Theory and Applications of CategoriesVol No pp

SYMMETRIC MONOIDAL CATEGORIES

MODEL ALL CONNECTIVE SPECTRA

R W THOMASON

Transmitted by Myles Tierney

Abstract The classical innite lo opspace machines in fact induce an equivalence of

categories b etween a lo calization of the category of symmetric monoidal categories and

the stable homotopy category of connective sp ectra

Intro duction

Since the early seventies it has b een known that the classifying spaces of small symmetric

monoidal categories are innite lo op spaces the zeroth space in a sp ectrum a sequence

of spaces X i with given homotopyequivalences to the lo ops on the succeeding

i

space X X Indeed many of the classical examples of innite lo op spaces

i i

were found as such classifying spaces eg Ma Se These innite lo op spaces and

sp ectra are of great interest to top ologists The homotopy category formed byinverting

the weak equivalences of sp ectra is the stable homotopy categorymuch b etter b ehaved

than but still closely related to the usual homotopy category of spaces eg Ad I I I

One has in fact classically a functor Spt from the category of small symmetric mo

noidal categories to the category of connective sp ectra those sp ectra X for which

i

X when ki Ma Se Th Moreover anytwosuch functors satisfying

k i

the condition that the zeroth space of SptS is the group completion of the classifying

space B S are naturally homotopy equivalent Ma Th

The aim of this article is to prove the new result Thm that in fact Spt induces

an equivalence of categories b etween the stable homotopy category of connective

sp ectra and the lo calization of the category of small symmetric monoidal categories by

inverting those morphisms that Spt sends to weak homotopyequivalences In particular

each connectivespectrumisweak equivalenttoSptS for some symmetric monoidal

category S

Received by the Editors April

Published on July

Mathematics Sub ject Classication Primary P Secondary C D D D

P

Key words and phrases club connective sp ectrum E space op erad lax algebra sp ectrum

stable homotopy symmetric monoidal

c

R W THOMASON Permission to copy for private use granted

Theory and Applications of Categories Vol No

Thus the category of symmetric monoidal categories provides an alternate mo del

for the connective stable homotopy category one which lo oks rather dierent from

the classical mo del category of sp ectra It is really co ordinatefree in that there are

no susp ension co ordinates at all in contrast to Mays co ordinatefree sp ectra Ma

I I which use all nite subspaces of an innite vector space as co ordinates and thus

are just free of choices of co ordinates The category of E spaces spaces with an

action of an E op erad Ma is similarly a mo del for connective sp ectra which

is really co ordinate free But a symmetric monoidal category structure is much more

rigid than a general E space structure and as a consequence can b e sp ecied and

manipulated much more readily As convincing evidence for this claim I refer to my

talk at the Collo que en lhonneur de Michel Zisman at lUniversiteParis VI I in June

There I used this alternate mo del of stable homotopytogive the rst known

construction of a smash pro duct which is asso ciative and commutative up to coherent

natural isomorphism in the mo del category This will b e the sub ject of an article to

app ear

The pro of that the functor Spt induces an equivalence b etween a lo calization of

the category of symmetric monoidal categories and the connective stable homotopy

category b egins by constructing an inverse functor The construction is made in several

steps First there are known equivalences of homotopy categories induced by functors

between the category of connective sp ectra and the category of E spaces Thus

one reduces to nding an appropriate functor to the category of symmetric monoidal

categories from that of E spaces AnyspaceX is weak homotopy equivalenttothe

classifying space of the category NullX of weakly contractible spaces over X When

X is an E space the op erad action on X induces op erations on this category For

example for eachinteger n there is an nary functor sending the ob jects C

Q

n

X C X C X to E n C C C E n X X where the

n n

Q

n

last arrow E n X X is given by the op erad action The internal comp osition

functions of the op erad induce certain natural transformations b etween comp osites of

these op erations on NullX Using Kellys theory of clubs Ke Ke one nds

that NullX has b een given the structure of a lax algebra for the club of symmetric

monoidal categories The next step is to replace this lax symmetric monoidal category

by a symmetric monoidal one There is a Go dement resolution of a lax algebra byfreelax

algebras yielding a simplicial lax algebra By coherence theory the latter is degreewise

stably homotopy equivalent to a simplicial free symmetric monoidal categoryTaking a

sort of homotopy colimit of the last simplicial ob ject Th yields the desired symmetric

monoidal category

The layout of the article is as follows The rst section is a review starting with the

denitions of nonunital symmetric monoidal categories and of lax strong and strict

symmetric monoidal functors I recall in the basic prop erties of the classical functor

from symmetric monoidal categories to sp ectra Next in comes a review of the

denitions of oplax functors into a category and left oplax natural transformations

between such The homotopy colimit or oplax colimit of a diagram of symmetric

Theory and Applications of Categories Vol No

monoidal functors is recalled in and its go o d homotopy theoretic prop erties stated

The rst section closes with a pro of in that all the variant categories of symmetric

monoidal categories considered have equivalent homotopy categories Section b egins

with a review of lax algebras over a do ctrine A Go dementtyp e simplicial resolution

of lax algebras by free lax algebras in given in This resolution is shown to b e left

oplax natural with resp ect to lax morphisms of lax algebras In I consider the

sp ecial case of lax symmetric monoidal categories The functor to Sp ectra is extended

to these in In I use the Go dement resolution and the homotopy colimit to show

how to replace a lax symmetric monoidal category by a symmetric monoidal category

Section reviews the notion of an E space and the equivalence of homotopy categories

between E spaces and Sp ectra Section contains the construction of a lax symmetric

monoidal category asso ciated to an E space Finally Section nishes the pro of by

showing various roundtrip functors made by joining the ab ove pieces are linked to the

identitybya chain of stable homotopy equivalences

Symmetric monoidal categories and homotopy colimits

I will need to use the homotopy colimit of diagrams of symmetric monoidal categories

The based version of the homotopy colimit do es not have go o d homotopy b ehavior

except under stringent nondegenerate basep oint conditions essentially one would

have to ask that the symmetric monoidal unit have no nonidentity automorphisms and

that every morphism to or from the unit b e an isomorphismSuch a symmetric monoidal

category is equivalent to the disjoint union of with a p ossibly nonunital symmetric

monoidal category and nally all homotopy colimit results are easier to state if one

works with variant categories of nonunital symmetric monoidal categories from the

b eginning Thus

Definition By unital symmetric monoidal category I mean a symmetric monoidal

category in the classic sense a category S provided with an ob ject and a functor

SS S together with natural isomorphisms of asso ciativity commutativity and

unitaricity for which certain simple diagrams are to commute For the details see for

example McL VI I x x

Definition A symmetric monoidal category is a category S together with a functor

SSS and natural isomorphisms

A B C A B C

A B B A

which are such that the following twopentagon and hexagon diagrams commute

Theory and Applications of Categories Vol No

A B C D A B C D A B C D

O

 

A B C D A B C D

A B C C A B A B C



A C B A C B C A B



The usual results of coherence theory that every diagram commutes continue to

hold for these nonunital symmetric monoidal categories indeed the precise statements

and the demonstrations b ecome easier without the additional structure of the unitaricity

natural isomorphisms Ep Ke

To compare with the existing literature it will b e useful to recall Ma and

McLa VI I x that a permutative category is a unital symmetric monoidal category

where the natural asso ciativity isomorphism is the identity It follows from coherence

theory that every unital symmetric monoidal category is equivalenttoapermutative

category Indeed the equivalence is realized by strong unital symmetric monoidal func

tors and unital symmetric monoidal natural isomorphisms See Ma or Ke

As to morphisms b etween symmetric monoidal categories one needs to consider

three kinds the lax the strong and the strict symmetric monoidal functors The usual

denitions adapt to the the nonunital case easily

Definition For S and T symmetric monoidal categories a lax symmetric mo

noidal functor from S to T consists of a functor F ST together with a natural

transformation of functors from SS

f FA FB F A B

such that the following two diagrams commute

f f

FA F B C F A B C FA FB FC

F

FA FB FC F A B FC F A B C

f  f

Theory and Applications of Categories Vol No

f

F A B

FA FB

F

F B A

FB FA

f

One will often abusively denote the lax symmetric monoidal functor F f as simply

F censoring the expression of the natural transformation f

A strong symmetric monoidal functor is a lax symmetric monoidal functor such that

