (11,-), (g), morphism and ((g,),, B) is an (extremal epi)-sink, m must be an isomorphism. Thus g must be a regular epimorphism. C] -»

o

=

o

o

=

Functors

Algebraic

algebraic category, then by definition the forgetful functor U and reflects regular epimorphisms and has a left adjoint. We will preserves between now see that each f unctor algebraic categories that “forgets part of the from Rng to Ab (“forgetting multisuch as the forgetful functor structure", plication”).from Rug to Mon (“forgettingaddition"),or from compact topological to Grp (“forgetting the topology"), has the same properties (32.20). On groups that has these properties is that each functor the other hand it will be shown of the i.e., is one that is faithful structure"; essentially one that “forgets part that plays a central role in (32.17). It is this concept of an "algebraic functor" of mathematics. categorically distinguishing algebra from other areas lf(.n’, U)

32.15

an

DEFINITION

is called

and preserves

32.17

an

Proof: Recall adjoint (27.8).

is

of algebraic functors

that

adjoint

the

algebraic.

of functors

composition

algebraic functor

is

that

have

left

adjoints

has

a

faithful.

that

G(f )

G: J!

that

Q

-.

IS

algebraic

and

A :1

-

,

A are

.d-morphtsms

9

G( g). Let (u, A’) be

=

I

.

.

a

left

[:1

Proof: Suppose is

a

PROPOSITION

Each

such

it has

PROPOSITION

The composition

left

that

algebraic functor provided and reflects regular epimorphisms.

A functor

32.16

is

h: A’

unique d-morphism

-»

G-universal

a

A such

that

the

map

G(A).

Then

there

triangle

G(A) ——>"G(A’)

IN

for

A'

n

l

I

I

Eco.) ih

+

G(A)

t A

commutes.

Hence

since

u

G-generates A’,

the

G(foh)cu=

implies

that

(l6.15 dual),

morphism;

f

o

h

=

so

that

hence

f

g

c

h. Since

since =

g.

G(golz)ou

GUI) is a retraction, it is a regular epimorphism regular epimorphisms, h must be an epi-

G reflects

[:1

equality

244

Set- Valued

32.18

Chap.

VIII

PROPOSITION

Each

(l) (2) (3)

FIIIICIOI'S

algebraic functor

preserves

and

preserves

and

preserves

and

32.19

reflects monomorphisms; reflects isomorphisms; reflects (regular epi, mono)~factorizations.

E]

PROPOSITION

has

lfs!

coequalizers

and if U : ss’

—r

Set, then the following

equivalent:

are

(1) (st, U) is an algebraic category. (2) U is an algebraic functor. 1:] THEOREM

32.20

If (s1, U) functor such that

and

(.68, V)

the

a

Set G is

Since

—>

33 is

any

-—G—->{E

(x then

G: s!

triangle .1/

commutes,

and

algebraic categories

are

/

algebraic.

U and

and reflect

V preserve

regular epimorphisms and limits, G has a left adjoint, and to do only this it is sufficient to show that each fl-object, B, extremally G-generates at most a set of pairwise non-isomorphic Ail-objects (28.9). Let (u, A) be a U-universal 6(3) extremally G-generate Z. Since .9! is regular map for V(B) and let g: B is sufficient to show that for some co-(well-powered), it morphism g, (g, X) is a the A. definition of of universal By regular quotient object map, there exists a A such that V(g) A U(§)o u. Let A 35—; unique sat-morphism g: A A L; A’ ——> I be the (regular epi. mono)-factorization of g (32.l3). Proof:

0 must

do likewise.

Thus

need

we

show

that

-+

-»

=

=

V(B);>U=(V°G)

where

regular separator,

a

G preserves

Q is is

complete and cocomplete and if B is regular-projective, and has a G-universal

Q-object

a

map ; then

limits.

Itom(B,_) is algebraic, it reflects limits (32.11), and since limits. Hence G must preserve representable (30.13), it preserves

Since

Proof: ham(B, _) limits. E]

G is

o

COROLLARY

32.24

G: s!

If

‘G preserves 32.25

Set and

->

limits.

if B is a non-empty set that has (Cf. Theorem 30.17.)

I]

a

G—universal map,

then

COROLLARY

Let d Then

Chap.

each

reflects

U

thctors

the

be

full subcategory of Set, containing at least one non-empty set. Set preserves limits. [I (Cf. Corollary embeddingftmctor E: d a

—r

30.18.)

EXERCISES

Suppose that .2! is

32A.

(regular epi, mono)-factorizab1e Um! and reflects monomorphisms. a preserves Prove that epimorphisms if and only if it reflects isomorphisms. a

category

328.

extremal

Prove

that

in any algebraic coincide.

epimorphisms

32C.

Prove

si-morphisms,

9,:

that X

if -¢

(M, U) is algebraic, m,: U(A.) and g2: X (1042) _.

U(m,) gl

extremally

d-morphism

the

category

=

cg,

U('":)

°

regular —b

A, are

and

U reflects

->

regular and

epimorphisms

A

and

functions

":2:

such

A;

that

—>

A

the

are

that

92.

U-generatcs Al, and m; is a monomorphism. f: Al A; such that the diagram

then

there

exists

a

unique

—.

X—’>

9

out)

QL,x’flm

U(m,)

k,

U019

——>

U(m,)

U(A)

commutes.

320.

pair (1;

Prove

A) has

an

that

it'(.af, U) is algebraic, then for each function f: X -> U(A), essentially unique (extremal U-gcnerating. mono)-factorization.

the

Sec. 32

and

Algebraic Categories

AlgebraicFrmctars

247

B is (.94, U) is an algebraic category and B is an d—objectrthen a of if it is a retract U-free only object. regular-projective 32F. Prove that in any algebraic category the pullback of a regular epimorphism is a regular epimorphism.

that

Prove

32E.

if

if and

326.

Fim'tary Algebraic Categories

algebraic category (511, U) [= (42¢,Itam(A, _))], show that the following equivalent: (i) (.a', U) is finitary (see 228). then each (ii) If (1,, L) is a direct limit, and all the ("s are monomorphisms, -> L from a finitely generated d object, B, into L can be factored morphism f : B

(a) For

any

conditions

are

through one of the ("5. (iii) If B is a finitely generatedsl-object, exists

finite subset

a

of I such

J

then

that

injection [[52JA —) ’A. (iv) A is “abstractly finite", i.e., for finite subset

[15:’A

J

that

I such

of

f

can

is

a

f

for each be

can

morphism f: through

B

factored

->

'A there

the

natural

morphism f: A -> 'A there exists a through the natural injection

each

factored

be

’A.

->

(v) If B is an d-object and M if (3,), is the family of all finite subsets M; of M, then by If (vi) ((k,), K) is a direct limit

U(B)

of

subset

that

extremally U-generates B, extremally U-generated e i covers 1} U(B). {U(B,) l in d, Bis an d-object. and f: U00 4 U(B) is a function such that for each i there exists a morphism f, with U(f,) f o (U(k,)) then there exists a morphism f: L —> B with U(]) fl are (b) In the case that .m’ is connected, show that the above conditions equivalent to: If B is a then for each Ll Ah '0‘) finitely generated d—object, morphism f: B and

of B that

subalgebras

are

=

=

—.

I

there

exists

natural

a

finite

subset

K of I such

that

A,

that

limit

if

U) is

(d,

of those

The Dual

32H.

9’: Set”

»

l

finitary algebraic category,

a

subobjects

((1) Show that the category but not finitary. Let

of A that

that

Y)(A)

[For =

that

(u, 9(X))

that

9’ preserves

and

reflects

that

(Set°’, 9) is

an

algebraic

a

of any or

not

g-univcrsal

concretizable

f"

=

each

{A |

show

(e) Prove that the dual whether (f) Determine

d-object

A is

a

U-gencrated by finite sets. spaces is algebraic,

Hausdorfi'

by:

{AIA CX}

(c) Prove (d) Show

is

defined

functor

9’ is faithful. 9'" has a left adjoint.

u(.t‘) and

compact

each

of the Category of Set:

Set be the power-set

9’th that

then

extremally

are

of zero-dimensional

90’):

(a) Prove (b) Prove

the

through

[I A,.

-.

x

direct

factored

be

can

injection:

”:5 L] (c) Show

f

map

X, define

set

.reA

[A].

c

for

u:

X

—>

9’(9(X))

by:

X},

X.]

regular epimorphisms. category. category

the dual of each

is concretizable

“algebrizable” category

(cf. Exercise is

IZL).

“algebrizable”.

Ser- Valued

248

FIIIICIOI'S

Chap.

VIII

boolean (g) Let (.51, U) be the concrete category of complete atomic algebras and boolcan Exercise Prove that the functor complete homomorphisms (sec l4l-l). G: Set” -> .2! defined by:

G(X)

boolean

complete atomic

the

=

Ger—L Y)(A) is

(h) Prove

[A]

categories. triangle

the

that

r1

of X.

of

equivalence

an

=

algebra of subsets

6

Seto-L—>.

v and commutes, Construct (i) is

a

that

functor

F: d

Prove

32J.

Let

the

that

(1) d

is

is

limit

direct

a

natural

isomorphisms

1] and

such

e

that

of finite sets.

limit

of

ail-objects

each

of which

is

cxtremally ‘

finitary. (5%. U) described

category

Prove

and

situation. inverse

of abelian

.11 be the category

forgetful functor.

the

Set”

—)

an

321.

Set

(.91, U) is algebraic.

conclude

n, a) equivalence (j) Prove that each set is an (k) Prove that each sat-object U-generated by a finite set. (l) Prove that (d, U) is not

(G, F,

W

in Exercise

torsion

31D

and

groups,

is

algebraic.

let U2“f

—b

Set

be

that:

complete and cocomplete.

(2) U preserves

finite limits.

(3) U preserves

and

reflects

regular epimorphisms.

(4) U reflects congruence-relations. (5) U does

not

preserve

products.

and Hausdorff spaces, category of compact (4!, U) be the concrete that let A be an d-object. Prove (a) hom(A, _) reflects regular epimorphisms if and only if U(A) ¢ 6. (b) ham(A, _) is algebraic if and only if U(A) ¢ Q and A is extremally disconnected cI(im B) for each subset B of A). (i.e., int(cl(int B)) Let

32K.

=

32L.

Let

(5/, U)

subcategory

of 51 whose

of finite

groups, Set denote

U

o

E: .9

abelian ->

be the

concrete

objects

category those

groups

of abelian that

can

let Q be the full groups, be embedded into products

93L. .9! denote the embedding forgetful functor. Prove that

let E: the

are

functor,

and

let V

=

(a) (ya, V) is algebraic. (b) (a. V) is finitary. (c) (9. V) is not strongly finitary. [Him: QIZ is a direct limit in d of its finite subbut does not belong to 96‘ since each homomorphic image of QIZ is divisible.] groups,

IX

Subobjects, Quotient and

Objects,

Factorizations

He

That He

thought he saw a Garden-Door opened with a key; looked again. and found it was Rule

A Double

of Three.

“And

all its mystery,” he said, “Is clear as day to me!" Lawns

CARROLLT

already seen that if a category ‘6’ has “sufficiently nice" smallness and is and completeness properties (c.g., if it is well-powered, has intersections, In this it is then chapter (extremal epi, mono)-factorizable. finitely complete), is actually uniquely (extremal epi, mono)we will show that such a category that it is also uniquely (epi, extremal factorizable mono)and, moreover, in each results will show that these factorizable. morphism together Putting extremal three-fold has an essentially unique such a category (extremal epi, bi. that the be noted It should (extremal epi, mono)-factorizamonoH‘actorization. in algebraic categories, but is considered tion is the one that is usually considered In the latter in categories such as Top or POS. to be of only limited interest are the (epi, extremal mono)-factorizacategories, the interesting factorizations In

§l7

have

we

tions. In order in each

to

study these

“reasonable”

two

category.

that exist distinguished factorizations of with a study begin general (6’. J!)-

of

kinds we

will

factorizations.

§33

(6’, JV) CATEGORIES

which is closed let 6“ be a class of epimorphisms Throughout this section under composition with isomorphisms and let .1! be a class of monomorphisms that is closed under composition with isomorphisms. ‘l From

Alice in Wonderland. 249

250

Subobjects, Recall

that

6’ and

e e

of

(if, .l/)-factorization

an

L)

o

where

Objects. and Factorization:

Quotient

m

e

o

Such

.11.

L)

I

=

morphism f

a

l)

o

is called

whenever

factorization

unique provided that

n‘:

E

I

a

IX

o

factorization

a

is

Chap.

0—)O=O—)O—)I

is also

(6, .l/)-i‘actorization off,

an

there

is

isomorphism

an

I! such

that

the

diagram

.

Recall

commutes.

provided 33.1

that

also

each

that

of its

7 w x; I.

6’ is called

category

a

.

has

morphisms

a

(uniquely) (6’, Jl)-factorizable

(unique) (6’,JI)-factorization.

DEFINITION

A

category

‘6 is called

and

(8, Jl)-factorizable The

an

both

(6’,.1!) category 6’ and

will turn

following property

all

are

out

to

provided

closed

under

be crucial

in

that

it is

uniquely composition. the study of factor-

izations. 33.2

DEFINITION

A

that

category

for every

‘6’ is said

to

commutative

have the (6’, .ll)—diagonalization property in g square

provided

e

0-,——>0.

f

eErT'and

with

g

me..//.

o————)or

there

exists

a

morphism

k that

makes

the

diagram

8 C

——)

C ’

III,

I

1”:

g

1” K .

.———)o m

commute. 33.3

THEOREM

For

(1) ’6’ is

(2)

%’ is

any

category

‘6, the following

(6”,./1) category. (6’, y//)factorizable

are

equivalent:

an

and has the

(6’, .l/)-diagonalization

property.

251

(6". .ll) Categories

Sec. 33

Proof: (1)

Let

(2).

=>

V

0———)o ”I

be

commutative

a

with

square

e

c

d“ and

Let

..//.

e

m

f

m'

=

o

e'

and

g

=

are m” (c"oc) and Then (mom')oe' (6’, J/)-factorizations. (6“,.ll)—factorizationsofg 0. Thus, by uniqueness, there exists an isomorphism I: such that the diagram

be

m"oe"

o

o

commutes.

"2’

Hence,

(2) exist

=>

II

o

(1). If f morphisms

e” is the

o

o

m

=

m’

=

e

desired

k’ such

k and

o

diagonal morphism. (6’,.//)-factorizations that the diagrams c'

—-—-)

o

O

.

———)

e I

’l’k'

m

”

I I

I

V; a

there

z' and

m

1/;

I

then

’1

1’

,

f,

o

I

I, e

of

are

I

—)

0

o

———)c

m'

m'

commute.

Thus k'ok

=

(k' l;

0

k)

0

i.c.. k'

k’

=

e

is

c

c'

=

c

=

retraction.

a

l

o

e,

so

However, that

.l/.

If

suppose

factorization

ml

of m:

and a

m:

ml,

belong then

there

to

exists

a

commutes.

,1 k

.l/ mo

m

is

cpimorphism, (being the first factor of a (3

an

is closed 0

=

morphism

1

ml

k'

since

Hence, k’ is

monomorphism) is also a monomorphism. To show ’6 is uniquely (6, ./l)-factorizablc. that

that

m2

an

isomorphism. Thus under compositions. oml

k such

is

that

an

(6, .//)~

the diagram

252

Snbobjects. Quotient Objects. and Factorizations Likewise

there

exists

k' such

morphism

a

that

the

Chap. IX

diagram

”1 1

commutes.

k'

Thus

c

=

e

1,

that

so

is closed

6” is

‘6’ is

‘6’,the following

category

epimorphism, composition under composition an

under

closed

hence with

an

iso-

follows

are

equivalent:

(regular epi, mono) category. (regular epi, mono)-factorizable.

a

‘6 is

Proof: This follows fact

and

PROPOSITION

For any

(1) (2)

a”

That

0

section

a

since

isomorphism. Consequently, morphisms, m; ml 6 .11. dually. D 33.4

is

e

that

immediately from the above theorem (33.3) and the the (regular epi, mono)-diagonalization property

has

category

every

(17.17). D 33.5

PROPOSITION

If the

g

is

(6’, .1!) category,

an

following

then

the (5,

~//)-factorizations

are

functorial

sense:

If —)o

o

f

g1 J" 0—).

fl is

a

of f

commutative and

square

f ’, then there

and

f

exists

=

a

’

and

e

o

=

e

l 0

o

e'

k such

((3, Jl);faetorizations

are

that

the

diagram

m

o—————)

9

m’

f unique morphism m

——-%o

fl

[It I

I

'

9

m

commutes.

Proof: By property. I]

the

above

theorem

(33.3),

W has

the

(3, JO-diagonalization

in

Sec.

(6“, J!)

33

33.6

Categories

253

THEOREM

If ‘6 is particular, .l/

(1)

‘6

Mar

{fe

=

then 6‘ and ./l

(6, J!) category,

an

| iff

h

=

o

and

c

uniquely determine

6, then

e e

e

is

each

other

.‘

in

isomorphism},

an

and 6

(2)

{f

=

6

‘6

Mar

| Iff

Proof: By duality

=

o

Since

and

m

only

to

e

need

we

f= hoe where e66". diagonalization property diagram

m

"6' is

6

then

there

m

is

isomorphism}.

an

(I). Suppose that

prove

(’6, .Il)

an

(33.3). Thus,

.ll,

it

category,

exists

a

has

fe the

k such

morphism

.l/

and

(6", .Il)‘ that

the

commutes.

Consequently, On

the other

with

f

=

is

e

hand,

a

section

and

that

an

has

epimorphism; the

hence that

f property e e 6’, then e an isomorphism. Then for the m e of f, e is an isomorphism. Thus. since J! is closed isomorphisms, f is in .II. [:1 suppose must be

o

with

isomorphism. II c, f (6’, s//)-factorization under composition an

whenever

=

a

COROLLARY

33.7

Let ‘6 be

an

(6', .1!) category.

