Categories, types, and structures : an introduction to category theory for the working computer scientist [Print ed.] 9780262011259, 0262011255

Table of contents :
Cover
Distribution
Half Title
Foundations of Computing
Title
Copyright
Table of Contents
Series Foreword
Introduction
Dedication
Book Title
PART I - Categories and Structures
1 - CATEGORIES
1.1 Category: Definition and Examples
1.2 Diagrams
1.3 Categories out of Categories
1.4 Monic, Epic, and Principal Morphisms
1.5 Subobjects
2 - CONSTRUCTIONS
2.1 Initial and Terminal Objects
2.2 Products and Coproducts
2.3 Exponentials
2.4 Examples of CCC’s
2.5 Equalizers and Pullbacks
2.6 Partial Morphisms and Complete Objects
2.7 Subobject Classifiers and Topoi
3 - FUNCTORS AND NATURAL TRANSFORMATIONS
3.1 Functors
3.2 Natural Transformations
3.3 Cartesian and Cartesian Closed Categories Revisited
3.4 More Examples of CCC’s
3.5 Yoneda's Lemma
3.6 Presheaves
4 - CATEGORIES DERIVED FROM FUNCTORS AND NATURAL TRANSFORMATIONS
4.1 Algebras Derived from Functors
4.2 From Monoids to Monads
4.3 Monoidal and Monoidal Closed Categories
4.4 Monoidal Categories and Linear Logic
5 - UNIVERSAL ARROWS AND ADJUNCTIONS
5.1 Universal Arrows
5.2 From Universal Arrows toward Adjunctions
5.3 Adjunctions
5.4 Adjunctions and Monads
5.5 More on Linear Logic
6 - CONES AND LIMITS
6.1 Limits and Colimits
6.2 Some Constructions Revisited
6.3 Existence of Limits
6.4 Preservation and Creation of Limits
6.5 ω-limits
7 - INDEXED AND INTERNAL CATEGORIES
7.1 Indexed Categories
7.2 Internal Category Theory
7.3 Internal Presheaves
7.4 Externalization
7.5 Internalization
Appendix
PART II: Types as Objects
8 - FORMULAE, TYPES, AND OBJECTS
8.1 λ-Notation
8.2 The Typed λ-Calculus with Explicit Pairs (λβηπt)
8.3 The Intuitionistic Calculus of Sequents
8.4 The Cut-Elimination Theorem
8.5 Categorical Semantics of Derivations
8.6 The Cut-Elimination Theorem Revisited
8.7 Categorical Semantics of the Simply Typed Lambda Calculus
8.8 Fixpoint Operators and CCCs
9 - REFLEXIVE OBJECTS AND THE TYPE-FREE LAMBDA CALCULUS
9.1 Combinatory Logic
9.2 From Categories to Functionally Complete Applicative Structures
9.3 Categorical Semantics of the λ-Calculus
9.4 The Categorical Abstract Machine
9.5 From Applicative Structures to Categories
9.6 Typed and Applicative Structures: Applications and Examples
10 - RECURSIVE DOMAIN EQUATIONS
10.1 The Problem of Contravariant Functors
10.2 0-Categories
11 - SECOND ORDER LAMBDA CALCULUS
11.1 Syntax
11.2 The External Model
11.3 The External Interpretation
11.4 The Internal Model
11.5 The Internal Interpretation
11.6 Relating Models
12 - EXAMPLES OF INTERNAL MODELS
12.1 Provable Retractions
12.2 PER inside ω-Set
12.3 PL-Categories Inside their Groethendiek Completion
BIBLIOGRAPHY
Index

Citation preview

, MIT LIBRARIES

is 3

9080

00720

of

Computing Series

0824

t t

t l t t t

t

C ategories Types ,

t

t t

and Structures

t

t t t

An

Introduction to Category Theory

for the

Working Computer

t t t

t

t

t t t t

t t

t l t

t t

Andrea Asperti and Giuseppe Longo

Scientist

The MIT Press

This

file

has been authorized and provided by the

publisher,

to

make

The MIT

available

Press, as part of

in digital

its

form older

ongoing efforts

titles

that are

no

longer readily available.

This

file is

lending,

provided through internet Archive for library

and no further reproduction or

distribution

allowed except as permitted by applicable law or writing

by the MIT Press.

in

is

Categories, Types,

and Structures

,

Foundations of Computing Michael Garey and Albert Meyer, editors

Complexity Issues

in

VLSI: Optimal Layouts for the Shuffle-Exchange Graph and Other

Networks Frank Thomson Leighton, 1983 Equational Logic as a Programming Language, Michael

J.

O’Donnell, 1985

General Theory of Deductive Systems and Its Applications,

S.

Yu

Maslov, 1987

Resource Allocation Problems: Algorithmic Approaches, Toshihide Ibaraki and Naoki Katoh, 1988

Algebraic Theory of Processes, Michael Hennessy, 1988

PX:

A

Computational Logic, Susumu Hayashi and Hiroshi Nakano, 1989

The Stable Marriage Problem: Structure and Algorithms, Dan Gusfield and Robert

Irving,

Realistic Compiler Generation, Peter Lee, 1989

Single-Layer Wire Routing and Compaction,

F. Miller

Maley, 1990

Basic Category Theory for Computer Scientists, Benjamin C. Pierce, 1991

Categories, Types,

and Structures: An Introduction

to

Category Theory for the Working

Computer Scientist, Andrea Asperti and Giuseppe Longo, 1991

1989

Categories, Types, and Structures

An

Introduction to Category Theory for the

Andrea Asperti Giuseppe Longo

The MIT Press Cambridge, Massachusetts

London, England

Working Computer

Scientist

© 1991

Massachusetts Institute of Technology

All rights reserved.

No part of this book may be reproduced in any form by any electronic or mechanical means

(including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.

This book was printed and

set in

bound

Times Roman and Symbol using

in the

electronic

media supplied by the authors.

It

was

United States of America.

Library of Congess Cataloging-in-Publication Data

Asperti, Andrea.

Categories, types, and structures scientist / P.

:

an introduction to category theory for the working computer

Andrea Asperti, Giuseppe Longo.

cm.

— (Foundations of Computing)

Includes bibliographical references.

ISBN 0-262-01125-5 1.

Programming languages (Electronic computers)

G. (Giuseppe)

QA76.7.A76

005.13—dc20

II.

Title.

III.

2.

Categories (Mathematics)

I.

Longo,

Series.

1991

89-49085

CIP

7c, then there exist

0

g:

if

g:

a-*b and

h:

a-*c; moreover, composition must be associative and an identity must

exist for all objects.

This

1.1.1 -

is

the formal definition:

Definition a collection

A category C is ObQ of objects denoted by a, b ... A, B More of morphisms (arrows), denoted by f g .

,

-

a collection

-

two operations dom, cod assigning

-

an operation id assigning

( source )

dom(id\j) -

and codomain

-

cod(idfr)

an operation

"

arrow f g such 8

-

identity

that

.

.

.

(target) of f to

each object b a morphism id ^ (the identity of b) such that

= b each pair f g of arrows with dom(f) = cod(g) an

dom( f°g) = dom(g), cod( f 8 g ) =

and composition, moreover, must satisfy

identity law: for

.

each arrow f two objects respectively called domain

to

” (composition) assigning to

8

.

any arrows f g such

cod(f)

the following conditions:

that cod(f)

- b - dom(g)

Categories

4

idb’f = f

g'idb = 8 associative law: for any arrows

f

g,

h such that dom(f) = cod(g) and dom(g) = cod(h)

(fog) •h=f*(g*h).

We write

f:

two objects

a -* b to denote a a

and

b, the collection

writing fEC[a,b]

moment we

is

of

way

thus a third

all

morphisms

table lists

f

some common

and

target are respectively a

such that

to express the fact that

one notation or the other

shall use

The following

morphism whose source and

f:

a -*

b

dom(f) =

is a,

b.

Given

denoted by C[a,b]; the

and cod(f) =

b.

For the

indifferently.

categories

by specifying

their objects

and arrows,

letting

the definition of their operators as an exercise for the reader:

The

Category

Objects

Morphisms

Set

sets

functions

Top

topological spaces

continuous functions

Vect

vector spaces

linear transformations

Grp

groups

group homomorphisms

PO

partially ordered sets

monotone functions

intuition of the notion of “category” suggested

objects as a collection of “structured” sets and the

functions with respect to the structure. This the definition

which may force

simple example

is

the

is

by the previous examples

morphisms

to

is

the category Rel with sets as objects and relations as

usually called

1.

Note

that,

the identity

law

id^'

,

= id^

0

id^'

= id^

.

of some object: in this case a category

is

made

in

be “single valued” or to be functions in extenso: a

The simplest category has only one object and one arrow category

consider the

too restrictive, though, since no requirement

morphisms

has an identity id^: b-*b. The identity

is to

as the “associated” or “acceptable”

by

definition, if

is

unique, since

C

is

a category, then every object b of

id^'

if

morphisms.

(the identity for that object): this

is

C

another identity for b, then for

A category is called discrete if every arrow is the identity is fully

determined by the collection of

its

objects. 1 is a

discrete category.

A category is called a preorder if for every pair of objects f:

a-*b.

relation

The reason

among

defined; thus are

C

its is

for the

known

is

a,

there

is

f:

b there

that a preorder category is fully

objects. Indeed, in a preorder C, there is only

known when

known. But every arrow

target are

name

the collection of

morphisms

one way

More

a-*b may be identified with the pair

no choice about what the arrow

is

is

at

most one morphism

determined by a preordering that

composition

may be

dom

and cod

and the operations (a,b), since

once the source and

to be; thus, all the information about the

5

Categories

category

C

is

relation. (Exercise:

the category 2,

=

prove that the relation

Every discrete category

is

which has two

and an arrow

idg, idi

Rc

given by the relation

(0,1):

{(a,b)

there

/

is

an arrow fEC[a,b]}, that

Rc is a preorder for every category C them 0 and

objects, let us call

1,

way we can

0-*l. In a similar

preorder category n, from the usual ordering on the set {0,1,

common

property: they

may have

a preorder

is

and three arrows: the two

.

.

is

identities

number n

define for each natural .

a preorder

).

The simplest nondiscrete category which

a preorder.

by

is,

a

n-1}. Preorder categories have a

plenty of objects, but given two objects, there exists

at

most one

morphism between them.

A

dual situation

is

given by monoids, viewed as categories.

associative binary operation and an identity element.

where composition of morphisms

is

A

monoid

a set having an

is

A category with just one object yields a monoid,

the binary operation. Conversely, any

monoid (A,

•

) is

a

category with just one object. For example, the category with the set of natural numbers as unique object and the recursive functions as

As

yields the

monoid of the

recursive functions.

well as preorders, another example where objects are not necessarily understood as

given by deductive systems as categories. In these categories propositions are

“structured sets”

is

objects and each

morphism

b

(a entails b).

i

:

a

morphisms

Observe

f

:

a-*b corresponds

to (a suitable equivalence class of) a

proof of a

that a category is obtained easily in the presence of the identical entailment

a-+a and the associative composition of proofs f

:

a-*b

g This approach to deduction

is

0

g

f

:

:

b-»c

a-*c

very relevant in the categorical understanding of logics of a

constructive nature, such as intuitionistic logic, where the intended interpretation of proofs

by

(effective) operations.

It

will be the

main paradigm

and objects investigated in the second part of

1.2

An

|-

this

for understanding the relation

is

given

between types

book.

Diagrams important tool in the practice of Category Theory

equations. In a diagram a

commutes

if

morphism fEC[a,b]

the composition of the

is

drawn

is

the use of

as an arrow

diagrams

from a

to

for representing

b labeled

morphism along any path between two fixed

commuting diagrams:

A diagram

objects

For example, the associative and identity laws of the definition of “category” visualized by the following

f.

is

may be

equal.

nicely

;

Categories

6

Diagrams

are a typical

be particularly effective

diagramj and

as “if

.

way,

when .

.

in

Category Theory, to describe equational reasoning and turn out to

dealing with several equations

at a

diagram n commute, then diagram

time. In particular, assertions such

commutes” express conditional

statements about equalities.

We hope that the reader, while using this book, will will learn this

how

go back and forth from diagrams

to

book comes from our

acquire

some

to equations.

familiarity with diagrams

and

Our extended use of equations

desire to stress the “computational” nature of

in

most categorical reasoning.

Categories Out of Categories

1.3

A main feature

of Category Theory

simpler ones. In this section

we

is

the facility to define new,

more

structured categories out of

consider only a few simple constructions; a

number of

other

examples occur throughout the book.

A

1.3.1 Definition i.

ii.

Obj)

for all

iii.

A

category

D

is

a subcategory of a category C,

a,

b

in Obj),

D[a,b]

CC[a, b] D coincide with those of C. ;

composition and identities in

subcategory

is full iffor all a,

b

in

Obj) D[a,b] « C[a,b],

A full subcategory is fully determined by

morphisms ofC,

that

in the

collection of objects.

C

id°Pfr

=

idfr

dom°P(f) =

cod(f), cod°P(f)

has the same objects and the same

= dom(f), and f

P

g - g °f

C°P[b,a] = C[a,b] and (C°P)°P = C.

Exercise Set°P

Duality

its

The dual category C°P of a category

1.3.2 Definition

Note

if

C ObQ

is

is

a subcategory of Rel, but not of Set. Is

it

a very powerful technique of Category Theory. If

language of Category Theory, the dual of P (P°P)

word “dom” by “cod,” “cod” by “dom,” “g

0

h” by “h

0

is

a full subcategory?

P

is

a generic proposition expressed

the statement obtained

g.” If

P

is

by replacing

true in a category C, then

the

P°P

7

Categories

C°P;

is true in

P

if

is

P°P

true in every category, then also

since every category

is,

the dual of

is

its

dual.

may be

Duality

C°P

is

obtained by simply inverting the arrows; of course, a dual diagram commutes

if

and only

if

the original one does.

(iff)

the pairs (a,b)

is

and

(fg) °(f,g') = (f of,

g

o

Given a category

C

and an object a

Ob£ j a

so defined:

composition in

C

is

Cja

g

')

g

/

it is

A under g). Then

A-*I

(denoted C/I). stage

I.

B-*B'

h:

is a

{g‘*(a)} a

f:

B

and

C

Set j

in

A

as an

(these sets are the inverse images of

£A

morphism from

B-»A

g:

to

induced by g and

B'-*A

g':

i.e., if

g',

if

and only

and only

if

if (iff)

h(b)Eg'"l(a).

as a collection {{i}xg‘l(i)}iei>

An

of objects

b-*a, g: c-*a, a

B-*A

useful to think of an object g:

disjoint sets, namely,

CJI

Since the intended meaning behind the construction of a category object g:

C la

the category

Obc,

in

cod(f) - a}; given two objects

an arrow hEC[b, c] such that g °h = f Identities and

is

Set in the above definition,

Vb bEg'l(a) =>

(fg):

Finally, id(a> b)

m

= {fEMorc

target

consistent with the “decomposition” of

it is

has for objects

are inherited from C.

A-indexed family of elements in

CxD

'

morphism with source f and

In case

product category

the

where a and b are respectively objects of C and D, and for morphisms pairs

1.3.4 Definition

over a

and D,

where f: a —*a and g: b-*b' are respectively morphisms of C and D.

(a,b)-*(a',bj (ida, idfr)

C

Given two categories

1.3.3 Definition

=

applied to diagrams as well: given a diagram in a category C, the dual diagram in

object

A section of

g:

g:

A-*I of the

A-*I

C|I

usual to call

is

slice category is then called a

a function

is

it

s:

I-*A such

that

g

0 s

=

is

that to consider an

category over

a slice

generalized object of the idea

idj;

is that

s

C

I

at

gives,

for each index iEI, an element s(i)Eg'l(i).

Exercise Define the dual notion, that the arrows with source

1.4

the category

Cfa

of objects under

whose

a,

objects are

Monic, Epic, and Principal Morphisms

A function f(a')

is,

a.

f

then a =

between two a'.

A

sets

and

In particular, given any

cEC

= h(c)

then for

all

behaves

like a left identity

pair of functions

g(c)

g,

h

:

or, also, if

(it is left

C-*A

,

f

B

is

called “injective”

two functions

f°g

=

cancelable). °

g =

f

°

h

f°h

g,

h

:

when, for

C-*A,

then g = h

The converse implies

is

g = h

.

all

if for all

,

cEC

a'

EA

f(g(c))

,

if f(a)

=

=

f(h(c)),

Thus, every injective function

also true: given ,

a

then

f

f:

A-*B,

is injective.

if for

any

For suppose

Categories

8

otherwise: then there are a and

= a for

cEC

all

and h(c) =

,

such that

a'

=

but a *

f(a')

a'

f°g

cEC; of course

for all

a'

f(a)

;

=

define then g and h by g(c)

f°h

g * h

but

,

that

a

is,

contradiction.

We have proved thus that a function similar

way

it

f is injective if

not difficult to prove that f

is

These considerations motivate the following

C be

Let

1.4.1 Definition

an arrow hEC[a,b]

i.

g-h=f-h h‘g

implies g = h

g°f=h°f

.

In a

implies g = h.

a,

b&Ob£. Then

an epimorphism)

iff

monic

(is

a

monomorphism)

iff

= h'f => g -f;

g °h =

is

iso (is

is

and h

id

objects a and b are

any isomorphism

(is

f°g = f°h

g=f;

is

an arrow hEC[a,b]

iii.

Two

=>

an arrow hEC[a,b]

ii.

epic

if

definitions.

a category and

is

and only

surjective if and only if

is

0

g=

an isomorphism )

gEC[b,a] such

that

id.

isomorphic

monic and

iff there exists

(a

* b)

there exists an

if

isomorphism hEC[a,b]. Clearly,

epic; the converse, though, does not

need

to

be true (see the example

and the exercises below).

A

monic

such that g

0

(or epic)

hEC[a,b]

h = id

0

(h'

=

g'

(or h'EC[a,b])

split if there exist

is

(or

g'EC[b,a])

id).