the natural transformation f is in fact a natural isomorphism

A strict symmetric monoidal functor is a strong symmetric monoidal functor such

that the natural transformation f is the identityThus F strictly preserves the op eration

and the natural isomorphisms and

The usual denition of a lax strong or strict unital symmetric monoidal functor

between unital symmetric monoidal categories imp oses the additional structure of a

morphism F resp ectively an isomorphism an identity sub ject to a commutative

diagram involving the unitaricity isomorphisms See eg Th

Definition A symmetric monoidal natural transformation F G between two

lax symmetric monoidal functors F f G g ST is a natural transformation

such that the following diagram commutes

f

F A B

FA FB



GA B

GA GB

g

A symmetric monoidal natural transformation b etween twostrictortwo strong sym

metric monoidal functors is a symmetric monoidal natural transformation b etween the

underlying lax symmetric monoidal functors

Note that suchan is automatically compatible with the asso ciativityandcommu

tativity isomorphisms and by naturalityof and On the other hand the de

nition of a unital symmetric monoidal natural transformation b etween unital symmetric

monoidal functors imp oses a new compatibility with the unitaricity isomorphisms as

in eg Th Thus a symmetric monoidal natural transformation b etween two

unital symmetrical monoidal functors need not b e a unital symmetric monoidal natural

transformation

Catalog of variant categories of symmetric monoidal categories

In order to mo del sp ectra I want to consider only small symmetric monoidal cate

gories those for which the class of all morphisms is in fact a set In order to b e able

Theory and Applications of Categories Vol No

to lo calize categories of such I supp ose Grothendiecks axiom of universes SGA I

App endice This axiom is that each set is contained in a set U the elements of which

form a mo del of settheory such that the internal p ower sets in the mo del U are the true

power sets For eachsuch universe U one has several categories of U small symmetric

monoidal categories A category is U small if the class of all its morphisms is an element

of U The class of ob jects is then also an elementof U Any of the variant categories

of U small categories b elow will then itself b e V small for any universe containing U as

an element The lo calization of any of the variants byinverting any set of morphisms

is then W small for any universe W containing V

The variants diering in unitaricity and in degree of laxity of morphisms are

SymMon

Ob jects U small symmetric monoidal categories

Morphisms lax symmetric monoidal functors

cells symmetric monoidal natural transformations

SymMonStrong

Ob jects U small symmetric monoidal categories

Morphisms strong symmetric monoidal functors

cells symmetric monoidal natural transformations

SymMonStrict

Ob jects U small symmetric monoidal categories

Morphisms strict symmetric monoidal functors

cells symmetric monoidal natural transformations

UniSymMon

Ob jects U small unital symmetric monoidal categories

Morphisms lax unital symmetric monoidal functors

cells unital symmetric monoidal natural transformations

UniSymMonStrong

Ob jects U small unital symmetric monoidal categories

Morphisms strong unital symmetric monoidal functors

cells unital symmetric monoidal natural transformations

UniSymMonStrict

Ob jects U small unital symmetric monoidal categories

Morphisms strict unital symmetric monoidal functors

cells unital symmetric monoidal natural transformations

Of these what the common man means by the category of small symmetric mo

noidal categories is UniSymMonStrong To express the universal mapping prop erty

Theory and Applications of Categories Vol No

characterizing homotopy colimits one needs instead to consider b oth SymMon and

SymMonStrict Adding the nonunital and unital analogs of all these pro duces the list

of six categories ab ove

One has the obvious diagram of forgetful functors

UniSymMonStrict UniSymMonStrong UniSymMon

SymMonStrict SymMonStrong SymMon

There is also a functor from the b ottom to the top of each column in

On ob jects it sends the nonunital symmetric monoidal category S to the copro duct

of categories S where is the category with one ob ject and only the identity

morphism The symmetric monoidal structure on the copro duct is determined bysaying

that the inclusion of S is a strict symmetric monoidal functor and that there are natural

identities of functors on the copro duct

Id Id Id

whichwe take as the unitaricity isomorphisms Thus b ecomes a strict unit This

construction on ob jects extends to a functor in the obvious way

The functor Spt SymMon Sp ectra

A slight elab oration Th App endix of either Mays or Segals innite lo op space

machines gives a functor into the category of sp ectra

Spt SymMon Sp ectra

Moreover symmetric monoidal natural transformations canonically induce homotopies

of maps of sp ectra Th

Let B Cat Top denote the classifying space functor By Th there is a

natural transformation of functors from SymMon to Top

B SSptS

where SptS is the underlying zeroth space of the sp ectrum SptS When S admits

the structure of a unital symmetric monoidal categorythis is a groupcompletion

Equivalently it induces an isomorphism on homology groups after inverting the action

of the monoid B S



H B S Z H SptS Z

 

Theory and Applications of Categories Vol No

When S is not unital SptS is a groupcompletion of the unital S since by

Th the inclusion of S into the this symmetric monoidal category induces a weak

homotopy equivalence of sp ectra

a

SptS SptS

This last assertion ultimately reduces to the observation that any monoid M has

groupcompletion isomorphic to that of the monoid M formed by forgetting there

was already a unit and freely adding a new one For the group completion pro

cess forces the identication of the new with all other idemp otent elements and in

particular with the old forgotten unit For details of the reduction see Th A

The pro of of Mays uniqueness theorem Ma Thm for functors dened from

the category of p ermutative categories easily generalizes to showanytwo functors from

SymMon to Sp ectra which satisfy the ab ovecited groupcompletion conditions of Th

and are connected byachain of natural weak homotopy equivalences See

Th pp In particular the two functors b ecome naturally isomorphic

after comp osition with the functor from Sp ectra into the stable homotopy category

The same statements hold for anytwo functors with the groupcompletion prop erties

dened on any of the six variant categories of symmetric monoidal categories considered

in

Definition A lax symmetric monoidal functor F ST is said to b e a stable

homotopy equivalence if SptF isaweak homotopy equivalence of sp ectra A morphism

in anyofthevariant categories of symmetric monoidal categories listed in is said to

b e a stable homotopy equivalence if the underlying lax symmetric monoidal functor is

such

If the lax symmetric monoidal functor F induces a homotopyequivalence of

classifying spaces B S B T then F is a stable homotopy equivalence This follows

from the group completion prop ertyofFor a map of connective sp ectra is a

stable homotopy equivalence if and only if the induced map on the zeroth spaces is a

weak homotopy equivalence Indeed the stable homotopy groups of the sp ectrum are

in negative degrees by connectivity and in nonnegative degrees are isomorphic

to the homotopy groups of the zeroth space And by the Whitehead theorem for H

spaces not necessarily simplyconnected this condition is in turn equivalent to the map

of zeroth spaces inducing an isomorphism on homology with Z co ecients

More generally this shows SptF is a stable homotopy equivalence if and only if BF

induces an isomorphism on the lo calizations of the homology groups byinverting the

action of the monoid



 

H BF B S H B S Z B T H B T Z

  

Theory and Applications of Categories Vol No

Left oplax natural transformations

I will need the notions of oplax functors and left oplax natural transformations

between them Theses concepts are ultimately derived from Benab ous work on bicate

gories and Grothendiecks theory of pseudofunctors I follow the terminology of Street

St suitably dualized from lax to oplax and generalized from Cat to an arbitrary

category

Let L b e a category and K a category

K consists of functions assigning to each Definition An oplax functor L



ob ject L of L an ob ject L of K to each morphism L L of L a morphism

  

L in L L of K to each comp osable pair of morphisms L L

 

L a cell in K and to each ob ject L of L a cell in K

These are to satisfy the following three identities of cells

L L L



L in L L L For each L




L L L L

O O C

C

C

C

C

C

C

C



C

C

 

C

C

C

C

C

C

C

L L L L



For each morphism L L in L

 

L L

L L

P E E

K

K

K

r

id

K

r

r

r

u

L L

A functor may b e considered as an oplax functor with identity structure cells A

pseudofunctor is an oplax functor where the structure cells are isomorphismsThe

denition of a lax functor is obtained byreversing the direction of the structure cells

in

Definition A left oplax natural transformation between two oplax functors

K is a function assigning to each ob ject L of L a morphism in K L L L

Theory and Applications of Categories Vol No



L and to each morphism L L a cell in K These are to

L L

satisfy the following two identities of cells

For each L L L in L

L L

 