(1) If 6' is the class of all epimorphisms in ‘6’,then J! monomorphisms in if, and (2) If all is the class of all monomorphisms in ‘6’,then epimorphisms in ‘6’. C] In this

section

we

have

studied

is the class

6' is the class

of

all extremal

of all extremal

(6, .ll) categories in general. In the next “reasonable” is simultaneously category

(§34), (extremal epi, mono) category and an (epi, extremal mono) category. In the last chapter (§39) we will show that a pointed category is a (normal epi, normal mono) category if and only if it is “exact”. In exact categories the techniques section

will show

we

that

each

an

involving

exact

be

disposal.

at

our

sequences

that

are

available

in

categories

such

as

R-Mod

will

EXERCISES 33A.

Show

that

(epi, regular mono) mono) category. an

each

of Set.

category,

an

Grp, and Top is an (epi, extremal mono) category. and a (regular cpi. (extremal epi, mono) category.

Subobjcrts. Quotient Objects.

254

3313.

that

Show

of the

none

and

Rng. POS.

SGrp.

categories

C Imp. IX

Factorizatimts

Top.

or

Top; is

an

(cpl. mono) category. of ( 6‘,.ll)-diagonalization

In the definition

33C.

k in the

morphism

(33.2).

property

that

prove

the

diagram

o——)o "I

of the upper triangle is Stimcient to unique. and also that commutativity versa. commutativity of the lower triangle. and vice guarantee then -l/ is Prove that if ‘6’ has the unique (epi. .//)-factorization 330. property, monomorphisms in ’6. precisely the class of extremal that for an (6”,.ll) category, ‘6. the ( 6’,..//)-factorizations 3313. Show that the fact F: ’6’2 -> ‘6”. are functorial means that the factorizations may be interpreted as a functor be

must

Let ’6 be

33F.

(a) Prove (b) Prove

that

6

that

if

n

f

o

(6, .II) category. .I/ is precisely the class of all isomorphisms g e .11, then 9 6 .II. an

in ’6’.

D

(e) Let .al D

(In):

->

E be

a

functors

be

33%

natural

(5. (1,.)

limits

with

and

transformation,

let

(if. q.) respectively.

and

let 1]

=

unique morphism that makes

fbe'the

the diagram

13———!-—+ E da

9.4 V

v

D(A)

-fl—->E(A) A

commute.

Prove

(d) Prove sections

that

1),. is in

ifcach

that

./l

is closed

continuous

(i. .//l

=

the

Among

products.

pullbacks.

and

inter-

of all extremal

the class

corresponding in

morphisms

to

the

(b),

monomorphisms

category

Top:

in ’6’.

(c) for the class

(c). (d). and of

l-lausdorll'

6'.

spaces

and

let

maps.

{fl [is dense:

{fl fis

=

of

formation

the

under

in .II.

ol‘.//-subobjccts [sec 34.2].

(e) Prove that ./l contains (l' J Provide the statements 330.

all. then fis also

[= {fl [is

closed

a

an

embedding:

(6.10t4))].

epimorphism:

[= {flfis

an

extremal

monomorphism}

(l7.l()(3))]. 6‘, .II;

=

=

o”, .l/,

=

=

{flfis

surjective}.

{j Ifis

an

{fl/is

a

{flfis

embedding}. quotient

injectivc:

map} [= {flfis

[= {flfisa

an

extremal

monomorphism}

epimorphism:

(6.3(2))].

(l7.|0(3))].

(Epi. Extremal

Sec. 34 Show

for i

that

ln

33H.

l, 2, 3, Top: is

=

let

CRegT,

(see DefinitiOn 37.8) and satisfies

of

Determine

the

that

(6", VIII) category.

functions the class of all dense, compact-extendable that all Show be the class of CRegTz perfect maps.

for being

which

condition

an

(3,17)

an

category

except

,7

that

does

not

alone.

monomorphisms

331.

drops

3 be let J?

all of the conditions

consist

255

Mono) and (Extremal Epi, Mono) Categories

of the

results

if (resp. .1!)

section (§33) remain only of epimorphisms

if

valid

of this

consists

(resp.

one

mono-

morphisms). that

Prove

331.

diagonalization

that

category

has

the

has

pushouts

(epi, extremal

mono)-

property.

33K.

Show

that

33L.

Let g

be

following

any

each a

algebraic category that

category

has

is

(regular epi, mono) category.

a

pullbacks

and

coequalizers.

Prove

the

that

equivalent:

are

(a) The class of regular epimorphisms in ‘6 is closed (b) ‘6’ is a (regular epi, mono) category [see Exercise

under

composition.

210].

MONO) AND §34 (EPI, EXTREMAL (EXTREMAL EPI, MONO) CATEGORIES

§I7

well-powered category (5 that has and equalizers is (extremal cpi, mono)-factorizable. Next we will an ‘6’ also has pullbacks, it is even (extremal epi, mono) category.

In

we

have

shown

that

a

intersections show

that

if

THEOREM

34.]

well-powered, finitely complete, (extremal epi, mono) category. If

‘6’ is

and

has

intersections,

Proof: Using earlier results (”.16 and 33.3), we has the (extremal epi, mono)-diagonalization property.

need

then

only

show

epimorphism

and

'6’ is

an

that

’6

Let

9

c-———)o

f

9

ofi. m

be

a

commutative

square,

where

e

is

an

monomorphism. Let

a

extremal

m

is

a

256

Subohjeets, Quotient Objects, and Factorization:

be the Since

of is

pullback the

square

and

m

a

g.

Now

a

there

pullback,

is

Chap.

monomorphism since exists a unique morphism h a

In

is

such

IX

(2|.l3). that

the

diagram e 0

\_)

AV "1‘31.

f

y

/b

0—).

m

commutes.

Since b

o

is

e

extremal

an

epimorphism, a must diagonal morphism. [:1

a‘1 is the desired Our

attention

focuses

be

Thus

isomorphism.

an

showing that each category satisfying the an (epi, extremal hypotheses mono) category. The next proposition indicates the crucial role of the (epi, extremal mono)-diagonalization The conclusions of the proposition should be compared with property. the analogous results for monomorphisms—6.4, 17.3, 2|.l3, and l8.16. now

of the above

34.2

on

theorem

is also

PROPOSITION

If

‘6 has the

(epi,

extremal

then in %:

mono)-diagonalization property,

(l) The composition of extremal monomorphisms is an extremal monomorphism. (2) The intersection of extremal subobjects is an extremal subobject. (3) The inverse image (pullback) of on extremal monomorphism is an extremal monomorphism. (4) The product of extremal monomorphisms is an extremal monomorphimt. Proof: Since monomorphisms are closed inverse images, and products, in each case we condition (l7.9(l)(ii) dual) is satisfied.

under need

composition, intersections, only verify that the extremal

h (I). If f and f’ are extremal monomorphisms and f f’ then epimorphism, by the (epi, extremal mono)-diagonalization exists a morphism k such that the diagram o

=

o

g where

property,

g is

an

there

._;>’o

commutes.

Thus

f

' =

k

(2).

Let

their

intersection,

(A,,f,

,

c

y,

be

where a

where

g is

an

family

of

for

each

epimorphism, so that g is an isomorphism. extremal subobjccts of B and let (D, d) be i, d f, odi. If (1 hog, where g is an =

=

Sec. 34

(Epi. Extremal

epimorphism,

then

Mono) and (Extremal

for each

Epi. Mono) C aregaries

257

i, the diagram

D—g>C d-

h

xii—>3 commutes.

for each i there exists a By the diagonalization property, morphism —> k,: C A; such that d,- k, 9. Hence since g is an epimorphism and since the intersection, being a limit, is an (extremal mono)-souree (20.4), g must be 0

=

isomorphism.

an

(3).

Let S

c—-—-—)o

f

m

o——-)0 t

be

pullback square, where m is an extremal monomorphism. We wish to show that f is also an extremal If f monomorphism. hog, where g is an epithen the there exists a morphism k such morphism, by diagonalization property, that the diagram a

=

Oa——)-o

commutes.

Thus, since pullbacks, being limits,

are

(extremal mono)-sources,

g must

be

an

isomorphism. (4). If (A; A) 8,) is a family of extremal monomorphisms, HA; "—I';HE; is their product, and Hf,Ii 9 where g is an epimorphism, then by the diagonalization property, for each i there is a morphism k, such that the diagram =

0

”A;

1'!

x’lki

A; '—‘_) commutes.

.

#118; B‘-

Subobjects, Quotient Objects. and

758

products, being limits, E] isomorphism. Since

‘6’ be

Let X

1—.Y

be

(extremal mono)~sources,

are

that

well-powered category ‘6’-morphism,and let 1/

a

a

IX

Chap. be

g must

an

LEMMA

FACIORIZATION

34.3

Factorizations

be

has intersections

equalizers. of ‘g-ntonomorphismswith

class

a

and

Let

the

following properties: ( 1)

under

.II is closed

i .e.,

intersections;

then d

family (A,, nt,), of .ll-subobjects, m qoh, where m e (2) If f =

m

o

(3) l,r Then

of a non-empt

y

.11.

e

and

is

q

regular monomorphism,

a

then

.ll.

6

q

J!

o

is the intersection

if (D, d)

.11.

e

there

(i) f (ii) if f diagram

exist m

=

o

morphisms

and

m

e

that

such

.11,

E

m

e

is

an

epimorphism,

and

e.

=

m'

=

m

oh, where

m'

e

.11, then there

exists

all, then

an

a

k such

morphism

that

the

commutes.

(iii) if

e

Proof

.'

o

y,

Since

where @ is

m

o

m 6

well-powered,

m is

exists

there

{x "—‘» A,- 1‘.)

Y

isomorphism. a

set-indexed

family

},

off, where each mi 6 .II and (A, m,-), is a representative of Y through which f factors. Let (D, m) be the interclass of all Jl-subobjects m 6 J1. is section of (A, m,), (which by (3) non-empty). By (I), By the definition m e. of intersection, there exists a unique morphism e such that f loss of generality we can assume If f m’ h, where m’ 6 J1, then without of factorizations

=

that

there

section

is

some

of (A‘,

m,),,

j

e

m’

I with

there

exists

a

=

m

and

morphism

h

=

k such

h].

X——e—->’D Ill h=h,.

/;c

is”

A;-—)'..— m

so

that

(ii) holds.

—mj

m

(D, m) is the inter-

Since

that

diagram

commutes,

o

o

=

m

=

m,

o

k. Hence

the

Sec.

(Epi. Extremal

34

If such

where

IT! 0 y

=

e

Mono) and (Extrema! o

m

.1], then

m e

259

Epi, Mono) Categories

by (ii) there

exists

a

morphism

!

that E

——>

0

I

I

I I

’,

1,"

I

I

g

[I

Iva

._).

O

3

m

commutes.

Hence n‘:

c

t

=

"—10!

mo

l. Thus

is

phism, it

:71 is

=

=

m

monomorphism.

a

1,

mo

that

so

and

retraction,

a

since

since

is

m

monomorphism,

a

it is the first factor m is

Consequently

of

monomor-

a

isomorphism,

an

so

that

(iii)

is established. To

complete

Suppose that of equalizer, By (2) m q

r

e

o

there

o

exists

need

only show that e is an epimorphism. (Q, equalizer ofr and 5. By the definition [1. Hence f m morphism [I such that e q q It. s. by (iii) (1 is an isomorphism. Thus r I]

proof, s

we

(1)be the

Let

e. a

o

=

“It, so that

E

o

the =

o

PROPOSITION

34.4

Every well-powered category ‘6 that has intersections the (epi, extremal mono)-diagonalizalion properly.

Proof:

a

commutative

diagram, Let

monomorphism. property

and

equalizers also has

Let

h

be

o

=

=

there

that

be

.1!

exist

g

where

c

the

class

is

an

of and

morphisms j;

epimorphism and f is an all g-monomorphisms n such that the g,l diagram

extremal with

the

commutes.

It is

a

exercise

straightforward

(2), and (3) of the Factorization factorization

f

=

m

o

e

of

f

to

Lemma

Since

m

show

that

J!

satisfies

conditions

(34.3), so that there exists is a monomorphism, f," =

an e.

(1), (epi, .1!)-

Hence

the

260

Subobjects,

Chap. IX

Factorizations

Quotient Objects. and

diagram ._‘__>.

.

51%

h

g

c

7‘ \ m

o

——)e

f commutes.

Now e

must

34.5

be

monomorphism, f is an extremal Thus e‘1 g," is an isomorphism. o

so

that

since

is

epimorphism, diagonal morphism. E]

the desired

e

an

THEOREM

‘6 that

Every Well-powered category

(epi, extremal

has

and

intersections

equalizers

is

an

mono) category.

Proof: By the above proposition (34.4) re has the (epi, extremal mono)diagonalization property. Thus by parts (1) and (2) of Proposition 34.2 and the fact that each regular monomorphism is an extremal monomorphism (l7.ll that the class of all extremal it follows monomorphisms of ‘6 satisfies dual), Lemma the three hypotheses of the Factorization (34.3). Hence, by that lemma, so it is an (epi, extremal and ‘6 is (epi, extremal mono) mono)-l'actorizable; category (33.3). [I 34.6

PROPOSITION

If ‘6’ is a well-powered, finitely complete category that has intersections, then each (if-morphism f has a factorization (which is unique up to isomorphisms) of m b e, where e is an extremal the form f epimorphism, b is a bimorphism, and m is an extremal monomorphism. Furthermore, this three-fold factorization that if is functorial in the sense o

=

o

f _)

e

o

0

_-)'

fl is

a

commutative

factorization: the diagram

off

and f and f’, then

square

=

m

o

there

e

o

exist

3r V

e

Q.

o

=

o

are

and

the

k2

three-fold such

that

m

v

v

Q

’

m' e' b' f unique morphisms kl

and

a.

'3 0

commutes.

b

(---—O oD

//

t‘,

’4.

h

’l K

A

7+3,

commutes.

Proof: Dualize

the above

E]

COROLLARY

35.4

Each

property

category that has and each category

35.5

a

B

that

has the

has

(extremal epi, mono)-diagonalization mono)pushouts has the (epi, extremal

1:]

THEOREM

If M is well-powered, finitely complete limits, then for functor that preserves

G(A) f: of (f, A). —>

there

We

Proof:

(28.6). To show (35.1). [:1 35.6

pullbacks

(33]).

diagonalization property

is

corollary (35.2).

exists

have the

a

already

any

unique (extremal established

uniqueness,

apply

and

9 if G: .2! sat-object A and any fi-morphism G-generating, mono)-factorization

and has intersections

the

existence

the

above

of such

a

—>

factorization

diagonalization

theorem

COROLLARY

If

s!

in ‘6’ has

then every well-powered, finitely complete, and has intersections, unique [(extremal epi)-sink, mono]-factorization. (cf. 19.14) E]

is a

sink

Subobjects,

270

Quotient Objects. and

Factorizations

Chap.

IX

COROLLARY

35.7

Every well-powered, finitely complete category (extremal epi, mono) category. (cf. 34.1) E]

which has intersections

is

an

PROPOSITION

35.8

Let

Gus!

—v

98 be

extremally G-generates

a

and

funetor the

A. Then

be

G(A) f: B following hold: _.

a

pullbacks, and h: A (1) If s! has pal/backs, G preserves epimorphism, then G(Ii) of extremally G-generates A’. (2) If Q has pal/backs, G preserves monomorphisms, and g: B’ epimorphism, then f g extrema/1y G-generates A.

—v

that

Q-nwrphism A'

—’

is

extremal

an

B is

extremal

an

o

Proof: (1). Since h is an epimorphism, it is clear that 601) of G-generates A'. To A is an let 601) f G(m) a, where m: A verify the extremal. condition theorem (35.1), d—monomorphism. According to the preceding diagonalization A such that the diagram k: A there is an sl-morphism —>

o

=

o

-’

f

3—1001) k

G(Z ) ———>

G(A')

G(m)

commutes. m k. Hence sinceh f G-generates A, h m must be an isomorphism. (2). Since 9 is an epimorphism, it is clear that f the extremal condition, let fog G(m) g be a Let A A is an d-monomorphism. m:

Since

is

o

=

=

o

o

g

an

extremal

epimorphism,

G-generates

factorization

of

A. To

f

o

verify

g, where

-»

be

pullback diagram a

commutes.

square.

Then

there

exists

a

morphism

h: B’

—+

P such

that

the

Sec.

35

Extremal

(Generating.

Since

271

Mono)-Factorizatians

G preserves

monomorphisms, C(m) is a monomorphism, so that pl monomorphism (21.13). Thus since 9 is extremal, pl is an isomorphism. Since I extremally G-generates A, f C(m) (p2 pl“) implies that m is an isomorphism. C] is

a

=

We

turn

now

attention

our

0

0

the consideration

to

of

(generating,

extremal

mono)-factorizations. 35.9

DIACONALIZATION

THEOREM

ll

is well-powered, has intersections and equalizers, G: s! Suppose that and i A are l, 2, j}: A,preserves monomorphisms, for .d-morphisms a!

—»

=

g‘:

B

—>

G(A,)

fi-morphisms

are

such

that

60]) my, monomorphism, then the diagram

generates A, and f2 is an extremal .tl-morphism k: A, A2 such that _.

9,

C(fz)ogz.

=

there

exists

lfg, a

—>

.48

and G-

unique

,

B——)G(Ax)

gal k”

’//’6(k)

161’”

GHQ—WOT,” commutes.

Proof: Let (X .-, nu), be the class of all subobjects of A for which there exist d-morphisms It}, 11}with f, m; oh}. Let (D, d) be the "130/1; and f2 intersection of (X i, m,), (l7.7). Then there exist d-morphisms (1., d; with fl do d, and f2 do (1;. Since G preserves monomorphisms, C(d) is a monomorphism, so that the diagram =

=

=

=

91

B

———>

GM.) G(d.)

9:

001:)

GU.)

0(0)

{:0

G(A-;) ——->G(A) GU)

commutes.