Although the

intuition of regarding

mono- and epimorphisms

many

interesting categories,

sometimes

correct for

gEC[b,a]

it

as injective

and surjective maps

Mon of monoids and the inclusion inc from co, the positive integers, into z, the relative ones. mono, inc

is

also epi, though.

\g(n) for g(-n).

Then g

a)).

As

inc = h

0

behavior of the monoids’

a matter of fact, take 0

we may

g,hEMon[z,a] for some monoid

inc implies g = h for g(-n)

homomorphism g

a side consequence,

an isomorphism:

As

is

can be misleading. Consider, say, the category

or h on z

is

=

\g(n) =\h(n)

entirely determined

= h(-n) by

also conclude that not every arrow that is both

a,

Clearly

and write

(that

is,

their behavior

the

on

monic and epic

is

this is clearly in contrast to the set-theoretic intuition.

Exercises 1.

Give an epi which

2.

Find a counterexample for the following assertion:

are 3.

mono, then

Prove

a

is

that a split

is

isomorphic to

monic

1.4.2 Definition Let i.

not surjective in Top.

C

is

an

b.

is

C be

(Note that the assertion

iso.

be a category and

an arrow hEC[a,b]

let

a,

bEObQ. Then

a principal morphism

Vf€C[a>.,b]3geC[a,a]

f=h-g;

iff

a category; if fEC[a,b] is

true in Set.)

and gEC[b,a]

.

.

9

Categories

ii.

By

a pair of arrows f&C[a,b] and gEC[b,a]

is

retract of b (a h°i°g = h°i°f => g = f

h°g. Just take

h°i 6 f

=

f.

2.

g°h = f°h => g°k = g°h°g' = f°h°g' = f°k (for a suitable

3.

Let a

Cja

common

target a, then

that in this case, the

f°k = f°k' => k =

unique k'.

determined by monomorphisms

11

Categories

When

f^g

and

gsf

we

write

monomorphisms with common

f

target

*

(prove

a

equivalence relation are called subobjects of

1.5.1 Definition Let a

class of a

Very

we

we

it

an equivalence relation

the

as an exercise); the equivalence classes of this

A

subobject

of a

[f]

*

b-*a, with respect to the equivalence relation

make no

among

a.

be an object of a category C. f:

is

between equivalence classes and

is

an equivalence

defined above.

their representatives,

and

should be clear that the categorical approach to “subsets” carries more information than the

set-

often,

shall It

monomorphism

Then *

g.

shall

distinction

denote a subobject with a single monomorphism.

theoretic one.

Monomorphisms,

like all

category. For example, in the category

must take the

identity to the identity

morphisms, preserve the structural information of the

Grp

of groups subobjects are subgroups: (mono)morphisms

and preserve the group operation. Similarly consider the category

of p.o. sets (partially ordered sets) with a bottom element. a structured subset as well,

and

References Any book

in

(1973), Arbib and specific notions settings,

Manes

it

A subobject of one such p.o.

must contain an element smaller than

all

set

must be

the others.

Category Theory, such as MacLane (1971), Herrlich and Strecker

(1975), Barr and Wells (1985), Rydeheard and Burstall (1988).

and categories introduced (such as retractions) will be used

with the appropriate references.

later in

more

The

structured

Chapter 2 Constructions

In this chapter

morphisms)

we

consider

language there

this

some fundamental

that satisfy a given set of

no way

is

must be defined by

categorical constructions,

axioms described

to look at the internal

their relations

in the

i.e.,

particular objects (and

language of Category Theory. Since

membership

structure of objects,

all

in

the concepts

with other objects, and these relations are established by the

existence and the equality of particular morphisms. This property of the categorical language,

compared

to the traditional set-theoretic jargon,

science; namely, as

we

may be

well understood by an analogy with computer

already mentioned, the categorical description corresponds to an abstract data

specification, while the traditional set-theoretic approach is

2.1 Initial

if

more

similar to a concrete implementation.

and Terminal Objects

2.1.1 Definition Let

C be a

category.

An

object 0

is initial iff for

any

bEObQ

there

is

a unique

f€C[0,b].

the 0 in Set; indeed the empty function function whose graph is empty) is the unique arrow with 0 for source. A more interesting example is the following. Let 2 be a signature. The class Alg£ of 2-algebras

The

typical

example of an

initial

object

is

the

empty

set

(i.e.,

with 2-homomorphisms as arrows forms a category. Alg£ has an word-algebra, or also Herbrand Universe for 2. The set of

set

all

well-formed expressions of

are nonterminals

sort

If

s.

2

is

derived by a context free

grammar from

the nonterminal

corresponds to the syntax of a language of signature 2.

Any

semantic domain; the semantic function (interpretation)

is

Initially

is

object

T^ which

grammar

and operator symbols are productions of the grammar), then

parse trees for derivations in the

unicity of

initial

s.

called 2-

other 2-algebras

the unique

satisfying certain properties. This

T^s

is

(that

is:

sorts

the set of

all

In general the initial 2-algebra Tj;

method

A in Alg^ is a possible

homomorphism from

the simplest universal notion in Category Theory, since

morphisms

is

T^ s (the carrier of Tj of sort s) is just the

is

it

is

Tj; to A.

given by the existence and

used everywhere

in

Category

Theory.

2.1.2 Proposition If

Proof Let 0

i:

then

i:

0—O', j:

0 and

O

O'—

O'

the

are two

initial objects in

morphisms

0-*0, but also idg: 0-*0, and since by 0 j

i

=

idg; in the

same way, by

initiality

initiality

a category C, then they are isomorphic.

respectively given

of

O'

by the

of 0, there

we

have

i

0 j

initiality

is

exactly one

=

idg'.

of 0 and

morphism

O'.

Then

j

in C[0,0],

Constructions

14

We will now show how duality can be used to define new concepts and to prove new assertions. such that dom(f) = c, cod(f) = b.” By Let P(c) be the property “for any bEObc there is a unique f,

definition c

P defines

is,

initiality.

The dual statement of P

c, bEObc there is a unique f, such Q defines a concept named by prefixing “co-” to the name of the property

“for any

cod(f) =

that

property

P°P

say that

P(c) holds; that

is initial iff

common

defines coinitiality.

An

object c such that P°P(c) holds,

is

known

as terminal object.

Note

P°P(c) =

Q°P

of a

we

Q. In our case,

Anyway

is called coinitial.

name which

practice to assign to every coentity an independent

properties; for example, a coinitial object

is

dom(f) = b.” Usually the dual

is

it

better expresses

its

that an initial object

is

coterminal.

Terminal objects are usually represented with the number

morphism from an object

Any

singleton set

terminal. If c

Consider dual

Its

to

is:

in C, then

“If 0

be an isomorphism

chapter

1

is

and

it

is

is

1

or with the letter

usually written

t.

The unique

a-*t.

!a:

is initial

and the other one

is

terminal in the dual category C°P.

P^ = “If 0 and

the statement

Pi°P =

t

terminal in Set. In the category 2 one object

is

is initial

now

a to the terminal object

O' are

the dual of

O' are

two terminal itself:

prove

two

initial objects,

then they are isomorphic .”

objects, then they are isomorphic.” it

as an exercise.)

By

and, since by proposition 2.1.2 Pi holds in every category,

(The property

our discussion of duality in

Pi°P

also does.

We conclude

the following:

2.1.3 Proposition

If

and

0

O'

are two terminal objects in a category C, then they are

isomorphic.

Proof By

An

duality and

by proposition

object c in a category

in this case,

it

C may be both

it is

quite

a.

and terminal.

the singleton {*} to a set

arrow from a terminal object or a point of

initial

An

example

is

the unit group in

Grp;

called a zero object.

is

morphism from

In Set, a

2.1.2.

t

to

A defines an element of A.

an object a in a generic category

In this case, however, the set-theoretic intuition

common

to

work

in categories

where

C

is

For

this

reason an

usually called an element

must be used very

carefully, because

the categorical notion does not reflect the behavior of

elements in Set. For example the set-theoretic intuition would suggest that every noninitial object

must have s

at least

2; clearly

0

one element: but consider the

is initial

two arrows are equal one has

f

* g

generic category

iff

iff

there

and 2

is

terminal,

they coincide on is

all

1

partial order category is

noninitial but has

points, or,

no elements. Similarly,

more formally, given functions

an element x of their domain such that

C with terminal object

t,

3 which has 3 objects 0 s

this is not necessarily true.

f (x)

*

g(x).

f

Set

in

and

However,

1

g,

in a

;

15

Constructions

C

2.1.4 Definition Let

fgEC[a,b], one has:

C

tEObQ

be a category.

is

=> 3hEC[t,a] f°h * g°h

f*g

has enough points (or

is

well pointed),

if

a generator

a,bEObc and

iff for all

all

.

there exists a generator

that is terminal in the

t

given category.

In short, a category has

enough points when the arrows from the terminal object allow

between morphisms, similarly as for elements over

Set.

Of course,

theoretic notions of “element” and of “extensionality” are

not a surprise that the set-

is

it

to discriminate

somewhat awkward

with in the

to deal

language of Category Theory.

and Coproducts

2.2 Products

The

categorical product

Given two

sets.

merely a “structural” generalization of the notion of Cartesian product of

A and B, their Cartesian product is

sets

AxB

is

= {

/

xEA, yEB}.

Associated with this set there are two special maps p^;

such that for every in

Pb( c )> = Let

C

=

f.

(c) = =

C

2.2.1 Definition Let

,

>c

PB^^)

g.

)

=

pg:

,

y-

AxB-*B

Note

= pa(

Conversely,

h(c), that

called projections,

that for every c in

AxB,

is,

let

=

0

pa

0

for every

cEC,

h.

the following definition:

be a category and a,bEOb£. The categorical product of a and b ,

an object axb together with two morphisms




cEC, pa( (c)

same way, we obtain pg

pb

=x

c.

be another

for every

AxB PA( )

is

p a axb ->a, p^: axb -*b, and for every object c an C[c,a] xC[c,b] -*C[c,axb] such that for all morphisms f: c-*a, g: c-*b, h:

:

:

c-*axb, the following equations hold:

Pa °c=f; ibii.

It is

Pb °c-g c =

common practice

The operation < is

,

to

>

:

h.

omit the subscript c in
c when

its

meaning

is

C[c,a]xC[c,b]-*C[c,axb] of a categorical product

the operation that takes every arrow

The proof that these operations

hEC[c,axb]

to the pair (pa 0 h,

are inverse of each other

is

stated

Pb

above

0

clear is

from the context.

a bijection:

its

inverse

h )EC[c,a]xC[c,b].

in definition 2.2.1.

Constructions

16

Conversely, given a bijective operation < (ib),

then

(ii) is

necessarily true. Indeed,

pair (f,g)EC[c,a]xC[c,b] such that

= Pb

The

last

°

thus

>

h =

C[c,a]xC[c,b]-*C[c,axb] which

:

h£C[c,axb]. Then, since


satisfies (ia)

is bijective,

there

and is

a

h and analogously, g = pb

0

,

h >.

consideration leads us to a

more compact but equivalent

definition of a categorical

product.

2.2.2 Definition Let

C be

a category and a,bEObQ. The categorical product of a and b ,

an object axb together with two morphisms

and gEC[c,b], there

exists exactly

pa axb-*a :

,

p^: axb-*b, such

that,

for any fEC[c,a]

one hGC[c,axb] such that the following diagram commutes: c

a

b

axb


also has

if it exists.

and qb-

axb

By symmetry, one

axb-* cxd.

e-*b, fxg° = .

alternative product with projections q a

< Pa>Pb >

•'

0

qb> =

id a

®b

-

is

.

17

Constructions

Exercises Prove the following 1.

a

2.

axb

a'

and b *

facts:

imply axb

b’

a'xb\

bxa.

2.2.5 Definition

A

C

category

contains a terminal object

L

it

ii.

every pair

Cartesian (C

is

is

a CC)

iff

t;

a,bEObQ has a

categorical product (axb,

p a,b,l

:

axb-*a,

p a> b,2

:

axb-+b).

Exercises 1.

Generalize the definition of a product of two objects to arbitrary products.

2.

Prove that a Cartesian category

3.

Let

C be a CC and let

Examples The

An

t

be

its

C

always contains

terminal object. Prove that for

categories Set, Top,

Grp

EN

are pairs a

enumeration of

a).

=

(a,e a ),

fEEN[a,b]

where

iff for

Theory

a

some

is

is

b

in

Ob^, b * txb

the category

a countable set

bxt

*

to

EN

and e a w-*a :

of numbered is

an onto

sets.

map

(an

f the following diagram commutes:

total recursive

f

e

all

are all Cartesian.

interesting Cartesian category in Computability

Objects in

products.

all finite

w

e

a

a

b

b f

We [,]:

say that f represents

The product

is

easily obtained

by using any

effective pairing of 0)2,

o)xa)-*a>.

A typical

numbered

set

with a Godel numbering

Of

f.

course, this

is

:

which

is

worth studying

is

PR

=

(PR,-*o) be an effective pairing. Define then, as usual,

function

3f EPR

iff

f(x,y)

= f ([x,y])

f:

partial recursive functions

coxod-^o)

is

f:

o)xo)-*o)

is partial

[,]:

a binary partial recursive

n *

(similarly, for n-ary functions,

s-m-n iteration theorem one immediately has

and

2).

recursive

By

iff

this

and by the

3sER

=

4>s(x)(y)

f(x,y)-

Thus, a two- (or more) argument function f

and the function x

|- f(x,_)

computability theory, f

also “computable,”

we need

C

is in

is

it

4> s

computable

(x )

=

in

each argument

f(x,_). In other

words,

in

cd-^cd-^o)). Similarly, the categoryf:

cxa-*b, of a morphism within

as s or x |- f(x,_) in recursion theory.

computable

We will call

it

A(f).

each argument needs

to

be

be a Cartesian category, and a,bEObQ. The exponent of a and b

is

computable. {Hint: take g

2.3.1 Definition Let

3sER

i.e.,

not every n-ary function which

2,

iff

concerns the existence, for any

which does the same job

Exercise For n *

computable

coxcu-^o) iff x |- f(x,_)

is in

theoretic closure property

the category,

is

is

total

is

in

nonrecursive and set f(x,y) = g(min{x,y})

).

an object b° together with a morphism evalajj: b°xa-*b (evaluation map), and for every object c an

Ac

operation

:

C[cxa,b]-*C[c,ba] such that for

all

morphisms

f:

cxa-*b, h: c-*b a the following ,

equations hold:

A q.

eva la,b °(A(f)xida)

=/;

Ac(evala> b °(hxida)) = h.

(We may omit

the indexes

when unambiguous, as

usual.)

A and B is B A = {f / f is a function from A to B}, thus B A = Set[A,B]. The function eval: B A xA-*B is given by the rule: eval() = f(x) A: Set[CxA,B]-*Set[C,B A takes every function CxA-*B to the function A(f): C-*B A defined by A(f)(c) = Xa.f(c,a), where Xa.f(c,a)EB A -Set[A,B] is the function which takes aEA to In Set the exponent set of

f:

]

The proof of (P) and

f(c,a)EB.

As

in the case

definition 2.3.1

hEC[c,b a ]

Indeed,

let

A(eval a> b

0

if

A

b

0

:

C[cxa,b]-*C[c,b a ]

(r|),

is

A‘l

is

the operation

a bijection and (p) holds, then

take fEC[cxa,b] such that

(A(f)xid a )) = A(f) = h is

by (p) and

which takes every

(hxida)EC[cxa,b].

hEC[c,b a ] and

The following

almost immediate.

a bijection. Indeed,

is

to eval a

Conversely,

(rj) is

of the product, observe that in general the operation A: C[cxa,b]-*C[c,b a ] in

h = A(f); then

.

thus an equivalent definition of “exponent”:

(r\) is

necessarily true.

A(eval a> b

0

(hxid a )) =

Constructions

20

2.3.2 Definition Let

C be

object b° together with a exists

a Cartesian category and a,bEOb£. The exponent of a and b

morphism evala

fr tflxa-*bf such that for all

morphisms

is

an

cxa-*b, there

f:

one and only one h: c-*ba such that the following diagram commutes:

Exercise

By

setting A(f)

=

h,

give the details of the equivalence proof between the two definitions.

The previous diagram should suggest

in

which sense b a “represents” C[a,b]. The eval morphism

generalizes the set-theoretic evaluation function eval(f,x)

Then

object.

C[t,b a

« C[txa,b] a C[a,b] as

]

This

sets.

=

is

Moreover, take c =

t,

the terminal

particularly significant if

C

has enough

f(x).

points (why?).

a Cartesian closed category

is

(CCC)

iff

C is Cartesian;

i.

ii.

Set

C

Definition

2.3.3

for every pair

a

is

CCC:

Another simple

known, given

a,bEObQ

there

is

an exponent.

the previous definition of exponents in Set clearly holds for every pair of sets.

CCC is CPO, a

c.p.o.’s

and

the category of complete partial orders and continuous maps. b,

CPO[a,b]

is

As

well-

also a c.p.o., with respect to the pointwise ordering.

Moreover, both eval and A(f), defined as for Set by using continuous functions, are continuous

and

satisfy the required conditions.

function

is

continuous

Actually, even

Among

A

is

iff

it

is

Note

that the proof uses the

so in each argument and the

|-

fact that in

Xy.f(x,y)

is

CPO

a

continuous.

continuous.

the various examples of categories mentioned in these notes, an important one does not

satisfy Cartesian closedness: the category

EN[ia,&] (= R, the recursive functions) a

well-known

map x

morphism eval with

is

EN

in section 2.2. Consider, say,

surely countable.

However,

the above properties existed, then u(x,y)

=

if a

& - (co,id).

numbered

eval(qp(x),y)

Then

set (co^qp)

would be

and

a universal

function for R.

Also, the (o-algebraic c.p.o.’s, that

approximating

form

a

all

is,

the others (see Scott

CCC. They

contain, though,

the c.p.o.’s with a countable collection of

compact elements

domains below) and continuous maps

morphisms do not

some fundamental sub-CCCs

as

for the purposes of denotational

)

)

)

:

21

Constructions

semantics of programming languages and higher type Recursion Theory. They will be explored in the

examples below.

