Theory and Applications of Categories, Vol. 1, No. 5, 1995, pp. 78 118
SYMMETRIC MONOIDAL CATEGORIES
MODEL ALL CONNECTIVE SPECTRA
R. W. THOMASON
Transmitted by Myles Tierney
Abstract. The classical infinite loopspace machines in fact induce an e*
*quivalence of
categories between a localization of the category of symmetric monoidal ca*
*tegories and
the stable homotopy category of 1connective spectra.
Introduction
Since the early seventies it has been known that the classifying spaces of smal*
*l symmetric
monoidal categories are infinite loop spaces, the zeroth space in a spectrum, a*
* sequence
of spaces Xi; i 0 with given homotopy equivalences to the loops on the succeed*
*ing
space Xi ~! Xi+1. Indeed, many of the classical examples of infinite loop spa*
*ces
were found as such classifying spaces ( e.g. [Ma2], [Se]). These infinite loop *
*spaces and
spectra are of great interest to topologists. The homotopy category formed by i*
*nverting
the weak equivalences of spectra is the stable homotopy category, much better b*
*ehaved
than but still closely related to the usual homotopy category of spaces (e.g., *
*[Ad] III ).
One has in fact classically a functor Spt from the category of small symmetr*
*ic mo
noidal categories to the category of 1connective spectra, those spectra Xi fo*
*r which
sskXi = 0 when k < i ([Ma2], [Se], [Th2]). Moreover, any two such functors sati*
*sfying
the condition that the zeroth space of Spt(S) is the "group completion" of the *
*classifying
space BS are naturally homotopy equivalent ([Ma4], [Th2]).
The aim of this article is to prove the new result (Thm. 5.1) that in fact S*
*pt induces
an equivalence of categories between the stable homotopy category of 1connect*
*ive
spectra and the localization of the category of small symmetric monoidal catego*
*ries by
inverting those morphisms that Sptsends to weak homotopy equivalences. In parti*
*cular,
each 1connective spectrum is weak equivalent to Spt(S) for some symmetric mon*
*oidal
category S.
_____________
Received by the Editors 9 April 1995
Published on 7 July 1995.
1991 Mathematics Subject Classification: Primary: 55P42 Secondary: 18C15, 18*
*D05, 18D10, 19D23
55P47.
Key words and phrases: club, connective spectrum, E1 space, operad, lax al*
*gebra, spectrum,
stable homotopy, symmetric monoidal.
cOR. W. THOMASON 1995. Permission to copy for private use granted.
78
Theory and Applications of Categories, Vol. 1, No. 5 *
* 79
Thus the category of symmetric monoidal categories provides an alternate mod*
*el
for the 1connective stable homotopy category, one which looks rather differen*
*t from
the classical model category of spectra. It is really "coordinatefree" in that*
* there are
no suspension coordinates at all, in contrast to May's coordinatefree spectra *
*([Ma3]
II) which use all finite subspaces of an infinite vector space as coordinates, *
*and thus
are just free of choices of coordinates. The category of E1 spaces, spaces wi*
*th an
action of an E1 operad ([Ma2]), is similarly a model for  1connective spect*
*ra which
is really coordinate free. But a symmetric monoidal category structure is much*
* more
rigid than a general E1 space structure, and as a consequence can be specified*
* and
manipulated much more readily. As convincing evidence for this claim, I refer *
*to my
talk at the Colloque en l'honneur de Michel Zisman at l'Universite Paris VII in*
* June
1993. There I used this alternate model of stable homotopy to give the first k*
*nown
construction of a smash product which is associative and commutative up to cohe*
*rent
natural isomorphism in the model category. This will be the subject of an arti*
*cle to
appear.
The proof that the functor Spt induces an equivalence between a localization*
* of
the category of symmetric monoidal categories and the 1connective stable homo*
*topy
category begins by constructing an inverse functor. The construction is made in*
* several
steps. First, there are known equivalences of homotopy categories induced by fu*
*nctors
between the category of 1connective spectra and the category of E1 spaces. *
*Thus
one reduces to finding an appropriate functor to the category of symmetric mono*
*idal
categories from that of E1 spaces. Any space X is weak homotopy equivalent to *
*the
classifying space of the category Null=X of weakly contractible spaces over X. *
*When
X is an E1 space the operad action on X induces operations on this category. *
*For
example, for each integer n 0 there is an nary functorQsending the objects C1*
* !
X; C2 ! X; : :;:CnQ! X to E(n)xC1xC2x: :x:Cn ! E(n)x n X ! X, where the
last arrow E(n) x nX ! X is given by the operad action. The internal composi*
*tion
functions of the operad induce certain natural transformations between composit*
*es of
these operations on Null=X. Using Kelly's theory of clubs ([Ke3], [Ke4]) one f*
*inds
that Null=X has been given the structure of a lax algebra for the club oe of sy*
*mmetric
monoidal categories. The next step is to replace this lax symmetric monoidal ca*
*tegory
by a symmetric monoidal one. There is a Godement resolution of a lax algebra by*
* free lax
algebras, yielding a simplicial lax algebra. By coherence theory, the latter is*
* degreewise
stably homotopy equivalent to a simplicial free symmetric monoidal category. Ta*
*king a
sort of homotopy colimit of the last simplicial object ([Th2]) yields the desir*
*ed symmetric
monoidal category.