We wish

morphisms exists

to

show

such

that

d2

rod;

.d-morphism

an

G(’°dl)°9| and

that e; =

such

=

is

Assume that r and s are d-epimorphism. 50:12 and let (E, e) z Equ(r, s). Then there that d, e hand e1. On the other an

=

G("°dz)°gz

=

0

C(S°dz)°92

C(5°di)°gi

=

s G-generates A“ implies that r d, d,. Hence, there with and .nf-morphism el (II eve]. Consequently, fl (doe)oe, (d e) e; implies that (E, do e) belongs to (X;. mi),. It follows that e is a f; retraction. Thus since it is also a monomorphism, it must be an isomorphism. Hence r s (16.7); so that (11 is an epimorphism. But since [1 is extremal. this

is

the

fact

that

an =

o

g,

=

o

o

=

=

o

=

Subohiects.

272

Objects. and

Quotient

k

Thus

implies that d2 is an isomorphism. morphism. Uniqueness follows from

the fact

'

(I;

=

C hap. IX

Factorizations

that

o

g.

(I, is the desired G-generates A,

diagonal E]

.

COROLLARY

35.10

Let

.1! be

((g,),, A) and for each i e I

well-powered category ((f,),, D) be sinks in s1,

that

has

and let

fit

a

equalizers. let .cl-morphisms suclt that

intersections

and h be

and

the square

commutes.

If exists

((g,),, A) is an epi-sink and k: A a unique morphism _.

m

Bt

I.

is

extremal

an

D such 9.

there

monomorphism, then is l, the diagram

for each

that

’A

——>

1 [kl ill I

in”

D

C

——>

m

E]

commutes.

COROLLARY

35.11

Let

.2!

be

co-(welI-powered) category let (A, (g‘),) and (D, (f,),) be sources equalizers. g; morphisms such that for each i e l, f,- e there then and e is an extremal epimorpltism, such that for each i e l the diagram a

o

commutes.

35.12

=

that

has in

o

is

h.

s1, and let is

a

and

cointersections

and

e

If (A, (99,) unique morphism a

co.

h be .9!-

mono-source

k: D

—'

A

59 is

a

[:1 THEOREM

and equalizers, and if G: d If .51 is well-powered, has intersections and equalizers, then for any sat-object intersections functor that preserves —o

A and

Extremal

(Generating.

35

Sec.

there

C(A), Q-morphism f: B mano)-factorization of (f. A). —*

any

273

Mono)-Factorizations exists

extremal

unique (G-generating,

a

m,-), be the family of all extremal subobjects of A for which G(m,-) 51,-. Let (D, d) be the interQ-morphism 9, such that f section of (X 1, m,-), (I 7.7). Since s! has the (epi, extremal mono)-diagonalization of A (342(2)). Since G must be an extremal subobject (34.4), (D, d) property Let (X i,

Proof: is

there

intersections,

preserves

(G(X,), G(m,-)),.

that

show

exists

1—)C(A)

intersection

the

$~morphism

a

g:

C(D) %

Bi.

=

D, let D

G-generates

g

is

(C(D), G(d)) there

Hence B

To

c

=

some

I:

A be

a

B

—.

of

the

C(D) such

family

that

C(A).

pair

of

d-morphisms

where

S

equalizers, there is C(r) g C(s) g. Let (E, e) z Equ(r, s). Since G preserves a 3-morphism G(e) 57.Since (1 e is an extremal monomorphism a such that g ins! (34.2(l)) withf C(d e) 17, (E, d e) must belong to (Xi, mi),. It follows be an it must so that e is a retraction. that since it is also a monomorphism, follows D. Thus r 3. Uniqueness Consequently, g generates isomorphism. theorem from the above (35.9). C] immediately diagonalization e

=

o

=

o

o

=

o

o

o

=

COROLLARY

35.13

If a! has

well-powered and has unique (epi-sink. extremal

a

and

intersections

equalizers, D mono)factorization.

.21 is

each

sink

in

COROLLARY

35.14

[fuel

in .m' has

35.15

COROLLARY

coequalizers, then {:1 epi, mano-source)~factorization.

and has cointersectians

women-powered)

is

source

unique (extremal

a

that is well-powered and has category (epi, extremal mono) category (34.5). E] Each

an

then

and

and

intersections

each

is

equalizers

EXERCISES

Let G: s!

35A. 9:

B'

B and

a

f:

B

->

->

.98 be

a

functor,

let h: A

C(A) be tie-morphisms.

Prove

-v

A’ be

an

.sJ-morphism,

and

let

the following:

A’. then h is an epimorphism. f Ggenerates epimorphism. (b) If C(h) f extremally G-generates A'. then h is an extremal o A. If then A. (c) f G-generates f g G-generatcs (d) If f g extremally Gcgenerates A. then [extremally G-generates A. (e) If g is an epimorphism and f G-generates A, then In 9 G-generates A. (f) lfh is an epimorphism andf G-generates A, then C(h) of G-generatms A’. (cf. 35.8) (a) If 60:)

c

a

c

358.

by

the

Prove

hypothesis

Theorem

that

it is

35.l2

after

extremally

replacing the hypothesis well-powered.

than!

is well-powered

274

Subobjects, Quotient Objects. and Factorization: 35C.

Prove

that if d

preserving functor, then for exists a unique factorization B where

g

extremal 35D.

separator

L)

is

well-powered and complete,

any

G(A)

=

41-013th B

Prove

that

‘6’,then

if g is

any

->

A'

is

a

and

G: .2!

fi-morphism f:

1» C(A') “i", our)

extremally G-generates A’, k: A’ monomorphism. for

A and

Chap. ->

B

Q is

a

IX

limit-

GM) there

->

E9. 0(4)

bimorphism,

co-(well-powered) and cocompletc lmm(S, _) reflects limits.

and

m:

A'

and

S is

an

-)

A is

an

extremal

X Reflective

Every theorem

in category

Subcategories

is either

theory

pushover

a

its

or

dual—a

put-0n.

W. E. Axon

In

VII

Chapter

mathematics results

claimed

we

and

that

situations

adjoint

that

adjoint functor theory. In Chapter

of category case of set-valued

theorems VIII

we

abundant

are

belong

to

concentrated

the our

throughout most

study

useful on

the

functors and their adjoints. The results of Chapter lX special now concerning (0‘. .11) categories, (extremal) subobjects and factorizations enable us to study in more detail the special case of embedding functors and their left and right adjoints.

§36 Definitions

GENERAL

General

and

SUBCATEGORIES

Properties

DEFINITION

36.]

Let 5! be

subcategory

a

(1) An E-universal (2) a! is called

map

reflective

an

ail-reflection

R: .46

—>

.91. In this case,

5

with

for each

or

a

embedding fi-object

functor

B is called

E: an

519+ .93: JI-reflection

of B.

reflective

subcategory of .98 if and only if there flit-object:i.e., if and only if E has a left adjoint, a

R is called

a

reflector

for d.

of fi-morphisms, then a! is called 6-reflective in .93 provided for each w-object B there exists an Jar-reflection (re, A,) such that each 6‘. For the case that 6 is the class of all epimorphisms [resp. mono-

(3) If 6' is that

ofw

(r3, A3) for in :58

exists

ru

REFLECTIVE

morphisms;

a

class

extremal

monoreflective;

epimorphisms]

of 53

(extremal epi)-reflective]in 275

we

.93.

say

that

.9! is

epirefleetive [resp.

Reflective Subcategories

276

(i.e., with respect

NOTIONS:

DUAL

co-universal

in £8 (or

of B; coreflective

E) .rl-coreflection

to E

maps coreflective

a

6-corefleetive for d; especially in .93. coreflective. (extremal mono)-coreflective]

eoreflector

in 3,

and

right adjoints

subcategory

monocoreflective

of

to

at);

[resp. epi-

EXAMPLES

36.2

examples of reflections given in 26.2(2) is in fact an epireflection. of the examples of coouniversal maps given in 26.2(4) actually de-

Each

of the

each

Also

scribes

a

(I) If

:3

monoocoreflective is the

fields, then is the

.2! is not

full

category

ordered

full

is the

d

In addition:

fields

and

subcategory

of

of all fields

and

order-preserving field homo36‘ consisting of all real-closed

in $. Field

subcategory

reflective

a

of

is reflective

a!

4% is the

(2) If

situation.

category

and

morphisms,

d

X

Chap.

in 53.

field

all

homomorphisms algebraically closed fields,

consisting of all (Why not?)

of 3

and then

homo(3) If .1? is the category TopGrp of topological groups and continuous of all Hausdorff ofg J! full is the consisting compact subcategory morphisms and called the Bohr is in .93. reflector a! is reflective then (The compactification groups,

functor.) 36.3

PROPOSITION

Each

full monoreflectivesubcategory of

Proof:

Let d

and let AB

:2,C

be monoreflective be

Q-morphisms

Ac is the .si-reflection

If C L

rcc5cra

in a,

a

is also

category

epireflective.

let B '—"»A, be the sat-reflection

with the property

3

o

r,

=

t

e

of B

rs.

of C, then =

rcctors.

in Since Ac belongs to .91, this equation together with the uniqueness property 0 s o monoI. Since is a that the definition of universal rc maps implies re rc s t. Hence is an that it follows D epimorphism. r3 morphism. =

=

36.4

PROPOSITION

If (I) .91

.91 is is

(2) For 36.5

a

subcategory

full subcategory of $. each .d-objeet A, the pair (1,4, A)

equivalent:

are

a

is

d-reflection of

an

A.

[3

PROPOSITION

Let .9! be

reflector R05

a

full reflective subcategory of

The a? with

with

3

embedding

E: a! (—>93 and

R: 939 —0 .11. Then

i

Imam! 550R. (2) (5010415012)

(1)

of as, then the following

[1

propositions show that if d embedding E and reflector R, then

is

last two

E

o

a

full reflective

R is

subcategory of “quasi idempotent" and

See.

A) is an sat-reflection approximates” B in false for reflective subcategories that are

“projects“a that

in

many

“best

sense

other

if (r,

.11. Also

onto

certain

a

All of this is and

Reflective Subcategories

General

36

results

that

reflective

subcategories in general, subcategories “occurring in nature” the following:

of B, then A is an d-object .53! (via the morphism r).

and are

full.

not

for full reflective

true

are

277

but not

subcategories fact

of these

Because

because

of the

full and

that

isomorphism-closed,

most

for

reflective we

adopt

CONVENTION

36.6

the

Throughout assumed assumed

remainder

be both

to

be

to

full,

a

dc»

embedding

E:

As will

become

of this

chapter, all subcategories will be isomorphism-closed. In particular, 41 will be isomorphism-closed subcategory of Q, with

full and Q.

evident

in that

which

reflective

about

subcategories. general epireflective subcategories has emerged. Relationship

follows, However,

relatively little is known satisfactory theory of

a

Subobjects

to

PROPOSITION

36.7

B be

Let are

equivalent:

(I)

r

is

(2) B

is

a a

w-object,

a

and let (r, A) be

an

.szl-reflectionforB.

Then the

following

monomorphism (resp. extremal monomorphism; section). subobject (resp. extremal subobject; sect) of some si-objeet.

Proof: Clearly (1) implies (2). That (2) implies (1) follows whenever goh is a monomorphism (resp. extremal f also then It is a section), monomorphism (resp. extremal section) (6.5, 17C, 5.5). E] =

from

the fact that

monomorphism; monomorphism;

PROPOSITION

36.8

If (r, A) (1)

r

is

an

(2)

r

is

a

is

ail-reflectionfor B,

an

then the

following

are

equivalent:

isomorphism. section.

Proof: lffo

r

I, then

=

the

diagram

B—:\-—>/ 1,.

A

commutes,

map,

r

so

of

=

that |. Thus

by the r

is

an

uniqueness

property

isomorphism.

[:1

in

the

definition

of universal

Reflective Subcategories

278

Chap.

X

COROLLARY

36.9

reflective in 38,

.94 is

If

Proof: Immediate

36.10

from in Q.

isomorphism-closed

then

s!

the

is closed

under

Propositions

the

formation

36.7, 36.8, and

of

the

in Q.

sects

fact that

d

is

1:]

PROPOSITION

If s! is epirefleetive in $3, subobjects in .9}.

then

.9! is closed

under

the

formation

extremal

of

A be an extremal Proof: Let f: B monomorphism in 3?, where A is an If is the fl-reflection of (r, A) B, then there exists some .d-object. morphism g such that f r. Since r is an epimorphism and f is extremal, r must be an g since 51 is B must be an Thus, isomorphism. isomorphism-closed, sat-object. [:1 —>

=

36.11

o

PROPOSITION

If Q is an (6, .1!) category the formation of .ll-subobjects Let

Proof: (r,

A) is

B

m:

—>

the w-reflection

A

and .24 is

£~refiectivein g5, then

d

is closed

under

in .3.

be

in J],

morphism

a

of B, then

there

exists

where

A is

sat-object.

an

9 such

morphism

some

that

If the

triangle

commutes.

Thus, since Relationship

to

“detects”

Recall

then

.31 is said

each

r

6.

e

r

be

must

in 3’8.the

embedding

an

functor

isomorphism (33.6).

E:

31¢» 33 has

it preserves limits. I.e.. if D: I the limit of E0 D: l —> 9?. Next

things)

is also

that to

an

if E: .319 be closed

:93 is

under

D: 1—» a!

functor

equivalent

conditions

C]

s!

-'

we

has will

a a

left that

see

adjoint, (L, (1‘)),

limit

are

and

full embedding functor the formation of [-limits

a

each

limit

(L, (1.)) of

and

I is

E also

in .2 En

a

category, provided that

D. the

following

fulfilled:

.si-object.

(2) (L. (1.)) is DUAL

and

limits.

36.12

L is

.l/

Limits

other

(among then (L, (1.))

(l)

e

is reflective

list so

for

m

a

NOTION:

limit

of D. closed

under

the formation

of l""-colimits

in 33

(23.5).

279

Reflective Subcategories

General

Sec. 36

THEOREM

36.13

If .31 is reflective formation of I-Iimits in

Q, then for each

in

the

under

I, .9! is closed

category

9.

Proof: Let D: l —> at be a functor, (L, (1‘)) be the limit of E D, and L r: AL be the eel-reflection of L. By the definition of universal map, for each D(i) such that I, 7,- r. From the uniqueness I,-there exists a unique li: AL for E D. Thus source is a natural it follows that of the morphisms i, (AL, (1‘)) L such that for each i, 7r h: AL there exists some l,- [1. Hence for each i, o

—>

—v

c

=

o

=

->

I;°/t°r=i,~or=l‘so

hence

an

that

so

o

r

since

=

1.

.2! is

Consequently, r is a isomorphism-closed,

COROLLARY

36.14

Each

reflective subcategory of

For

epireflectivesubcategories

only “initially in will

at“

then

its limit

later, this property

see

of “nice”

I-complete category

an

know

we

more.

even

in .68 must

object

actually characterizes

is

l-complete. Even

if

a

E]

functor

is

belong to .m’ (36.15). As we the epireflective subcategories

categories (37.2).

DEFINITION

36.15

A functor i there

I-object

be

.5! is said

to

for each

functor

the

(36.8),

an

licl,

=

(20.4), h

mono-source

a

isomorphism sat-object. C]

section, L is

(L, (1‘)) is

since

that

o

object

L is

provided that for each is an l-object j such that D(j)e 0b(.s!) and hom(j, i) aaé Q. formation of I-litnits in Q provided that strongly closed under the D: I .93 that is initially in .2! and each limit (L, (1.)) of D, an .d-object. D:

NOTIONS:

DUAL

1

93 is said

—¢

to

be

initially

in .5!

—r

finally

in of;

strongly

closed

under

the

formation

of I”-

in 9’9.

colimits

THEOREIH

36.16

If 51 is epireflectit‘ein .98, then for in .9. the formation of “mm:

each

category

I, .311 is strongly closed under

.90 is initially in .21, (L, (I,)) is a limit of D and Proof: Suppose that D: I is the .d-refiection of L. Let I be the class of all I-objects j for r: L AL for each j 6 J there exists a unique morphism which DU) is an Ail-object. Then r. such that D(j) ii: AL [j 71 Since D is initially in .d, for each l-object i i. Now for each Lobject i, let there is some j,- e .l and somefizji _.

—.

—.

=

o

—>

7,.

=

pm)

”1.-

it can be shown without difficulty epimorphism that is a natural choice and and of the off} j,(7‘)) (AL. pendent

Since

r

is

an

that source

I} is

inde-

for

D.

Reflective Subcategories

280

Hence

there

is

a

h:

morphism

AL

L such

-»

Chap.

that

for each

i, I;

o

h

=

X

7,. Con-

sequently

lior=

lie/tor: so

that

since

(L, (I,)) is

isomorphism (36.8),

(20.4), h

mono-source

that

so

owe/h

sincea!

is

o

r

I, =1,-ol,

=

1. It follows

=

isomorphism-closed,

L is

an

that

r

is

d-object.

an

D

to Collmits

Relationship 36.17

a

th.)oljlor=

PROPOSITION

If .3! is reflective in a? ( with embedding E : .n/ C_, a? and reflector R: .9? d), d is afunctor, ifD: I ((ki), K) is the colimit ofE D, and r: K AK is the d—rcflectionof K, then ((r k,), A“) is the colimit of D. —’

and

—v

c

—'

0

This

Proof:

is immediate

since

the sink

((r

‘7

ki)7 AK)

is the sink

((lei». and

R. having

36.“?

a

right adjoint,

must

R00).

preserve

colimits

(27.7).

[:1

COROLLARY

reflectivesubcategory of

Each

an

l-cocomplete category

is

l-cacomplete.