Given a

CCC D,

A

collection

it

subcategory such that

Exercise Prove that

In definition 1.4.2, 0

may be

interesting to consider specific "structures of types" in

of objects in D,

=

id a

ACD^ any

in

we

be the

let

full

and a,bED^ =>

CCC one has

a^ xc

sub-CCC generated by

(a^) c

turns out to be

mono and

a subobject of b in the sense of section 1.5. In the case of Set,

is

happen

is,

for a

the least full

introduced the notion of “retract”: in a category C, a

i.e.,

A function f: X->Y is

set

{y

of finite elements

|

y £ xq}

compact and y £ xq then y

Let

order preserving,

finite set

compact then the

xq

2.4. 1.2 Definition is

a Scott domain. This topology

|

usually called the Scott topology.

i

*Y)

(Y*

*x J implies

continuous

if for

f(i)

be

is

*Yf(j)

finite, if

function

will often

2.4.1.3 Definition

The category

morphisms. Let X,

Y

X

to

Y

f:

X->Y is monotonic

DCX, f(UD) =

In order to

is

show

is

D has

be objects ofD.

Ud(=Df(d).

f:

X-»Y

is

continuous

continuous with respect to the Scott topology.

Scott domains for objects

Y%

is

and continuous functions for

just the collection of the continuous functions

directed set

U

thus,

D

in

We have to prove that

a c.p.o.

Y^

that

then h by h(x) =

by g(x) and,

is

X

from

algebraic.

an upper bound

well defined since the set {fi(x)}j(=i is

g. is

Define

bounded

continuous because for every

=U ieI {fi(UD)}

= easy to check that h is

is

that {fi)i(=i has

one has:

=

Y^

bounded complete and

has a least upper bound in Y. Moreover, h

=

that

it is

bounded complete, assume

iElffiW}- The function h it

h(UD)

To show

it

if it

X in *x)

ordered pointwise.

Of course

It

iff

exists.

omit the subscript

Exercise Let (X,^x)> (Y, ^y) be Scott domains. Prove that a function according to the previous definition

it

is finite;

A

(We

every directed

always

compact.

c.p.o.’ s. •

is

is

U iei U X ED (fj(x)} U xED U iei {fi(x)> U xED {h(x)}-

a least upper

algebraic,

2.4.1.4 Definition

A

defined by: step-a,b

(x)

we

bound

if a

for {fi}i£I-

explicitly define the set

step function from

=

by the continuity of fj

zx then

X to Y is

b else

1

.

(Y^)q of its compact elements.

a function

step-a,b

where oEXq, bGYg,

25

Constructions

We claim that the compact elements of

are exactly the least upper

bounds of finite bounded

some

of step functions. In other words, for every fQG(Y^)o, (*) fo = Ui 6Ei{step-aj,bj} for

Let us prove

that every function f = Uiei{step-ai,bj}

first

Ui£i{step-ai,bi} exists,

(We

exists

Then, in

J

then say that

when

compatible

set

let

kED

for

,

=> bj = Uj£j{bi}

f

£ UheDigh}- 1°

particular, for every

u hED

{gh( a j)}- Clearly, for each

£ gh(j)(aj) for some h(J)ED. Since

be such

,

I.

and

exists

= Uj 0 j{aj}

aj

such that

directed. Let then bj

{ghlhED is directed, we are done.

I,

finite

is finite

I

of indexes.)

Y^

= bj £ (UheD{gh})( a j) =

compact and {gh( a j)}hED and

a

is

I

for each subset J of

be a directed family in

{ghlhED

let

as above, f(aj)

I

i.e.,

compact, when

is

sets

for all J in

* gh(j)

that

J,

bj

is

is finite

I

Clearly f £

I.

gk and

Prove

now

as an exercise that for every continuous function

1.

2.

f

{ UjQ{step-ai,bi}|

that

compatible

By

f

is

every

the exercise, f

element

finite

is

Interesting

and

bj

f:

£

X-*Y, one has f(aj) }

need

= UF,

f=

for

UF ^

to

is

directed;

F

prove that f = Ui£i(step-ai,bj}

directed; thus, there exists

:

Y^xX-*Y

is

some

for

finite

such that U^i{step-

I

UiQ{step-ai,bi}. In conclusion f

has the form (*) and, in particular,

in

The function evalx,Y continuous

We

compact.

ai,bj}EF and Ui(=i(step-ai,bi} ^ is,

I finite,

= UF.

Suppose then I.

F=

the set

= Uj£i{step-ai,bj};

that

X^ is a Scott domain.

defined by evalx,Y(f> x ) =

f( x )*

The proof

eval

that

is

straightforward.

examples of Scott domains may be found everywhere

of denotational

in the literature

semantics. Indeed, the Cartesian closedness of the category allows you to construct plenty of them as

products and exponents over

commonly used ground

of data (integers, booleans, strings,

and

set

x £

x'

iff

x =

1

or x

=

etc.).

x'.

That

types.

Organize them as

is,

consider your preferred types

add a

flat p.o. sets, i.e.,

These are clearly objects of

D

least

1

element

as well as their products and

exponents.

Other relevant examples are given, for example, by the co,

the natural numbers, and Pco, the powerset of

inclusion, f(n)).

exercise, the reader

Interestingly enough, these this for Pco

,

>

:

,

may check that both P and

two familar

see section 9.6-2 for

P of

p.o. sets

the partial

i.e.,

f

£ g

iff

Vn

(f(n) j

co to

given by set

=

g(n) =

Pco live in D.

structures are also reflexive objects in

more on

maps from

partial order, in these cases, is

D

.

We sketch the

P.

be a canonical (bijective and effective) enumeration of the

Let




BEX.

Let X,

Y

be two coherent domains.

A

function F:

X -* Y

is

stable

iff

continuous;

BEX A fB

Definition

=> F(ADB) = F(A)DF(B).

The category Stab has coherent domains as objects and stable functions as

morphisms.

Given two coherent domains 1.

X

|XxY| = {(0,z)

2. (a,z)t(a',z')

/

and Y, zE|X|

}

[mod XxY]

their

U

product

XxY

is

defined by the following:

{(l,z) / zE|Y| }; iff

a

a'

=> z

f z'

[mod

D(a)],

where D(0) =

X

and

D(l) = Y.

Exercise Define the projections and check that they are stable,

There

is

a simple

way

to obtain stable functions over coherent

i.e.,

prove that Stab

domains.

is

Cartesian.

28

Constructions

2.4.2.4 Definition Let to

X,Y be coherent domains. Let one has

such that, for all x,x'E\X\,

|y]

x}EX

{x,

also

f be an

{f(x),

injective function

f(x')}EY. Define then

from

\X\

f+:X-+Y

and /-: Y-*X by t

f+(a) = {f(z)/zea};

it

f(b) =

It is

{z

/

f(z)Eb }.

a matter of a simple exercise to prove that both f+ and

f'

are stable functions.

We need to construct next an exponent object out of the set of stable maps over coherent domains.

a

zE|y|, zEF(a),

is finite,

F

is

X-*Y

Let F:

2.4.2.5 Definition

be a stable function. The Trace of

(Va'Ca zEF(a') => a ,

completely determined by

its

trace

by means of

F

is

Tr(F) = {(a,z)

/

aEX,

a')}.

the following equation:

F(A) = {zE|Y|

/

3a

CA

(a,z)ETr(F)}.

Exercise Prove that the correspondence between stable functions and their traces

Notation The symbol ft

to represent strict

Definition Let \Y%\ = {(a,z)

2. 4. 2. 6 i.

a ff a'

ii.

a

f a'

Yx

Then

used

is

is

[modX] => z

[modX] => the

z

f f z'

f z'

/

aEX, a

coherence,

is finite,

zE\Y\

}.

i.e.,

A| TB

Moreover,

is bijective.

AfBandA^B.

iff

let

(a,z)

f (a',z')

iff

[modY], and

[modY].

arrow domain (exponent

object).

Exercises 1.

Prove that conditions

2.

Prove

=

(a,z)

3.

X—Y

is

X

Y

Let f,g

Define

f

:

sg

may be

stable then

stated equivalently as

not a|a'. is

a trace of

tr(F)EY x

=»

ftg

X,Y be

i.

a

U bEX

ii.

f(

0

)

=

0

(f

sB g

f

s

p

.

from

iff

Vx,yEX

f(a

xCy

=>

f(x)

X

to

Y

,

and conversely

= f(y)Dg(x)

Let moreover £p be the pointwise order. Prove that

g).

coherent domains.

=>

stable function

be two stable functions.

if and only if Tr(f) C Tr(g).

Let

some

.

that f £3 g fs B g => fspg;

ii.

4.

or

g (Berry's order)

Prove i.

(ii)

element of

that every

that if F:

and

z\\z'

or

(a',z')

(i)

A stable function

U b) = f(a) U

f(b);

f:

X-*Y

is

linear

iff

f

29

Constructions

Prove that singleton.

X-*Y

f:

Observe

linear iff

is

its

trace

is

f+

and

f"

maps

that the

formed of pairs

where the component

(a,z),

in definition 2.4. 2.4 are actually linear. Call

a

is

Lin

a

the

category of coherent domains and linear maps. 5.

Let

in

X

X-*Y,

f:

f(x)

X-*Y

g:

be two linear functions. Prove

£ g(x). Deduce as a corollary

that

on

that Tr(f)

linear functions

C Tr(g)

if

and only

if

for all x

between coherent domains the order

of Berry coincides with the pointwise order.

2.4.2.7 Definition

The function evalxy Y :

VA Y(A,B) = fy

stable. Continuity is trivial.

[mod Y^] then evalx,Y( (A,B)Pl(A',B')

defined by the following equation:

is /

3(b,y)eA, b

We

must only check

= evalx,Y( (A,B)

)

C B}.

Cl

)

(A,B)t(A',B')

that if

evalx,Y( (A',B')

).

The

inclusion

C is immediate by continuity. Take then z

evalx,Y( (A,B)

in

C

Note

3(b',z)EA',

b'

hypothesis,

Af A' and C BOB'

B'.

In conclusion, the category

f+

and

f~

Prove that f’°f+ = idx,

By

this

evalx,Y(

(A',B')

by definition of consistency

be defined as

X

0

h:c'-*a) such that f

exists a

unque

k: c'-*c

33

Constructions

Pullback

2.5.7

Lemma

(PBL)

If a

diagram of the form

commutes, then i.

ii.

if the

two small squares are pullbacks, then the outer rectangle

if the

is

a pullback;

outer rectangle and the right-hand square are pullbacks, then the left-hand square

is

a

pullback.

Proof Exercise.

2.5.8 Proposition

is

a pullback and g

If the square

is

monic, then

p

is

monic as

well.

Proof Exercise.

The previous property suggests an If f is a

under It is

f,

function from a set

denoted f“l(C)

easy to

show

that the

A

is that

interesting generalization of a

to a set

subset of

B, and

A

C

is a

* _1 1

f

(C)

C in

in’

f

Ais

a pullback square in Set.

set-theoretic construction.

subset of B, then the inverse image of

defined by f"l(C) = {x/

diagram

f

common

B

xEA, f(x)EC

}.

C

:

Constructions

34

monic

In general, given a is

the subobject of a (if

required,

There actual

is

exists) obtained

in the introduction, the

Category Theory

when

morphism

c -*b and a

f:

a-*b, the inverse image of g under f

by pulling back g along

f.

Morphisms and Complete Objects

Partial

2.6

As mentioned

it

g:

due

is

is

to the priority

common

perspective of

Programming Language Theory and

given to “functions” with respect to “sets.” The

latter notion,

a derived notion of the former, in a sense.

an other aspect, though, that should be considered. In Computation Theory, as well as

programming, diverging computations cannot be avoided unless a

restriction is

made

in

to a

subclass of the computable functions.

The notion of partiality has domain of convergence of moreover the

f,

a natural interpretation over sets. Let call

restriction of f to

it

f j,

fj,

is

If

map

we f

f,

h

is

just a subset of its

a total

represented by a pair of total functions

convergence of

is

(i:

map

the restriction of f to D, and

try to simulate the

f|(f|)

D-»A,

h: i:

D-»A

is

in a

C

category

we

to

are forced to identify partial

morphisms up

Given a category

C and two objects

R (m

map

partial

DC A

the

is

a and

to

d'->a, h': d'->b)

m

is

iff

b,

f

maybe

domain of

natural to define a partial

it is

d-»a

h:

,

choose d

d-*b), where

in a canonical

isomorphic variations of d (see section

equivalence class of pairs (m: d-»a, h: d-*b), where (m: d-*-a, h: d-*b)

the broader sense;

the canonical injection.

as a pair (m:

we have no way

a partial function; the

A in

fi~*B. Thus a

:

previous definition in categorical language,

between two objects a and b

relation R:

A-*B be

D-*B), where

monic. However, as in the case of subobjects,

2.6.1 Definition

f:

“domain”

a partial

map

m

is

way, so

1.5.)

[m, h]: a->b

an

is

monic, with respect to the following

3k: d-*d', k

m' =

iso,

m

°k, h' = h

0

k.

Taking some license, actually

mean

Our next aim composition.

we

will often speak about a particular pair

(m: d-*a,

h:

d-»b) when

we

[(m: d-*>a, h: d-*b)]R.

If

is

C

to define a category

pC

of partial

maps on C. The main problem

has pullbacks for every pair of arrows, then the problem

is

is

in defining

resolved in the

following way.

Given

(n:

e-*b,

k:

e-*c)

and

(m: d-*a,

h:

d-*b), define

[n, k] 0

[m, h]: a-*b

equivalence class determined by the outermost sides in the following diagram, k°h':

dx^e-^c)

i.e.,

as the

(m°n': dx^e-*a,

35

Constructions

e

dx^e

c

k n

n'

b

d

h

m a

where the square Let us see partial

how

{(x,y)

by proposition

works

}

=

{(x,y)

domain of convergence of

canonical injection

= k(h(x))

i:

as

,

2.5.8, since

in Set.

{x

we

/

{x

/

the

/

xEd, yEe

composed

xEd, h(x)Ee}

pC

map

in

pC

is

of total maps.

will

is

monic,

n'

is

monic, too.

,

e-*b as canonical

h(x) = y } « {x

function.

xEd, h(x)Ee}-*>d. In conclusion,

for all

The

h|{ x

/

injections.

xEd, h(x)Ee

/

projections

}, that

and

h'

n'

a °d

X Ed, h(x)Ee}

xE{x / xEd, h(x)Ee

Then

},

one has

wanted.

total iff

As

n:

are respectively

Every arrow fEC[a,b] has a natural associated arrow that a

n

For the equivalence relation defined on

suppose dCa, eCb and take m: d-»a,

associated with the pullback

k(h'(x))

that

xEd, yEe, h(x) = n(y)

/

the expected

is,

Note

the previous definition

we

morphisms,

dxbe =

a pullback.

is

it

in

pC,

that is (id: a-*a,

has the above form (up to equivalences). (pC)t

become

clearer in the next chapter,

is

f:

a-»b).

We

say

the subcategory of

C is “isomorphic” to (pC)t-

Exercises Prove the following assertions: 1.

id is total.

2.

f,g total

3.

f°g total =>

=>f°g

Example pSet

total.

g

total.

is

the category of sets with partial

the partial recursive functions. that defines the

commutes

iff

(pSet) t = Set,

pEN

is

PR t

=

Remark Sometimes,

R

fEPR

as morphisms.

pR

=

PR

is

the

monoid of

defined by using partial recursive functions in the diagram

morphisms, instead of the

eb°f = f°e a for

maps

total

ones (see section

2.2).

Observe

and, hence, e a (n) = e a (m) and f(n)j

that the

diagrams

=> f(m)|. Clearly,

and (pEN) t = EN.

in the construction of the category

pC

from C,

it

may be

intersting to restrict

our attention to only a subset of the class of monies of C. Consider, for example, the category p.o. sets:

an interesting definition of a partial

be defined

in

an upward closed subset. Let

map

C be

f

between two

p.o. sets

would

PO of

also require f to

a category with pullbacks for every pair of arrows.

Constructions

36

An admissible

M

family

of monies of C,

a family of

is

monies closed under

identities,

M on a category C gives rise to a different category

compositions, and pullbacks. Every admissible

(M)(pC) of partial maps.

As

already recalled, diverging computations play an essential role in the theory of computation.

They

also have an obvious interpretation in Set Theory.

computations

may

However, the

not suffice. Type-free languages and

set-theoretic understanding of

many typed

we

ones, as

shall see later,

escape a “naive” interpretation as sets and functions.

As need

days of denotational semantics of programming languages, the

a matter of fact, since the early to give

meaning

to (possibly) diverging

computations over nontrivial mathematical structures

suggested the introduction of various categories of p.o. sets with least elements. The naive understanding of this

them

is

immediate: add to

a p.o. set structure as flat p.o. sets

all

(i.e.,

required data types, as sets, an extra element and give

1

is

the least element and

all

others are incomparable).

We already dealt with this in practice, when presenting various categories of p.o. CCCs.

In a sense, the bottom element

1

provides the

first

sets as

examples of

hint of the introduction of approximation

and continuity notions to the mathematical semantics of programs.

As we have

demonstrated, this concept

is

very clear mathematically,

at least in several specific

categories, such as continuous or algebraic lattices, c.p.o’s, and Scott domains. interesting categories for computations such as setting

and avoid any abuse of

this

EN,

It is

not so simple in

though. In order to understand

simple concept, a category-theoretic perspective

it

in a general

may

provide a

sound mathematical frame.

In the rest of this section, C is a category of partial maps, i.e., C = pD for D with pullbacks for every pair of arrows. Ct is the associated category of total maps.

Notation

some

category

Since

Ct

is

a subcategory of

C,

we compose

composition. For typographical reasons,

The

set-theoretic idea

we

total

we

and

partial

write a

0

adding an extra least element, then any hom-set of

Definition The lifting of

exa EC[a°,a] and for every

L

exa oxc(f) =f;

H Tc( exa°g) = i

same operation of

total

when an

maps with

object a

target

a

is

0

is

“lifted” to a

isomorphic

0

by

to the

.

aEObQ

is

an object

cEObQ an operation rc

and for every gGC([c,a°] one has

the

1

try to formalize categorically is that,

corresponding hom-set with target a

2.6.2

morphisms by using

instead of a- -.

