Topological Hochschild homology

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Topological Hochschild homology

Marcel Bokstedt

Fakultat flir Mathematik Universitat Bielefeld

4800 Bielefeld, FRG

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§ 0

Inlroductlon

In this paper we construct a topological version of Hochschild homology. One motivation for doing this , is the relation of K-theory to ordinary Hochschild homology. Recall that the Hochschild homology of a bimodule ring

R

can be described as the simplicial group [i]

M ® R

�

r.) �

{

=

over a

HH(R)

®i

mr 1 d. (m ® r ® ... 1 J

M

® r2 ®

m ® .

.

.

r.m ® r 1 �

.

.

r.

.

® r r + j j 1

�

j r.

®

j

�- 1

.. . This functor satisfies Morita equivalence, the matrix ring of

R

i. e.

, then

HH(M (R) , M (R) ) n n

�

HH(R , R)

if


( '

-======----

·

-

Example 1 .3. 1 .

The category

7

I

-

of finite ordered sets and ordered

injective maps is a good index category.

u

is given by concatena-

tion,

and F.I is the full subcategory of objects with cardinality l --=./' :-: i . grea te � equal to

Example 1 .3.2.

The category

I

cornm

of finite sets and injective

maps is also a good index category. This category has an extra structure, a natural transformation between If

u(A, B) C

and

is a good index category, -

defines a functor to

G

u(B,A)

f.!

cn � c

:

defines maps

Lemma 1.4.

If

C

is a good

.

index category,

equivalence

by

Let

A * (B)

=

A E F.C . l u(A, B) .

n . C

F. C l

c

There is a functor

C

G

This functor induces a map

\

\

I

thi s

C � simplicial

A* : C

c

F. (C) � C l

A*

so in

and the iden-

A* to L G �L G F. (C) F. (C) l l By the homotopy extension

property, we can extend this homotopy to a homotopy of

to a map,

i ,

map is hom o t o pic t o the identity.

is also homotopic to the identity.

(G ) ( G) � L F.C l which is a retraction *

given

•

By the same argument, the restriction of

A

u

induces a homotopy

since there is a natural transformation between Lc(G)

:

This functor raises filtration by

particular it factors over

tity on

Iteration of

A natural transformation from Fou �

sets a functor, then the inclusion

P roof.

so is

:

L

C

Lc (G) � L . (G) F C l

- 8 -

Composing homotopies, retract of Now,

let

we see that

is a deformation

(G) L F.C l

L (G) C

be a good index category,

C

G

C -+ simplicial

sets a functor with the property that

is a

r�

G

: �G ext � G ex · )

A. (i) -equivalence for� X�X'

Theorem 1. 5.

Let

X E F.(C) l

cons isting of the identity of

c:

F. (C) l

The inclus ion of the category

.

X

in

defines a

C

A.(i) -equivalence

G(X ) -+ L (G) C Proof.

By lemma 1. 4,

it s uffices to

is a A.(i) -equivalence. that it suffices to

Replacing

G

prove

are homotopy equivalences,

G(X )

-+ L (G) F. (C) l

by a cosk eleton of

that if all maps

pr ove

that

then

G (f)

:

G , we s ee

G(X)

G(Y)

�

(G) G(X ) -+ L F.C l

is also a homotopy equivalence.

This statement is equivalent to the statement that contractible,

B(F C) i

is

since we have a fibration

L

(G) -+ BF (C) i

F.C l

with fibre homotopy equivalent to

G(X)

induces an H-space structure on

But the product and by conditions 2

°

and 3

°

it is connected.

using the homotopy equivalence

I

BF.C l

A well kn own trick,

BF C X BF C -+ BF.C X BF.C l i l i ....,

( a , b)

s hows that

BF.C l

By condition 2

°

( ab, b )

'

th e

following maps are homotopic

pr , � : BF.C 1 l

x

BF.C -+ BF.C l l

Composing with the skew diagonal

BF.C we obtain that

>

has a homotopy invers e . ,

l

BF�C

�

1)

c�...,�z"')

/

BF . C x BF. C l

is contractible.

��..... �:-' ;).

l

-...:._

and noting that the inclusion of this subspace ]:._� highly connected>., The [ 3 ] . cyclic

r-space

f

··...

I

..

p- .

...

... .

·

·

THH(F)

�··--

......,__

·

has a cyclic structure in the sense of \..

,...) . -'- - - · rs

c o m �·�·t· at i v e,

-:�·.:

t hen

the

_··-

·--·

-

--

-

··hyper-f-spac e

THH"(F)

structu r e.

The same statements are true for the (hyper-)r-space The inclusion GL (F) A

�

X

LI n

has

..... --------

(

a

y Nc (F) .

X M (F)(S ) A

defines a map of (hyper-)r-spaces compatible with the cyclic structure:

y NC (F)

�

THH(F)

We finally examine the relation between Let

X

be a monoid,

acting on itself. If

X

y

Nc (X ,X )

K(F)

and

y Nc (F)

.

the cyclic barconstruction of X

This is again a cyclic object.

is a group, we can include

BX

in

C N Y (X, X )

as the

-f --

.

15 -

- ·

cyclic subobject consisting in degree

n

of

{ (x ,...,x )E(N CY) n o n The isomorphism to the standard barconstruction is given by n

projection onto the last

coordinates.

This object is not equivalent to the constant cyclic object cy N (X,X)

In order to relate

to this constant object,

BX .

we replace

by an equivalent category. In fact,

for any category

C ,

we can define

cy N (C, C)

as the

simplicial set [n]

�

{(f0,

.

.