D

reflective

subcategory .n’ of a complete and cocomplete .93 is both complete and cocomplete (36.14, 36.18) and that the limits category in .sv' are formed in the same way as the limits in 33 (36.13). However, 51 is not of colimits in Q; i.e., if ((k;), K) is a necessarily closed under the formation D for some colimit ofE D: l is not d. ((k‘), K) necessarily the colimit of D. To see this. let 01,), be an infinite family of compact Hausdorff The spaces. of this is the disjoint topological sum, which is not coproduct family in Top; even is in The of the compact. though Comp'l'z epireflective Topz. coproduct in family (A9, CompTz is actually the Stone-Cech compactification of the disjoint topological sum (as is evident from Proposition 36.17). We have

seen

that

a

c

~>

EXERCISES 36A. Consider .n/

Non-Full

Subcategories following two (non-full) subcategories .n/ and those sets which have J!) are partially ordered

the

(resp. (resp.

subset

non-empty

subset)

has

a

supremum.

d—objects (resp. fi-objects) belongs tosv' (resp. J?) all subsets

(resp. non-empty

subsets).

Prove

.36 of

Objects

the

A monotone

ifand

POS.

only ifit the following:

that property function between preserves

suprema

of

each two

of

in POS. but neither is epireflective in (a) .d and a? are each (extremal mono)-reflective POS. (Does this contradict Proposition 36.3?) A is the pair(l,.. of A. (What is thew-reflection A) and-reflection (b) For nod-object of a complete lattice?)

(c) (18, B) is

a

Q-reflection that

Prove

36B.

of 3-Reflectt've Subcategories

and Generation

Characterization

Sec. 37

of the

for

complete

a

only if Bis inversely well-ordered.

B if and

92.0bject

{B the following

category

is

strongly (a) For each small category 1,5! of products in (E, ands! (b) d is closed under the formation of equalizers in .48 (cf. Theorem formation 23.8).

proof of Theorem

36C.

In the

36D.

Let ‘6 be in its

category

own

@—objectsfor which of‘g

gory

36.16

show

exists

there

an

of l-limits

in 9.

strongly closed under the a

natural

for D.

source

subcategory a! that is cocomplete as a subcategory of {9 consisting of all those Show that Q is a cocomplete subcated-reflection.

has which category right. Let .93 be the full a

is

(211,00) is

that

equivalent:

are

the formation

under

closed

281

a

(23.5).

§37

CHARACTERIZATION K-REFLECI'

GENERATION

AND

IVE

OF

SUBCATEGORIES

of reflective give a satisfactory characterization such nice categories as Top, we will in this section of even characterize the 6-reflective the (in particular epirefiective) subcategories of the smallest of “nice" categories. This will naturally lead to the concept of that a class of objects of 6-reflective a contains given category subcategory the category (called the 6-reflective hull of the class). Recall that throughout this chapter a! is considered to be a (full, isomorphism-closed) subcategory of the category Q. Also in this section 6" (resp. .11) will denote a class of epimorphisms (resp. monomorphisms) that is closed under composition with isomorphisms. Even

though subcategories

we

able

not

are

CHARACTERIZATION

37.1

If .99 is following are

an

6-reflective

d

is

(2)

a!

is closed

(36.13 and

$~object. powered),

under

I

THEOREM

J-co-(weII-powcrcd) equivalent:

(1)

Proof:

to

(6‘, .1!) category

the

and

formation of products

.

is

a

then

products,

ll-subobjects

from

the

representative

set

(fl, A3),

of

(2)

in Q.

of the

results

36.] I). To show the converse, that suppose We need only find an .nl-reflection for B. there

has

in .fi.

(1) implies (2) is immediate

That

that

is satisfied

Since

.93 is

section

last

and

a

6”-co-(wellof 3

quotient objects (TIA... tr.) be the product

form

B is

of the

6 and A e 0b(.el). Let in 53 of (f. A) wherefe is an and of the definition family (Am. By hypothesis TM, sit-object. by product there is a {if-morphism h: B f... [1.4, such that for each i, n,- h

the

—~

Now

o

=

let

B—SxtBLH/li

ELI-D15:

be the (6’, .//)-factorization of II. We claim that (r. —> To see this. let g: B A’ be a 38-morphism, where

Bis/1’

=

BLHiL/t'

AB) is the sci-reflection of A’ is an .csl-object. If

B.

Reflective Subcategories

282

Chap. X

is the (6, Jl)-l‘actorization of g, then there is some j —> A such k of}. Hence the diagram that e

k: A j

l and

e

isomorphism

some

=

AB

m

\HA; 8

B

—_7’ 3

g

A

A

AIK

m

and since

commutes, x

from

r

is

CHARACTERIZATION

37.2

e

epimorphism,

an

A a to A' for which

o

x

r

=

THEOREM

is

If complete, well-powered, equivalent : (l)

.2! is

(2)

s!

is

epireflectivein a. strongly closed

under

the

strongly closed under is strongly closed under is strongly closed under

.9! is

the

d

the

d

o

(r,

k

c

it]

A a) is

o

m

an

is the

unique morphism

sat-reflection

for B.

[I

II

and

ca-(well-powererl), then

formation

of I-limits

the

in 33

following

for

each

are

small

I.

category

(3) (4) (5)

g. Thus

ti:

formation of products and pullbacks in Q. formation of products and inverse images in Q. the formation of products and finite intersections

in Q. in 93. strongly closed under the formation of products and intersections s! is strongly closed under the formation of products and inverse images of extremal monomorphisms in Q. (8) d is strongly closed under the formation of products and finite intersections of extremal subobjects in 5:3. (9) d is strongly closed under the formation of products and intersections of extremal in £3. subobjects (10) at is strongly closed under the formation of products and equalizers in Q. (ll) .2! is closed under the formation of (extremal mono)-sources; i.e., if (B, U,»

(6) (7)

is

d

is

(extremal mono)-source such

an

then B is

(12)

an

that

the codomain

of

each

j}

is

an

sat-object,

d-object.

.21 is closed

Proof: The

the

under

proof of

formation of products the

and

extremal

subobjects

in 93.

equivalence of conditions (2) through (10) is in a manner accomplished analogous to the of completeness (23.8). It remains to be shown proof for the characterizations = = that (2) (12) (1) (II) (ID). => B A be a Let (1). (2) f: w-morphism with A e 0b(.9!), and let J! be the class of all d-morphisms that are monomorphisms in .429.Since .2! is strongly left

as

exercise

an

which

=>

can

be

=

_.

Sec.

37

Characterization

closed

under

conditions

the

of g-Reflective Subcategories

and Generation

formation

for .1! in the

of

and

equalizers

Factorization

Lemma

(34.3).

Thus

f

has

let (B 1) A;), be a set-indexed B and codomain with domain some

Now

factorization.

satisfies

all

intersections,

283

the

(epi, J!)family of all

an

representative sat-object. By the factorization A I-),is a solution set for B. Also (by (2)) the inclusion functor limits. Hence Theorem preserves by the First Adjoint Functor (28.3), the inclusion has a left adjoint; i.e., .52! is reflective in 35. To show that it is epimorphisms property proved above, (ei,

epireflective,suppose

that

B

r:

->

A B is the al-reflection

of B and

BL)AB=BL>ZL>AB is its

By the universality there

(epi, .ll)-factorization.

such

h

that

o

r

mohor=rnce=r= so

that

by the uniqueness condition and

retract

[1: A a

some

I

—>

lor

for universal

hence

monomorphism,

a

exists

Hence

e.

=

an

maps,

isomorphism.

oh

m

=

1. Thus

Consequently,

m

is

r

is

a

an

epimorphism. (1)

Let

(11).

=>

Hence B is

(II)

=

of

cases

A

so

source,

B

1—)A,-) be

If B —'—s A, is

whobject. morphism ff:

an

the

(B,

B

_.

an

A; such that

that

since

r

is

(extremal mono)-source where each Ai is d-epireflection for B, then for each i there is a for f,- f,- r. This then provides a factorization an it must be an epimorphism, isomorphism. an

o

=

d-object.

an

(12). This is trivial since products and extremal (extremal mono)-sources. This

(12)=(10). extremal.

immediate

is

since

every

subobjects

regular

are

special

monomorphism

is

E]

The above

characterization

theorems

guarantee

the existence

of most

of the

reflections

given in the examples 26.2(2) (independently of any special constructions). In particular, a subcategory of Grp (resp. Top, Topz, etc.) is epireflective if and only if it is closed under the formation of products and subgroups and and closed subspaces, etc.). (resp. products subspaces, products In many cases in all coreflective (e.g., Top) subcategories are automatically monocoreflective. The reason for this is shown in the following theorem. CHARACTERIZATION

37.3

If

.93 is

(l) (2) (3) (4)

93, then

the

coreflectire in

a!

is

d

is both

s!

is

category

lll

cocomplete, well-powered,

for

separator

THEOREM

following

are

and

co-(weIl-powered)

contains

3?.

I.

under

if .2!

equivalent:

monacoreflective and epicoreflectivein 33. strongly closed under the formation of I-colimits

.2! is closed

and

the

formation

of

colimits

in 33.

in

.93

for

each

small

a

Reflective Subcategories

284

(5)

.2! is closed

(6)

.2! is

(7)

M

is

under

the

Chap.

X

of coproducts and coequalizers in .93. strongly closed under the formation of coproducts and coequaiizers in Q. closed under the formation of coproducts and extremal quotient objects formation

in 3.

Proof: We will show (1) => (2) => (3) = (4) => (5) => (6) => (7) => (1). That Furthermore the implications (2) => (3), (3) => (4) and (4) => (5) is immediate. => of the (6) (7), and (7) => (1) follow immediately from the dual statement II (37.2). Thus we need only show Characterization Theorem that (1) => (2) and (5) ==- (6). then each Q-object is a (1) => (2). If S is a separator for 3 that belongs to .5241, of some ’S ofS that (19.6 dual) quotient copower again belongs to .2! (37.2 dual). Hence 5! must be epicorefiective in a (36.7 dual). Thus 5/ must also be mono-

coreflective

in 93

(5)

Let (c,

(36.3 dual). 1

(6).

=

C) be

of B .:;

coequalizer

a

A, where

A is

an

9

is sufl‘icient to show for .123,then

there

exists

(19.6 dual). Since

E;

’S

A

C is

that

an ’

(163 dual). Since

fl-object that is a separator ’S of S and an epimorphism e: ’S B copower must be the of (c, epimorphism, C) coequalizer Jul-object.

an

S and

A both

We

turn

now

belongs

to

at.

an

to

s!

(37.2 dual), the hypothesis

of

generation

C]

attention

our

of “nice”

subcategories 37.4

C

If S is

belong

9-:

(5) implies that

It

—>

some

is

e

sat-object.

the

to

notion

of d—reflective

categories.

THEOREM

If .93 is

(6”,.ll) category

an

that has

products and

is

6-co-(well-powered), then

(1) The intersection of any class of 6-reflective subcategories of .93 is also g-reflective in 33. (2) Each subcategory s! of 96‘ can be embedded in a smallest 6"-rcflectivesubcategory £092!)of Q, the objects of which are precisely the .ll-subobjects of products of sad-objects in :93. contains

first assertion, all JI-subobjects of

from

the

Characterization

from

the

fact

The

Proof:

formation 37.5

products

of

61.91), and the fact that as!) products of std-Objects in 3? follow immediately I (37.1). The reverse Theorem containment follows

these

under

and

existence

circumstances

compositions

(33F

the

class

and

33.1).

all

is closed

under

the

D

DEFINITION

If .52! is

contained

in

the g-reflective

of 93, as!) DUAL

d

that

of

the

in .43.

of .46, 6' is a class of epimorphisms of .43, and .2! is g-reflective smallest subcategory 6°(ss’) of 9’3,then 60:!) is called hull of a! in 99. For the case that 6’ is the class ofall epimorphisms

subcategory

a a

is called NOTIONS:

the

epireflective hull

.ll-coreflective

hull

of d of d

in (E. in

3?; monoeoreflective

hull

of

285

of & Reflective Subcategories

and Generation

Characterization

37

Sec.

COROLLARY

37.6

If

.9 is

(I) The

complete, well-powered

intersection

of

any

and

co-(well-powered),

then

of epireflectice subcategories

class

of

{3

is

epi-

reflectivein 559. (2) Each subcategory d of Q has an epireflectiuehull in cisel y all extremal subobjects of products of d -objects. Immediate

Proof: extremal

(epi,

the

from

fact

the

objects

are

pre-

:3 is

given hypotheses,

an

D

(34.5).

mono) category

under

that

93 whose

EXAMPLES

37.7

(I) CompTZ is the epireflective hull of the closed unit interval [0, I] in Topz. (2) CchTz is the epireflectivc hull of [0, l] (or of R) in Top. (3) If X is the topological space with two points and three open sets, then the epirellective hull of X in Top is the category of all To spaces. monocorefiective

(4) The

G with

groups

the

Zn in Ab that r10 {O}.

hull

of

is the

subcategory

of all abelian

finite

groups

=

property

hull of the category (5) The monocoreflective torsion Ab is the category of all abelian groups.

of all

abelian

in

DEFINITION

37.8

A

is called

C ail-morphism f: B A. where fi-morphism g: B such that the triangle -»

A is

—>

an

d-extendable

.nl-object.

provided exists

there

that

some

for each

g7: C

—»

A

commutes.

PROPOSITION

37.9

Let

a

let

as!)

if

and

be

an

(6’, .ll) category

that

be the 6-reflective lmll

only if

it is

of d

has .

products Then any

and £~co-(well~powered), morphism f in 6“ is tai-extendable and

is

6(d)-extendable.

morin £(ssl). each 6(d)-extendable Proof: Clearly since a! is contained C is a that f: B To show the converse. suppose phism is also si-extendablc. X is a and B of Xis an 6 is that .nl-extcndable. (its!) g: object morphism in 37.4, there is a set-indexed family (xii), of (ii-morphism. According to Theorem .II. to Since f is sim: X that and a I'IA; belongs .cl-objects morphism is a morphism g5: C for each ie I there extendable. A; such that g,- of is a of there m Now the definition I'IA, morphism g’: C by product g. in -+

-—»

—>

—>

o

o

=

_.

Reflective Subcategories

286

such

that g;

the

diagram

=

g' for each

0

nt

(l’IAi, in.) is

i. Since

X

Chap. a

the square

mono-source,

in

B—/>C i

9

'9 i

x—>nA‘.——+Ai m E‘Hence

commutes.

C

phism g:

there exists a (6, .l/)—diagonalizationproperty, that a of 9. Consequently f is £(d)-extendable.

the

by

X such

—.

mor-

E]

=

COROLLARY

37.10

If a is complete, well-powered and co-(weII-powered) and if ‘6’ is the epireflective hull of .9! in .933,then a Q-epimorplzism is d—exrendable if and only If it is ‘K-extendable. [:J

EXERCISES 37A.

of items

equivalence

(2) through

(10) in the Characterization

II (37.2).

378.

if it is

the

Prove

Theorem

that

Prove

cocomplete

37C.

as

a

a

subcategory of CompTz is reflective in CompT, if and only in its own category right. full

Filling Properties QC» %’ be full embeddings

LetJlC»

lets!

and

be monoreflective

that the following are equivalent: (i) {B is reflective in %. in $2 (ii) 9 is monoreflective 3 is in Q)”. (iii) epireflective (b) Show that if ‘6 is finitely complete. then (I) and (2) below

in ’6.

(a) Prove

are

equivalent:

If

(I)

P—>A

l

i

B—>C is

a

pullback

A is

square,

gal-object,

an

and

B is

a

vii-object, then

P must

be

a

Q-

object. (2) (a) If X 1+ be

a

B is

fi-object,

([3) If (If a: has these (0) NOW let J!

A is

an

regular

d-object

properties. be

a

monomorphism

and

B is

a

fi-object, then

X must

and

some

and

B is

it is called class

of

Q-object, then d—fitting.) a

monomorphisms

A

in %’ that

x

B must

be

a

iii-object.

is left-cancellable

(i.e.,

Sec.

C haraclcrimtion

37

whenever

f

c

g

6

and

.11, then

9

of g-Refleclive S ubcalegories

Generation

.11) and equivalent e

is Jl~reflective

a!

that

assume

to the following: (i), (ii), and (iii) above are in ‘6’. (iv) .93 is .Il-reflective If, in addition, % is complete, well-powered, and co-(well-powered),

the formation

under

fo

9 is

and

an

of intersections

epimorphism and f are equivalent

in ‘6. Prove

and

J!

that

is closed

has the property that whenever pullbacks show that (i), (ii), (iii), .11, then 9 is an epimorphism; the following:

e

(iv) above

to

and

287

and

of products and extremal (v) Q is closed under the formation subobjects. .4! is and closed under the formation of intersections. (vi) d-fitting strongly of intersections. (vii) Q is fl-fitting and closed under the formation of finite products and arbitrary intersections. (viii) Q is closed under the formation that d, e. and .1! have the above properties, Q is xii-fitting, and 53 is the (d) Assume epireflective hull of 58 in ’6’. Prove that the Q-reflectionof 3 @-object C am be obtained as

the

Q and

intersection contain

(e) Apply J!

the

=

37D.

the

of all those

of the M-reflection

Jl-subobjects

belong

to

above

results

the

to

.d

where

case

Comp'l‘z. ‘6’

=

=

CRegT2. and

topological embeddings. (3) Prove

simultaneously

that

reflective

and

full, isomorphism-closed

a

coreflective

in

of Top that is subcategory with Top. subcategories of Ab that are simulta-

Top

all full. isomorphism-closed (b) Characterize in neously epireflective and monocoreflective

coincides

Ab

(Cf.

Exercise

37E. Reflective Hulls ‘6’ be complete, well-powered and co-(well-powercd), .2! subcategory of g, and 5? the epircflective hull of a! in ‘6’. bet

Prove

of C that

C.

23C(c)]. full isomorphism-closed

a

that:

(a) :59 is complete and well-powered. If 98 is co-(well-powered), then .2! has the epireflective hull old in Q.

(b)

“reflective

a

in ‘6’ that

hull"

coincides

with

Let .2! be a reflective subcategory of a complete, well~powered, co-(well3!. let R: E —v .9! be the reflector, and let r,: B -> R(B) be the category d—reflection of B. Show that for any fi-epimorphism f: B —> C. the following are

37F.

powered)

equivalent: (a) f is d-extendable. (b) R(f) is an isomorphism. (c) There exists a 38-morphism (d) There exists a Q-morphism

g:

C

-»

g:

C

—>

R(B) such R(B) such

that

g

that

the

=

f

=

r5.

diagram

f B

—>’.C ,

/

//

’u

/

’1:

9

’1 u

R(B)

WINCH

commutes.