:

a°EObc (

together with a morphism

C[c,a]-*Ct[c,a °] such

that,

for every fEC[c,a]

37

Constructions

The operation x c C[c,a]-»Ct[c,a°] Conversely,

xc

if

bijective

is

and

Thus, the following

h.

2.6.3 Definition The lifting of

equation: exa

°g-f

we

x c (ex a

g)

= x c (ex a

0

];

wanted, for

may be

b,

all

0

(g).

since x c

is

x c (f)) = x c (f) =

lifting object:

an object a °EOb(j together with a morphism exa E

is

c

that,

fEC[b,a]

aEOb£

is,

0

0

for every fEC[c,a], there exists one and only one gEC([c,a j

C[a°,a] such

Thus, as

That

g.

an equivalent definition of the

is

x c ’l(g) = ex a

defined by

is

(1) holds, then also (2) does. Indeed, let gECt[c,a

such that x c (f) =

surjective, then there is fEC[c,a]

t c _1

as

is bijective,

:

satisfying the

C[b,a] and Ct[b,a°] are isomorphic, since any partial morphism

when

uniquely extended to a total one

the target object

is lifted

(and the

lifting

exists).

Exercise Prove that the

2.6.4 Proposition

C

(notation:

Let a

a

f °h = f oh.

2.6.8

Theorem Let

0

a

be the

lifting

of aEObQ. Then a < a°

Vb, VfEC[b,a],

3fECt[b,a]

such that f extends f Proof(=>) Set f = out

°

Tb(f).

Then, for every hECt[c,b], such that

(f 0

h)ECt[c,a]

f°h =out°Tb(f)°h

= out°rc(ex a oTt,(f) o h = outor^foh)

by

(ii)

by

(i)

= out^c( ex a oTa(id a)°f°h

by

(i)

= outox^id^ofoh

by

(ii)

=

in def. 2.6.2

f°h

( b


where mg:

B-*A is the

subset inclusion, and true: {*}-*{true, false} takes

The above diagram suggests

2.7.1 Definition

object

Q

C

Let

{true .false}

to “true.”

the following definition in Category Theory:

be a category with a terminal object

together with a morphism true:

arrow cfr a-*Q such

*

t->Q such

that for every

t.

A

subobject classifier

monic

i:

b-*a there

is

is

an

a unique

that the following is a pullback diagram:

a

b i

cb

!b

true t

Exercise

A subobject classifier, when

The subobject It

classifier

exists in

it

C,

is

unique up to isomorphisms.

Q plays a central role in the translation of Set Theory into Category Theory.

allows the simulation into the categorical language of concepts like intersection, union, and

complement, by the definition of a Hey ting algebra of truth-morphisms over Q.

2.7.2 Definition

A

topos

is

a category

C with

a terminal object, a subobject

classifier,

pullbacks

for every pairs of arrows, and exponents for all pairs of objects.

It

will turn out that a topos

confidence as

we do

is

in Set.

a “universe”

where we can carry out constructions with almost

Of course, Set

itself is a topos.

Exercises 1.

Prove

2. In

that if

a topos C,

C is

a topos, then

what

is

C

is

a

the arrow eval:

CCC.

Q a xa-» Q

?

the

same

Constructions

40

In the spirit of a categorical description of set-theoretic concepts,

one may also

talk in topoi of

“relations,” “power-sets,” and so forth. Quite generally, given objects a and b, a relation on a

and

b, is

simply a monic

given by a relation

2.7.3 Definition

object of

which r:

is

C

Ea Let

:

r:

d-*axb. In particular, then, one

may

consider power-objects P(a) as

d-*axP(a) with the following universal property:

C

be a Cartesian category and

together with an object

universal in the sense

that,

b-*P(a) and a (forcedly unique)

d and a monic

aEObc

Ea

:

for any object b and monic

map

g:

e—>d

to

make

•

The power-object P(a)

d-*axP(a)

m

:

(a

is

an

membership-relation)

e—*axb, there

is

a unique

map

the following square a pullback:

m

€

a

idxr

axb

As

Q

is

axP(a)

the truth value object, the set-theoretic intuition suggests that, in a topos, the object

should represent exactly the power-object of

2.7.4 Proposition In a topos

Proof Let

Ea

:

C every

Qa

a.

object a has a power-object

Qa

.

d-*axP(a) be defined by the following pullback square:

true

aXQ a

Q eval

Then

the required universality of

1.

of the subobject classifier,

2.

of the

3.

of the above pullback

map

eval

,

i.e.,

Ea

i.e.,

is

given by the properties

the existence of the characteristic

the existence and unicity of

map

c e of m;

A(c e );

;

and by an application of the pullback lemma diagram, where the squares are pullbacks:

2.5.7. All this is described

by

the following

commuting

41

Constructions

!e

m

truue

aXQ a

aXb ldXA(c

^

e

eval

)

c

The reader who has completed

e

exercise (2) above

compare

the construction in definition 2.7.3 and

eval says whether “an element of a

is in

it

may

easily understand the intuitive

to his set-theoretic understanding.

a given subset of a

meaning of

In particular,

.”

Exercises any topos has

1.

Prove

that

2.

Prove

that a category

C

lifting. is

a topos

if

and only

if it

has a terminal objects, and

all

pullbacks and

power-objects.

References The though

general notions can be found in the texts mentioned at the end of chapter

their presentation

and notation may be

different.

For example, in the case of CCCs,

also because these categories play a greater role in the categorical approach to

Type Theory,

denotational semantics, than in other applications of Category Theory. Notice that not

CCCs

and consider

that the origin of

Category Theory

is

1,

so

this is

or to

Grp and Top

are

largely indebted to algebraic geometry.

Applications of universal concepts from Category Theory to Programming Language Theory have

been developed by several authors, the

ADJ group

data types; see categories

aims of

is

explicitly

Goguen

et al.

may be found

this

(1973/78)

in Burstall

for the

category

used

in

program

specification. For instance, the

).

An

(initial

early insight in the connections

examples

EN was

first

we

work by

algebras of abstract

between programs and

and Thatcher (1974). All these aspects go beyond the limited

book, which privileges “theories of functions” over “algebraic theories.”

notions with an algebraic flavor

As

in particular for

based on universal conditions for specification

may be found

Some more

in chapter 4.

have provided, they originated entire areas of research. In

particular, the

introduced by Malcev, as a general setting of Recursion Theory, and widely

Ershov (1973/75). Scott domains and related categories are broadly treated

in several places,

42

Constructions

e.g.,

Scott (1981/82), Scott and Gunter (1989), or

many books

in denotational

semantics. Scott

(1976) presents Poo as a reflexive object and discusses categories of retractions. The (generalized) Myhill- Shepherdson theorem and related matters (1984), Berger (1986), Rosolini (1986), and

Coherent domains are given introduced (with

There

is

some

in

may be found

in Scott (1976), Giannini

Longo (1988a), among

others.

Berry (1978) and used in Girard (1986), where linear maps are

relevant consequences, see section 4.4).

a broad literature

on categories with

DiPaola and Heller (1984), and

in

partial

Longo and Moggi

maps.

We

partly followed the approach in

(1984), where the notions of “complete object”

and partial Cartesian closed category were introduced (and applied

The

properties of complete objects carry on also

shown

in

Moggi

(1988), which

is

when using

a

in Asperti

more

and Longo (1987)

).

topos-theoretic perspective, as

devoted to a deeper insight into the concepts just sketched here.

New results and surveys on categories with partial Moggi

and Longo

morphisms may be

also found in Rosolini (1986),

(1988a), Robinson and Rosolini (1988) and Curien and Obtulowicz (1988). Independently of

the category-theoretic approach, Plotkin (1984) used the category

pCPO

for the purposes of

denotational semantics.

Topoi originated by works of Lawvere and Tierney.

may be found

in

Johnstone (1977). The reader

(1985) on this subject.

may

A (difficult)

introduction to

Topos Theory

also consult Goldblatt (1979) or Barr

and Wells

Chapter 3 Functors and Natural Transformations

The

starting point of

object

Category Theory

comes equipped with

morphism functor

is

is

the premise that every kind of mathematically structured

a notion of “acceptable” transformation or construction, that

that preserves the structure of the object. This

the “natural” notion of

morphisms between

is,

a

premise holds for categories themselves: a categories.

By

a further step, the question

about what a morphism between functors should look like suggests the notion of natural

transformation.

Of course,

this is not a

matter of arbitrary generalizations:

may

it is

mathematics

a habit of

to build

in this

way, by the uniformity of

methods and language, the reasonableness of specific constructions. And

this theoretical unification,

constructions on top of constructions, since one

suggesting power and intellectual unity,

is

understand

one of the major merits of Category Theory and

its

applications.

we

Moreover, as essential

way

shall see in several

that are easily

met

3.1

Functors

If a

transformation

of D,

it

in

an

or even suggested by the categorical language of functors and natural transformations.

Indeed, there are even too

few

examples, there are notions which are clarified

in

many examples

for our purposes, and

we

will

be able

to

mention only

a

computer science.

F between two

C and D must map the categorical structure of C to that must of C to objects and morphisms of D; moreover,

categories

must take objects and morphisms

it

preserve source, target, identities, and composition. Such a transformation

F:

C-*D

is

called a

functor.

C

3.1.1 Definition Let

F0 fr ObQ Fmor(f) Fob(a)

operations "

"

"

•’

D

and

be categories.

Fmor MorQ Fob(b)

-* Obj)

,

:

A

(covariant) functor

-* Morj) such

that,

for each

F C :

f:

-*

D

is

a pair of

a-*b g: b->c ,

in C,

= FmoAs) °^moAf) Ffnor(ida) = ^F0 b(a)

FmoAs

°f)

*

It is

usual practice to omit the subscripts “ob” and “mor” as

whether the functor

is

meant

to operate

it

is

on objects or on morphisms.

always clear from the context

Functors and Natural Transformations

44

Examples

C

1. If

C

and

D

are preorders, then a functor F:

D, indeed

to the objects of

F(f)ED[F(a),F(b)]

between two

f

among

partial orders

C

the objects of

F(a)

and D, there

the categories associated with

C

is

D and iff

G

be functors and t:

aEObQ,

ra

0

:

F - idc and F

F-+G

E D[F(a),G(a)]

°

G

:

G-*F by

x"l

)‘l. a = (x a

x‘l

is

idc

idj)).

be a natural transformation from is

an isomorphism.

natural isomorphism.

Proof Define x"l

-

are

natural, since

VfEC[a,b].

Then t

is

a

t

51

Functors and Natural Transformations

= t(j -1

t-ib’G®

®

0

G(f) » a

ta'

= tb_lo tb°F(f)«V = F(£)«x-l a

A simple

Example

little

One may

xc

Inc:

Ct~*C be

C be

The

lifting

When

.

writing this

a Category of partial maps,

is

the object a

0

is

by

hidden behind definition

down

explicitly, the definition is tidier

C* be

the associated category of total maps, and

the obvious inclusion. Thus, definition 2.6.2

of aEOb(j

“liftings” (see

actually understand the general notion better,

exercise on natural transformations. Indeed, what

2.6.2, is the “naturality” of

expressive. Let

.

example of natural transformation may be given by studying

section 2.6), in various categories.

completing a

1

1

may be simply

and more

restated as

such that the functors C[_,a] °Inc, Ct[_,a°]

:

C(-*Set

are naturally isomorphic.

Then, by definition of natural transformation and horn-functor,

this requires the existence

function x such that the following diagram commutes, for any object b and c in

Tb

C[b,a;

C

C and

of a

fECt[c,b].

[b,a°]

t

_ °f

_of

C

Clc,aj

That

is,

With

this definition, to

(x c) l(h°f)

isomorphism. Set

=

'(id): a

Then a

0

,

liftings is

Let

we

are

C be a pC

(j>

=

(

f (n)(0))

else _L

fER

.

defined, then (J>f(n)(0) converges and, hence,

x a b(f) = f

For each

.

f(n) =

f(e'(n))

=

x a gives the required natural

a,

g'(n)(0).

the fact above, this defines the lifting functor in

Exercise Define the category

as for

.

is

2.6.2.

Thus,

isomorphism, as VgEEN[b,al] 3!fEpEN[b,a] diagram.)

to the set a

o)-*a°

:

such that f

sets in section 2.2. Let

example before propoition

e°(f(n))

= e(f(n))

f( e '( n ))-

.

a

which represents

(iteration) theorem. f(e'(n)) =

have a bottom element. As

numbered

the category of

sets in the

by adding

fEEN[b,al] of exists

EN,

given by

fEpEN[b,a]. By f

to

pPo.

numbered

(a°,e°)

do not need

with open subsets as domains. Clearly, the

(Exercise: check the due

pEN.

of equivalence relations on

and the morphisms are induced by

co

and effective maps

{hint: the

total recursive functions similarly

ER and EN are equivalent, but not isomorphic.

Indeed, one

is

small, while

is not.

next discuss ways to derive natural transformations from a given one and, finally, the notion of

horizontal composition.

Let

H: A-*B, F: B-*C, G: B-*C, K:

transformation as

shown

C-*D

in the following diagram:

be functors, and

let

x:

F-*G

be

a natural

53

Functors and Natural Transformations

H

K

B

x induces two natural transformations Kx: (Kx)b = K(xjj) (tH) a = XH( a )

We

:

:

KF-»KG, and xH: FH-»GH,

respectively defined by

KF(b)-*KG(b) FH(a)-*GH(a).

have, for every fSB[b,b'] and every geA[a,a'],

(Kx) b '° KF(f) = K(xb ) (xH) a

'°

o

FH(g) = x H (a')

that proves the naturality

K(F(f)) = K(x b ° F(f)

F(H(g)) = G(H(g))

•

)

= K(G(f)

«

xb ) = KG(f)

x H(a = GH(g) )

of

Kx and xH

—

KF(b)

b

«

KF(f)

KG(b)

in

Consider

now

*

FH(g)

KG(b')

B

o

(Kx) b

(xH) a

FH(a)

a

KG(f)

KF(b')-

Kxb = KG(f)

.

Kt V b'

•

•

inD

in

GH(g)

—

FH(a')

a'

A

GH(a)

in

.

GH(a')

C

categories, functors, and natural transformations as described in the following

diagram:

F

H

G

K

B

Then, for the naturality of for every

a

with respect to the arrow T5, the following diagram

bGObg: HF(h)-

H(x b )

'G(b)

'F(b)

KF(b)

HG(b)

K(x b )

KG(b)

(in

D

)

commutes

o

(

.

Functors and Natural Transformations

54

The horizontal composition of o and t for every

b€ObB,

crtb

oq

=

We check the naturality of crtb' o

b)

or

Let f£B[b,b’]

.

that if

we

0

=

0

=

0

= K(G(f)

»

HOty

0

= KG(f)

K

identify the functors

K(xb)

H

and

H(xb') ° HF(f)

)

by

the naturality of x

xt, )

by

the diagram

Tb

F(b)

«

o CTTb.

0

° F(f) )

H( G(f) Xb)

defined by,

op(b)-

°

•

HF-*KG

then

,

H( G(f)

= KG(f)

Kx and xH may be understood

idjj,

°

= aQ(b ')

HF(f)

= °G(b ')

Note

the natural trasformation or:

is

H(tb) = K(tb)

»

oF(b )

•

with the identity natural transformation idj£ and

as particular cases of the horizontal application

between

natural transformations (why?).

Exercise Prove the following equality (v

3.3 Cartesian

By

p)(x

among °

natural transformations (interchange law):

o) « (vx)

pa)

° (

and Cartesian Closed Categories Revisited

C is

definition, a category

there

o

Cartesian

iff

contains a terminal object

it

an object axb together with two morphisms p^: axb-*a

is

an isomorphism
c

:

C[c,a]xC[c,b]-*C[c,axb] such that for

,

t

,

and for every a,bEOb^

p2: axb-*b, and for every object c all

morphisms

f:

c-*a, g: c-*b, h:

c-*axb, the following equations hold: i.

ii.

Pi

°

c = f

P2

°

c =

We want now to show that (nat)

Indeed,

we have

The previous A:

C-*CxC

then


is


c

g-

is

also “natural in c”,

for every k: c-*c' c

c

*

0

'

the functor defined

by A(c) =

(c,c),

,

>

'

1

°

k,

c

is

A(f)

isomorphism from CxC[_,(a,b)]

0

0

k >c

k >c

is

:

by

(ii)

by

(i).

actually a natural isomorphism. Indeed, let

= °

'

g

(f,f)

A

(A

is

called the diagonal functor);

to C[_,axb].

Note

C[_,axb]: C-»Set are contravariant functors. Conversely suppose

and b an object axb and a natural isomorphism x

of x

0

satisfies the property:

i.e., it

k = b: b a xa-*b (evaluation map) and, for every object c a function A c C[cxa,b]-»C[c,b a such that, for all morphisms cxa-*b, h: category

and b

it

in

,

c-*b a

,

:

p. eval a T|.

It

f:

]

one has

turns out that

Ac

is

check

°

(A(f)xida)

=

f

h.

“natural in c”, in the sense that for any fEC[cxa,b], gEC[d,c] Ac'(f)

In order to

b

A(eval°(hxid)) =

this, recall that (|3)

°

g = Ac(f

o

gxid).

corresponds to f

exa

(Diag.Eval)

A(f)Xid

b

a

xa

Thus, by twice (Diag.Eval), the following diagram commutes:

b

.

Functors and Natural Transformations

56

Moreover,

set

A‘l = eval°(_xid).

Then

A'1°A =

id

by

and

A°A"1 =

id

by ( q ).