L L L L

x x x

x x x

x x x

x x

x

x x x

x x x

x x x

x x x

L

ks ks

L L L

F F F

F F F

F F F

F F F

F F F

F F F

F F F

F
F
F

L L L L

L L

And for each ob ject L of L

L L

L L L L

ks ks

L L L L

L L

Any natural transformation b etween two functors may b e considered a left oplax

natural transformation with identity structure cells There is a notion of rightop

lax natural transformation obtained byreversing the direction of the structure cells

For a left lax natural transformation b etween two lax functors the cells go in the

same sense as for a left oplax natural transformation but of course the structure two

cells of and go in the opp osite sense The conditions to imp ose on the cells of a

left lax natural transformation are analogous to and See St x

Definition A modication s between two left oplax natural transfor

a cell s of K These mations is a function assigning to each ob ject L of L

L L L



are to satisfy the cell identity that for each morphism L L in L

Theory and Applications of Categories Vol No

L

L

s

L

L L L L

L

L

   

s

L

L L L L

L

L

Notation Let L b e a small category and K a category Denote by

CatLK

the category of functors from L to K Its cells are natural transformations and its

cells are mo dications Denote by



K opLaxL

to K Its ob jects are oplax functors the the category of oplax functors from L

morphisms are left oplax natural transformations and the cells are mo dications

Denote by



FunL K

the sub category whose ob jects are functors whose morphisms are left oplax natural

transformations and whose cells are mo dications

Homotopy colimits

I recall from Th the homotopy colimit of a diagram of symmetric monoidal cat

egories This is a sort of oplax colimit which turns out to have go o d prop erties with

resp ect to stable homotopy theory More precisely

The homotopy colimit is a functor



SymMon SymMonStrict ho colim FunL

L

which is left adjoint to the comp osite of the forgetful functor SymMonStrict



SymMon and the functor SymMon FunL SymMon sending a symmetric mo

sending each L to S Thus there is noidal category S to the constant functor from L

a natural adjunction isomorphism of categories for each functor L SymMon and

each symmetric monoidal category S in SymMonStrict



SymMonStrictho colim S SymMon S FunL

L

Theory and Applications of Categories Vol No

This adjunction isomorphism is equivalent to the universal mapping prop erty stated

in Th Prop as follows by a straightforward calculation on expanding out the

various denitions and using Th

I call this oplax colimit a homotopy colimit b ecause of its relation to the homotopy

colimit of diagrams of sp ectra Recall BK XI I Th x the latter homotopy colimit

is a functor

ho colim CatL Sp ectra Sp ectra

L

which sends natural stable homotopyequivalences to stable homotopy equivalences

equivalences It induces a total derived functor of colim on the homotopy categories

L

Among its other go o d prop erties is a natural sp ectral sequence for the stable homotopy

groups whose E term is expressed in terms of homology of the category L

H L E ho colim

p q pq L

pq

But by Th Thm there is a natural stable homotopy equivalence b etween

functors from CatL SymMon to the category of Sp ectra

ho colim Spt Sptho colim

L L



SymMon In particular there is a sp ectral sequence natural in FunL

E H L Spt Sptho colim

p q pq L

pq

forq the sp ectral sequence lives in the rst quadrant Since Spt

q

and converges strongly eg Th Since a map of sp ectra is a stable homotopy

equivalence if and only if it induces an isomorphism on stable homotopy groups



this sp ectral sequence directly gives go o d homotopytheoretic control of the symmetric

monoidal category ho colim The extended naturality of the sp ectral sequence

L

for left oplax natural transformations is proved using rectication of oplax functors as

in the last paragraph of Th x cf Th

Comparison of variant homotopy categories

As a rst application of the homotopy colimit I will now pro ceed to show all the vari

ant categories of all have equivalent lo calizations on inverting the stable homotopy

equivalences

As explained after the diagram for each of the vertical forgetful functors in

this diagram there is a functor going in the opp osite direction which freely adds a unit

There are natural transformations b etween the identity functors and the comp osites

of these vertical forgetful and freeunit functors The comp onents of these natural

Theory and Applications of Categories Vol No

transformations are the inclusion SS for S nonunital and the map sending to

the old unit S S for S unital By Spt is a stable homotopy equivalence

Since Spt is a rightinverse to an instance of Spt it is also a stable homotopy

equivalence Thus these natural transformations b ecome natural isomorphisms in the

lo calizations and the lo calizations of the unital and nonunital variants in each column

of are equivalent

It remains only to see that the forgetful functors b etween the nonunital variants

in the b ottom row of induce equivalences of lo calizations But one has another

functor in the opp osite direction given by



Cat SymMon SymMon Fun SymMon SymMonStrict

where the last functor is the ho colim of Let this comp osite functor b e denoted

d

by Using the universal mapping prop erty of ho colim one gets a natural

b

transformation of functors on SymMonStrict with comp onents S ho colim SS

By these comp onents are stable homotopy equivalences Moreover this is still

d

a natural stable homotopyequivalence after pre or p ostcomp osing withanyof

d

the forgetful functors in the b ottom rowof Thus these comp osites with

induce inverses to the forgetful functors after lo calization This has proved the following

reassuring principle

Lemma The forgetful functors in diagram al l induceequivalences of cate

gories between the localizations of these variant categories by inverting the stable homo

topy equivalences

Lax symmetric monoidal categories

In this section I will study the category of lax algebras in Cat over the do ctrine of sym

metric monoidal categories that is the category of lax symmetric monoidal categories

Following Kelly I give generators and relations for a club whose strict algebras are the

lax symmetric monoidal categories The functor Spt of extends to a functor

on these A Go dement construction shows that each lax symmetric monoidal category

admits a simplicial resolution by strict symmetric monoidal categories Using this and

op

the homotopy colimit along I dene a functor from the category of lax symmetric

monoidal categories to that of symmetric monoidal categories which commutes up to

natural stable homotopy equivalence with Spt

Clubs and do ctrines

I will supp ose the reader is familiar with Kellys theory of clubs an ecient means

of describing algebraic structures imp osed on categories A club prescrib es certain nary

functors which are to b e op erations on a category and natural transformations b etween

them These may b e considered as generated by iterated substitution and comp osition

Theory and Applications of Categories Vol No

of a smaller basic set of op erations and transformations The club structure enco des

this substitution pro cess I will consider only clubs over the skeletal category of nite

sets with p ermutations as morphisms That is eachoperationinaclubwillhave a nite

arity n and natural transformations b etween op erations are allowed to sp ecify a required

p ermutation of the order of inputs b etween the source and the target op erations The

typ e functor from the club to the category of nite sets sp ecies the arityofthe

op erations and the p ermutations asso ciated to the natural transformations The reader

may consult Ke x and x for a quick review of club theory See also Ke Ke

Denote by the club for symmetric monoidal categories The underlying category

of this club has as ob jects a T of typ e n N for eachway to build up an nary op eration

n

by iterated substitution of a binary op eration into instances of itself This includes

an empty set of substitutions which yields the identity functor as a ary op eration