376.

Let .9? be

an

(6, J!)

category

that

has

products

and

is

6-c0v(well-powered).

Reflective Subcategories

288

let are

80:4) be the vii-reflective equivalent:

hull of .a’ , and

let B be

Chap.

{la-object. Prove

a

that

the

X

following

object of 60%). B is an l-subobject of a product of .aI-objects. Each d—extendable dimorphism in Q is {B}-extendable. Each d-extendable 8-morphism f: B -> C is an isomorphism. Each .d-extendable (e) morphism f: B —> C is an .ll-morphism.

(a) (b) (c) (d)

B is

an

e

and 37H. if a? is complete, well-powered, epireflective hull of a! in Q, then show that for equivalent:

(a) B is

a

co-(well-powered), each

Ji-object

if %’ is the

and

B. the following

are

g-object.

subobject of a product of .saI-objects. (c) Each d-extendable epimorphism in {E is {B}-extendable. .aI-extendable Each (d) epimorphism f: B -> C is an isomorphism. monomorphism. (e) Each .d-extendable morphism f: B -> C is an extremal (b)

B is

an

extremal

§38 In this

section

we

concern

ALGEBRAIC

ourselves

SUBCATEGORIES

with

the

algebraic (or varietal (38.3)) category (Recall that in this chapter all subcategories isomorphism-closed.) of

an

38.1'

question of when a subcategory is itself algebraic (or varietal). are

assumed

to

be both

full and

THEOREM

If (93, U) is an algebraic category bedding E: .n! C» 98, then the following

and are

s!

is

a

subcategory

of

33 with

em-

equivalent:

(l) (d, U E) is algebraic. (2) d is reflective in Q and contains with each morphism its (regular epi, mono)factorization in Q. each Q-object that is simultaneously a (3) .a/ is reflective in 33 and contains subobject of some .sJ-object and a regular quotient of some d-object. o

Proof: Clearly (2) and (3) are equivalent since each algebraic category is (32.13). To show that (l) implies (2), uniquely (regular epi. mono)-factorizable E must be an assume that (d, U E) is algebraic. Then algebraic functor d in and E preserves is reflective E has a left that Hence at) (32.20). adjoint (so in .9! (32.18). Thus 5! contains with each the (regular epi, mono)-factorizations .519. in its morphism (regular epi, mono)-factorization that To show (2) implies (I) it is sufficient (since the composition of is algebraic) to show that a! has coequalizers and that E is algebraic functors regular epimorphisms and since 51 algebraic. By hypothesis E preserves is is reflective in u E has a left adjoint. Also .98, being an algebraic category, be .sa’ is reflective in it must .90, cocompletc cocompletc (32.14). so that since c

38

Sec.

a!

(36.18). Thus epimorphisms. in 3.

289

Algebraic Subcategories

Since

has

Let g:

9? is

It remains

coequalizers. A

3 be

—>

complete (32.12),

we

form

can

show

to

.nl-morphism

an

that

that is

E reflects

regular regular epimorphism

a

the congruence

relation

of g in .93.

P

B—>A

.1 l. 1—»;

.4

(g, 21)is the coequalizer in .9? ofp and in 9?, E must reflect limits (36.13). Thus

Thus reflective

Consequently (9, xi) is reflects regular epimorphisms. 1:] belongs

to sf.

the

(21.11).

q

the

above

in s!

coequalizer

Now

since

pullback

ofp

and

q.

is

a!

square

Hence

E

COROLLARY

38.2

If (a. ding

E

then

the

:

(1) (d, (2)

a!

is

(3) (4)

a!

is

a!

is

is

U)

.n/ L,

9?, such

.9! is closed

are

U

algebraic.

is

E)

is

under

Proof: By the (3), (4) follows It should

theorem

subcategory then (d, U

from

o

(1) and the

(2)

E) is

are

in 51?.

that

it is not

The

equivalent.

Characterization

algebraic category algebraic.

an

subcategory of .93, with embedformation of subobjects in 9?,

equivalent:

be remarked of

a

the

reflective in .9?. a complete subcategory of 51?. closed under the formation of products

and

38.3

that

following a

and s!

algebraic category

an

Theorem

always

true

11

that

(.99, U), with

equivalence (37.2). C]

if a! is

an

embedding

of

(2),

epireflective E: sic»

55,

DEFINITION

An

algebraic category

congruence 38.4

(at. U) is called

varietal

provided

that

U reflects

relations.

EXAMPLES

The

following algebraic categories are varietal: Set, pSet, SGrp, Mon, abelian and Grp, Rng, R-Mod, R-Alg, BooAlg. Comp'l‘z, compact groups, with the unit disc functor (commutative) C*-algebras [together (30H)]. The following algebraic categories are not varietal: torsion-free abelian zero-dimensional

groups. 38.5

spaces.

THEOREM

If (.93. U) then

Hausdorfi

compact

the

is varietal

following

(1) (at. U

o

are

amiss!

is

equivalent:

E) is varietal.

a

subcategory of

.9? with

embedding

E: .215» .93.

290

Reflective Subcategories J! is

(2)

reflective

in 93 and

Chap.

X

if A

—>71

Z—~> —->B is

pullback square d-objects, then B is a

Pro

(I)

-—~

an

f

is

a

regular epimorplrism

and

A and

Z

are

.d-object.

of: Since

(2).

in Q and

where

in 3

(.131,U

E: dc-v

3

o

E)

must

is varietal, and hence

algebraic,.si must be regular epimorphisms (38.1). If

preserve

reflective

AL»?

1

[I

278 is

pullback

a

gruence preserves in d. If

in 93 and f is a regular epimorphism, then (p, q) is a consquare for f and (f, B) is a coequalizer of (p, q) (21.16). Since U o E reflects congruence relation relations, (p, q) is a congruence

relation and U

(e, C)

2

Coeq(p, q) in .xs’,then

AL»?

X—c’c pullback square in .91 (and hence in 93) and c is a regular epimorphism in a! (and hence in w). Consequently (c, C) is a cocqualizer of (p, q) in 93 (21.16). B and C are isomorphic, which implies that B belongs to .24. Therefore (2) = (I). We first will show that (.124,U E) is algebraic by showing that it contains with each morphism, f, its (regular cpi, mono)-factorization in :3 To see this let (38.1). is

a

o

be

a

Now

pullback

square

let (c, C) be the

in Q.

Since .9! is reflective

cocqualizer

of p and

in .93, P must be in .9! (36.13). let II: C —» B be the

q in (B and

See. 38

Algebraic Subcategories

unique 9-morphisrn factorization off in

with a

291

h c. Then f ho c is the (regular epi, mono)f the of Theorem (see proof 32.3) and the square =

o

=

P

P—->A

‘11 1c is

pullback square in Q (21.16). Thus by (2) C is an Jul-object, so that h and c belong to s1. Consequently, (sat, U E) is algebraic. It remains to be shown that E reflects congruence relations. Let (p, q) be a congruence relation in Q where each of p and q are d-morphisms and let (c, C) be the coequalizer of p a

o

q in 9.

and

Then

A—p—)B

B—c—IvC is

pullback square in a (21.16) so Hence ([2, q) is a congruence relation a

that in

by (2) it is .51. [:1

a

pullback

square

in set.

COROLLARY

38.6

(HQ, U) is varietal ifs! is closed under equivalent: and

and .m’ is the

subcategory of Q formation of subobject: a

embedding E: d L» {B Q, then the following are

with in

(1) (s1, U E) is varietal. (2) s1 is reflective in 33 and is closed under the formation of regular quotients (3) at is closed under the formation of products in g3 and regular quotients o

‘

in Q. in Q.

Proof: (1) => (2). Clearly sat regular epimorphism in

must

9

be reflective

and A is

an

in 9.

eel-object.

Suppose

that

f:

A

—»

B is

a

Lct

P—p>A

l ’

l——>B r

[I

pullback square in d. Then by the canonical (21.3), Pis the object part of a subobject of A x in .93, 1’ must be an sat-object (36.l4). Hence by d—object. be

construction

a

A. Thus

the

since

theorem,

for

pullbacks

.91 is reflective B

must

be

an

Reflective Subcategories

292

(2)

This

(3).

a

is immediate

of I-limits,

formation

is clear

(I).

=

from

that

U

(d,

o

the theorem.

closed

are

E) is algebraic. [:1

under

the

That

it is

is not subcategory of a varietal category necessarily closed formation of subobjects of regular quotients. (See Exercise 38E.)

varietal

A

the

under

subcategories category I (36.13).

discrete

By Corollary 38.2 it also varietal follows immediately (3)

reflective

since

for each

X

Chap.

EXERCISES Prove

38A.

varietal

is

finitary if and only if it is strongly finitary

that

a

that

in mach finitary varietal

category

(see 22B and 326). 388.

Show

(cf.

commute

that

and

CompT2

(considered

38D.

direct

category

limits

finite limits

and

320).

that

Show

38C.

Top,

253

as

a

is the

concrete

full epireflective subcategory only non-trivial? is varietal. category)

of

Identities I

For

category

any

5!,

a

pair of sat-morphisms

A

3

B with

quasi-identity ind.

It

i: called

domain

common

and

identity provided that B A quasi-identity is said to hold in an .szI-objeet C if and only if is regular-projective. k of k g for each morphism k: B —> C. If 9-" is a class of quasi-identities in 5!, for which then the full subcategory of .9! whose objects are precisely those d-objects .9; in is denoted each quasi-identity holds, by deflf). that (.91, U) is algebraic and 33 is a full subcategory of .321. Now suppose is called

codomain

common

a

an

a

=

.

(a) Prove that the following are equivalent: of products and subobjects. (i) Q is closed under the formation #67). (ii) There exists a class 9 of quasi-identities in.saf such that Q are Prove that the following equivalent: (b) of products, subobjects. and regular quotient (i) a? is closed under the formation =

objects. (ii) There exists 38E. least

two

continuous

a

class

.“7 of identities

(Q, U) be CompT2 and let A be a compact that the identity on A points and the property self-map of

that if E: (c) Conclude (d) Show, however, that quotients in Q.

T A subcategory

points.

.93

that

Let

A.

(Such

a

space

(a) Prove that the only non-constant projections. (b) Show that the full subcategoryd 38.5. condition (2) of Theorem

two

in .s/ such

ol‘

51¢» Q

CompT,

d

is

not

strongly

continuous

maps

of 9? whose

objects

is the closed

is non-trivial

is called

then

embedding, under

if it has

the

a

Hausdorff is the

with

space

only

at

non-constant

rigid.)

from

are

a

power

the powers

(.ss’. U

formation

space

.1109").

=

whose

o

of

A’

A

to

are

the

A' of A satisfies

E) is varietal.

subobjects

underlying

set

or

has

regular

at

latst

XI Pointed

Categories

in category Virtually all algebraic notions theory the most “classical" of categories the category .

.

parodies of their parents

are

in

of left A-modules.

.

H.

East

The

doubt the most (for various rings R) are without categories R-Vlod Because of their nice thoroughly investigated categories. properties, they provide a useful tool for the study of other categories as well. For example, algebraic of functors topology is essentially the study of topology by means (homology. cohomology, higher homotopy) from topological categories into categories of R-modules. The reason that the categories R-Mod are so nice is surprisingly simple. Besides the fact that they are (considered as concrete categories) finitary are other more algebraic (as many complicated categories such as SGrp and Mon). they are distinguished by the fact that they have finite “biproducts”; i.e., finite products and finite coproducts whose corresponding object parts coincide. and that they are (normal epi. normal mono) categories. (This latter condition allows one to define the extremely useful concept of “exact sequences") In addition, for the categories be R-Mod. each morphism set Itom(A, B) can of an abelian in such a way that uniquely supplied with the structure group acts on the leftand onthe distributively morphism composition right with respect to the group addition. As we will see. the categorical properties mentioned above For example. the existence of a group independent. to the existence of finite morphism sets is closely related the above will be studied relationship among properties

are

not

Categories "locally"

of R-modules like

categories

will be characterized of

R-tnodules

introduced. i

From

The

Murilu

‘l‘Iu-on'rm.

293

and

(called

those abeliatt

structure

on

biproducts. in

this

categories

that

categories)

the

The

chapter. behave will

be

294

Pointed

Categories

of this chapter indicates

As the title

we

will

each

and pointed. category is non-empty usually be denoted by 0 (as will the zero

§39 In this

EXACT

throughout it that morphisms will objects, when they occur). zero

CATEGORIES

will

we

X1

assume

The

investigate the question has especially nice factorization properties. This when the category is “exact" (Theorem 39.17). Exact

section

AND

NORMAL

Chap.

of when will be

pointed category occur precisely

a

to

seen

Categories We

which

results begin by restating some (see 168, 16L, and 27R).

established

have

essentially already been

PROPOSITION

39.1

A

lf f: following

monomorphism equivalent:

are

(1) (K, k) (2) K is a

B is

—»

z zero

a

KerU). object,

0.

k: K

and

—>

A is any

morphism,

then

the

[:1

COROLLARY

39.2

Each

that

category

has kernels

or

cokernels

also has

a

zero

C be

a

monomorphism

object,

0.

[:1

PROPOSITION

39.3

Let

f

:

A

—r

B be

an

d—morphismand

In:

B

->

in d.

Then:

(1) (K, k) z Kerlj) ifaml only if(K, k) 0. [:1 (2) Ker(Ker(f))

z

Ker(m of).

=

PROPOSITION

39.4

(resp. Let

39.5

an

object of

.92)be the

G: cf

let F: 2 Then

.51 that has kernels aml cokernels. Let J the category class all of snbobjects (resp. quotient objects) of A. quasi-ordered 3 be the map that semis each subobject (S, m) of A to Cok(m), and rf be the map which sends each quotient object (q, Q) of A to Ker(q).

A be

Let

—r

->

(d, 2, G, F)

is

a

Galois

correspondence (see 270).

E]

COROLLARY

If

.14 has

kernels

aml cokernels.

then

(i) For each .d-morphimn f, Ker( f ) z Ker(Cok(Ker(f))) Cok(l\’er(Cok(f))). (2) An xii-morphism f is a normal monontorphism if and only if f (3) An .sI-morphism f is a normal epimorphism if and only if f z

and

z

Cok(j)

z

Ker(Cok(f)). Cok(Ker( f D.

Sec.

39

Normal

and

Exact

Categories

each

For

cal-object A, the quasi-ordered anti-isomorphic with the quasivordered ofA. E]

(4)

class

is

39.6

295

of all normal of all normal

class

subobjccts of A quotient objects

DEFINITION

A category

is called:

(1) normal provided that it has kernels and cokernels, is (epi, mono)-factorizable, and each of its monomorphisms is a normal monomorphism.

(2)

eonormal

factorizablc,

(3)

Note are

that

duals

of each

the

each

a

that

case

39.7

normal

is

exactness

that

out

and

a

a

and

self-dual

normal

cpimorphism

need

has

conormality concept. with kernels

category

monomorphism also

(epi, mono)epimorphism.

conormal.

is

the category

normal

a

and

equalizers

or

notions

which

cokernels

that

monomorphism

and

(sec Exercise

393).

be exact

not

are

it must

be

the property

in

coequalizers,

39.19).

table

following “—"

question and

means

(l) R-Mod

—

(5) The full subcategory

at

property

that that

epimorphism

each

pTop consisting

of all

pointed compact

ofAb

consisting is

of all abelian

whose

groups

a

underlying

exact.

of the

categories (I). (2), (3), and (5) above monomorphism is a regular monomorphism regular epimorphism.

in it each

is

Hausdorfl‘

conormal.

27 elements

most

Notice

of

but not

(6) The full subcategory has

it.

—

—

is normal

have

+

+

(4) Mon

not

has

+

—

(3) pSet

the category

that

Conormal

+

(2) Grp

spaces

“+" means that it does

Normal

Category

39.8

epimorphisms

is

cokernels,

normal

a

EXAMPLES

In the

set

and is

each

(see Proposition

exact

kernels

categories normality

pointed is

epimorphism

However, in

for

that

property

has

it is both

other,

be

it

of its

that

since

It should has

each

provided

exact

that

provided and

has and

the each

PROPOSITION

lfaf intersection.

has kernels.

than

each

pair of normal

subobjects of

any

d-object

has

an

Pointed

296

X1

(B, n) are normal subobjects of C, then (A, m) z morphism f. Let (D, ii) z Ker( f n). Then there exists a unique

(A, m) and

If

Proof:

Chap.

Categories

Ker( f) for

some

morphism

171: D

o

A such

—’

the square

that

or

D——->A

:l l", 3—H“

such

that

m

39.9

n

=

r

o

k such

morphism and

above

The

commutes.

is

square

a

thenfo

as,

that

s

n

F:

=

pullback s fa o

k. Thus

o

since if

square m

=

(D,

n

0, Ft) is an

so

=

r

o

0

and

morphisms a unique of (A. m) intersection

r

that

s are

there

is

E]

(B, n).

PROPOSITION

Every normal

has

category from

Proof: Immediate

a

object and has finite

zero

Corollary

39.2 and the above

intersections.

proposition (39.8).

[3

PROPOSITION

39.]0

If a! is exact, then for each doobjeet A the quasi-ordered classes of all subobjects and of all quotient objects of A are (up to equivalence) anti-isomorphic and largest members. lattices (possibly on a class) with smallest Proof: By Proposition 39.8, the class r! of all subobjects of A has finite and by its dual, the class of all quotient objects of A, infima (= intersections), has finite infima ( cointersections). Thus since :J and :2 are anti-isomorphic as [I quasi-ordered classes (395(4)), .‘2 and of each must have finite suprema. =

COROLLARY

39."