That

is,

cEOb^, A c C[cxa,b]-*C[c,b a

for each

:

isomorphism, by proposition

Note

A

that, formally,

to the contravariant functor

We

(p)

abbreviate C[_,b]

0

A

is

a natural

3.2.2.

a natural transformation

is

,b a

C[

_xa

Conversely, suppose

an isomorphism. Thus

is

]

]

(where _xa

as C[_xa,b].

is

a

0

_*a

the product functor defined in the obvious way).

Note also

that, for all objects

from the contravariant functor C[_,b]

that C[_xa,b],

and

an

b, there is

A

isomorphism A: C[_xa,b] a C[_,b a ]. Then the naturality of

C[_,b a ]: C-*Set.

"1

is

object

ba

and a natural

expressed by the following

commutative diagram:

A' 1 c

C[c,b a

C[cxa,b]

c

]

ba

ofxid

A-l,

C[c'xa,b

Set

now

eval a> b

= A’l^a

C[cVb a ]

]

(id^a) and note that

exponent with eval a> b as evaluation

map

.

eval^: b a xa-*b

Indeed,

let

f

=

c'

.

We want

A

to

a c (g): c-*b in the

prove that b a

is

an

above diagram. Then

we have eval a,b

0

(

Ac(g) xid ) =

°

Mg)

)

=

This argument gives the following equivalent characterization of

3.3.2

Proposition

a,bEObQ

Observe

there

is

C

is

a

Cartesian closed category

an object b° and a natural isomorphism

that propositions 3.3.1

and 3.3.2 require

A

g-

CCC:

iff it :

is

Cartesian and for every

C[_xa,b] -* C[_,ba]

that the category

based on horn-functors. Thus, in a sense, the equational definition

C be locally is

more

small, since they are

general.

.

57

Functors and Natural Transformations

Remark

It is

easy to prove that the following (natural) isomorphisms hold in

all

CCCs,

for any

object A, B, and C:

A « A;

1.

2.

txA»A;

3.

AxB * BxA;

4.

(AxB)xC * Ax(BxC);

5.

(AxB)-*C « A-»-(B-»C);

6.

A— (BxC) * (A-*B)x(A-*C);

7.

t-*A «A;

8.

A-*t «

What i.e.,

is

worth mentioning, though,

no other isomorphism

application of

by the

3.4

t.

Its

More Examples

that these are exactly the

valid in

is

lambda calculus

of this book!).

title

is

all

CCCs. The proof of

to categories (an application in

key idea will be mentioned

of

isomorphisms

that hold in all

this fact is nontrivial

and

the other direction from what

CCCs,

is a is

nice

meant

in chapter 9.

CCCs

Both examples here derive from bordering areas of (generalized) computability and Proof Theory.

The

first,

in particular, like

many

aspects of these theories,

is

widely used in the type theoretical

understanding of programming languages constructs.

3.4.1

Partial Equivalence Relations

A very relevant example of CCC is suggested by

a long story.

interpretation of intuitionistic logic and continued with the

Proof Theory. The idea

is

to

look

but by a simple and insightful

at

way

As

K

=

usual, let

Cartesian closedness

(to,-)

defined by

semantics of polymorphism in cp:

is

work of Kreisel,

obtained.

cp

m (n) A partial

nRm

3.4.1. 1 Definition

iff

n relates

to

The category

m

iff

Girard, and Troelstra in

The approach

co.

is

EN,

also used in the

will give us a paradigmatic structure

•

:

partial recursive functions. Let

coxco-^co

R

on a

is

set

the partial application

V

is

a symmetric and

on V.

in R;

PER

symmetric and transitive relations on

fePER[A,B]

realizability

PART II.

equivalence relation

.

transitive relation, not necessarily reflexive,

(Notation:

It

co-*PR be an acceptable Godel numbering of the

be Kleene's applicative structure, where

m-n =

began with Kleene’s

functions as computable ones, as in the case of the category

“quotient-set” semantics of types, in functional programming. in the categorical

It

{p}R = {q

|

qRp};

Q(R) = {{p}R

of partial equivalence relations on

a)

|

p

Rp

has as objects the

Morphisms are defined by

f : Q(A) -» Q(B) and 3n Vp (pAp =>

f({p}A)

})

= {np} B ).

)

Functors and Natural Transformations

58

PER

Thus, the morphisms in recursive functions,

Let

AxB

is

now
: (dxcd-*o)

,

are

when

defining the computable elements in Scott domains

The

D

idea

is to

generalize the

(see section 2.4.1), in the

explained below (in short, the computable elements are the limits of recursively enumerable

indexed sequences.) This suggests several CCCs, such as limit spaces (L-spaces) and

(FIL) with examples

3.4.2. 1

their relevant subcategories.

in sections 5.3

Definition

A

and

8.4, together

They

spaces

with other ideas for higher type computations.

limit space (L-space) (X,i)

between countable sequences {xJiEw

filter

will be introduced here and discussed also in the

—X

is

and elements

a set

xEX

X

and a

relation

(convergence

(notation: {xj} lx) such that

,

59

Functors and Natural Transformations

if all

i.

if

ii.

if not({xj} lx),

iii.

h(0)

An

but finitely

many

X[ are x,

{x \}[x and k(0)
dl

:

is

also a Petri net, since

f:

where

category).

:

As

(3'q,

T'-*V'),

:

(S+S')©,

i.e.,

:

N

and

N'.

is

their

Petri

is,

TxT'-»S©xS'©)

the product of

two

free

and S'© coincides with the monoid freely generated by (S+S'). The synchronous product of the nets

g:

the careful reader

d';[:

N' = (d'o,d'i: T'-*S'®), their product as graphs, that

S©xS'© «

S©

T-*T' and

with the extra condition that g

= (do>dl T-*V) and G' =

NxN' = (doxd'o^ixd'i is

comma

as a

,

are pairs

4.1.4.),

as objects,

GxG' = (do x d'o>dl x

I

)

A—oB, A

r 2>

ri,

A, A'

I-

r2 b

Ml- a, a

—°» r

r

AUB

,

,

A—oB

I-

a’

A, A'

I-

3. additive rules

r|-T,

(T,r)

(

r|- a,

a

0

,

r,

1

)

o

r|-B,A (n,

I-

AflB,

r,

ADB A I-

b

r, i,

a,

I-

a

2) r,

I-

l-

i)

i,

A

(n,

r

a a

r,

(n,r)

r

a

I-

AHB A I-

a

(®,r,l)

r

I-

A©B, A

r

I-

b,

a

r,

(®,r,2)

r

4. linear

a

I-

A©B, A

r,

r,

b

|-

a

A©B A I-

r l- a, a

I\A|-A (

suggested by the rule (®, as

disjunction

while the

which

is

1

,1

)

r,Ai|-A

rl-A-1-A

meaning

a

negation

CM)

As

I-

(®,1)

1),

the

commas

dual to ®.

comma

in the left

in the right

hand side of a sequent has the same logical

hand side correspond

to the tensor

sum U,

the

A

)

.

87

Categories Derived from Functors and Natural Transformations

Observe

that the rules (1, r)

and (1,1) are equivalent

to

r,A|-B,A (contrap) T,

1,1

the rest

From

is

1-

gives both

1

’

steps,

(cxabc)

Exercise (A-L)J-

and

1

and

1- 1-*-

1

|-

A—oB

1- -

while 1

1

,

1,1 gives both l- 1

|-

1-

1

and

l-1-

1-

1;

|-

B

by

(

by

(

®

r)

>

—

o,

1

one also obtains

(A®B)®C

Show

I-

(a-iABC) A®(B®C)

A®(B®C)

the logical equivalence of

and (A- -)-!1

1-

may be enough

A,

)-*-

i.e.,

(

is

(A®B)—oC

to suggest that the categorical

=> given

and the bifunctor

and

|-

(A®B)®C

A—o(B —oC).

Derive also that

®

meaning of linear logic may be found and

—o

are interpreted

in definition 4.3.5, respectively.

too, so to provide an interpretation for

D

and

its

Observe

|-

identity

T

by the

in particular

tensorial product

that the cartesian product is

(see later). Negation and

constructions are taken care of by a “dualizing” endofunctor (-)*, which

is

all

dual

closed in the sense of

Indeed, theorem 4.4.5 below shows that this functor, given

definition 4.3.6.

A

“dualizing.”

symmetric monoidal closed categories, where

needed

l-l-l-l

(1,2)

easily derive the following sequents:

a®b

I'

A,

(eval^B)

This

may

the rules one

few

,

obvious).

Oab) A B

In a

A1

1-

1|-±J-

(1,1)

(Hint:

B1

®

and

fl

,

immediately yields their duals.

4.4.3 Definition

A *-autonomous

contravariant closed functor (-)*:

category

K —> K

-

there exists a natural isomorphism

-

the following

K

is

a symmetric monoidal closed category and a

such that

d : Id a

diagram commutes (where

(-)**

(-)* is the action

of the functor on exponents)

.

Categories Derived from Functors and Natural Transformations

88

—

(-)*

A =>B

>B*=>A*

d o_od-\.

(-)•

^ A

*-autonomous category

is

linear if it

is

A** =>B**

also Cartesian.

In other words, a linear category has both a

monoidal and a Cartesian

structure,

©

and

PI

plus a

,

dualizing functor (-)*.

Both the vertical and the horizontal arrows

4.4.4 Proposition

in the

diagram

in definition 4.4.3

are isomorphisms. Moreover

,

L

A* sB*

iff

A e B;

A* sA=>l*.

ii.

Proof By

definition, the following

diagram commutes and d

0

_

0

d* 1 is

an isomorphism:

f**

Then

the horizontal (-)*, in the diagram in definition 4.4.3,

vertical

one

split epis as

implies

B*

iff

4.4.5

is

is

a split

mono

(see section 1.4) and the

a split epi. Since the latter is an instance of the former, they are both split

well and, thus, isomorphisms. In conclusion,

A=>B

K be a

A*

a B*

a 1=>A*. Thus,

A* a

a B*^>A*. Finally,

A a A** a B** a B and, by exercise (3) below definition 4.3.5, A a B and, moreover, A* a A 1*.

Theorem Let

monos and

A*

linear category. Define

AUB = (A*®B*)* A®B = (A*DB*)*. Then

U

is

a dual to the tensor product and

0 = T*

respectively.

Proof

U

is

a well-defined dual to

®:

®

is

just

the coproduct in K.

Their identies are

compare with definition

AU1 - (A*®1)*

-

(A*®B*)®C* a A*®(B*®C*)

a A** a

A

(similarly for the identity to the left),

implies

((AUB)*®C*)* a (A*®(BUC)*)*

(AUB)*®C* a A*®(BUC)*

implies

(AUB)UC a AU(BUC)

= 1* and

4.3.1 (and the

afterwards) and note that -

1

implies

remark

Categories Derived from Functors and Natural Transformations

As

images (pa*)* and (PB*)* v ^ a the

(A*®1)* «

The of

pa* and pg*

for the Cartesian coproduct, note that, if

their

A

and

A©0 = (A*OT)*

1=

1*. In particular,

when

A*OB*,

are the projections for

A©B.

functor are the injections for

(")*

Finally,

then

AU1 =

« A.

and theorem 4.4.5 prove the “dualizing” role of the

results in proposition 4.4.4

linear logic,

89

theorem 4.4.5

is

a version of

De Morgan

functor and

(-)*

rules in the semantic structures for

described in categorical terms. Note also that

A*UB

a (A®B*)* a (A®B*)=*1 a A=>(B*=>1) a A^(1=>B) a

A=>B

which gives the “classic” flavor of this fragment of linear

The meaning of linear |-

B

as a

logic as a deductive system

morphism between

interpretation,

is

logic.

then given by interpreting each entailment

the interpretation of the formulas. Linear categories yield such an

which we sketch

here,

by induction on

rules.

crucial ones are simply the properties corresponding to

are interpreted by

for 1 and

1

Here are some of the basic cases. Some

(mab)> ( eva lAB)>

given before definition 4.4.3, in monoidal closed categories.

the rules

As

(

a ABC)

anc*

1

and

1

are (the unique) identies for

by commas, on

®

and U, which

the left or right, respectively,

|-

(®,r) and (U,l)

®

and

U

are bifunctors

(H,r) and (©,1)

0

and

©

have pairing and sum of morphisms

there exist projections for

(©,r,l) and (©,r,2)

there exist injections for

(contrap)

(-)*

(-U-L)

1

(By the argument before definition

a “^ABC)»

the fact that

are expressed

and (0,1,2)

(

for the others,

of

(0,1,1)

A

is

O

©

so defined in linear categories

= 1* and, thus, 1* «

4.4.3, (contrap)

1

.

and (1,1) are equivalent

to (l,r)

and

(1,1)

.)

Categories Derived from Functors and Natural Transformations

90

Example The

category Lin of coherent domains and linear maps, in section 2.4.2, not only

provides an example of linear category, but

it

has even been the source of inspiration for linear logic,

since linear functions depend on inputs “additively” as deductions from assumptions in this logic.

(There

is

We

more than

example and section 5.5 should

that analogy, though, as this

noticed that

Lin

is

Cartesian in 2.3.7(11). Lin

is

X©Y

be coherent structures. Define then the coproduct

(|Y|,t)

zE|X|}U

with the coherent structure ({(0,z)

{(l,z) |

|

and x|y

.

maps

,

category. Indeed,

initial in the

domain associated

is

iff

a

=

a'

the same, but the coherence relation

as an exercise.)

The singleton coherent domain T = 0 = {0} both terminal and

as the coherent

zE|Y|},|®), where (a,x)t®(a',y)

(Note the difference with the product: the support

changes. Give the embedding linear

clarify.)

also co-Cartesian. Indeed, let (|X|,f) and

associated with the it

is

empty coherent

structure, is

the identity for both the product and the

coproduct. In the previous section

exercise

below

we

turned Lin into a symmetric monoidal closed category (see the

coherent domain associated with the coherent structure

(|A|x|B|, f®),

A@B

product

Recall, in particular, that the tensor

definition 4.3.5).

is

the

where (x,y)f®(x',y')

iff

xfx' and yty'.

The dual of

the tensor product, is

domain associated with (x',y')

or xf fx' or

yt

U

given by the tensor

the coherent structure ty')-

Also * n

sum AUB,

product and

this case, the tensor

that

is,

(x,y)t u (x',y')

u ), where

(|A|x|B|, f

sum have

the

the coherent iff

same

(

(x,y)

=

support,

but the coherence relation changes.

Notice that the identity for both the tensor product and

{0,{1}}, associated with ({1},

As

coherent domain associated with f*:

As one =1

.

B*-*A*

is

(-)*

,

is

the coherent

(|A|, f *),

given a coherent structure

where xf*y

~(x|y)

iff

defined in the following way: (y,x)ETr(f*)

domain 1 =

in

iff

define

(|A|, t)>

A

.

On

in

1-

theorem 4.4.5 are realized

For the reader's convenience,

we summarize

A*

as the

a linear function

(x,y)ETr(f)

f:

.

could expect (and easily compute), in coherent domains one has T* = 0 =

Moreover, the equations

Memo

dual

=).

for the contravariant functor

A-*B,

its

T

and 1* =

1

in Lin.

the identities in linear logic (and linear

categories) 1

which

are interpreted in

Lin

® 1 1 = 1-

for

as

;

for

D

{0,{1}} and

T

for

U

;

T

;

0 for

©

= 0 = {0}, with the obvious coherent

structure.

References Textbooks, which

stress the applications of

Category Theory

to

computer science

with an algebraic perspective, are Arbib and Manes (1975) and Rydeheard and Burstall (1988).

Goguen and

Burstall (1984) contains a survey and references to part of this area, broadly construed,

Categories Derived from Functors and Natural Transformations

which ranges from the work

in Elgot (1971), to the contributions of the

volume, the many references

this

91

that include the

Besides the references to general Category Theory, the reader

and Wells (1985) for monoidal categories, monads, and references,

The

which

ADJ group

their

may

the understanding of programs as “functions

Montanari (1988) or Degano and

al.

we

consult Barr (1979) and Barr

described are borrowed from

from values

(1989), as for the

end of

or Wright).

mathematical applications. Further

also discuss linear logic, are Barr (1990) and Marti-Oliet and

recent applications of “monoidal notions”

(see, at the

names of Goguen, Thatcher, Wagner,

to computations,"

examples on

Meseguer (1990).

Moggi

(1989), as for

and from Meseguer and

Petri nets (see also Marti-Oliet

and Meseguer (1989)).

The main reference in Girard

for linear logic is Girard (1987). Further

work may be found, among

others,

and Lafont (1987), Lafont (1988), DePavia (1987), and Seely (1987), where *-autonomous

categories are proposed for

its

semantics.

Lambek

(1968) contains early work on the categorical

significance of logical systems and discusses weakenings of the structural rules.

Chapter 5 Universal Arrows and Adjunctions

In chapter 3,

we

The reader may have noticed

constructions in Category Theory. construction F:

way

introduced the notion of functor as a uniform

C-*D,

there exists a particular object c in

C

that in

u

,

i.e.,

some

and an arrow

a “universal behavior” with respect to d, in the sense that every other

factorizes through

to describe “acceptable”

u:

arrow

there exists a unique f c-*c' such that f = u :

0

cases, for a given

F(c)-*d f:

which has

,

F(c')-*>d

uniquely

F(f) or the following

diagram commutes: f -

F(c')

c'

1

2.

c*c'

be a functor and dEObj)

.

Assume

that

is

universal

is

universal from

=>

via (h,h~l)

d

F

to

=> etc';

u'= F(h°h>u'. But u = F(id) (

Proof

1.

By

unicity, h°h'

2.

Exercise.

By

C-+D

Then

=

id

.

Similarly, h'°h =

id

°u'.

.

the proof of proposition 5.1.3, one even has that the

property of universal constructions as a very

common

isomorphism

tool in

is

unique.

This

a strong

is

mathematics (see the examples below).

hard to think of a more suitable language than the categorical one for expressing properties like

It is

this.

We

next give two alternative characterizations of the notion of universal arrow: the

theorem

5.1.4, is of

first

one, in

an equational nature; the second one, in theorem 5.1.5, makes use only of the

notion of a natural isomorphism. In a sense, the definition of universal arrow given in definition 5.1.1 is in the middle

way:

it is

based on one equation and on the existence of an isomorphism (does

we

the reader see any analogy with the various characterizations of products and exponents

have

given?).