For each n the typ e functor is an equivalence of categories b etween the sub category

of ob jects of typ e n and the symmetric group of order nThus anytwo ob jects of the

same typ e are isomorphic and the group of automorphisms of any ob ject of typ e n is

the symmetric group All this follows from standard coherence theorygiven that in

n

this pap er symmetric monoidal categories are not assumed to have units

In order to construct the Go dement resolution needed to pass from lax symmetric

monoidal categories to symmetric monoidal ones I will need to consider the do ctrine

asso ciated to a club The reader may consult KS x for a review of do ctrines I recall

the denitions for its convenience

Definitions A doctrine in a category K is an endo functor D together with

natural transformations j Id D and DD D satisfying the dening identities

D D Dj id jD

together with a morphism a DA A the action A D algebra is an ob ject A of K

satisfying the dening identities

a Da a A DDA A id a j

A

A lax morphism between two D algebras A a and B b consists of a morphism in

f A B together with a cell f K

Df

DB DA

a

b

f

A B f

Theory and Applications of Categories Vol No

which is to satisfy the following two identities of cells

DDf DDf

DDA DDB DDA DDB

A B

Da Db

D f

Df Df

DA DB DB DA

a a

b b

f f

A B A B

f f

f

B A

f

jA jB

A B

Df

DA DB

a

b

A B

f

f

B A

f

f is A strong morphism between D algebras is a lax morphism such that the cell

an isomorphism A strict morphism of D algebras is a lax morphism suchthat f is the

identity

A D cel l between two lax strong or strict morphisms of D algebras f g A B

is a cell of K s f g such that one has the identity of cells

g Ds s f

Denote by D Alg the category whose ob jects are D algebras whose cells are lax

morphisms of D algebras and whose cells are D cells The variant sub categories

Theory and Applications of Categories Vol No

whose cells are the strong or strict morphisms of D algebras are denoted D AlgStrong

and D AlgStrict resp ectively

The do ctrine D on the category Cat asso ciated to a club sends a category

C to the underlying category of the free algebra on C From Ke x and Ke x

or Ke x one derives the following description Denote by n the category whose

ob jects are T where T is an ob ject of typ e n in the club and A morphism

n

   

T T consists of a morphism u T T in whose typ e satises

nisthus roughly sp eaking the category of op erations induced by elements of the

club and p ermutations of inputs Then the do ctrine corresp onding to is

a

n

n n C C

n

n

n

by p ermuting the factors and on nbythe Here acts on the nfold pro duct C

n

free right action T T

The natural transformation C C is given as part of the club structure

and enco des the substitution of op erations into other op erations It is induced bya

collection of functors

n j j j j

n n

C is induced by the inclusion The natural transformation j C

of the distinguished ob ject corresp onding to the identity op eration

Assigning to each ob ject and morphism in the club an op eration and natural trans

formation of op erations on C of the appropriate arity corresp onds to giving a morphism

c C C If the assignment is compatible with substitution of op erations and the

identity then c is the structure of an algebra for the do ctrine

In the case of the club of the structure of a algebra is exactly the

structure of a symmetric monoidal category The lax strong and strict morphisms

of algebras corresp ond resp ectively to lax strong and strict symmetric monoidal

functors The coherence theorem for symmetric monoidal categories noted in is

equivalent to the fact that for each n there is a unique isomorphism b etween any

two ob jects of n

Go dement enriched and lax

op op

are the simplicial ob jects of Recall that the category such that functors C

C is the opp osite category of the skeletal category of nite nonempty totally ordered

f ng and monotone increasing maps Following Th let sets n

Theory and Applications of Categories Vol No

b e the category with ob jects n f ng for n and morphisms

the monotone increasing maps sending to There is a standard inclusion functor

op

from to sending n to n A functor X C is a simplicial ob ject X

together with an augmentation d X X and a system of extra degeneracies



s X X for n These must satisfy certain relations extending the usual

 n n

simplicial identities That is one requires on X that

n

d d d d if ij n

i j j  i

s s s s if i j n

i j j i

d s s d if ij n

i j j  i

d s id if j n and i fj j g

i j

d s s d if j and j i n

i j j i

The extra degeneracies imply that the augmentation X X is a simplicial homo



topy equivalence even that there is a simplicial homotopyonX that is a deformation

retraction to the constant simplicial ob ject on X Cf eg Ma x



Recal l the notations of Proposition Let D bea doctrine on the category K

and of xabove

Then there is a natural functor the Godement resolution



op

R D Alg Fun K



n

such that R D for n while R is the functor sending a D algebratoits

n 

underlying object in K



op op

After restriction of the values in Fun from to the standardsubcategory R



lifts canonical ly to a functor



op op

R j D Alg Fun DAlgStrict



Considering R Id D Alg D Alg as taking values in constant simplicial D





op

algebras the augmentation induces a natural transformation d R j R

 



op

of functors into Fun DAlg



The restriction of R to D AlgStrict factors through the sub category of Fun whose

cel ls are strict natural transformations

op

R D AlgStrict Cat K



Thus for a D algebra A R A is a simplicial free D algebra with a D algebra augmen



tation to A the augmentation b eing a simplicial homotopy equivalence after forgetting

the D algebra structure The whole thing is strictly natural for strict morphisms of D

algebras and is left oplax natural for lax morphisms of D algebras If the category

Theory and Applications of Categories Vol No

K has only identity cells the do ctrine D is just a monad and all reduces to the clas

sical Go dement resolution Go App endice or to Mays reformulation as a twosided bar

construction B D D Id in Ma x



op

Proof For A aaD algebra dene the functor R A afrom by giving values



op

on on ob jects and on the generating morphisms d and s of On ob jects R A

i i n

n

D A for n For the morphisms

i ni n n

D D D A D A for in

d R A R A

i n n

n n n

D a D A D A for i n

i ni n n

s R A R A D jD D A D A for i n

i n n

Naturality and the dening identities for the structure maps j anda of do ctrines

and algebras give that these d and s satisfy the extended simplicial identities

i i

op

and so dene a functor on All d and the s other than s are strict maps of

i i 

D algebras

n

Given a morphism of D algebras f A B let R f D f Note that for a strict

n

algebra morphism f this formula denes a natural transformation R f R A R B

  

op

of functors on For suchan f strictly commutes with j and a For a D

n n n

cell b etween two morphisms s f g let R s D s D f D g Routine

n

verication now yields the results of the last paragraph of Prop osition

The essential p oint remaining is to dene the structure of a left oplax natural trans

f A B formation R f R A R B for each lax morphism of D algebras f

  

n

The structure cells of this R f are the R f D ab ove The structure cell

 n

n

D f

n n

D A D B

Rf

q q

D A D B

q 

D f

n in is derived from the structure cells of the asso ciated to a morphism q

lax algebra map as follows

If f qgf ng preserves the maximal element

op

q n then in can b e written as a comp osite of s and those d for which i

i i

 i k i k

is not maximal Thus can b e written as a comp osite of D D and D jD without

i

using any action maps D a In this case the diagram of cells in commutes

and one takes Rf to b e the identity cell In the other case where do es not preserve

the maximal element it factors uniquely in as

f qg f q qg f pg

Theory and Applications of Categories Vol No

where is the inclusion of the initial segment and is determined by with

  q

q n Thus preserves the maximal element As for d D a

q

One sets the structure cell Rf to b e that induced by the structure cell of the lax

algebra morphism

n

D f

n n

D A D B

q 

D f

q q

D A D B

q  q 

D a D b

q 

D f

q q

D A D B

q 

D f

That these Rf satisfy the dening identities and for an oplax natural

transformation follows by routine case by case analysis of these sp ecications and the

identities and of lax morphisms of algebras

Finally for s a D cell R s is a mo dication of left oplax natural transformations



The required identity results from

Addendum Preserving the hypotheses and notations of suppose that

E K Q is a functor which is a right D algebra That is suppose thereisa

natural transformation e ED E such that

e e eD EDD E

and

e Ej E ED E

Then there is a functor



op

ED D Alg Fun Q



specied as fol lows

op

is denedby Given A a a D algebra the simplicial object ED A Q



n

n ED A

Theory and Applications of Categories Vol No

i ni n n

s ED jD ED A ED A

i

n

eD for i

n n

i ni

ED D for in d ED A ED A

i

n

ED a for i n

Given f f A a B b a lax morphism of D algebras the associated left oplax

n n n

natural transformation ED A ED B has structure cel ls ED f ED A ED B

 

and structure cel ls given by the obvious generalization of on replacing al l

n n

D A by ED A etc

Given s f g amodication of lax morphisms of D algebras the associatedmodi

cation of left oplax natural transformations has component cel ls

n n n

ED s ED f ED g

Verication of all these p oints results from calculations which are trivial mo dications

of those required for the verication of

LaxSymMon

Definition The club for lax symmetric monoidal categories is the club over

the category of nite ordinals and p ermutations with the following presentation

T for eachobject T of the The ob jects of are generated under substitution bya

club for symmetric monoidal categories The typ e of the generator T equals the typ e

n

of T No relations are imp osed on the ob jects In particular if denotes the ob jects

n

in and corresp onding to the identity op eration one do es not have

The morphisms of are generated by

i A morphism

u T R

for each morphism u T S in The typ e of u is the p ermutation which is the typ e

of u

ii A morphism

a T S S T S S

n n

T S S  S

n

of typ e the identity p ermutation for each ob ject T of and each ntuple of ob jects