(I)

An exact

(2) Erery concretizable

exact

wish

we

especially

nice

this

is

category that

demonstrate

factorization

For

sequences. but not

to

only If it is co-(well-powered). well-powered and co-(well-pon'ered).

is

category

Proof: Each concretizable Next

and

well-powered if

is

category

purpose, Herc each

consider

categories

exact

This

properties.

[3

regular co-(well-powercd) (16N).

lirst

the

has

are

enable

will later

category

distinguished us

to

Grp that

define

by

exact

is conormal

unique (normal

epi, mono)Also the Cok(Ker(f)). cg. factorization,f the and relation of is determined by Ker( f ), question of f completely congruence can be decided a monomorphism whether or by knowing only Ker(f). notfis of the fact that Grp is conormal. All of these facts are consequences Indeed, it normal.

=

can

be

shown

equivalent

to

that

each

morphism

Moreover

m

under

other

certain

and

to

g

a

be chosen

can

conditions

the fact

that

on

s!

a

as

category

is conormal

d, these (see Theorem

are

all

39.13

Normal

Sec. 39

and Exercise

39C).

(normal epi,

normal

39.12

297

Categories

fact, the exact categories will mono) categories (see Theorem

be shown

be

to

precisely the

39.17).

LEMMA

m

=

has

M

If f

In

and Exact

o

kernels

g, and g is

a such that

a

and normal

Cok(Ker(j))

then

epimorphism, a

z

o

and

g,

Jul-morphisms such that there exists a unique sal-morphism m

are

9.

Since

Proof:

f there

cokernels, f;

exists

Ker(g)

°

m

=

[1 such

unique sat-morphism

a

Ker(g)

up

Ker(g)

that

Ker-(f)

=

0,

=

h.

o

Ker(g)

l

._—)

O

—fif

.

KerU)

X /: Hence

Cok(Ker(f)) Since g E with 39.13

Ker(g)

Cok(Ker(f))

=

Cok(Ker(g)) (395(3)), this implies 57 g. I] Cok(Ker(f)) z

=

oh

Ker(f)

o

the existence

of

a

0.

=

unique morphism

o

THEOREM

If

s!

morphism, (I)

o

s!

OF CONORMAL (CHARACTERIZATION CATEGORIES) has kernels and cokernels and each si-epimorphism is a normal then the following are equivalent:

is conormal.

(2) If f is an d—morphismsuch that Ker( f) (3) If f is an ssf-morphism such that f morphism. =

(4)

.sf is

(5)

For each

a

either then

Proof: We will show that (I) (I) =~ (2). Suppose that Ker( f) imtion of f. Then according to morphism a such that g e

is

0, then f is

=

m

a

Cok(Ker(j)),

o

monomorphism. then

is

m

a

mono-

(normal epi, mono) category.

ail-morphism j; if of Cok(Ker(f)) exists,

relation

Hence

epi-

a

section,

so

o

e

z

that f

congruence they both exist

(3) 0. Let f preceding (2)

==

=

the

m

o

e

is

and

(4)

a

=

Cok(Ker(f)) =

relation

a

=

a

m

o

or

a

congruence

coincide.

(l)

=

be

e

lemma

Cok(0)

off

=

and

(3)

=>

(5)

=>

(2).

(epi, mono)-l‘actor(39.12), there exists a an

1.

monomorphism.

Pointed

298

(2) = (3). morphism

that

Suppose such

that

f

Chap. Then

moCok(Ker(f)). Cok(Ker(m)).

=

171'o

=

m

Categories let

m

be

the

XI

unique

f

m

Cok(Ker( f))

711

Cok(Ker(m)) Now g a normal

Cok(Ker(m)) Cok(Ker(f)) is an epimorphism cpimorphism. Thus according to the lemma unique morphism 5 such that

and

(by hypothesis) (39.12), there exists a

o

=

Cok(Ker(j))

=

5

o

g

=

5

Cok(Ker(m))

c

so

Cok(Ker(j)).

o

5 Cok(Ker(m)). Cok(Ker(f)) is an epimorphism, this implies that l a section an and, hence, Consequently, Cok(Ker(m)) is isomorphism. Thus m 0 is a which that by (2) implies Ker(m) monomorphism. = the definition of cokcmel each morphism f has a factor(3) (4). Clearly by ization f mo Cok(Ker(f)). By (3) m must be a monomorphism. Hence a! which implies it is.a (regular epi, mono) is (regular epi, mono)-factorizable, category (33.4). from the definition of a conormal category. (4) => (I). Immediate mo of f induced Cok(Ker(f)) be the factorization by the (3) => (5). Let f If (p, q) is a congruence relation of fl then the diagram of cokernels. definition Since

°

=

=

=

=

1’ \

.

°

Cak(Ker(f)) 1!

Cak(Ker(f))

if

'\ f

commutes, (p, q) is gruence

so

that

the “inner

a congruence relation for

square” is a pullback square (21.10(l)) and hence relation for Cok(Ker(f)). Conversely if (p, q) is a conBy (3) m is a Cok(Ker(f)), then the above square commutes.

monomorphism. Thus the “outer square” is relation off. (p, q) is a congruence (5) => (2). If Ker(f) =. 0, then Cok(Ker(f)) so that by (5) it is relation of Cok(Ker(f)), sequently f is a monomorphism (21.17). [:1

a

pullback

square

(2|.IO(2)),

l. Hence

(1, l) is

a

congruence

relation

so

that

congruence of 12 Con-

COROLLARY

39.14

If (1) (2)

=

a

has kernels

s!

d

is conormal.

d

is

a

balanced

and cokernels,

then the

following conditions

(normal epi, mono) category.

[:1

are

equivalent:

Normal

39

Sec.

Exact

and

for

299

Categories

THEOREM

39.15

then

5% is exact,

If (1)

and

.5! is

(epi, mono) category,

an

f

Ker(Cok(f))

=

Cok(Ker(f))

c

of f. Consequently,

unique (epi, mono)factorization

is the

fi-morphism f,

each

Im( f )

Ker(Cok( f ))

z

and

Coim( f )

Cok(Ker( f )).

z

(2) For each tel-morphism f, is

f

f

.91-morphism f,

is

epimorpht‘sma

an

each w-morphism

For

(4)

a

each

For

(3)

monomorphism

a

f

is

Ker( f )

Cok( f )

from

Proof: Immediate dual, and the definition

=

0

0

f

¢>

f

a

1m( f )

a:

Coim( f )

z

Coim(f)

a.

Im( f )

c:

1.

=

the

Ker( f )

a

Cok(f)

=

theorem

characterization

above

0.

=

its

(39.13),

1—:

of exactness.

LEMMA

39.16

If

m

is

normal

a

monomorphism

and g

then

Cok(m),

z

m

Ker(g).

z

Proof: Since m is normal, m z Ker(f) for some morphism fl and g z Cok(m), there exists some f morphism f such that f r 0, then morphism such that g o

that

exists

there

a

r

THEOREM

For any

fog

=

unique morphism m

39.17

z

(CHARACTERIZATION

category,

of, the following

(1)

.2! is exact.

(2)

.27 is

(normal epi, normal

(3)

42¢ is

a

(normal

Since

f

9. Now

=

o

oh.

Thus

o

m

if

=

r

is

=

fo so

1.

=

f,

isomorphism

an

=

epi, normal

o

r

=

[1 such

0

that

r

=

m

Ker( g).

EXACT

OF are

CATEGORIES)

equivalent:

mono)-factorizable. mono) category.

Proof: (1)

=>

(2)

=:-

category

Immediate

from

the above

theorem

(39.15). Immediate from the fact that every (regular epi, mono)-factorizable is a (regular epi, mono) category (33.4).

(2). (3).

0 a

300

Pointed

Since

(3) is self-dual,

Categories

Chap.

XI

has kernels and each only show than! is normal. Let f be an .nl-morphism and let f m e be its d-monomorphism z normal (normal epi, mono)-1‘actorization.Then e Cok(g) for some morphism m E be the (normal epi. normal of y. We wish 9. Let g mono)-factorization

(3)

(1).

=

need

we

o

=

c

=

to

show

that

6 is

Since

m

m is

n7

it follows

epimorphism,

an

Cok(g)

z

e

Since

Ker(f).

z

normal

a

J!

Consequently

Coho?!

a:

e)

o

has

Kcr(e)

=

=

Comm)

z

monomorphism, by

Ker(Cok(tTt))

z

that

(39.3(1) dual).

the lemma

Ker(m

o

e)

=

Ker(f)

(393(1)).

kernels.

fit é is its (normal epi, normal and h d-monomorphism mono)-l'actorization, then é is a monomorphism and a normal epimorphism; hence an isomorphism. Consequently, h is a normal E] monomorphism.

If h is any

=

a

PROPOSITION

39.18

If .a/ has kernel: then a! epimorpht’sm,

and is

coequalizers conormal

a

and

If

each

aI-epimorphism

is

a

normal

category.

for (2) of the above characterization Proof: We will establish condition mor0 be a of and let that (r, s) pair Ker(f) conormality (39.13). Suppose z exists a Thus there r 5. Now let for which 3). C) Coeq(r, f (9, f phisms Lemma 39.12 that there It g. Consequently, implies morphism [1 such that f ]. exists a Cok(0) unique morphism g7 such that g cg z Cok(Ker( 1)) r s. so thus an that Hence g is a section and an epimorphism; isomorphism, is a Therefore E] monomorphism. f =

o

=

o

0

=

=

=

PROPOSITION

39.19

Suppose that a! has equalizers and coequalizers, each .nl-monomorphism is a monomorphism. and each d-epimorphism is a normal epimorpht’sm.Then

normal d

is exact.

Proof: Immediate Exact

Sequences

39.20

DEFINITION

Let a!

be exact.

infinite) interval each

n,

n+1

(l) cod(f,,) (2) MU“)

e =

3

of

from

the

A sequence

above

proposition

Ml)”,

of

integers is said

I

dom(j;,+l), firm“)-

and

to

be

(39.18) and its dual.

sat-morphisms an

exact

by a (finite or provided that for

indexed

sequence

[1

Sec.

39

Normal

39.21

and

Exact

Categories

301

PROPOSITION

If —f-> L» equivalent: .

is

(1) (f, g)

(2) Cok( f) (3)

g

of

exact

an

in

exact

an

i.e., [mm

sequence;

then the

category,

following

are

Ker(g).

x

Coim(g).

z

0 and

=

morphisms

are

.

.

Cok(f)

Ker(g)

o

0.

=

Proof: (1)

(2).

=.

and

By (I)

Theorem

39.]5(l),

Ker(Cok(f))

Im(f)

z

2:

Ker(g).

Thus

Cok( f )

Cok(Ker(Cok( f D)

z

Cok(Ker(g))

z

(2) =. (3). (i) g °f (1M9) C0im(g)) °f 1M9) (ii) Cok(f) Ker(g) Coim( g) Ker(g) = Since (3) (I). 0, we have g of °

=

o

=

o

Caim(g)

(Coka) °f) Cok(Ker(g))

°

=

z

=

=

o

lm(9) Ker(g)

(39.5(l)). °0 =

=

0.

0.

=

g

Hence

lm( f)

0

9

(Im(f)

o

0,

=

Thus

Ker(g)

39.22

0

and g

—.

.

that

.

=

Ker(g)

z

Im( f).

(3)

0

(4)

.

L»

(5) 0

-+

The

exact

Im(f)

o

k

(39.15(1)).

E]

and

only if f

is

0 is

exact

if

and

only if g

is

—v

L)

.

i»

—’

.

L,

.

(i) 0—> 0L) is

is exact

.

0 is

exact

0 is exact

—.

following

.i»

.

g is

.

(iii) 1‘

(8)21;

=

.

.

-

epimorphism.

[fond only ifg

z

Cok(f).

and

only iff

is

an

isomorphism.

—t0isexact.

an

.

an

Ker(g).

a

.

monomorphism.

a

z

if equivalent:

are

then

ifand only iff

monomorphism and g (iii) epimorphism and f z (7) The following are equivalent: is exact. (1) —[-9 is i» -f—> is exact. (ii) (ii) f

category,

if

.

.

an

exact

L»

.

in

—f-> is -

—.

morphisms

are

.

i»

(2)

(6)

so

k

o

PROPOSITION

If f (1)

Im(f),

S

Coim(f).

o

o

Ker(Cok(f))

=

0

=

by the definition of kernels there exists a morKer(g) [1. Hence 1m(f ) s Ker(g). Similarly since exists a morphism k such that

=

=

Ker(g)

0

=

that

so

phism 11 such that 1m(1') 0, there Cak(f) Ker(g) o

Coim(f))

o

z

CokU'). Ker(g).

0.

AL>A isexactifana'onlyifA

=

O.

E]

Pointed

302

C Itap. XI

Categories

Functors

Exact

DEFINITION

39.23

A functor

F: d it

that

provided

r

—(f—’> fl»

exact,

then

39.24

PROPOSITION

i.e., whenever

sequences;

.

functor

exact

an

l»

.

—’-> is .

.

.

.

is called

categories

exact

exact

preserves

r

Each

.93 between

—>

exact.

is

.

functor preserves zero objects, zero morphisms, kernels, cokernels, epimorphisms, monomorphisms, images, coimages, and (epi, mono)-factorizations. exact

Immediate

Proof: 39.25

from

39.22.

Proposition

[:1

PROPOSITION

If

F is

a

functor

between

then

categories,

exact

(I)

F is exact.

(2)

F preserves

(3)

F preserves

(4) (5)

F preserves

of the form 0 kernels and epimorphisms. cokernels and monomerphisms.

F preserves

kernels

exact

sequences

and

—»

the

following

i» —f—>

.

.

.

equivalent:

are

—»

0,

images.

Proof: Clearly (I) implies (2), and (5) implies (I). By Proposition 3922(6), (2), (3), and (4) are equivalent. Thus we need only show that (2) implies (5). If

kernels by the equivalence of (2), (3), and (4), F preserves z Thus since for each morphism f, 1m(f ) Ker(Cok(f)) (39.15(1)), also. [:1 images then

(2) holds,

and cokernels. F must

preserve

EXERCISES

39A.

Prove

that

39B.

Letss’

be the

whose

spaces,

(I) (2)

a

pointed

each full

only objects one-element

that

that

39C. the class has

.91 has kernels

epimorphism

equalizers

or

topological

and

in which

space

the

and

the

distinguished cokernels,

in .n/ is normal,

following

point, and

that

each

but

that

sets

the

are

set

monomorphism .ss’ is

not

open: Q, the entire consisting of the two

ins!

is normal,

and

exact.

that has kernels and cokernels and that ind category is under Prove that either closed ifs! composition. epimorphisms is (epi. mono)-l'actorizable, then the following are equivalent:

Suppose of normal

pTop of pointed

of the category

are:

a

each

is balanced.

category

subcategory

space,

pointed three-element of set, the set consisting points. non-distinguished Prove

normal

thats/

is

(a) Each regular epimorphism (b) For each .sl-morphismflfis

a

insal a

is normal.

monomorphism

if and

only if Ker(f)

=

0.

Sec.

39

Normal

(c) For

and

Exact

Categories

303

each

tel-morphism f, the unique morphism m with f monomorphism. (d) .91 is (normal epi, mono)-factorizable. (e) a! is a (normal epi, mono) category. relation of f (i) For each d—morphism f, if either a congruence tion of Cok(Ker(f)) then both exist and coincide. exists, they

m

=

39D.

Prove

that

Cok(Ker(f)),

o

or

a

congruence

is

a

rela-

if q

C ——)A

if

n

—g>D

B is

in

pullback square monomorphism. a

39E.

Prove

that

a

normal

for any Ker

0 39F.

then

category,

fin

morphism

(IV)

Suppose that

in

an

exact

a

with

diagram

(3) There (b) There

is

a

is

a

39G.

exact

a

sequence

.

>0156xact.

C

>

0

;

l

B'——>C’-——)0 that

morphism

->

A’ that

morphism

C

->

C

in

the

category,

>-

> B

A

’

the following

are

makes

the above

makes

the above

that

equivalent:

diagram diagram

commute.

commute.

Lemma

Nine

Suppose that

exact

only if p is

category

Prove

rows.

if and

monomorphism

Cok(f)

0—-)A'——-> is

a

>-

7‘ A

0

an

I

:n

;.

f is

an

exact

category ()

l

A!

O

B'—-—)

——)

uu—kzé— e—ue— l

l

0*——->C'———>C*—>

(—— 0

0 is

a

diagram

exact

that

and

commutes

exact

rows

and

columns.

Prove

that

there

sequence 0

which,

has

o

A

——>

D

—'

Q

—)

0

0

—>

C

—-)

Q

—)

E

—>

0

and that

such

the

diagram

o

‘1

O

v

commutes.

that

Prove

391.

o

.

__)

o

—)

o

Fm

FU) .

is exact.

-a—)

then .

.

0

sequences

of the

form

Sec.

Additive

40

C alegories

305

functor zero (a) Prove that every half-exact preserves objects and between exact (b) Prove that a functor categories is if and only if it preserves kernels. (i) left-exact if and if it cokernels. (ii) right-exact only preserves

morphisms.

zero

if and only if it is both left-exact and right-exact. (iii) exact Prove that: (c) Let A be an R-module. » Ab is left-exact. (i) Hom(A, _): R-Mod » R-Mod Ab is exact if and only if A is projective. (ii) Hom(A, _):

®

Ab is

right-exact.

J3 be

exact

categories.

(i) [.21.:9] is exact. (ii) the full subcategory

of

(iii)

A

R-Mod

_:

Leta!

39M.

and

»

Determine

[5%.93] consisting

whether

of

all

exact.

or

not

0-preserving

functors

is

.

(iii) the full subcategory

of

§40 As

has

been

mentioned

the

morphism

sets

in such

[.m’,.93]consisting

ADDITIVE

of all exact

functors

is exact.

CATEGORIES

in any category of (left or right) R-modules, be supplied with the structure of an abelian

before.

Irom(A, B) can way that morphism

composition acts distributively from the categories finite products (= direct coincide with finite products) coproducts ( direct sums). In this section we will how these two seemingly unrelated see properties are linked. In particular it will be shown that a category 5! having finite products has biproducts if and only if there is a (unique) semiadditive structure on M. Recall that throughout this chapter. all categories are assumed to be pointed. group left and

from

a

the

right.

in these

In addition.