5.1.4

Theorem

Let G:

uED[d,G(c^)] such

that

C->D

be a functor

operation tc : D[d,G(c)] -*C[c^c] such

Proof ()

let

xc

only prove that

:

D^G^J-^Qc^c] it

is

be an operation which

satisfies

and (2) above.

(1)

an isomorphism. Define then, for every hECIc^c]

x c "l(h) = G(h)

We 0

u

;

must thus,

we have

V^c©) =

G(x c (f)) "u =

x c(Tc' 1 (b)) = tc (G(h)

5.1.5

Theorem Let

there exists

C,

D

•

u)

by (1)

f

=h

by

(2).

be locally small categories, G: C-*D, dEObj) and

uED[d,G(c cj)] such

that

Proof (D, dEObj) and

uED[d,G(c ci)] such

exists

5.2

D

F:

we assume

that

a functor such that

D-*C such

that

we

are dealing with locally small categories.

VdEObj) 3

universal from d to

.

Universal Arrows and Adjunctions

98

F(d) = cd ;

L ii.

(Note that

C[F_,J “D[_,GJ

D[_,G _/

C[F_,_],

:

D°P xC -*Set).

Moreover, the functor

F

is

unique, up to

isomorphism.)

Proof By assumption we know

fED[d,d']

that, for all

,

F(d)

G(F(d))

3!g

G(g)

F(d')

Set then F(f) = g, that

is,

u^'of

G(F(d'))

= G(Ff))°Ud

.

By

the uniqueness property, G(id)

=

id.

Moreover, by twice the definition of F,

G(F(d))

G(F(h)oF(f))

And

again by unicity of F(h

We need now to define for a suitable

qp,

0 f),

a natural

one then has F(f°h) = F(f)

isomorphism

that for all gED[d',d]

qp:

0

F(h).

D[__,G_] « C[F_,_].

Thus we

first

need

and hEC[c,c'] the following diagram commutes:

to check,

.

99

Universal Arrows and Adjunctions

i

D[d,G(c)]

C[F(c),d]

ho_oF(g)

G(h)o_og

?

D[d',F(c')]

C[F(d'),C']

Equivalently,

VfED[d,Fc] qp(G(h)°f°g) = h°qp(f)°Fg

1.

Now write = G(f)

0

u

u .

(u')

for

Define

u^ cp(f)

(u^', respectively).

= f

,

that

is,

By

G

the definition of the functors

and

we have

Moreover, for

all

only to prove the naturality

F, the following

diagram commutes:

G(F(d'»

d'

2.

cp(f « g)

.

VfED[d,G(c)] 3!fEC[F(d),c]

= G(cp(f))°u (compare with the definition of F

f

clearly a set-theoretic isomorphism; thus,

We know then that

=

cp(f) °

F(g)

fED[d,G(c)], G(F(d'))

(1).

).

qp

f is

)

)

.

Universal Arrows and Adjunctions

100

G

by the definition of

.

Therefore,

(G(h)

This proves

Dually,

5.2.2

0

f 0 g)

=h

0 qp(f 0

=h

° cp(f) • F(g)

(1), i.e., the naturality

we have

of

by

g)

cp

dc

i.

G(c) =

ii.

C[F_,J *D[_,GJ

is

completed.

G:

C—>D

VcEObg 3

that

universal from

F to

c.

such that

;

result follows

.

by

duality;

technique from the one used above. she or he

is

the following:

exists a (unique) functor

Proof The

(2).

and by proposition 3.2.2 the proof

,

Theorem Let F:D-*C be a functor such

Then there

the diagram and unicity

by

As

anyway we

explicitly reprove

it,

but by using a different

is

essentially notational, but

the reader will see, the difference

good examples of two common proof styles

invited to study both proofs since they are

in

Category Theory. Let

Gob Obc-*ObD :

universal arrow.

be the function defined by

GQb( c) =

define,

For every

VgED[d,GQb(c)] T d>c (g) = u c

dEObj) and cEObc,

isomorphism. Note 1-

By

that,

^d',c(g

T d;C -l(f)

we now

For simplicity, Let Gjy^or

1

T d,c

:

More

o

•

o

F( g

h

)

we have

is

the

F(g).

D[d,Gob( c )] -*’C[ F ( cO,c]

= uc

F(g)

°

o

F(h) = x d>c (g)

x d c ( x d>c ‘l(f) «

is

0

h

)

=

f

0

°

F(h).

Morj) be the function defined by

GMor( k) =

= x-l(u c

'

' t 1 (k °

«c)

is a

eD GOb(c G Ob(c')]).

[

functor

« id

F(G(c))

= t!(u c ° F(id G c)) ( = f 1 (x(id G c )))

)

(

=

id G(c)

c'-*c", k: c-*c', °

clearly a set-theoretic

F(h), or equivalently,

).

G(id c ) =x-!(u c )

G(f

°

F (g)-

°

h = Td>c-l(f»F(h)

We want to prove that G = (GQb» GM or)

f:

= uc

omit the indexes of x and x'l

VkeC[c,c']

and, for every

f

VhED[d',d],

h) = u c

0

taking x d?c "l(f) for g in (1), 2.

:

,

We have VfGC[F(d),c] 3!gED[d,G 0b (c)]

Now

d c where u c F(d c )-*c

k)

= x“l( f

°

k

°

Uc)

= x*!(f °u c '°F(T-\k

°

u c)

)

d

)

101

Universal Arrows and Adjunctions

= x-!(f = G(f) (1)

o

U C ')

°

r !(k

°

by

u c)

(2)

G(k).

-

proves the naturality of x d c in the component d

We have still to prove the naturality in

.

c, that

VgED[d,G 0b (c)] VkEC[c,c']

is,

f

G ( k)

3- *d,c'(

»

g

0

= k

)

x d)C (g).

•

We have

By

x(G(k)°g) =x(x-!(k

0

u c)

= x(x-!(k

0

uc

=k

°

= k

o x(g).

taking x d c "l(f) for g in (3)

and

Note

we

(4) state the naturality

obtain x d>c '(G(k) °

x^c"

,

(2)

1

®

xd c“l(f)

°

= x d)C --l(k

)

= k

0

f

,

or equivalently,

o f).

then

G(k) = G(k)

which shows

by

) )

of x“l.

G must satisfies (4)

that since

g F(g)

F(g)

®

G(k)

4.

(2)

uc

0

o

id

= G(k)

that the adopted definition for

G

x-!(u c) = x’l(k

o

was

°

u c)

actually forced. This proves the unicity of the

functor G.

Example An

interesting

By

categories.

the previous section,

axb-*a, p2 axb-*b) :

to b.

Then

is

to

we know

universal from

the functions _x_:

axb and b

to

example of application of theorem 5.2.2

A

to

b a can be extended ,

definition is the following: for every

to f:

and _a

a-»c

,

g:

a

(_ )(g) = A(eval a> b

may be

T:

C-*l

1-*C

to

t.

is trivial,

is

aEObc

°

g)

g

ba

:

0

—

c-*t

!c:

it

Obc^Obc

Cp-^Cj. The

is

as in the second

there exists the lifting a

which takes every object a

explicit definition of the functor (_°)(f)

where x(f°exa)

is

a,b

all

=

in

Ob£,

:

P2 >

axb-*cxd

:

a .

may be

,1]

seen as universal arrow from the

lEObj

example below definition

0 .

By example Cj-^Cp

to its lifting a°,

5.1.4

to a.

5.1.2,

we know

By theorem

may be extended

_° on a partial arrow

0

to

in

t

t

C

a C[c,T(l)=t].

f =x(foexb)eq[bV]

the only arrow such that ex c

(p:

_xa

universal from

interesting that the existence of the terminal object

an universal arrow from the embedding functor Inc:

_°:

is

Obc-*Obc which respectively take (a,b) CxC-*C and _ a C->C. The explicit

:

expressed by the natural isomorphism l[!C(c)=l

each object

then for

In this case, the extension of the function that takes

but

Example Consider C, Cj and Inc that for

CCC,

b-*d

f 0 pi,

For every object c in C, even the unique arrow

to a functor

a

two functors _x_:

(_ x _)( f>g) = fxg =
cp D[f(c),d]-»>C[c,G(d)] be the function defined by

Equations (1) and (2) express exactly the fact that x c>cj and x c>cf * define an

We have still to prove their naturality.

Let kED[d,d'], hEC[c,G(d)], h'EC[c',c], and

then

x _1 (G(k) «h) =

r !(G(k)

= x‘l(G(k = k nat-2.

°

0

unit°k)

x"l(h

°

k)

0

0

°

G(x'l(h)) -

x l(h))

0

°

unit c )

unit c

)

_ x l(h)

= x _ l(G(x"l(h)

o )

unit

0

k

by

(1)

;

.

107

Universal Arrows and Adjunctions

= x -1 (h)

°

_ T l(unit

°

k)

by

(nat-1)

by

(nat-2)

= x-l(h)oF(k); x(k'

nat-3.

0

F(h')

)

=x

(x"l(x(k')) ° F(k) )

=x

(x"l(x(k')

= x(k

nat-4.

= G( k

° k' )

rj :

0 h' 0 k') °

= G( k

)

= G( k

)

°

unit

G(k')

°

unit

o x(k').

:

5.3.6 Proposition Let

transformations

x(k')

0 h')

C-*D

be an adjunction. Then there exist two natural

Idc~*GF and e:FG-*Idj) such

e(d) are respectively the unit

and counit of the

that,

for every c in

C and d in D,

r](c)

and

adjunction.

Proof Exercise.

In other words, that, if y\

:

one may construct the unit and counit “uniformely” and naturally.

Idc“*GF and

FG-*Id]) are the natural transformations

e:

Observe also

in proposition 5.3.6, then the

following diagram commutes:

The previous diagrams

fully characterize an adjunction: as a matter of fact

define an adjunction between two categories

C-*D

and G:

D-*C

are functors, and

r\

:

C

D

and

Idc“*GF

,

e:

as a quadruple

many (F,

G,

authors prefer to r\

,

e)

where

F:

FG-»Idj) are natural transformations

such that

We leave

it

adopted.

We

(Ge)

°

(eF)

° (Ft))

=

(riG)

id

G

= idp

as an exercise for the reader to prove the equivalence of this notion with the

one

we have

shall use the definition of adjunction as a quadruple in the next section, since

simplifies the investigation of the relation

it

between adjunctions and monads.

Exercises 1.

An

adjointness (F, G, r\

,

e)

from

C

to

are natural isomorphisms. Prove that given is

2.

D

is

an adjoint equivalence

two equivalent categories

C

if

and

and

e

(see section 3.2)

it

and only

D

if

r)

always possible to define an adjoint equivalence between them.

Given an adjointness

statements: i.

tiGF = GFri;

(F,

G,

T],

e)

from

C

to

D, prove the equivalence of the following

.

Universal Arrows and Adjunctions

108

an isomorphism;

ii.

r]G

iii.

eFG = FGe;

iv.

eF

is

Example

we

In section 3.4

respectively,

example

that

an isomorphism.

is

defined the

CCCs

H(f) = f by the definition of continuity.

in this case, filter continuity

easily define the natural

H

:

={01 O

has a

where OEtop

T((X, F)) = (X,top)

may

H Top

are actually adjunctions. Recall that

H((X,top)) = (X,F) where F(x)

Also

of limit and

which generalize topological spaces, Top. The

is

filter

functorial

-*•

FIL

a filter

left adjoint

is

“embeddings” mentioned

T FIL :

-*>

Top

corresponds to topological continuity,

isomorphism x

in

given by

and VOEtop (xEO

VxEO VOEF(x) OEO

iff

L-spaces and FIL

spaces,

OEO)}.

=>

defined by .

i.e.,

T(f) = f

.

The reader

.

A stronger notion of convergence, to be used for

In general, limits are not unique in filter spaces.

computability (see the final remark in section 8.4),

may be

given as follows. For (X,F) in

FIL

consider (X,top) = T((X, F)) and define (s-conv.)

Then top Let

Tq

iff

s x iff

s-convergent

xEO

{xj}CO

eventually).

Adjunctions and Monads

In this section

monad. As

we

study the relation between two seemingly distant concepts as adjunction and

a matter of fact, every adjunction immediately defines a

monad, and conversely every

,

.

Universal Arrows and Adjunctions

monad can be thought of

109

by an adjunction, called

as generated

definition 5.4.2 below). Resolutions for a given

a natural notion of

morphism between them;

Categories associated with the

and

which

presentation

is

a

C and

monad on

GF,

we

=

p

0

Tr|

p

0

rjT =

eFG =

Indeed, for any

Then one has

GeF

D' such

,

a category C.

C to D e): C->D

from

resolutions (F, G, r]

that F'

=

A

resolution for

such that

and

(F',

H °F, G = G'

\

resolutions for that

monad.

r]

p)

is

a

,

°H and He ,

prove that the Eilenberg-Moore and Kleisli categories associated with a they are respectively the terminal and

(T,

T = GF and p = GeF. A G', r e'): C-+D' (for the

proved that resolutions with the associated morphisms form a category.

rise to resolutions. In particular, all

is

.

,

5.4.2 Definition

easily

e)

first that

FGe

e 0

= n

category

be an adjunction from

GeF ° r|GF = (Ge

= GeF

rest is

e)

GFr| - G(eF

0

dEObj) and

p

to

theorem,

at the first

We must prove the unity and associative laws for the monad.

e d ° e FG(d)

It is

may just look

have

For the associative law, note e 0

,

- GrjF,

and

r|,

GFGF-*GF.

for the unity laws

tj

(T = FG,

first that

and p - GeF:

The

then happens that the Eilenberg-Moore and the Kleisli

will be applied in the next section.

Proof Note

As

(see

by introducing

(see definitions 4.2.3 and 4.2.4) are respectively the terminal

rather technical; at first reading, the reader

5.4.1 Proposition Let (F, G, is

it

monad

a resolution for the

build up in a category

object of the category.

initial

The

monad

monad can be

monad

initial

=

e'H.

Now we (T,

r

\

,

are going

p) both give

objects in the category of

F

Universal Arrows and Adjunctions

110

5.4.3 Proposition Let (T,

Moore category

rj

monad over

p) be a

,

morphism of algebras h

which

is

an

C-*CT

C. Let F^:

in

and every morhism

f:

be the natural transformation defined by e^(c,a) = (F^, l/T

UT o Ft = T UT e T FT = p, since for any object .

ri

e^)

,

a

a resolution for (T,

is

be the

c-*c' to (note

ri

,

p).

c one has

=UT( e T fT(c)) Ut(et(T(c),

=

of FT

by

def.

= llT(nc )

by

def. of

= pc

by

def.

have to prove that (pT,

UT

^

(U T e T) (e

e T)

>

js

(r|UT)

o

TFT) o

hc ))

(fT-t,)

an adjunction from

of U^-

to

C,

that is,

= idpjT

= idpT

has, for every T-algebra (c,a),

(U T£ T

•

UT(eT(c>a) ) • TluT(c ,a)

riUT) (c,a) =

a 0 »l c

=

= idc for every (e

TFT.FTr,)(c)

= eTpT(c) 0

l^c

=

We say that the resolution by proposition

5.4.3, is associated

77

(F,

G,

tj

,

e):

C-+D

unique arrow from

such an arrow

is

(F,

a functor H:

Define, for any object

d,

G,

r\

def. of T-algebra.

by

def. of

,

any

to the

monad over a

Eilenberg-Moore category C^, and given

category

monad (T,

C .Then

,

e) ,

to (F^,

U^,

such that

and any morphism )

cEObc, any hEMorc,

is

the resolution

(F^, U^,

a terminal object in the

rj, pt).

be another resolution for (T,

H(f) = G(f). has, for

C

e^) from

the Eilenberg-Moore Category C?,

r|

e^ and F^

with C^.

D-^C^

H(d) = (G(d), G(e(d))

Then one

by

by the unity law of the monad.

be a

p)

,

category of all the resolution for the

exists a

uT

def.

c)

T(i1c)

U^,

C--CT, associated with

Proof Let

T(r|

•

idT(c)

(F^,

5.4.4 Proposition Let (T,

E?):

of eT and

by

cEObc,

=

r\y

fi)

,

morphism

as a

We want to prove that

T(c)-F^ljT(c,a)-»c).

(UTeT FT)(c)

And

rj

(c,a) to c, and every

to its free algebra (T(c),p c ),

c

F^uT-Md^T

F^(f) = T(f). Let e^:

One

be the Eilenberg-

that takes every algebra

same h regarded

to the

functor which takes every object

We still

CT

let

C? to C.

Proof Let U^: cT-*-C be the forgetful functor

Obviously

and

,

associated with the monad. Then there exists a resolution for (T}

adjunction from

that a:

C

a category

f

r\

,

F^ = H of D,

e^). 0

F,

r\

,

p).

We

Remember

G = uT o h

must prove

(cf.

,

that there

definition 5.4.2) that

and He -

e

TH.

,

Universal Arrows and Adjunctions

111

H(F(c)) = (G(F(c)), G(e(F(c)))

)

=

(T(c),

nc ) = FT(c)

H(F(h)) = G(F(h)) = T(h) = FT(h) that

H°F= F^.

proves the equality

Moreover, for any dEObj), and any fEMoq),

uT(H(d)) = uT(G(d), G(e(d))

)

= G(d)

uT(H(f)) = uT(G(f)) = G(f), as IjT

G

That proves the equality Finally, for e

that

have

o

h

the identity

is

on morphisms.

.

any dEObi),

TH(d)

proves the

We

= IjT

= eT(G(d), G(e(d)) equality He - e^H.

still

prove that

to

H

= )

G ( e (d ))

the unique

is

= H(e(d))

morphism from

(F,

G,

r\

e) to

,

(F^,

U^,

ri

,

e^).