S S of Heren N is the typ e of T

n

iii A morphism

a

of typ e

The relations imp osed on these generators are ae b elow

Theory and Applications of Categories Vol No

a

u v uv

id id

T

T

b

T a a

T T

O

O

O

O

O

O

a

O

T  

O

O

id

O

O

O

O

T

c

a S

S S

D

D

D

D

D

a

S 

D

D

id

D

D

S

d

a

T S R S R T S S R R

n n n nq

T a a a

T S R S R

T S R S R

n nq

n n

a

e

a

T S  S 

n

T S S

T S S

n

n

uv  v

uv  v

n

n



  

 

T S S

S T S

n

n

a

T S   S

n

 

for each ordered set of morphisms in consisting of a u T T and of v S S

i i

i

where is the typ e of u

Comparison with Ke shows that this club is indeed the club

whose do ctrine has as strict algebras the lax algebras over the do ctrine of symmetric

monoidal categories

There is a map of clubs

s

Theory and Applications of Categories Vol No

sending the ob ject T to T the morphism u to u and the morphisms a anda

T S  S

n

to identity morphisms This map of clubs induces a morphism of the corresp onding

do ctrines

By Ke Thm there is also a lax do ctrine map going in the opp osite direction

h h h

As in Ke h is induced by the functor b etween the underlying categories of the

clubs which sends the ob ject T to T and the morphism u T S to u This functor

is not a strict morphism of clubs since it do es not strictly preserve the op erational

substitution which is part of the club structure

The following facts are immediately deduced from the cheap coherence result of

Ke and the description of ab oveinFor each ob ject U of there is an ob ject

V hV Given U any V for which there exists a V in and a morphism inU

V is isomorphic to sU in Two morphisms U V in are equal if morphism U

and only if their images under s are equal that is to say if and only if they havethe

same p ermutation as typ e

Rephrasing this we get that there is a cell of lax maps of do ctrines hs such

that s and h In particular for any category C the functor h C C is

right adjointto sC C

Note that these results of coherence theory together with and the ele

ments of the homotopy theory of categories Qu x imply that one has the following

homotopy equivalences of classifying spaces naturally in any category C

a

n

E B C B C B C

n

n

s

typ e

n

Definition The category LaxSymMonStrict has as ob jects the U small alge

bras for the club ietheU small lax symmetrical monoidal categories The cells

are the strict morphisms of algebras those functors which strictly preserve the op er

ations T and the natural transformations ua anda The cells are the mo dications

of cells

There are variant categories LaxSymMonStrong and LaxSymMon whose cells are

resp ectively the strong and the lax morphisms of algebras and whose cells

are the mo dications of such

To preserve compatibility with the more common terminology of x I will use the

term lax symmetric monoidal functor only for a cell in the category SymMon of

A cell in LaxSymMonStrict will b e a strict functor b etween lax symmetric

monoidal categories There is a forgetful functor

Theory and Applications of Categories Vol No

SymMon LaxSymMon

induced bythemapofclubss

Spt on LaxSymMon

The functor Spt SymMon Sp ectra of extends to a functor on

LaxSymMon Indeed the construction in Th App endix generalizes easily The only

dierences arise b ecause now for A a lax symmetric monoidal category which has a strict

op

unit one step of the construction gives a lax functor or St A Cat

instead of a pseudofunctor To construct this cho ose for each nite ordinal n one

of the isomorphic ob jects T of typ e n in the club Let T denote the strict unit of

n

Q

p

A The lax functor A sends p to AFor each morphism p q in the category of

Q Q

p q

op

nite based sets let A A A b e the functor sending A A to

p

 

 

T A i T A i q

i i

q





Here in each T A i j one orders the A by increasing order of the indexes

i i

j

i The structure natural transformations of the lax functor AA A and

id Aid have comp onents induced by the unique tuple of morphisms in the club

that have the appropriate typ e to universally dene such a natural transformation

The essential ingredients are thea anda of The identities required by the lax

functor axioms hold by the coherence result mentioned in In the case where A

is a strict symmetric monoidal category this lax functor is in fact a pseudofunctor

that asso ciated to A in Th App endix A lax morphism of lax symmetric monoidal

op

categories induces a left lax natural transformation of lax functors Cat by trivial

generalization of the formulae in Th To the lax functor A one now applies Streets

op

second construction of St to convert this into a strict functor from to Cat which

p

sends p to a category related to A by an adjoint pair of functors From this it follows

Q

p

that its classifying space is homotopyequivalentto B AQux Prop Cor

Thus on applying the classifying space functor B Cat Top to this one has a sp ecial

space in the sense of Segal Se to which his innite lo op space machine asso ciates

a sp ectrum By St Thm Streets second construction is in fact a functor



op op

Lax Cat Cat Cat

and thus a left lax natural transformation yields a map of sp ecial spaces

The rest of the argument including generalization to the nonunital case now pro

ceeds exactly as in Th

In the case where A is a strict symmetric monoidal category the construction coincides

op

with that given in Th except that instead of considering a pseudofunctor on as an oplax functor and applying Streets rst construction for oplax functors one considers

Theory and Applications of Categories Vol No

it as a laxfunctor and applies the second construction The pro of given in Th shows

that this variant also yields a functor Spt SymMon Sp ectra satisfying conditions

and of Th and thus Th p is linked byachain of natural stable

homotopy equivalences to the original version Spt

By an argument identical to that in Th one obtains the analogs of the group

completion results Similarly as in these imply that if F ST is

a cell in LaxSymMon such that BF is a homotopy equivalence of spaces then SptF

is a stable homotopy equivalence of sp ectra

Proposition There is a functor

S LaxSymMon SymMon

and a chain of natural stable homotopy equivalences of spectra

op

SptS S hocolim SptR S SptS



It fol lows that S preserves stable homotopy equivalences between lax symmetric monoidal

categories

Denote by I SymMon LaxSymMon the inclusion functor There is a natural

transformation to the identity functor

S I

SymMon

such that Spt is a stable homotopy equivalenceofspectra

If U is a Grothendieck universe insp ection of the construction in shows that the

functor S preserves U smallness

Proof Recall that LaxSymMon is the category of algebras for the do ctrine

The strict map of do ctrines s of induces the structure of a right

algebra on the do ctrine for symmetric monoidal categories The action map is given

by s Thus the addendum to the Go dement resolution



op

yields a functor from LaxSymMon to Fun SymMonStrictsendingS to



n

the simplicial symmetric monoidal category n S

Let S LaxSymMon SymMon b e the comp osite of this functor with the homotopy

colimit functor of



op

op

ho colim Fun SymMon SymMonStrict SymMon

To construct the chain of stable homotopy equivalences of sp ectra let R b e the



n n n n

Then s gives a simplicial Go dement resolution functor n

Theory and Applications of Categories Vol No

 op

map R which is a stable homotopyequivalence in each degree n by

By and this map induces a natural stable homotopy equivalence

 

op op op

ho colim SptR S ho colim Spt S Sptho colim S SptS S



On the other hand since R S is a resolution of S the augmentation map R SS

 

op

induces a stable homotopy equivalence ho colim SptR S SptS Indeed bya



standard reasoning on connective sp ectra using the group completion prop erty

cf Th it suces to show that the map induced by the augmentation from the

n

geometric realization of the simplicial classifying space kn B Sk B S is a weak

homotopy equivalence of spaces But this map is such since the resolution simplicially

deformation retracts to S b ecause of the extra degeneracies s cf eg Ma



This completes the construction of the chain of stable homotopy equivalences

It remains to construct the natural stable homotopyequivalence S I Id But

there is a natural transformation



op

I R SymMon Fun SymMon

 

to the Go dement resolution for algebras The comp onentat S is the natural

op op n n

transformation of functors on whose comp onentatn is s S S

n

S The natural transformation induces a map of homotopy colimits

n n

op op

SI ho colim n S ho colim n S

By the universal mapping prop erty of homotopy colimits the augmentation of

the Go dement resolution for algebras SS yields a natural cell



n

op

ho colim n S S

The natural transformation SI S S is the comp osite of and To

show it is a stable homotopy equivalence it suces to show and are such

From the homotopyequivalences it follows that sC C induces a

homotopy equivalence on classifying spaces of categories and that b oth functors

op

and preserve homotopy equivalences of categories Thus in eachdegree n

the comp onent of is a homotopyequivalence of categories Then it is a stable

homotopy equivalence for each n andsoby the induced map of homotopy

colimits is a stable homotopy equivalence

As for the map to show it is a stable homotopy equivalence it suces by

to show that the augmentation induces a stable homtopy equivalence

op

ho colim SptR S SptS



But as ab ove this holds since R S is a resolution of S 

Theory and Applications of Categories Vol No

From Sp ectra to E spaces

In this section I will recall Mays notion of E op erad and his result on the equivalence

of the homotopy category of connective sp ectra with a lo calization of the category of