=

Biproduets and Semiadditive 40.]

Structures

DEFINITION

(I) An

additive

function

domain

+

A and

codomain

associates

that

(Al’), (A2). and (A3)] (Al)

For

(Al') structure

each

of

structure

For

an

each

ofa

with

each

codomain

common

B such

semiadditive

[resp.

structure

that

the are

an

following

ail-morphism] conditions

(Al),

+

.9! is

a

common

A and g with domain (AZ), and (A3) [resp.

satisfied:

pair (A. B) abelian

of

Jul-objects.

+

induces

on

Itom(A, B) the

of

.nl-objects.

+

induces

on

hom(A. B) the

group.

pair (A. B) commutative

(A2) Composition

B,

structure] on a category with pair (f, g) ofd-morphisms

monoid.

is left and

right

distributive

ALB%CL»D

over

+

:

i.e.. whenever

Pointed

306

d—morphisms, it

are

C alegories

Chap.

XI

that

follows

j¥(g4-h)=(f°g)+(f°m and

=(gck)

(51+ Ii)ck

(A3) The

zero

i.e., for each

morphisms of .szl-morphism f,

5! act

+

(lick).

monoid

as

identities

with

respect

to

+ ;

0+f=f+0=f (2)

is

If +

additive

an

call (5!, (resp. semiadditive then

we

(resp. semiadditive

structure

+) [and by category).

an

abuse

structure)

of notation

also

5!]

an

on

a

category of, category

additive

Concerning the above definitions. it should be noted that if a! has finite then + is completely products or finite coproducts and (d, +) is semiadditive, This tends to the above notational abuse. .s/ determined (40.13). justify by It is also worth mentioning that (A3) above follows from (A I) and (A2) but does not follow from (Al’) and (A2) (see 40A). EXAMPLES

40.2

(l)

R-Mod

is additive

(for

every

ring R).

(2) Grp is not additive. (3) If R is any ring). then R can be regarded as an additive category with exactly one object. Conversely, each additive category having only one object can be regarded as a ring (cf. 35(7)). (4) All full subcategories, all quotient categories. and all product categories of (semi)additive categories are (semi)additive. (5) it's! is ('semi)additivc, then so are 51"” and .sa’“,for any category ’6. REMARK

NOTATIONAL

40.3

If

(A),

is

a

family

of

ale/H"

for each j, k

then

d-objects, ”‘* be

1

t

0"

if if

I

e

we

let

'=k

j»:

k.

DEFINITION

40.4

(1) Let (21,-), be a family of .cl-objects. Then the family (11,-,B, m), is called hold: biproduct of (A), provided that the following conditions

(i) (B, in), IS a product of(Ai),. (ii) (m. B), is a coproduct of(A,),. (iii) 7:, a): = 6“ for each j,_k e l. t Recall

identities.

our

convention

that

all

rings

have

identities

and

ring homomorphisms

preserve

a

Sec.

(2)

40

Additive

A category

C ategarier

307

(finite) biproduets provided that has a biproduct.

as! has

family of d-objects

each

(finite)

The

the

(objecbpart of a) biproduct is usually denoted by 911,-. biproduet of a pair of objects. (A, B), is often denoted by (I‘m “.99 A e B!

40.5

“A:

set-indexed

In

particular

d

has

“3)-

EXAMPLE

The empty

of

family

has

xii-objects

biproduct if

a

and

only if

a zero

object.

[—>8,),

family of morphisms and ”A“ LIA“ 113,-,and LIB,- are the products and coproduets of (A1), and (83),, then we have defined Hf,- and 11f,-to he the unique morphisms which for each j e I make the squares Recall

that

if (A ,-

"I;""""

'-

is

a

’n

«l

Aj———+B n-

A‘-

U

Bi

”I;

"""

l» t

Iii—)3]r,-

"

"

LIB.-

t

m) and Hf,[vi of,-] (18.5 and I8.lS). Since we are now assuming that all categories are pointed, we also have the following naturally occurring morphism:

commute;

i.e., "fl

=

(f,

=

o

DEFINITION

40.6

(A; [—53,), is a family of morphisms and (m, 11/10, and (118‘, p,the coproduct and product of (Ai), and (85),, respectively, then If

69]}: LIA,is the

unique morphism

from

LIA .-

lo

-»

are

113,

“B; such that for each j, k

e

I, the square

"Bl. ”AI. "““@‘/-;-‘-')

commutes. 40.7

are

same;

PROPOSITION

If (A , [—5B‘), is a family of morphisms and (m, 921;. n;) and (vs. GB]. pi) bipmducts of (A i), am! ( 13,-)“respectively, then I'll}, Llj}, am! 63f,- are all the i.e.,

rm

=

U];

=

on.

Pointed

308

C hop. X I

Categories

Proof:

=fi°m°m

WWW"!

=k1°“°fj

since

products

5n

=

if} iae {0,-

=

=5J-x °fj=

if

k

ir

1'95 k

pig—{O

Inca!“ Thus

'=

if

””"fm‘j

are

if

fj

_

0

j

k

=

ifjgék.

and

mono-sources

coproducts

epi-sinks,

are

D PROPOSITION

40.8

Let

(3!, +) be

a

semiadditive

let (A 1), be

category,

finite family of d-objects

a

and let

A‘. i) be

d-morphisms.

Then

the

following

B l.

A,

conditions

equivalent:

are

(I) (m, B, m), is a biproduct of(A,~),. (2) (B, m), is a product of(A,), andfor allj, k e l, 22‘ p, (3) (pl, B), is a coproduct of (A ,), and for j, k e I, n,‘ 31,(4) in): I, andfor allj, k e I, 1th on}(SJ-k. o

o

=

=

6n. (SJ-k.

:

210m:

ie

Proof: Clearly each of (l) and (4) is self-dual, (2) and (3) are dual to each other, and together are equivalent to (1). Thus we need only show that (2) and (4) are equivalent. => (4). By distributivity of composition over addition (2) (A2) it follows that for each

A- e l

“NZWN’W=Z(nls°fli°ti) Thus

since

products

are

=Z(6ik°ni)

=

”t

=

“1°18-

mono-sources,

£01,421“)13. =

(4):: Then

Let

(2).

for each

(C —‘>A), be a family of morphisms. k e I we have, by distributivity

nk°z(l‘t°fi) =Z(nk°#i°.fi)

nk°f= i.e., for each

k

e

I the

triangle

=

Definef:

2(61t°fl) =fi

2.-(ui of)

Additive

See. 40

Also

commutes.

n.

o

g

fl,

=

a

we

log

=

Notice

with

unique

this

to

respect

property

if for each

since

k,

then

(B. n‘), is

Thus

is

f

309

Categories

of

the above

=f-

21(wa

[:1

(A;),.

morphism

zero

c

valid

remains

proposition

([1,. 12,-)to be the

X

interpret

=

=

product

a

that

(Elk-WOW thcnioy)

=

in the

0: B

—»

l

that

case

E. if

=

B.

lel

PROPOSITION

40.9

category and (B, m), is a product If (5!, +) is a semiadditive finite family (A,),, then (B, 11,.)can be completed in a unique way to (”iv B, Ht)! 0f(Ai)l‘

Proof: By the definition of product. for each j e I there B such that for each k e I the triangle morphism u}: A,

of the biproduct

in d

exists

a

a

unique

—'

"I

Aj

>3

""""

l”:

5,}

At:

which

by the above proposition (408(2)) (11,-),is the unique family for (m. B. m) is a biproduct of (A;),. E]

40.10

COROLLARY

But

commutes.

If (sf, +)

is

a

semiadditiue

then

category,

the

following

are

equivalent:

(I) sf has finite products.

(2) s! has finite eoproducts. (3) s! has finite biproduets.

[:1

LEMMA

40."

lf(tt.,

pz.

A, @ A2, It], n2) artd(v1. r2, BI 69 32. p,, p2) are biproducts and It: A, B2 med-morphisms, then BI and k: A2 82. 9: A2

f: A‘ 8,, the morphisms —v

—v

—»

x

=

-.

[]1A1

9

A2

—‘

81$

3;,

([f, 9]» [”9I‘D: Al

9

A2

—’

31$

3:

and .l' are

the

=

same.

Proof: moxou.

=

p.°

=f=

l’2°v“'"l‘i

=

P2°

THEOREM

40.16

.13 is a functor between senziadditive If F : s! then the are products, following equivalent: —r

categories

and s!

has

finite

( I) F is additive.

(2)

F preserves

finite products.

(3) (4)

F preserves

finite coproducts.

F preserves

finite biproducts.

Proof: The equivalence of (2), (3), and (4) follows immediately from Propositions 40.8 and 40.9. That (1) implies (4) follows immediately from the characterization of biproducts (403(4)) and the fact that each additive functor must zero preserve morphisms. To see that (4) implies (I), note that since F zero preserves empty biproducts, it preserves morphisms. Also by the uniqueness of the semiaddilive structure defined in terms of biproducts (40.l2), F must addition. preserve [3 Module-Valued

Functors

pair (.21, a?) of additive categories one can define Add[.d, a] to be the full subcategory of [$1, at] whose objects are the additive functors from .n/ to 38. Likewise one can define the quasicategory of all additive categories and additive functors, and the category of all small additive categories and additive functors. into analogous Many of the results of general categories translate results in the realm of additive We leave the task of such translations categories. to the reader. and restrict ourselves to pointing out the important fact that the role Set-valued functors play in the study of arbitrary categories is played by Ab—valued (or more in the study of additive generally Mod-R-valucd) functors categories. For

PROPOSITION

40.I7

If groups.

any

s!

is

then

an

.4”

additive x

s!

and (Ab, U) is the concrete category is additive and there exists an additive functor

category

Harms!”

x

.91

—~

Ab

of

abeiian

Additive

Sec. 40

such

that

313

Categories

the triangle

5/”

x

xvii—Nu;

MN/ Set

commutes.

In

fact,

Proof:

(Al)

enables

group

Ham 40.18

The

preserves

verification

supply (A2) guarantees homomorphisms. us

to

and

group, as

Ham

must

them

preserve

limits.

that

at

each

set

that Since

x

.2!” is additive

is straightforward

with

the structure

limits

and

hom(A, B) morphisms homo", g)

the

hom preserves

can

of

(403). abelian

an

be considered

U reflects them

(32.12),

additive

functor

E]

(24E).

COROLLARY

lfsi is additive and A is an d-object, then there Hom(A, _): d —> Ab such that the triangle 9/

Hom(A. _)

exists

an

Ab

homl/lmxA / Set

conmmtes.

E]

PROPOSITION

40.19

If a! is additive and A is an .5! -object, then the fill! subcategory of .m’ whose role object is A can be considered as a ring R, and there exists an additive functor Hom(A, _): d —> Mod-R such that the triangle

/Hom(A._) —>-Mod

R

1'0"“:fo et/ :ommutes

(where (7 denotes

the

forgetfltlflmctor). be considered

right group (Al) the distributivity of composition that the morphisms also guarantee )ver addition (A2). The last two conditions as linear transformations. Hom(A, _) is additive iom(A, f ) can be considered limits and (7 reflects them. ;ince hom(A, _) preserves El Proof:

That

for

each

d-objcct

B, hom(A, B)

fi-modulc follows from the fact that it has the 1nd from the associativity of composition and

can

structure

of

an

abelian

as

a

314

Next

this,

We

40.20

Chap.

XI

would like to be able to characterize those .sd-objects A for which functor constructed Hom(A, _) preserves coproducts. To accomplish first need

the

following lemma.

LEMMA

Let :5! be a

additive

an

set-indexed

coproduct

and

that

category

has

and

family of d-objects,

let

fill/owingare

A

products and coproducts, let (A,), (11,-, A,) and (n A“ to) be the

III

Then

product, respectively, of (A ,), f:

the

Categories

we

the above

be

'

Pointed

fbr each morphism

HI A1,

—)

equivalent:

(1) f can be factored through a finite coproduct; i.e., there is a finite set K c I, a coproduct (vk,H A k) of (210,-, and a morphism f: A H A,‘such that the triangle —v

x

K

A

4’11“

flu

f

3’“ commutes.

201,0 I

(2)

a

o

a,

I,“of)

f; i.e.,for all butfinitely many

=

lit

and the remainder

have

a

°

that

sum

631.4.°f=

77; ° is

is

l,

0.

12

Proof: (I)

the finite

Complete

(2).

a

(th

coproduct (vb

H

At:

=

K

(40.9 dual). Then

for each

a

biproduct

>

I’M,

k, 15 K, the diagram "'

>11“, [#1:]

7

to

HX A," Pk) @l

f

A

A.) IE1

W

.

R/

Ak—ak'—> A;

(91,)

l

pt

"I:

2a

EA]:

.

commutes.

Letj

e

l— K. Then

#1" =

for each

k

"1°elm°[l‘k]°“k Oovk. pjoO =

e

K, =

we

have

HIM)”

elmam

=

”1°15”

See. 40

Since

coproducls

are

epi-sinks,

this

°

°

implies

9'4.

1‘," 71'; so

Categories

Additive

°

that

315

for each j

1—K,

5

[”15] 0. =

that

.“j°nj°@],h°f=

elA.°[Pk]°f=

.“jonj°

0°]:

0-

Thus

elmf) ;01i°ni°@1113!) ;(fle°“k° 20% 7‘1: 91,“ =

°

=

the

Using

of the above

commutativity

;(i‘t

°

9.7)

Pt

Assuming (2), notation

above

°

Elma] "1

=tut1elct 2) =— (1). Using the

and

diagram

°

=

°

97)

Pt

[#1:]°j)-

°

40.8, this becomes:

”1.1“?“Pt) °f

=

6

=.r.

{is Ilflz" 7:; @lm °f95 0} is finite. for this K, letf Z (1-,; m. (B I,IIof). Then the

K

set

=

0

=

o

o

K

[M] °f

; (Vs. "k @ln. °f) Eadie]"L- ”k @111;“1) =f~ D ;(l‘k°nk°®lzh°f);(l‘i°fli°®lm°f) [M]

=

=

°

°

°

°

=

°

°

=

THEOREM

40.21

(.111,Itom(A, __)) be an algebraic category. If .9! is additive, R Mod-R is the functor constructed Hom(A, A), and HomM, _): 53’ then the following conditions are equivalent: Let

is the

—->

Hand/i. _) preserves (I) F coproducts. is afinitaryftmctor (see 22E (2) ltom(A, _)

in

ring 40.19,

=

and

326).

Proof: (I)

(2).

=

Suppose

that

R so

that

g:

R

—»

since

Mod-R

"R such

that

is

f: =

A

—>

’A. Then since F preserves

FM) M» F(’A)

finitary. there

the

exists

’F(A)

=

a

finite

=

set

coproducts

F”)

'1‘:

=

X

c

I and

F(’A)

=Flvt/ \ /l'utl KR

COHH'IIUICS.

=

PIN/1)

have

'R,

triangle

r(,t)=t‘2

We

a

morphism

Pointed

316

Define

5

g(l,.):

=

[Vt] °£7

=

=

(A, —)

ham

Thus

Then —>VKA.

A

[Vt] °g(l,«)

F([Vz])(y(l,4)) [flk](g(14))

=

=

([1113°9)(1A)

=f°

F0304)

=

1,4=f-

finitary (32G(iv)).

be

must

Chap. X I

Categories

(1). Let 01,, U A,) be the coproduct of (11,),in .9! and let (v,, 1] F(A,)) be the coproduct of (HAD), in Mod-R. Then there exists a unique linear transformation f such that for each i, the triangle

(2)

=>

F(A.-)

4meta-n if

Fox.)

I V

F(HA,-) commutes.

Ifg,

F(A,)

e

Hom(A, A:). then

=

f("i(gt)) Sincc

f is

linear

a

whenever

f (2 v,(g,))

that

for

each

i, g;

by Exercise

then

I;

0(401).

=

and

(320)

the

Thus

f

have

2 (Hi °!It)-

=

0

=

Thus

is

injective. If keF(l_I A,), k 2 (men, $14.01;), (510.20) Then f(k) k, so that f is surjective. isomorphism. Consequently F preserves f

lemma

2(vi(n,o®lA‘ok)). bijective and so is an coproducts. [I Define

we

have

we

20mm) so

ungr-

v,-(g,)e LlF(A,)

Zfb’tigt»

=

0,

=

=

for each 2

transformation,

“2 v.-(g;)) Thus

F(u:)(g:)

=

0

=

=

=

is

EXERCISES

40A.

Leta!

be

a

category with

(I; g) of d-morphisms morphism f + g with domain that

(a) Prove imply condition (b) Prove that

for

such

a

and let + bea A and

function

function

domain

common

A and

codomain +,

which

associates with each pair codomain

common

B,

an

.91-

B.

conditions

(Al)

and

(A2) of Definition

40.1

(A3). conditions

(Al’)

and

(A2) do

not

imply (A3).

[Hirm

Consider

the

Sec. 40

Additipe Categories

following category that has precisely one fix, 0, b} and composition defined by

Define

a

function

+

317

object X, with morphism

0

0

0

0

b

b

0

b

+

1x

0

b

set

hom(X, X)

=

byi

beOIX 0000

51x01)

(.91, +) satisfies

that

Show

408.

Show

that

(Al’)

for any

and

(A2) but

not

(A3).]

categoriesaal and Q

(a) s! is additive if and only it‘d” is additive. (b) If .9! is additive, then so is .913. (c) If M and Q are additive, then so issl x .9. 40C.

Ker(f

—

Prove

g)

40D.

z

that

Eun;

Prove

that

if

f,g: A a B are morphisms in an additive g); i.e., if either exists, then they both do, and if A

B, A i»

L)

—"-> D, and

C, B

C

L»

D

are

are

then category, the same.

morphisms

in

a

'

scmiadditive

category,

then

Ada)

B®

CflD=

hef-l-

keg.

40E.