Let H' be another morphism; then, for any fEMorj), H'(f)

and, for any

=

U T (H'(f)

= G(f) =

U T (H(f)

= H(f)

dEMorp, (

U T (H'(d), e TH'(d))

by def of uT and eT

=

(

G(d), H'(e T (d))

as

=

(

uT(H(d), H(eT(d))

H'(d) =

*

)

G

=

UT ° H' = H(f)

as H'(f)

)

= (UT(H'(d), eT H d )) ( = H(d). This completes the proof.

The unique functor p,)

is

called

to (F^, IjT, r\

eT) in the category of

,

comparison functor and

The category of

resolutions of a

it

is

all

usually denoted by

monad has

monad

resolutions for a given

(T,

r]

kT.

also an initial object,

which

is

based on the Kleisli

category associated with the monad.

5.4.5 Proposition

Let

(T,

rj

}

n)

be a

monad over a

category C, and

category associated with the monad. Then there exists a resolution for

from

Cj

Cj-^C

any object c of

Ux(h) =

for

Cj (i.e., of C ),

that

is

an adjunction

and any morphism hECj[c,c'] (and thus hEC[c,T(c')])

Pc’

0

T(h).

C-*Cx be functor defined by the following:

any object c of C, and any morphism fEC[c,c'], Ft(c) = c

FT(0 = ’1c’°f Let ej:

fi)

be the Kleisli

be the functor defined by the following:

Uj(c) = T(c)

Let Fj:

rj,

Cj

to C.

Proof Let Uj: for

(T,

let

FjUj-^id be

(

= T(f)

° t) ). c

the natural transformation defined

by the following:

Universal Arrows and Adjunctions

112

for

any object c of Ct, ®T( c) = id T(c)

(inC).

ej): C-*Ct We want to prove that (Ft, Ut, Obviously, Up 0 Ft = T Moreover, Uj ej Ft = p, since for any object c r|

is

,

a resolution for (T,

r\

,

p).

.

(Uj et Ft)(c) = Ut(et (c)

We have still to prove that

One

by

def. of

Ft

= UT(idT(c))

by

def. of

et

= pc

by

def. of

Ut.

(Ft,

Ut,

ti

(Utet)

°

(etFt)

0

has, for every object c of

(Ujst ° nUj)

)

et)

,

(F-m) =

an adjunction from

Ct,

that

is,

Ut~*Ut

id: id:

C to

Ft~*Ft.

Ct,

= He

6

= He

0

=

is

(tiU T) =

= Ux(id T c)) (

(c)

one has

•

rix(c )

T(idT(c))

6

nT(C )

by

def. of

by

def.

as

^TCc)

T is

et and

Ut

Ut

on morphisms

a functor

by the unity law of the monad.

id c

And, for every cEObc,

(ejFx

°

Fxn)

(c)

= ex(c) =

by

def. of

by

def.

by

def. of

’Ic

by

the unity

id c

by

def. of the identity in

°

OlT(c)

idT(c) °

= hc

°

0

(mcc)

Tlc) 0

n c)

T(idT(c)) 0 nT(c) ° n c

Ft

of et

composition

° in

Ct

= HC ° ,1T(c) ° Tlc = —

5.4.6 Proposition Let (T,

C-^Cj associated with the

monad

Proof Let exists a

(T,

rj

(F,

,

rj,

n) be a

the Kleisli Category

F=

over a category C. The resolution (Fj, Uj,

Cjis an

initial object in the

r\

,

C-*D

e):

be another resolution for (T,

K 0 Ft, Ut = G K 0

t]

,

et) to (F, G,

Ket

and

,

Define, for any object c of Ct, and any

K(f) = ef(c')

Then one

r\,

ej):

category of resolutions for

r\

,

r|

,

p).

e), that is

We

must prove

f are

»

morphism fECT[c,c'],

F(f)

regarded as object and morphism of C.

has, for any

cEObc, any

hEC[c,c'],

K(Fx(c)) = K(c) = F(c)

K(Fx(h)) = K(r) c

'

»

h)

that there

a unique functor K:

eK.

K(c) = F(c)

where c and

Ct-

p).

G,

unique arrow from (Ft, Ut,

such that

monad

law of the monad

by

def. of

Ft

Ct^D,

)

.

Universal Arrows and Adjunctions

= =

113

by

E F(c') ° F(r)c' ° h) 0

EF(c')

FOlc’)

°

F(h)

as

= F(h)

as

def. of

F

K

a functor

is

G,

(F,

r\

,

e )

an

is

adjunction.

K 0 ¥j = F

This proves the equality

.

Moreover, for any object c of Cj, and any morphism fECT[c,c'], G(K(c)) = G(F(c)) = T(c) = Ut(c) G(K(f)) = G(e F(c ')

=

Me'

°

by

° F(f))

= G(e F(c .))

»

G(F(f))

as

as (F, G,

T(f)

= Ux(£) that

that

,

e) is a resolution

Uj

def. of

K(ex(c)) -K(id T(c) )

by

def.

of ej

=

E F(c')

by

def.

of

-

e F(c')

as

F

-

E K(c')

by

def.

Uj = G 0 K

any dEObj),

proves the equality

Kej -

0

F(idT c)) (

Op-^D be

K

a functor

is

of

K

eK.

We have still to prove that K is the unique morphism from Let K':

ri

by

proves the equality Finally, for

def. of K G is a functor

(Fj,

Uj,

r\

,

ej) to (F, G,

ri

,

e).

another morphism; then, for every object c of Cj,

K'(c)

= K'(Ft (c)) =

by

F(c)

def.

as K'

of

0

F^

FT = F

= K(Fi0

is

exactly what

is

1-

!A

|-!(ATIB).

needed

to recover the intuitionistic calculus:

it is

an embedding of intuitionistic (and classical) logic into linear logic. The

intuitionistic

A A

formula

A=A

if

to a linear is

A

formula

in the

following way:

an atomic formula

=An B A vB = !A® !B A =>B = A —o B

AaB

!

(

!

is

supposed

to bind tighter than

The absurdum

F

—o

of intuitionistic

is

U

and

).

translated into 0, the identity for

©

.

Thus

zA

=

!A—oO.

In other words, the iterated use of the premises, in a linear implication, gives exactly the intuitionistic

implication. Then,

Our aim now

T

if,

is to

|-j

A,

in intuitionistic logic, then

T

|-

A

in linear logic.

give categorical meaning to the connective

we

categories (see definition 4.4.4),

We

!

have already remarked

also obtain an interpretation for

that a resource

6:

®!A

!A-*!A

?,

why

duality, in linear

not.

such as !A can be duplicated and erased, and

sense these properties characterize the meaning of the connective

expect to have two morphisms

By

of course.

!,

monoidal category. (Commutative) comonoids

,

and

in 4.3.4

characterized by a sort of diagonal map, such as

6,

e:

seem

and a

!

Thus,

.

where

!A-*1

at the

1

is

in a

sematic level,

we

the identity of the

the right structure for this, as they are

map

e

which dualizes

the

map

r|

in

definitions 4.2.1 and 4.3.3.

We

start

then with a monoidal category C.

isomorphisms, given in 4.3.1, which 1.

assoc:

2.

ins-1:

3.

exch:

4.

ins-r

we rename

By

definition,

C

must

satisfy certain natural

in this section, for convenience, as follows:

X® Y® Z) « (X® Y)® Z; (

X « 1 ® X; X ® Y * Y ® X.

Let also

As mentioned

= exch

0 ins-1

:

X « X ® 1.

in section 4.4, the connective of linear implication

adjoint to the tensor product

®

,

when

C

is a

—

o

is

interpreted

monoidal closed category, as defined

by the

right

in section 4.3.

.

!

117

Universal Arrows and Adjunctions

Recall that the category

C,

as objects, for c in

arrow

f:

c-*c' in

C

,

CoMon^ c-*c®c,

(c, 6:

f

e' °

f-

- (f®f) e:

(e®id c)

exch

°

assoc

As

c-*l), and morphisms

e:

0

0

6

for each

c-*c'®c'

:

c-*l

(c, 6:

6 =

6 = 6

° (id

:

c-*c®c,

ins-1

c-*l) observe that the following equations hold:

e:

c-*l®c

:

c-*c®c

c®6)

6 - (6®id c)

°

pointed out below definition 4.3.6,

Cartesian product

6, e)-*>(c', 6', e'),

f: (c,

such that 6' °

Given a commutative comonoid

Exercise

of commutative comonoids over a monoidal category C, has

0

6

:

C

if

c-*(c®c)®c

.

and the isomorphisms are the canonical ones, then

fl

isomorphisms above can be constructed for each object canonically given; namely, 6 = the terminal object). Prove that, if

C is

c-*cflc

:

is

C

in

all

e:

c-*t

is

is

the

(in particular, recall that

the diagonal and

Cartesian, then

®

Cartesian, in the sense that

is

actually a

maps and

6 and

the unique

C is actually isomorphic to

e

are

map

to

CoMon^;. Does

the converse hold?

5.5.3 Definition exist natural

A

! -

model

is

a linear category

C

and a comonad

(!,D,E)

such that there

isomorphisms

I: !(ADB) * !(A)®!(B)

J:!t*l where

t

and

1 are the identities for tl

and ®,

the Cartesian

and tensor products

in C.

Indeed, by definition, in a linear category one has both a monoidal and a Cartesian structure. The relation established

5.5.4

Lemma

d':!c-*!c(8)!c

Proof

by the

natural isomorphisms gives the

Let be an ! -model.

and

e':

!c-*l, such that

6'« I°!6

Just set

e'-

where 6 = to the Cartesian

:

:

Me:

!c

!:

C

C-*C, -*

is

in C, there exist

maps

a comonoid.

c-*t are the monoidal

e:

D:

!

is

CoMonc !(c)

=

maps

in the

remark above, with respect

easy.

—

0

!,

E:

!-*Idc

ingredients for the interpretation of the connective

define the functor

,

!c-» !t-»l

c-*cflc and

(!:

Then for each object c

-*!(cDc) -*!c®!c

product D. The rest

Thus, the comonad

(7c, 6', e')

monoids we need.

!,

)

associated with a !-model gives

of course. In view of the above lemma,

by (!c, 6':!c

—*-!c®!c,

e':

!c

-*1)

.

all

the

we

can

A

Universal Arrows and Adjunctions

118

D

This gives the required monoids, while the natural transformations

Dc

:

!c

We

Ec

-*!°!c and

!c —*c, which

:

are

needed

and

E

to interpret the rules in (!,r)

uniformly yield maps

and

(!,1).

already mentioned in section 4.4 that the idea of the interpretation relies on viewing

entailments as morphisms. In short, observe that, with an informal blend of syntax and semantics,

are interpreted by

the rules

the fact that

each morphism

(weakenings)

gives a

each morphism

(contractions)

gives a

gives a

gives a

for the rules :

which contain

?,

their

meaning

is

that is

5.5.5

!a-»A

:

!a®!a-»A f°6 a !a-*A :

:

-

*

f:

morphism

!c-»a !f°D c !c —*•! :

easily derivable

by

duality.

The idea

is to

define a

C-»C by ?

why

f°e a

f: a-*A morphism f°E a !a-*A

each morphism

(!.r)

functor ?

f:

morphism

l-*-

each morphism

(U)

As

f:

morphism

?A =

Then

(!A*)*.

the following

o

.

*

j

theorem gives the categorical meaning of the modality

?,

not.

Theorem Let be an

'.-model.

Then there

exist a

monad

(?,

D': ?° ?-*?, E':

Id-*?) and natural isomorphisms

where 0 and

1

/':

?(A®B) * ?(A)U?(B)

J':

?0*1

are the identities for

®

and

U, the duals of the Cartesian

and tensor products

in

C.

Proof Set D:

!-*!!,

?

=

*

°

• !

*

:

C

-*

C

and, for each object A,

D'a = (Da*)* and E'a = (Ea*)*- As

one has

Da*: !A*-*!°!A* (Da *)*: (!°!A*)*-(!A*)* (Da*)*: D'

Each of these Similarly,

steps

from

is

E:

a

by by

def. of (-)* id

»

(-)**

?°?A— ?A.

an isomorphism, uniform in A, and gives a natural transformation D': ? 0 ?-*?.

!-*Idc one has Ea*: !A*-*-A* and,

properties required for a

As

:

(!°* 0 *°!A*)*-*(!A*)*

monad follow by

for the natural isomorphisms,

compute

duality.

thus,

E'a = (Ea*)* A-*-?A. The :

119

Universal Arrows and Adjunctions

?(A@B) =*°!°*(A®B) -

Finally, ?0

is

Next we

0 !

0

by theorem 4.4.6

!(A*DB*)

« (!(A*)®!(B*))*

by

- ?AU?B

by theorem

*0 »

0

Endow

Exercise product

=

*

*

*

0

* 1* *

t

!

a structure of

1

find, within fl

Then give

to intuitionistic logic,

from

we

use

its

categorical

once the connective

is

!

?.

logic, an interpretation for the intuitionistic

As

in the

!

-model. Namely, given an

the purpose of the iterator

meaning

we

a matter of fact, in the previous remark,

linear ones,

4.4.6.

!-

“and” and “implication” by Cartesian product and

exponential in a suitable category derived from C.

As

4.4.6.

the details of the interpretation of the rules for

any categorical model of linear

interpret intuitionistic

!-model

monoidal category whose tensor

in a

and =>, by using the comonad construction

model C, one may

back

theorem

definition and

monoid over each object

actually a Cartesian coproduct.

connectives

by

,

def. of

to construct this

hinted

available.

how

new

!

was

to take us

category.

to derive intuitionistic connectives

The following

result gives the categorical

counterpart of that construction.

Observe

comonad

that in general, given a

(T, 6, e)

category whose objects are those of C, and the set K[A,B] of morphisms from

C[T(A),B]. The identity in K[A,A] in

K is the B in K

over C, the co-Kleisli category

A

to

is

T(A)-*A. The composition of fEK[A,B] and gEK[B,C]

is

K is gof=g° T(f)

(see definition 4.2.4

5.5.6

where

Proof

is

8

A T(A)-T2(A)-T(B)-C :

Kleisli categories over

Theorem IfC be an

(!,D,E)

o

monads were

defined).

Then the co-Kleisli category

! -model.

K associated with

the

comonad

Cartesian closed.

(hint)

The exponent of two

objects

B

and

C

(!B

is

—

oC).

We

then have the following

chain of isomorphisms:

K[ADB,

C] * C[!(A(TB), C]

by definition of

* C[!(A)®!(B), C]

—oC

a C[!(A), !B

a K[A,

Example

In section 2.4.2

we

!B—oC

]

]

as

C is monoidal closed

by

(see section 4.4),

it

definition of K.

defined the category Stab of coherent domains and stable functions.

In that section (see exercise 4) the subcategory Lin, with linear maps,

was given

coherent domains as follows:

K

as !(AflB) a !(A)®!(B)

if

as an

example of

X is a

linear category.

coherent domain, then !X

We is

was

also introduced and, later

also defined a function

the coherent

!

on

domain defined by

Universal Arrows and Adjunctions

120

determined by

map

f

Tr(!f)

=

Next,

we

to a functor

!

:

is

aUbEX.

iff

Stab-*Lin. Recall

map may be

that a linear

equivalently described in terms of

map

i.e.,

g:

Z-*Z'

is

uniquely

on the elements of

its trace.

|Z|.

Set then, for each

X-*Y,

:

bEY, b

{({a}, b) |

finite,

aEX,

define an adjunction between

Stab. This

a finite};

behavior on the points of the coherent domain Z,

its

Moreover, any stable stable

it

aEX,

/

afb [mod !X]

ii.

We need now to extend

= {a

|!X|

i.

a finite and least such

b

C f(a) }.

and the obvious inclusion functor Inc from Lin into

!

given by a natural isomorphism

(iso)

qp

:

Lin[!A,B] * Stab[A,B]

where the inclusion functor

Once more we use

omitted.

is

traces, that

Tr()

= Q(

= Q(

0

snd

B °!snd = e (E

!fst

EAn B

>)

•

(!fst

0!snd)

o

=

(!fst

0!snd)

- (

=

(!fst o

b

by

naturality of

by

def.

of

E

Ap| B ® Q(k) Q(k)0Q(k)

Q(k) )0(!snd

®

Q(k)

) )

• »

b 6

0

b^s >)

°

!snd >)

and !snd are comonoid morphisms

Q(k) =

=

I

AnB = £A°- lst

B0id B ) 0 bs

Eahb)

= i d !(AflB)

k 2 and

by properties of comonoids

!snd >) °

0

def. of k\,

as e

1 0

k

by (hom-1)

E

0

0

))

)

Universal Arrows and Adjunctions

=

(

Q(

k) )®(

fst °

Q(

123

fst 0

k)

)

°

6

-(Q(ki))®(Q(k2))°6

Ea

° ins-r' 1 °

-

(

-

1 ° ( ins-r' (id;

-

1 ( ins-r"

-

1 o ( ins-r'

=

1 ( ins-r'

Q(

(idiA

®

eg) )®(

Q(Eb

0 ins-1' 1 °

a ® eg) )®(

ins-1" 1

(ea ® idg) )°8

(idiA

® eb) )®(

ins-1' 1

(ea ® idg)

0

(idjA

® ea) )®(

ins-1' 1 °

® idg)

0

by

(nat-1)

by

def. of

(ea ®

0

6

by

def. of

by

(nat-2) and def. of

and k 2

E

niix ° (6a®$b)

by (Eg

k

idg)

def. of

6

(6 a®&b)

by application of mix

a ® ea) &A )®(

(id

°

!

ins-1" 1 °

(eb

® idg)

0

&b)

by properties of comonoids = id!(A)®idi(B)

= id!(A)®!(B)

The construction

As

is

•

clearly uniform in

for the natural

isomorphism

J,

A

and B.

note that !A » !(ADt) * !A®!t

,

!A

at

!(tHA) s !t®!A and that

the right and left identity, in a monoidal category, are unique.

References

Universal arrows and adjunctions are fundamental notions in Category Theory.