E spaces This is an essential link in the chain of equivalences connecting the former

with the lo calization of SymMon I b egin by recalling some denitions from Ma since

one will need to have the details available for x

Definition An operad in the category of compactly generated Hausdor spaces

consists of the following data

i For eachinteger n a space E n and a right action of the symmetric group

on E n

n

ii A p oint E

iii For each n and each sequence j j j of nonnegativeintegers a

n

continuous function

E n E j E j E j E j j j

n n

These are to satisfy the following conditions

a The space E is a single p oint

b The distinguished p oint E is an identity for the comp osition law That

is for any f E nandg E j one has

f f

g g

c The comp osition law is compatible with the action of the symmetric group

That is for any any sequence of for k n and any

n k j

k

f E ng E j one has

k k

 

f g g f g g j j

n n

n

f g g f g g qq

n n n n

that p ermutes the n Here j j denotes the p ermutation in

n j j

n

blo cks of j successiveintegers according to leaving the order within

k n

eachblock xed q q is the p ermutation leaving the n blo cks invariant

n

th

and which restricts to on the k blo ck

k

d The comp osition law is asso ciative That is given f E n g E j for

i i

i n and h E for k j one has

ik k i

f g h f g h

i ik i ik

Theory and Applications of Categories Vol No

Definition An E operad is an op erad such that for each n the space E nis

homotopy equivalenttoapoint and acts freely on E n

n

Definition An E space for E an op erad is a based space X together with con

tinuous functions for each nonnegativeinteger n

n

Y

E n X X

n

such that this action is based unital and resp ects the p ermutations and the comp o

sition law of the op erad More precisely one requires that

a is the basep ointof X

b xx for the distinguished p oint E

c

f x x x f x  x 

n n n

n

for any f E nand

n

d

f g g x x x x

j j n j nj

n n

f g x x g x x

n j j j n n nj

n n

for any f E n and any sequence g E j fori n

i i

A morphism of E spaces is a continuous function X Y such that for all n the

following diagram commutes

Q

n

Q id Q

n n

E n X E n Y

n n

X Y

A morphism of E spaces is a weak homotopy equivalence if it is a weak homotopy

equivalence on the underlying spaces

These denitions are due to May Ma inspired by earlier work of

Adams Beck Boardman MacLane Stashe and Vogt

One xes an E op erad E in a Grothendieck universe U and considers the category

of U small E spaces For anytwochoices of such E there are functors b etween the

two categories such that the comp osites eachwayarelinked to the identitybya chain

of natural homotopy equivalences of E spaces MaT x x cf Ma x up to

Thus these categories are essentially interchangeable I will therefore conform to

the standard abuse and sp eak of any of them as the category of E spaces

Theory and Applications of Categories Vol No

Mays approach to innite lo op space theory pro duces an functor Spt from the

category of E spaces to the category of connective sp ectra Ma In fact he

shows this functor induces an equivalence of a lo calization of the category of E spaces

and the stable homotopy category of connectivespectra The inverse functor is

given by imp osing a natural action of an E op erad on the zeroth space of a sp ectrum

Proposition May The functor Spt E spaces Spectra induces an equiva

lencefrom the localization of E spaces by inverting al l maps that Spt sends to stable

homotopy equivalences to the ful l subcategory of the stable homotopy category consisting

of connective spectra The inverse equivalence is induced by the zeroth space functor

This is just a combination and slight reinterpretation of parts of the statements

Ma Thm Cor Thm to which I refer the reader for the pro of

The morphisms of E spaces X Y that Spt sends to stable homotopy

equivalences are precisely those which induce isomorphisms on homology after lo calizing

byinverting the action of the ab elian monoid of the E spaces

 

X H X Z Y H Y Z

  

This follows from the groupcompletion theorem Ma iv Se x

Let b e a club such that the categories n of havecontractible classifying

spaces Bn Supp ose further that is isomorphic to the category with one mor

phism By coherence theory the club for strict unital symmetric monoidal categories

is such a club Then setting E nBn for n yields an E op erad For the

free action of on n induces a free right action on Bn The distinguished

n n

ob ject induces a distinguished p oint B Since the classifying space

functor B preserves nite pro ducts the comp osition law of the club expressed in form

induces a comp osition law on the Bn for n The conditions ad

for an op erad hold as a consequence of the similar conditions satised by a club

Moreover an action of such a club on a category C induces an action of the op erad

B on BC For the action map of the club

a

n

n n C C C

n

n

induces an action of the op erad on applying B

This pro cedure is natural for strict morphisms of actions In particular taking

to b e the club for unital symmetric monoidal categories this pro cedure gives a functor

B UniSymMonStrict E spaces

The May functor Ma from UniSymMonStrict to Sp ectra is the comp osition of

this functor B and the Maymachine Spt As noted in this functor is linked by

achain of natural stable homotopyequivalences to the restriction from SymMon to

UniSymMonStrict of our functor Spt

Theory and Applications of Categories Vol No

From E spaces to lax symmetric monoidal categories

In this section I construct a functor from E spaces to LaxSymMonStrict that will b e

the essential constituentofaninverse up to natural stable homotopy equivalence to the

functor Spt of x The strategy of the construction is rst to show that the category

of contractible spaces over a space X has a classifying space weak homotopy equivalent

to X and then to use the action of the E op erad on X to pro duce a lax symmetric

monoidal structure on this category

I b egin by recalling some wellknown facts from the theory of simplicial sets Denote

by Top the functor sending p to the standard top ological psimplex Recall

that the singular functor

op

Sing Top Sets

sends a space X to the simplicial set which in degree p is the set ToppX There

is a natural weak homotopy equivalence from the geometric realization of the singular

complex

jSingX j X

Denote by

X

the category whose ob jects are the singular simplices of X cp X and whose



morphisms from c to c q X are morphisms p q insuch that the following

diagram commutes in Top

p

C

C

C

c

C

C

C

C

C

X

c

q

This construction yields a functor from Top to Cat

Proposition Quillen There is a natural weak homotopy equivalencefrom the

classifying space of the category X to X

B X X

Proof Given the weak homotopyequivalence of it suces to nd a natural weak

equivalence of simplicial sets from the nerveofX to SingX For the geometric

Theory and Applications of Categories Vol No

realization of this weak equivalence is then a weak equivalence of spaces and can then

b e comp osed with the map of to yield the equivalence of

By denition a psimplex of the nerve N X is a sequence of p comp osable mor

phisms in X

p

k

k k

p

W

W

R

W

W

R

W

R

W

W

R

W

R

W

W

R

W

R W

W

R

W

c

R W p

W

R

W

R

W

c

W

R 

W

R W

W

R

W

R W

W

R

W

W

X

One sends this to the psimplex of SingX sp ecied as follows The sequence of

morphisms determine a map in f pgf k g in the category of

i p

nite ordinals and monotone maps by i k fori p

p p i i

For i p one considers the comp osition of an empty set of to b e the identityso

pk The desired psimplex of SingX is the comp osite

p

c p k X

p p

One easily checks this denes a map of simplicial sets

N X SingX

It remains to show this map is a weak homotopyequivalence

This could b e deduced from trivial mo dications to the argument given in Il VI I x

for the weaker formulation of the prop osition given there I prefer an alternate pro of