•

, f )E(Morph C) n

n+1 ;

f

i

and f

i+1

f and f n o It is clear that a functor

�

C

Ncy (c , c )

V

composablef

induces a map

y Nc (v , v)

�

composablel

.

It is not true that a natural transformation induces a homotopy. Remark 2. 6. morphisms,

If

F, G: C

�

V

are naturally equivalent through iso-

then the induced maps are homotopic after realization.

To see this,

consider the category

I

with two object and

exactly one isomorphism between them. The natural equivalence defines a functor I

X

c

�

v

inducing Cy N (IxC, IxC) The simplicial set

�

N

NCY(IxC,IxC)

Cy

(VxV)

is isomorphic to the diagonal

of the bisimplicial set N

A path in

I NCy(I, I) I

homotopy between

N

cy

cy

(I, I) x N

cy

(c ,c )

between its two zero simplices defines a (F)

and

N

cy

(G) .

This path is given by 1 cy (f, f- ) E N (I, I) 1

X

16 In pnrlicular, the remark shows that equivalent categories have c homotopy equivalent realizations I N Y(C,C) I

The application is to Let in

a

certain bicategory.

be the bicategory consisting of commutative diagrams

Sq(C)

C

y

X

-+

Z

-+ T

� � [ 1 0]

.

There are two ways of forming a nerve in this bicate-

gory, yielding two different (but abstractly isomorphic) categories.

Let us denote them by Nerve1 (S qC))

There is an inclusion of Nerve 1

Sq (C)

simplicial

and Nerve 2 (SqC))

•

C

in the simplicial category

C

are isomorphisms, this inclusion

.

In case all morphisms in

is an equivalence of categories in each degree. There is also an inclusion of the simplicial set objects

of the simplicial category

Lemma 2 . 7.

If all morphisms of

C

Nerve

1

BC

as the

Sq (C) · •

are isomorphisms, then the two

maps Nerve( C) Nerve( C)

-+ -+

Nerve (Sq (C) ) 1

Nerve (Sq (C) ) 2

are homotopic. P roof.

-+ -+

Nerve

1

Nerve (Sq (C) ) 2

Nerve2 Nerve 1

( Sq ( C ))

We can explicitely write down a binatural transformation

between the functors

id

�

id

as follows:

f

H>

f

)

-

17 -

id -

f

---+

--

-

·-·--

·--·-··· �

f

l . h-1 f

f id

We can now for any monoid

where

cyclic objects, In case

that

(X)

BX

where

X

to a group. n 0

consider

is the inclusion of the objects. Then

i

follows,

X

is a map of

has the trivial cyclic structure. f

is a group,

f

i

is a homotopy equivalenceo

is a homotopy equivalence when

X

It

is equivalent as a monoid

This again is equivalent to the statement that

is a group.

In order to apply this to the r-spaces above, we note that the construction

C

�

1s q ( S )

N e rv e

form the (hyper-)-r-space

commutes with products,

so we can

NCY(Nerve1sq(F)) .

There is a diagram of r-spaces and cyclic maps K(F) i

�

NC y (F)

�

_L NC y ( N e r ve l Sq (F)) t.J :· /·:;

�;:

:

- '·

".:

THH(F) Since equivalence.

lim n

is a group,

f

is a homotopy

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§ 3 In this paragraph,

we will make miscellanous remarks on the

constructions we made. � we will use the cyclic structure to deiine a map

In particular,

1

n n lim n F(S ) � THH(F) � n This map will be essential in a later paper, s

the homotopy type of

x

1

THH(F)

where we compute

in cer tain case s.

The first remark is that in some cases our constructions agree with known concepts. Example 3.1.

F(U) = U . Then

Let

K(F) = A (*) See [ 4] . Slightly more generally, let X G(X)

the loopgroup of

X

be a simplicial set,

and

then

,

the f unctor

F(U) is a

FSP

=

U

A

G(X)+

satisfying the stability condition. We have K{F) = A(X)

The topological Hochschild

homology is given by the simplicial

object [i]

H

L i

i+1

11

H

X ll... X X ll. o i "" o

•

.

4

X

i

(G(X)

i+1

+)

with the usual structure maps, modelled on Hochschild homology . This can be rewritten as the diagonal of a bisimplicial set whose realization in one simplicial direction is X JJ. X.� X lL X. � 0 0 [i],... L (AX+ ) i+1 >I

\

•

•

•

•

•

•

All structure maps in this object are homotopy equivalences, so THH(F) n

- 19 There is a map

defined as in [3]. I claim that this map agrees with the map K(F)

-+

THH(F)

This follows from the fact that lemma 2.7 provides a homotopy commutative diagram c

N Y(Nerve1 SqF)

K(F)

I

Cy N (F) where

K(F)

-+

N

Cy (F)

is given by the inclusion

BGL (F) -+ A BGL (F) A A followed by the identification

A BGL (F) A

Cy N (F)

�

It is not difficult to check that this map followed by N

Cy

(F)

-+

THH(F)

agrees with the map of Waldhausen. It is possible to define a stable version of 3.2.

D-efinition

is the

limit

-+

�mX

of

The map

K(F)

THH(F)

Finally,

consider the space F

Then

F

m

,

factors over

=

L � I

X

Xm

where

FSP

We can define a map

s1

x

F

-+

r-space

X F(S )

F(-) .

A

is the

8 K (F) .

is a (hyper)-r-space. We say that

r-space of the

K(F) .

THH(F)

F

is the underlying

- 20 -

1 8

There is a simplicial model of [i]

�

{0, 1,

.

0

. ,i} .

with the usual cyclic structure maps ,

I�




j

i J,L

in degree

k