Their treatment, in various forms, and references to their origin

mentioned

in the previous chapters.

may be found

in all textbooks

References for linear logic have been given in chapter

4.

we

Chapter 6

Cones and Limits

In chapter 2,

we

how common

learned

constructions can be defined in the language of Category

Theory by means of equations between arrows of given definitions

we saw

objects. In chapter 4,

were based on the existence of an universal arrow

given functor. The category-

to a

theoretic notion of limit is merely a generalization of those particular constructions, as

common

universal character.

From

another point of view, the limit

of universal arrow, where the involved functor see.

To

help the reader

become confident with

we

is

this

is

that those

it

stresses their

a particular and important case

a “diagonal,” or “constant” functor, as

new

notion,

we begin

this

we

chapter by looking back

regard them as particular instances of limits.

Then we study

at the

constructions of chapter 2 and

some

relevant properties concerning the existence, creation, and preservation of limits.

computer science,

limits

have been brought

to the limelight

shall

As

for

mainly by the semantic investigation of

recursive definition of data types: this particular application of the material in this chapter will be

discussed in chapter 10.

6.1 Limits

and Colimits

The concept of

limit

embodies the general idea of universal construction,

has a privileged behavior

among

is,

essentially

is,

of an entity which

a class of objects that satisfy a certain property.

define a property in the categorical language

arrows, that

that

is

The only way

by specifying the existence and equality of

by imposing the existence of

a particular

to

certain

commutative diagram among

objects inside the category.

6.1.1 Definition

A

diagram

nor must

More

it

in

C

is

a category

C

is

a directed graph whose vertices

similar to a subcategory of C; however,

iEI are

it

does not need to contain

identities,

be closed under composition of morphisms.

formally, a diagram in a category

from an index graph I”.

in

and whose edges eEE are labeled by morphisms

labeled by objects di

A diagram D

D

I

to the (graph

C

should be defined as a graph

underlyng the) category

For the adjunction between graphs and categories,

category I freely generated by the graph the index category

is

small.

I

C

.

Such

a

this is exactly the

(the index category) to C.

homomorphism

diagram

same

is

D

called “of type

as a functor

from the

A graph is called small when

Cones and Limits

126

6.1.2 Definition Let is

C be

a category and

D

a diagram with objects

c and a family ofmorphisms {fiEC[c,dj]

an object

Vi,jG VeEE fe eC[db dj]

Cocones

Note now

hj 0

Clearly,

that,

given a diagram D, the cones to

g=

f[.

That

to

D

=> fe

°

=fj.

fi

Cones^D

Cones(j

j) f

|

D

form a category,

call

it

ConesQD

iEI}) any gEC[c,c']

iEI}) to (c',{hjEC[c',di]

•

Just take

such that

|

is,

is

6.1.3 Definition. Let in

Then a cone

.

In a partial order P, cones correspond to lower bounds, cocones to upper bounds.

morphisms from (c,{fj£C[c,di]

ViEI

iEl

dj,

i£7 } such that

are defined dually.

Example

as

1

a category.

C be a

Dually one defines the category

category and

D

CoconesQD

a diagram. Then a limit for

D

is

•

a terminal object

Colimits are defined dually.

.

iEI})

(c,{fiEC[dj,c] |

commutative diagram:

is

the initial object in

CoconesQD, ^ ma y

visualized by the following

Cones and Limits

127

Limits are also called universal cones, as any other cone uniquely factorizes via them. Dually, colimits are called universal cocones.

Examples 1.

Let

P be

Then

a partial order.

limits correspond to greater lower bounds, while colimits

correspond to least upper bounds. 2.

Let

D

= ({djjqeoj, {fjESet[dj, dj+iJIieo)

Then

(the set-theoretic inclusion).

(Exercise: what

is

the limit of the

now

a

of the category

be a diagram

same diagram?)

Exercise Prove that the colimits in

Consider

)

C

are the limits in

C°P

is

the

image of

degenerate diagram of type

in

I

natural transformation

from

D

to

Once diagrams

D'

is

for a

diagram

Kq

D to

of type

I

K c (i) = c

K c (e) = id c

K c (j) = c

and

d[+i -*

is

fj

=

incl

U{dj}

I to

C. Note YLq

first that

D

any object c

can be regarded as it

makes sense

and D' are two diagrams of type f|

indexed on objects

in

I

such

I,

a to

a

that,

I),

from an object c

D as follows:

C d[+\,

are defined as functors,

a family of arrows

for each arrow e in I (each edge of the graph of type

constant diagram

-*>

I-*C, and so

a constant functor

C.

dj

of the dual diagram.

diagram as a functor from an index category

C

such that dj

.

consider natural transformations between diagrams. If

A cone

in Set

do -*

the colimit of

is

then a natural transformation from the

Cones and Limits

128

D

Dually, a cocone for a diagram the constant diagram

6.2

of type

I

to

an object c

a natural transformation

is

from

D

to

.

Some Constructions

Revisited

D be an empty diagram, that is a diagram with no objects and no arrows. By definition, a cone C to D is then just an object c of C, with no other structure (and every object of C can be seen

Let in

as a cone).

A limit for the empty diagram is then an object

exactly one arrow from c to

i.e., it is

t,

a

t

such that for any other object c there

terminal object. Dually, the

is

initial object is the colimit

of the empty diagram.

A graph is called discrete if A diagram of type

it

has no arrows. For example the set {1,2} can be regarded as a {1,2} in a category

discrete graph.

limit for such a that for fi

0

But

diagram

is

an object

d,

any other cone (d',{giEC[d',q]

C

is

A

an ordered pair of objects, (ci,C2).

together with two arrows i\: d-»ci and f2: d-*C2, such |

iE{l,2} }) there exists exactly one arrow

h: d'-*d,

with

h = gi for iE{l,2}.

this is just the definition

of product d of c\

and C 2 with

fy.

d-*ci and

f2

:

d-*C2

as

projections.

Dually, the coproduct q#cj,

The product of any indexed

n^Di,

I

kEC[d,b] such for 1

=

that

any other cone k'.

I

I

in a category

for this

g

0

is

the index set considered as a discrete graph. This product is

is

often omitted.

with two vertices and two edges:

C

is

a pair of objects,

a

diagram consists of an object

h = k and

diagram {q,cj}.

defined analogously as the limit

is

although explicit mention of the index set

Consider the following graph

A diagram of type f,gEC[a,b]. A cone

the the colimit of the

collection of objects in a category

of the diagram D: I-+C where usually denoted by

if it exists, is just

f 0

h =

k.

and

(d',{h',k'}) there exists exactly

one arrow

(d,{h,k}) that

d —d f

1:

and a pair of parallel arrows

and two arrows hEC[d,a]

d,

A limit is a cone

b,

is

and

universal, that

such that h

0

1

=

h',

is,

and k

0

129

Cones and Limits

f

ZZZTtb

a

Note now

that the existence of

two arrows, h and

equivalent to the existence of an arrow h such that g k',

k°l = f°h°l = f°h' =

since

k',

now the

k,

h =

thus the above limit

Dually, coequalizers are the colimits for the

Consider

0

such that f is

°h = k and

g

Moreover, h

° h.

° 1

=

h =

implies

h'

just the equalizer of f

f °

and g

k,

k

°

is 1

=

.

same diagram.

following graph:

A diagram of this type in a category C is a given by three objects, a, c, and b, and two morphisms, fEC[a,c] and gEC[b,c]. A cone to this diagram is an object d, together with three morphisms h a EC[d,a], h c EC[d,c] and ht,EC[d,b], such

A cone arrow

(d,

k:

{h a hb, h c }) ,

d'-*d such that

is

that the following

a limit, if for any other cone

h'{

=

Thus, the diagram for the cone

commutative “square”,

i.e.,

(d', {h' a , h'b, h' }) c

there exists a unique

h[ 0 k, for iE{a,b,c}.

The commutativity of the previous diagram implies arrows h a and hb such that f

diagram commutes:

0

ha = g

(d,

,

(d,

ha = g

0

h^; conversely, given

hb, one obtains a cone by defining h c =

0

{h a hb, h c })

by giving

that f 0

is

f 0

ha = g

usual,

two

h^

equivalently expressed by giving only the outer

{h a hb}). In conclusion, a universal cone for a diagram ,

of this type turns out to be just a pullback.

As

0

by taking the colimit of the same diagram we obtain

the dual notion of pushout.

.

Cones and Limits

130

Existence of Limits

6.3

we

In this section,

study the important question about the existence of limits in a given category.

Starting with the familiar category of sets,

we

common

generalize a

construction that allows the

existence of complex limits to be states, provided that simpler ones exist.

D

every diagram

Note

first that

Let

{Dj}i 0 i be a family of objects in

indexed by

I.

n^Di

The elements of

equivalently functions f

:

I

has limit in Set.

:

pj

* fe°Pi

condition xj

pj

•

The

idea

= fe °Pi- That

n^iDi-^Di

for

= fe (xj). Let then

the limit (prove

it

to take the subset

is

be the projection

Yi

n(Dj

projections

/

jij:

X2,

i.e.,

the product

such that xfEDi, for

...}

all

iEI, or

L if

defined by Pi({xQ,xi,X 2

is, if

fe

:

Di~»Dj

fl^Di

of

and only

if,

of

...})

,

=

X[.

an edge of D, one

In

may

the tuples that satisfy the

for all edges fe

Then

pi restricted to L.

all

is

(L,{ Yi

:

D^Dj,

E C[L,Df]

|

one has

iEI})

is

as an exercise).

This set-theoretic construction Let

xj_,

iEI,

all

{xq, x\, X2, ...}EL

is,

n^Di,

that f(i)EDi

general these projections do not form a cone on D, that

have

obtained as follows.

It is

and consider the object

are tuples {xq,

-*Ui£iDi, such

riieiDi has projections pi

D

3iEI

3eEE

n(Dj

3iEI

/

fe

better formalized in

is

)

:

3eEE

fe

Df-*Dj

:

Category Theory in the following way.

be the product of -*• )

Dj.

By

codomains of edges

all

definition of product, there

in

D, with

is

a unique

function \pl

such that

pj

=

Jtjoapi

for

This

is

fe °pi

=

Jtj °\p

2

ni eI Di— n(Dj

any edge \\>

such that

:

2

:

fe

:

/

3iEI

3eEE fe

)

:

Dj-»Dj of D. Analogously there

HEIDI- n(Dj / 3iEI 3eEE fe

f° r an Y edge ^e

:

is

a unique function

)

:

D.

Di-*Dj

visualized in the following diagram:

Note now

that, in set-theoretic terms, for all the tuples

{xq,

xj_,

X2,

...}

in

properties are equivalent: 1.

for all edges fe Dj-*Dj,

2- i(>l({xo.

:

*1. x 2>

•••})

xj

=

fe (xj)

= ^2({ x 0- x l> x 2>

•••})•

the following

131

Cones and Limits

Then, what

we

aleady

we

are looking for

know

that this is

maximal subset L of nj^jDj whose elements

the

is

none other than the equalizer of

and

\p2*

By

satisfy (2), but

a diagram,

We are now ready to generalize to every category C the previous construction of limits in Set. 6.3.1

Theorem Let

D

be a diagram

in

C

with sets I of vertices

and

E

of edges. If every

/-

indexed family and every E-indexed family of objects has a product and every pair of morphisms ,

has an equalizer, then

D

has a

limit.

Proof Exercise (use the previous diagrams).

6.3.2 Corollary

If a category

C

has arbitrary products, and equalizers for every pair of

morphisms, then every diagram has a

6.3.3 Corollary

If a category

C

limit.

has

all finite products,

morphisms, then every finite diagram has a

The relevance of theorem

and coequalizers for every pair of

limit.

6.3.1 is that, in general,

it

is

simpler to check the existence of products and

equalizers than to prove directly the existence of limits.

Example

a family of c.p.o.’s, let

is

structure Pi

Corollary 6.3.2

:

may be used

n^jCj

by the componentwise

prove that every diagram has a limit in

be the product indexed by

order, that

HiEi(Ci)-*Ci are defined by

c.p.o., that the projections are

to

pj(

(q)j£i s (dj)iei

is,

(q)jei

)

=

q

.

It is

I

.

iff

may be

ITj^jCj

V

iEI

CPO.

q

s;

If

{Cj}jei

given a c.p.o.

dp The projections

easy to prove that rij(=i(Cj)

is

indeed a

continuous, and that rij(=i(Ci) satisfies the universal property of the

product.

Given

f,g

:

A-*B

,

their equalizer is

ordering inherited by A, and h

is

h:

A'-* A, where A' =

the injection. A'

is

{aEA

a c.p.o. Indeed, let

/

f(a)

D

= g(a)} with the

be a direct subset of

.

Cones and Limits

132

then

A';

D

is

UDEA'. The

this,

and thus f(UD)=

also a direct subset of A,

U a ei)f(a)

=

u aEDg( a)

= g(UD). By

continuity of h and the universal property for equalizers are easy to prove.

and 2.5.6

In propositions 2.5.5

we showed how

products and equalizers from terminal

to define

objects and pullbacks. This suggests an even simpler sufficient (and necessary) condition for the existence of

all finite limits.

6.3.4 Corollary If

C has

a terminal objects and pullbacks for every pair of morphisms, then

it

has

all finite limits.

Exercise State the dual versions of theorem 6.3.1 and corollaries 6.3.2 to 6.3.4.

and Creation of Limits

6.4 Preservation In this section

we

some cases of functors which “preserve”

study

the property of objects to be limits

of a diagram.

6.4.1 Definition Let

G:

A-*X be

from a on the diagram

D

iEI}) if and only if (Ga,{G%[

G(D)

in

WiEI

>

a functor, and

We

A.

EX[Ga,Gdj]

A

in

has

X, and the limit

iEI})

G:

limit

is

A -*X

has a

in

Proof By

the properties of functors, ({Gdi}j£i,

it

is

universal.

left

{TiEA[a,dJ

(a>

adjoint F:

(a,{x[

E A[a,d\]

\

and

the

diagram

D

=

G(D) = ({Gdj} iEj, {Gfe} e eE)

then

iEI}), iEI}). \

{Gfe } e £E

is

)

a cone;

iEI}) be another cone, and

let

|

the

be an universal cone

iEl}) |

X -*A,

\

(Ga,{GxiEX[Ga,Gdj]

Let (x,{ajEX[x,Gdj]

EA[a,di]

preserves the limit

defined dually.

has a limit

is

G

an universal cone from Ga on the diagram

is

\

If the functor

{fe}eEE)

(a,{Tj

let

then say that

Preservation of colimits

.

Theorem

6.4.2 (

X

in

isomorphism of the adjunction. Then (Fx, {cp~l(oi)EA[Fx,dj]

we qp:

iEI})

only need to prove that

A[Fx,dj] * X[x,Gdf] be is

a cone. Indeed, for

all

|

fe dj :

—dj

one has fe»(p-l(oj)

=qr l(G(fe)°Oi)

by

= (p“l(oj)

By

the universality of

(a,

because (a[)

{xjEA[a,dj]

iEI} |

ViEI Ti°h =

qp“l(oi).

a mediating

morphism between

Take then

qp(h):

naturality

)

=

x-*>Ga. Since Grj°cp(h)

the cones

(x,

is

a cone

on G(D)

there exists a unique arrow h: cp(xi°h)

=

)

at first sight.

.

Note

A(f): A(s')-»A(s)

o x(s')

and B(f) o

H(s)

:

:

B(f)

A(f) o

0 x(s')

that

x(s):

H(s)-*K(s),

and B(f): B(s')-»B(s)

— K(s)

o

H(s') -> B(f)

are natural transformations of the following type:

A(f) o K(s').

But, according to the definition of S-indexed functors, for any

and K(s)

0

A(f) = B(f)

0

K(s'), thus equation (t) is

f:

s-*s',

one has H(s)

S and any object

a in A(s'),

T ( s )A(f)(a) =

where the previous equation holds

The previous

situation can

B(f)(x(s') a )

in the category B(s).

be summarized

0

A(f) = B(f)

0

well typed.

Spelling out the composition of natural transformations and functors in (t), s-*s' in

°

We are thus composing natural transformations and functors in the way described at the

end of section

H(s')

o

f: s->s' in S,

x(s) °A(f) = B(f) ox(s')

H(s')-»K(s')

are functors.

H->K consists

for any

(f)

t(s'):

f*

Let H: A-*B, K: A->B, be two S-indexed functors.

7.1.4 Definition

transformation

the identity natural transformation

=

composition

component-wise

is

a

collection of functors H(s): A(s)-*B(s), for s object of

(being the composition of natural transformations),

identity id a:

is

B: S°P-*CAT.

to

A(f) = B(f)

A-*B

Given two indexed functors H:

An S-indexed functor H: A-*B

in the following

diagram:

we have

for

any

f:

139

Indexed and Internal Categories

componentwise,

(Vertical) composition of S-indexed natural transformations is defined

H, K,

good

L

:

A-»B,x:

H-*K

(P

0

rj:

idj^-^K°H,

e:

K-*L, p

p: f:

s-*s' in

x )(s )A(f)(a)

7.1.5 Definition Let A,

and

and

definition since, for any

B

° x:

H-*L

given by (p

is

S and any object a °

=

(P(s)

=

P(s)A(f)(a)

0

x)(s)

=

p(s)

given

This

is

a

x (s))A(f)(a) °

T (s)A(f)(a)

=

B(f)(p(s') a ) o B(f)(T(s')a)

=

B(£)(p(s') a ° T(s’) a )

=

B(£)((p ° T)(s')a).

be S-indexed categories

if and only

is,

in A(s'),

,

H: A-*B, K: B-+A be S-indexed functors,

H°K—*idft be S-indexed natural transformations.

:

A—>B

is

an

if

K

(Ke) o(rjK) = id

(eH) o(Hr]) = id

H

The notion of indexed adjunction particular

it

is

is

.

the obvious generalization of the usual notion of adjunction. In

easy to check that for any object

s

of S,

:

A(s)-*B(s)

is

an

adjunction in the usual sense.

The main problem with

the definition of adjunction as a quadruple

generalization of the case with parameters

adjunction of this kind).

As

(remember

:

A-*B

is

in its

exponents requires an

a triple, an indexed adjunction can be defined in the following,

somewhat

informal, way:

7.1.6 Definition functors. i.