X is isomorphic to the opp osite category of the Grothendieck construction Th

R

op op

SingX on SingX considered as a functor Sets Cat Thus one

may conclude by dualizing the homotopy colimit theorem of Th that there is a

natural weak homotopy equivalence of simplicial sets

Z

op

op

N X I ho colim p Sing X p Sing X

p p

Here I is the version of the nerve functor used in BK XI and Th which has

the opp osite orientation to the nerve functor N of Qu x used in this pap er cf

The two are related by

op

NC IC

By BK XI I there is a natural weak homotopyequivalence from the homotopy

op op

colimit of a F Sets to the diagonal simplicial set of F considered as a

bisimplicial set Applied to the simplicial set SingX considered as a bisimplicial set

constant in one direction this gives a weak equivalence

Theory and Applications of Categories Vol No

op

SingX Diag p q Sing X ho colim SingX

p

Explicit formulas for the equivalences and may b e deduced from Th

and BK XI XI I A routine calculation then yields that these equivalences

t into a commutative triangle with the map which is therefore also an

equivalence as required

op

ho colim N SingX

G

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

w

G

G w

w

N X SingX

In verifying the commutativity it is imp ortant to note that BK XI I is incorrect

op op op

in stating that obviously np p What is correct is that np p

When this correction is fed through BK XI I the result is that the description given

ab ove of is right whereas the erroneous formulae in BK XI I and XI would

lead one to exp ect a description using minima instead of the maxima k

i

Notation For X a top ological space let NullX b e the category whose ob jects

are maps of spaces c C X where C is weak homotopyequivalent to a p oint A

   

morphism C c C c is a map of spaces C C suchthat c c

Lemma The obvious inclusion of categories X Nul lX induces a natural

weak homotopy equivalence of classifying spaces

B X B Nul lX

Proof By Quillens Thm A Qu x it suces to show for eachobject c C X

that the comma category C ciscontractible But this comma category is just

C Thus Prop gives that its classifying space is weakly equivalentto C hence

contractible as required

It follows immediately from and that the functors B Cat Top and Null

Top Cat induce inverse equivalences of lo calized categories on inverting the weak

homotopy equivalences Mo dulo the usual precautions to take the categories of U small ob jects

Theory and Applications of Categories Vol No

Proposition The functor Nul l Top Cat lifts to a functor E spaces

LaxSymMonStrict

Proof This claim is that one can construct a natural lax symmetric monoidal struc

ture on NullX from an action of an E op erad fE ng on X By the presentation of

the club for lax symmetric monoidal categories given in and the description of

a club action in terms of its presentation Ke this amounts to giving the

following data




i For each ob ject T of typ e n in the club for symmetric monoidal categories ie

for each nary op eration built up by iterated substitution of a binary op eration

into itself a functor

n

Y

T NullX NullX

ii For each morphism u T S in the club a natural transformation with

comp onents

 

u T c c S c c

n

n

Here is the p ermutation which is the typ e of u

iii A natural transformation

a

iv For each T in of typ e n and each ntuple S of ob jects in a natural

i

transformation of typ e the identity

T S S T S S a

n n

T S S  S

n

such that the diagrams abcd and e commute

I construct all this as follows Note that the category Top is a symmetric monoidal

category under pro duct of spaces Thus for each T of typ e n in the club andeach

ntuple of spaces Y Y one has the pro duct space T Y Y usually denoted

n n

Y Y with the choice of parentheses censored

n

T will b e the functor sending an ntuple of ob jects c C X in NullX to the

i i

contractible space over X given by the pro duct of the C and E n

i

T c  c

n

n

E n T C C E n T X X X

n

The natural transformations of ii u T S for u T S in will have as comp onents

the maps induced by the symmetric monoidal structure of Top

Theory and Applications of Categories Vol No

 

u E n T C C E n S C C

n

n

This map is compatible with the structure maps to X by naturalityof u and the

compatibility c of the op erad action with the symmetric group action on the

E n Thus it is a map in NullX

The natural transformation of iiia willhave comp onents

C C E C C

induced by the inclusion of the op erads distinguished p oint E

This is a map over X by b

The natural transformation of iva will have as comp onents the maps

T S  S

n

induced by the comp osition law of the op erad

E n T E k S C C Ek S C C

k n n n nk

n

E n E k E k T S S C C

n n nk

n

id

E k k T S S C C

n n nk

n

This is a map over X by d

These data satisfy the conditions imp osed b ecause of the conditions satised by

the structure of an op erad and of and op erad action Thus condition a holds

by naturality conditions b and c by b d by d and e by

naturality with c and c

Finally the construction is functorial that is it takes maps of op erad actions to

strict morphisms b etween lax symmetric monoidal categories

Lemma The functor Nul l E spaces LaxSymMonStrict of preserves

stable homotopy equivalences

Proof By group completion it suces to showthatif f X Y is

a map of E spaces which induces an isomorphism on homology lo calized by the action

of the monoid

 

f X H X Z Y H Y Z

  

then is also an isomorphism the map

 

B Null f B Null X H B NullX Z B NullY H B NullY Z

  

Theory and Applications of Categories Vol No

But by and there is a chain of natural homotopy equivalences b etween the E

space Z and B NullZ inducing an isomorphism on H Z and on the set



Thus it suces to show this isomorphism resp ects the actions of of on the homology

groups and so induces an isomorphism of the lo calizations For z Z considered as

a representative of a class in Z and for anychoice of m in the contractible E

the action on H Z Z is the map on homology induced by the homotopy class of the



endomorphism Z Z given by

Z m z Z E Z Z Z

By naturality of the chain of homotopy equivalences this map on homology agrees with

the endomorphism of H B Null Z Z induced by Z Z This is the map on homology



induced by the endofunctor of NullZ sending C Z to

C m z C E Z Z Z

On the other hand the action of z Z B NullZ for the lax symmetric monoidal

structure is the map on homology induced by the endofunctor sending C Z to

Z T z Z C Z E z C E Z Z

for anychoice of T in the contractible But the two endofunctors are homotopic

since they are linked by the natural transformation with comp onents

C m z C E z C

This shows the actions on homology byanelementof are compatible under the

isomorphism induced bythechain of and Similarly the isomorphism of Z

with B NullZ resp ects the translation action of on itself and so is an isomorphism

of monoids Thus the two actions are isomorphic

The main theorem

Recall that one has xed a Grothendieckuniverse U and by convention considers only

symmetric monoidal categories and sp ectra whichare U small

Theorem Let Spectra be the category of connective spectra and SymMon the



category of symmetric monoidal categories Then the functor

Spt SymMon Spectra



induces an equivalencebetween their homotopy categories formed by inverting the stable

homotopy equivalences

Theory and Applications of Categories Vol No

The inverse equivalence is induced by the functor

Spectra E spaces LaxSymMon SymMon



which is the composite of the zeroth space functor Spectra E spaces of



the functor Nul l E spaces LaxSymMonStrict of and the functor

S LaxSymMon SymMon of

Proof The functors of and all preserve stable homotopyequivalences

Thus the comp osite functor do es induce a functor on the homotopy categories It

remains to show the two functors are inverse on the homotopy categories

One already knows by that Sp ectra and E spaces are linked by functors



and Spt inducing inverse equivalences of homotopy categories Similarlyby

the inclusions b etween all the variants of SymMon listed in induce equivalences

of homotopy categories Moreover one knows that the restriction of Spt to

UniSymMonStrict is linked byachain of natural stable homotopy equivalences to the

comp osite of a lift of the classifying space functor B UniSymMonStrict E spaces

and the May machine functor Spt E spaces Sp ectra



In light of this to prove the two functors of the theorem are inverse to each other on

the homotopy category it suces to show

a The comp osite functor

S Null

B

UniSymMonStrict E spaces SymMon

is linked to the inclusion functor by natural stable homotopyequivalences

b The comp osite functor

Sp ectra SymMon Sp ectra

 

is linked to the identity functor by natural stable homotopy equivalences

These will b e proved in the course of this section The statement a will b e proven

by direct construction of the link and b will result from the uniqueness theorem for

innite lo op space machines of MaT

pro of of a

In order to prove a it suces to construct a chain of functors and natural stable

homotopy equivalences linking the comp osite

Null

B

UniSymMonStrict E spaces LaxSymMon

to the inclusion I of UniSymMonStrict into LaxSymMon For then on applying the

functor S LaxSymMon SymMon of to this link one obtains a link of S NullB

Theory and Applications of Categories Vol No

to SI which in turn is linked to the inclusion of UniSymMonStrict in SymMon bythe

natural stable homotopy equivalence of

I will need to dene a functor

Null UniSymMonStrict LaxSymMon

which will b e the analog for categories of Null for spaces First I will construct

the underlying endofunctor of Cat and show there is a natural homotopy equivalence