40.| I and Exercise Using Lemma 40D, represent morphisms between of biproducts pairs objects in any categorysl as 2 x 2 matrices and show that if d is semiadditive, their composition is usual matrix multiplication; i.e., of

A,

9,4ng,

GI! in

@

(gal:13 Br£+cx

c2

a

Iiu+9|rfz 0! 0! (oil-ruwuoh: ailori§+a§id§9 9

=

40F.

biproducts semiadditive,

Generalizc of

Exercise in any

A: 6 A2 40E

a

objects category their composition is usual

as

finite

matrix

C1 @ C2

finite morphisms between and showing that it‘s! is multiplication.

by representing 4!

+9

all

matrices

Painted

318

Let F —{

406.

G:

(.94, a) be

Chap. X l

Categories .whered

adjoint situation.

an

and Q? are

additive

categories. is additive.

(a) Show that F is additive if and only if G (b) Show that if d has finite products, then

tion. U

o

both

are

as! be

Let

->

=

exists

there

that

Show

17

G

additive, A an Jul-object, Gm! Set the forgetful functor, and t]: hom(A, _) ->

40H. U: Ab

F and

natural

a

transformation

->

U

additive.

Ab

additive

an

G

o

natural

a

17: Hom(A, _)

functor,

transforma-

->

G such

t].

Lets!

40!.

be additive,

A

and

d-object,

an

G: .2!

->

Ab

additive

an

functor. the abelian

(30.6) induce an isomorphism between (a) Prove that the Yoneda functions C(A) and [Hom(A. _), G]. groups a that the Yoneda full additive embedding (30.8) induces (b) Show E:

that

embedding

d°’C-> Add[.a!, Ab]. Let .137 be

401.

(11,,HA.) be are

is

a

semiadditive

has

and

products

coproducts.

for

=

=

l

I.

this

section

previous

sections

two

Definition

to

the

obtain

and General

notions

of

exactness

additivity of

and

important concept of abelian chapter all categories are assumed the

this

throughout

CATEGORIES

ABELIAN

combine

we

Recall that

the

category. to be

pointed.

Properties

DEFINITION

4].]

is called

A category

finite

biproduets.

41.2

EXAMPLES

is abelian

R-Mod

an

(for

abelian

category

provided

it is

that

and

exact

ring R). (2) The full subcategory of Ab consisting of finite abelian groups is abelian. (3) The full subcategory of TopGrp consisting of compact abelian groups (l)

let

that coproduct of (Am, and let (A 3L) A,), be a family of sl-morphisms that if then 0 for all is I. Show 0, finitely many g, 2 (It: Rat)

a

0 except

§41 In

that

category

has

every

is

abelian. full

(4) The set

has

at

subcategory

consisting

is

abelian

27 elements

most

(5) The opposite of 41.3

of Ab

any

not

abelian

of abelian

whose

groups

(even though it is

exact

underlying

and

additive).

is abelian.

category

THEOREM

If

.e’ is exact

am! has

conditions

are

( I) There

exists

an

(2) There (3) .ss’ is

exists

a

an

finite products

or

finite coproducls,

equivalent:

abelian

additive

structure

unique additive category.

on

structure

d.

011.91.

then

the

following

Sec. 41

Abelian

Categories

319

Proof: In view of Theorem 40.13, it is sufficient finite biproducts, then the unique semiadditive structure

to

induces

the structure

ltom(A, B). That

is,

wish

to

show

that

each

additive

Given

-v

B, consider

the

A

f:

of

abelian

an

Let

(K, k)

Ker(lt).

z

(0, 0)

lick

=

°

But since .9! is semiadditive

if

1:4 ok

(c, C)

O and

=

[0,0]

Thus

Cok(lt)

1A But since

1493

=

0

=

by

track

f

6‘, #2

=

[mac

+

0.

Consequently, I:

=

Irma

=

”A

=

°

°

k)-

truck)

(40D).

0. Hence

=

Ker(lt)

0.

=

Similarly

+

o

°

c] (11.1"). [m o

0 +

[120cc];

nA°h°ll-l

=

21,, +

Ms°7ra)°h-l

°l14

°

6,;12

[11.0]°h“

=

Now

nu.

this is

“4°11” °l‘.4

=

61° (0.13)]

isomorphism (39.15).

an

Mu

(40.8),

113° n,

°

uzoc].

be

must

15)]

co(0,

+

0

=

“A

”1—1°l‘A-

=

fiBOhOh-l °l14

equality,

0,

so

that

f

=

[f, 13]olt"

[fs la]°(llA°“A

=

0,1,, +

A“n°“n)°h-l

°l1A

nae/1'10”.

+

on.

0”,.

this

is

has

an

+

g

=f+

additive

9-

inverse.

[:1

COROLLARY

41.4

A category

(1) (2) (3)

We

011A. Then

the above

g

inverse.

B.

°

[co(lA,f),

=

U; 131° I °I‘_‘

+

(40.12) actually

k): U; 18] (“A 1‘: “3

°

k

so

fo 14 Thus

7‘3

that

0,

=

colt

=

°

=f°1rdolt-' But

A $

+

[[111

1:30:14

=

an

0, fonAOk

=

ch"

as

+

0

[1,4,0] °(ll.4°nA Lctg

°

=

=

B has

has

then

Cok(lt),

z

—*

if d

(40.12), this is

(190/: Thus

set

that

([IA, 0]ok, Lf, 1,]ok)

=

(DA: 0] (“.4 k:

=

each

h=Ag 5‘:

’1

'_“)Aqu2

A1

I

commutes.

the lemma, k = 0. that Cok(lz) = be shown

by part (3) of

Hence

Dually, it (39.15). Hence

can

.1! has finite

0,

so

that

[1 is

isomorphism

an

E]

biproducts.

PROPOSITION

41.10

Let

B

(11,, #2, 8,1:1, n2) satisfied:

(I) (2)

7‘1 ° ”1

be

:23::3:A;

A

is

IA, ‘7’"! “2

=

°I‘2

an

abelian

Then

category.

only if the following conditions

and

biproduct if

a

in

morphisms

are

IA;-

=

The sequences

{1.31) 132» A2

31) A2-"—’»

and

A1

exact.

are

and

(l)

have

a

Corollary biproduct satisfied. Then clearly, for k, j

41.8.

Conversely,

(2)

are

and

(I)

the definition

from

satisfied

we

That

Proof:

(2)

are

if

and

of

“k011i

=

is immediate

biproduct

that

suppose

i, 2,

61*.

=

Let

In —(#1°“1)—(I‘2°7rz)-

f: Then

(“1°ln)

nl°f=

(711°!11°1Tr)—(711°!12°nz)

-

=

—

"I

7‘1

-

0

0,

=

and

7‘2°f= By (0-11, is

In

“2° a

(“2 °l‘1°7‘1)_(7‘2°l‘2°7‘z)=

=

section,

so

that

11,

111

Hence

since

7:;

o

f

Imam.

Im01,)

exists

0, there

=

z

z

a

z

Thus

J"z

"

0

‘—

7‘2

0-

=

by exactness,

Ker(1t2).

unique morphism

In such

that

11,

o

h

=

f.

Now

IM°l=7T1°p,°/l=n10f=0.

h=

#1011 =f= Consequently, ln product (40.8). 1:]

ThusO

1;;

=

=

0‘1““)

—

111

o

n.

+

p;

—

01:2,

(112°7rz)so

that

(111,112, B,

11,,

n2)

is

a

bi-

See. 4]

Abe/ion

Exact

Functors

41.1]

THEOREM

Every (half)

F

functor

exact

:

d

Categories

323

1? between

_.

abelian

categories

is additive.

Proof: Let (P, 7“, n2) be a product in .9! of the fl-objects A! Complete this to a biproduct (#1. #2, P, n1, n2) (40.9). Then in op, o

it:

1,4,and

=

p,

and

A2.

I",

=

the sequences 0

—.

A! "—3 P is

0

—»

A2 "—5 P ’—'»Al

A,

-»

0

—»

o

and

exact.

are

since

Therefore,

F is

functor

a

F(ni)°F(I11) FOB) and

since

F is

(half)

°

the

exact,

F012)

=

F04.)

=

=

FHA)

=

1m.) IRA»

sequences

F(A1) “L". F(P) 1‘2 F(A2) and

PM» “#1

mom

FM.)

exact.

are

Thus

by the above

proposition (41.10) (F011). F012). RP). Fm).

is

F(1I2))

biproduct.

a

Hence

finite products

F preserves

so

that

it is additive

(40.l6).

D

THEOREM

41.12

Let Then

the

F

:

.91

.3

—r

following

(1)

Fis

(2)

F preserves

(3)

F preserves

(4)

F preserves

are

be

zero-preserving

a

functor

between

abelian

categories.

equivalent:

exact.

finite

finite colimits. pill/backs and epimorphisms. kernels and epimorphisms. limits

and

Proof: (1)

s

(2).

By

the

above

theorem

finite products and finite preserves also preserve kernels and cokernels and

exact

=

(3).

Immediate

category

f

is

an

f unctor

is

additive,

and

coproducts (40.16). However,

exact

functors

hence, in this case,

also

equalizers

and

exact

(39.24); (40C and its dual). Consequently (23.7 and its dual).

coequalizers

finite colimits

(2)

(4l.l l) each

F preserves

finite limits

pullbacks are particular finite limits and epimorphism if and only if f z Cok(Ker f) (39.15). since

so

and

in

an

Pointed

324

(3)

Since

(4).

=

F preserves

(23.7) and, hence, kernels.

(4)

Immediate

(1).

=>

three

Hom(A,

A) is considered introduced in

be the functor

to

is

Hom(A, _)

f

preserve

1:]

propositions

.52! is

that

assume

an

abelian

with

category

ring R and Hom(A,.__):

be the

to

d

—»

Mod-R

40.19.

Proposition

PROPOSITION

41.13

left-exact (39 L).

Proof: Suppose Ker(g). We wish

a:

it must

object,

zero

39.25.

Proposition

the next

A.

object

from

the

XI

Functors

Module-Valued

For

and

pullbacks

finite limits

Chap.

Categories

that

0

to

show

B

—>

—[-> C i»

D is

exact

an

in d;

sequence

i.e.,

that

Hom(A, f)

Ker(Hom(A,

z

g));

i.e., that o

Hom(A, 3)

—»

Horn(A./)

——__..

Hom(A, C) M

Hom(A, D)

IS exact.

Let h

Hom(A. B). Then

e

Thus

0.

gof= morphism

Hom(A,g)o f is. Now

Hom(A, g)

Hom(A,f)

Hom(A, f)(h)

c

=

Also

0.

=

g

a

f

Hom(A,j)

o

h

0, since

=

is

a

mono-

k: M that Hom(A, C) such that suppose o 0. Then sincef z Ker(g), x e for each 0; i.e., M, g (k(x)) HomM, g) B such that f o y, for each x e M there exists a unique morphism y,: A it is easy to and f is a monomorphism, k(x). Since k is a linear transformation transformation see that y: M —~ Hom(A, B) defined by y(x) y, is a linear to this with such that Hom(A, f) k, and is unique E] respect property. y since k

o

—.

=

=

=

=

o

41.14

=

PROPOSITION

faithful if

is

Hom(A, _)

am!

only if

A is

a

separator

for

d.

[:1

PROPOSITION

41.15

The

following

equivalent:

are

(1) Hom(A, _)

is exact.

(2) Hom(A, _)

preserves

A is

(3)

projectire

a

object

epimorphisms. in .nl.

Proof: (1)

c:

Since

(2).

if it preserves

2) and

c:

reflects

(12.14).

preserves

kernels

(41.13), it is

=

[:1

exact

if and

only

epimorphisms (39.25). U: Mod-R Set obviously preserves Since the forgetful functor epimorphisms if and only if epimorphisms, Hombl, _) preserves if and U Hom(A, _) does; i.e., only if A is projective —»

(3).

Izom(A, _)

Hom(A, _)

o

Sec. 41

Abe/ion

It should

be remarked

that

Categories

the small

abelian

325

categories

be characterized

can

those

categories .91 for which there exists an exact embedding there is or, equivalently, precisely those categories .2! for which some ring R and a full, exact embedding E : d "—>R—Mod. The proofs of these assertions are not within the scope of this presentation. However, we are now able to give a categorical characterization for the categories of R-modules. as

precisely

E:

#9» Ab;

41.16

THEOREM

U) be

Let (M,

(1) (5!, U) (2) (d, U)

concrete

a

Then

category.

the

following

equivalent:

are

equivalent'l' to (Mod-R, (7)12"some ring R. a finitary algebraic abelian category.

is is

Proof: (1) => (2). Clear. (2) =~ (I). Since (5!, U) (30.20). Let R be the ring

is of

algebraic, U z Itom(A,_) for some morphisms Hom(A, A) (40.19). Then H

.d-object A the triangle

Ham(A._)

=

.9!————-—->

Mod-R

[MINA / Set

(where (7 is the usual forgetful functor for Mod-R). It remains to be that H shown Hom(A,_.) is an equivalence. Since (.91, hom(A,_)) and (Mod-R, L7)are algebraic, H must be algebraic (32.20); in particular, it is faithful (32.17) and preserves (regular) epimorphisms; i.e., is exact (41.15). Also, since (if, U) is finitary, H must preserve coproducts (40.21). To show that H is full, let X be any d-object and let 9x be the class of all of H to d-objects Y for which the restriction commutes

=

Itomd(Y, X) is

ItomMM,§(H(Y),H(X))

—>

surjective. First

formation.

of all, A Consider

fix. To f (l A): A

5

H(f(14))(/I) Thus

H( f (l 4))

be the

11 Therefore

=

1th copower

of A. Then

this, let f: H(A)

see

X. Then

—r

fa.)

=

A

be

a

TI.e.,

linear

transformation.

1th copower there

exists

isomorphic (14.1).

an

of

H(A).

equivalence

fix.

h

=

Next

’A belongs {1:

is the

6

°

for each

H(’A)

Since Now 11:41

H

since —.

—>

to

be

H(X)

—>

h: A

a

f(l.t °/1)

=

let I be

an

index

fix.

see

this, let

To

linear

trans-

and

([1,, ’A)

A,

->

f0!)set

H(X)

preserves A 6 fix

Mod-R such

coproducts, it follows that

U and

that Us

(H011). H('A)) for each H

are

i

e

naturally

l

exists

there

is

a

morphism

coproduct,

a

for each

i

exists

there

H(Q)

"

that

X such

_.

My.)

=

unique morphism 57: ’A

a

for each

1. Hence.

e

A

9‘:

Chap.

C alegories

Pointed

326

HUM)

i

e

—~

0

(#5, ’A)

”(pi). Since

X such

that

g

o

p,-

=

g,-

l.

mg em)

=

g

XI

”(9.)

=

=

9°

”0“)

’216 fix. Finally g. Therefore coproducts are epi-sinks, H(Q) Y of fix, then Q belongs to 3xelement if (q, Q) is a regular quotient of some Since is a linear transformation. that g: H(Q) To see this, suppose H(X) X such that H(Q) Y 6 fix. there is some g H(q). Thus, morphism g: Y X such that a exists there since H is faithful. H(Q) morphism g: Q since H H(g) g. epimorphisms, preserves H(g) H(q). Consequently, Now since A represents U, A is free over a singleton set (31.4), so that the objects of the form ’A are exactly the free objects of d (3|C). Thus the members of 33x are precisely the d-objects (3L9). Hence HomM, _) is full. that H is dense; i.e.. that each right R-module M to be shown It remains is isomorphic to H(X) for some sci-object X. To see this, observe that there so

since

that

=

—»

=

4

o

=

_.

=

o

exist

transformations

linear

ngmLM 9

where

(q. M)

products,

z

there

’A

H(f)

such that

H is exact.

Since

morphic

to

M.

since

Coeq( f, g). Furthermore, are d-morphisms

=

[and

H(Q)

it preserves

=

5a'

H

is full

and

preserves

co-

“A

((7, Q) be the coequalizer offand {7in .91. coequalizers (4H2) so that H(Q) must be isog. Let

[j

EXERCISES

4lA.

Prove

that

In any

abeliun

a

category

5!

is abelian

if and

only if

products. 4| B.

and

the

calegoryal.

consider

the

square

sequence A

?(A) e 47/. “2/is a function =f[1] e W. (iv) 16 4?! andf:1 and

categories,

quasicategories

so

used

are

—»

(v) aeAefl-naeall. of the

( 1) Every

For is

set

foundational

of the

features

easily verified.

‘1’!and

e

(3) If

x

and y

c

x

z

e

x

and y

x

and y

Proof: (6) If]

is

a

x

set

is

set

a

{x, y} fl

6

WI: hence,

x

511/.

c:

[:]

a set.

90»)

e

x

z

is

a

byf(0)

e

“11«in»,

that

so

x

e

51! ((v)).

D

set.

xandf(a)

=

then

(x, y)

is

=

yifa

9e 0.

Thenf[N]

=

set.

a

{{x}, {-V,J’}}-Apply (3)-

=

y

—*

by (v)

[:1

then

sets,

are x

N

sets,

are

Pr00f: (x. .t') (5) If

then

sets,

are

then

x,

oil, then

6

y

Proof: Definef: {x, y} is a set ((iv)). (4) If

be

class.

(2) Every subconglomerate of

Proof: If

all

example:

a

Proof: Ifx

subconglomerates of all “classes”, system required in Chapter II can now

of ‘71 “sets” and the

the members

Calling

x

x

99°(U

r:

l

andf:

->

y is

{.r, y}).

J21, then

a

C]

53!.

:J

f(i) 1U,

and

[I]f(i)

are

sets.

Proof:

U,f(i) f(i) fl is I 1‘ Adding

c

9(1

=

x

Uf[l]

6

Uf [11);

4,

by (ii) and (iv).

((6), (5). (iii). and (2))

Cl

the prospect an a new axiom, of course, poses ofintroducing inconsistency. However, of a universe is essentially the same as hypothesizing the existence hypothesizing the existence of a strongly inawessible cardinal. This is generally felt to be free of inconsistencies.

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