A b e the category from the identity to this endofunctor For A a category let Null

whose ob jects are C c where C is a U small category suchthat BC is contractible and

 

c C A is a functor A morphism in NullA from C ctoC c is a functor C

 

A b e the full sub category of those C c such that C has C such that c c Let Term

a terminal ob ject For eachsuch C chose a terminal t out of the isomorphism class of

C



terminal ob jects Then one may construct a functor Term AAsending the ob ject

  

sends the morphism C c C c C c to the image of the terminal ob ject ct

C

 

to c of the unique morphism in C from the image of t to the terminal ob ject t This

C C



functor is not strictly natural in A b ecause of the choice of terminal ob ject Consider

now the full sub category A whose ob jects are those C cforwhich C is the total

order f ng for some n Since here the terminal ob ject n is unique the



A A is strictly natural in A Moreover given restriction of gives a functor

anyobject A A the comma category A has a terminal ob ject fgc c A

and so is contractible By Quillens Theorem A Qu x it follows that is a homotopy

equivalence of categories This statement is the analog of Prop As in the pro of

of Lemma the comma categories C c of the inclusion ANullA are

isomorphic to C and hence are homotopyequivalent to the contractible C Then by

Quillens Theorem A the inclusion is a homotopyequivalence Similarly the inclusion

A into TermA is a homotopyequivalence Thus NullA is linked byachain of of

homotopy equivalences of categories to A In fact the following natural functor

ANull A

is a homotopy equivalence This sends an ob ject A to the canonical forgetful functor



from the comma category C AA A sends a morphism A A to the canonical



map of comma categories AA AA As each AA has a terminal ob ject is the



A and the homotopyequivalence given bythe comp osite of a functor into Term

  

inclusion of the latter in NullA But and since is a homotopyequivalence



so is and This completes the pro of of the chain of homotopyequivalences of

categories for a general category A

Now return to the case where A runs over UniSymMonStrict Let b e the club for

strict unital symmetric monoidal categories Then as in the natural action of on

the unital symmetric monoidal A is expressed by action maps The action of

the E op erad E nBnonB A is given by applying the classifying space functor

B to the categorical action Then replacing everywhere in the construction

Theory and Applications of Categories Vol No

of the lax symmetric monoidal structure on NullX the op erad E nBnbythe

categories n of and Top by Cat one nds a construction of a lax symmetric

monoidal category structure on Null A strictly natural for A in UniSymMonStrict

There is a natural strict morphism of lax symmetric monoidal categories

B NullA NullB A

This B sends the ob ject C c C AtoBC Bc BC B A Moreover this B

induces a homotopy equivalence on the underlying categories For since B f

AB A and one has a commutative ladder ngn B restricts to a functor

of classifying spaces whose sides are given by the homotopyequivalences of and

and the maps induced by their ab ove categorical analogs

o

B Null A B A

B A

BB

o

B NullB A B B A

B A

This diagram shows that up to homotopy equivalence the B of is indentied

to the identity map of A A fortiori is a fortiori a natural stable homotopy equivalence

A of For A in UniSymMonStrict the natural homotopyequivalence A Null

is a lax morphism of lax symmetric monoidal categories To give such a structure it

suces considering the presentation of the club for lax symmetric monoidal categories

and the description of lax morphisms in terms of presentations Ke Ke

to give for each ob ject T of typ e n in the club for symmetric monoidal

categories a natural transformation

T A A T A A

T n n

in Null A satisfying the following compatibilities with the natural transformations which

are part of the lax symmetric monoidal structure

a For each morphism u T R in of typ e the following diagram

n

commutes

T

T A A

T A A

n

n

uA uA

 

RA A

 

RA A

n

n

R

Theory and Applications of Categories Vol No

b For each natural transformationa as in iv the following dia

T R R

n

gram commutes

T A

R A

R ij

T i ij

i

T R A

T R A T R A

i ij

i j i ij

a A

j

T R 

i

T R A

T R A

i ij

i j

A

j

T R 

i

c For the natural transformationa of iii the following diagram

commutes

A A

a

A

A A

But for T of typ e n T A A is the contractible category over A

n

n AA AA n A A A

n

Since T is a terminal ob ject in n and A is terminal in AA the ab ove functor

i i

factors canonically through the comma category AT A A T A A

n n

Then letting b e this factorization yields the required natural transformation It is

T

routine to check the prop erties hold

Thus the of is a natural stable homotopy equivalence b etween functors

SymMonStrict LaxSymMon Comp osing this with the natural stable homotopy

equivalence of yields a natural stable homotopy equivalence from the inclusion

functor I to NullB as required to complete the pro of of a

pro of of b

It remains to link the functor Sp ectra SymMon Sp ectra to the identity

 

functor bya chain of natural stable homotopyequivalences

After comp osing with the zeroth space functor Sp ectra Top one has the

chain of homotopy equivalences of spaces all naturally in X Sp ectra

SptNullX B NullX B X X

The second and third maps of are the natural homotopyequivalences of and

The rst map is the canonical group completion map which is a homotopy

Theory and Applications of Categories Vol No

equivalence in this case For as in the pro of of B NullX is isomorphic as a

monoid to X and so is already a group

It would b e quite dicult to sp ecify by hand a chain of E homotopyequivalences

between the two ends of Instead I will circumvent the need to do this bythe

following Lemma Its pro of will complete the pro of of b and hence of the Theorem

It is essentially an avatar of the MayThomason uniqueness theorem for innite

lo op space machines MaT

Lemma Let F Spectra Spectra be an endofunctor of the category of





connective spectra Suppose there exists a chain of functors G Spectra Top and

i



of natural homotopy equivalences of spaces

F G G G G

n n

Then there is a chain of endofunctors of Spectra and natural stable homotopy



equivalences linking F to Id

Proof Recall that a map of connective sp ectra is a stable homotopy equiva

lence if and only if it induces a homotopy equivalence on the zeroth space From this and

the chain of homotopy equivalences of zeroth spaces G one sees that F preserves

i

stable homotopy equivalences Also all the G send pro ducts of sp ectra to pro ducts of

i

spaces at least up to homotopy

Consider the category of sp ecial spaces of Segal Se the category of functors

op

from the category of nite based sets to Top that takewedges to pro ducts up to

homotopy Segals innite lo op space machine Sg is a functor from the category of

sp ecial spaces to Sp ectra By Se x there is a functor is the opp osite direction



with the two comp osites linked to the identityby stable homotopy equivalences From

this it follows that it suces to link FSg to Sg byachain of natural stable homotopy

equivalences of functors from sp ecial spaces to Sp ectra



But this can b e shown to hold by a slight mo dication of the pro of of MaT In

more detail one has a functor given by smash pro duct of nite based sets

op op op

op op op

If A Top is a sp ecial space then the induced A Top is such that

for each p q Apq is a sp ecial space Indeed as p varies it is a sp ecial sp ecial

op

space so Sp ectra sending p to Sgq Apq is homotopyequivalenttothe



pfold pro duct of the sp ectrum Sgq Aq Applying the chain of pro duct

preserving natural homotopy equivalences of spaces to the sp ectra Sgq Apq yields

achain of homotopy equivalences of sp ecial spaces linking p FSgq Apq to

p Sgq Apq The Segal machine gives a group completion map natural in the

space Ap Ap Sgq Apq Thus it induces a homotopy equivalence of

asso ciated sp ectra Sgp Ap Sgp Sgq Apq Applying Sgp

Theory and Applications of Categories Vol No

to the chain of homotopy equivalences of sp ecial spaces gives a chain of natural stable

homotopy equivalences from the latter sp ectrum to Sgp FSgq Apq By

the over and across theorem MaT for bisp ectra asso ciated to sp ecial sp ectra

there is a natural chain of stable homotopyequivalences linking Sgp FSgq

Apq toFSgq Aq FSgA Combining the ab ovechains give the required

chain of stable homotopy equivalences linking Sg to FSg

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