The layout of the article is as follows: The first section is a review, star*
*ting with the
definitions of nonunital symmetric monoidal categories, and of lax, strong and*
* strict
symmetric monoidal functors. I recall in 1.6 the basic properties of the classi*
*cal functor
from symmetric monoidal categories to spectra. Next in 1.7 comes a review of t*
*he
definitions of oplax functors into a 2category and left oplax natural transf*
*ormations
between such. The homotopy colimit, or oplax colimit, of a diagram of symmetr*
*ic
Theory and Applications of Categories, Vol. 1, No. 5 *
* 80
monoidal functors is recalled in 1.8, and its good homotopy theoretic propertie*
*s stated.
The first section closes with a proof in 1.9 that all the variant categories of*
* symmetric
monoidal categories considered have equivalent homotopy categories. Section 2 b*
*egins
with a review of lax algebras over a doctrine. A Godement type simplicial reso*
*lution
of lax algebras by free lax algebras in given in 2.2. This resolution is shown *
*to be left
oplax natural with respect to lax morphisms of lax algebras. In 2.3 I conside*
*r the
special case of lax symmetric monoidal categories. The functor to Spectra is ex*
*tended
to these in 2.4. In 2.5 I use the Godement resolution and the homotopy colimit *
*to show
how to replace a lax symmetric monoidal category by a symmetric monoidal catego*
*ry.
Section 3 reviews the notion of an E1 space and the equivalence of homotopy ca*
*tegories
between E1 spaces and Spectra. Section 4 contains the construction of a lax sy*
*mmetric
monoidal category associated to an E1 space. Finally Section 5 finishes the pr*
*oof by
showing various roundtrip functors made by joining the above pieces are linked*
* to the
identity by a chain of stable homotopy equivalences.
1 Symmetric monoidal categories and homotopy colimits
I will need to use the "homotopy colimit" of diagrams of symmetric monoidal cat*
*egories.
The based version of the homotopy colimit does not have good homotopy behavior
except under stringent "nondegenerate basepoint 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 be an isomorphism. Such a symmetric mon*
*oidal
category is equivalent to the disjoint union of 0 with a possibly nonunital sy*
*mmetric
monoidal category, and finally all homotopy colimit results are easier to state*
* if one
works with variant categories of nonunital symmetric monoidal categories from *
*the
beginning. Thus:
1.1 Definition. By unital symmetric monoidal category, I mean a symmetric monoi*
*dal
category in the classic sense, a category S provided with an object 0 and a fun*
*ctor
: S x S ! S, together with natural isomorphisms of associativity, commutativity*
*, and
unitaricity for which certain simple diagrams are to commute. For the details *
*see for
example [McL] VII x1, x7.
1.2 Definition. A symmetric monoidal category is a category S together with a f*
*unctor
: S x S ! S and natural isomorphisms:
ff: A (B C) ~! (A B) C
(1.2.1) ~
fl: A B ! B A
which are such that the following two pentagon and hexagon diagrams commute:
Theory and Applications of Categories, Vol. 1, No. 5 *
* 81
(1.2.2)
A (B (C D))) __ff_//_(A B) (C D) __ff_//_((A B)OOC) D
1ff ff1
fflffl 
A ((B C) D) ______________ff_____________//_(A (B C)) D
(1.2.3)
A (B C) __ff_//_(A B) C __fl_//_C (A B)
1fl ff
fflffl fflffl
A (C B) __ff_//_(A C) B _fl1_//_(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 sta*
*tements
and the demonstrations become easier without the additional structure of the un*
*itaricity
natural isomorphisms ([Ep], [Ke1] 1.2).
To compare with the existing literature it will be useful to recall ([Ma2] 4*
*.1 and
[McLa] VII x1) that a permutative category is a unital symmetric monoidal categ*
*ory
where the natural associativity isomorphism ff is the identity. It follows from*
* coherence
theory that every unital symmetric monoidal category is equivalent to a permuta*
*tive
category. Indeed the equivalence is realized by strong unital symmetric monoida*
*l func
tors and unital symmetric monoidal natural isomorphisms. See [Ma2] 4.2 or [Ke3]*
* 1.2,
4.10, 4.8.
As to morphisms between symmetric monoidal categories, one needs to consider
three kinds: the lax, the strong, and the strict symmetric monoidal functors. T*
*he usual
definitions adapt to the the nonunital case easily.
1.3 Definition. For S and T symmetric monoidal categories, a lax symmetric mo
noidal functor from S to T consists of a functor F : S ! T together with a na*
*tural
transformation of functors from S x S:
f: F A F B ! F (A B)
such that the following two diagrams commute:
(1.3.1)
F A (F B F C) _1f__//_F A F (B C)___f__//F (A (B C))
ff Fff
fflffl fflffl
(F A F B) F C _f1__//_F (A B) F C___f__//F ((A B) C)
Theory and Applications of Categories, Vol. 1, No. 5 *
* 82
(1.3.2)
f
F A F B _____//_F (A B)
fl Ffl
fflffl fflffl
F B F A __f__//_F (B A)
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 s*
*uch
that the natural transformation f is the identity. Thus F strictly preserves th*
*e operation
and the natural isomorphisms ff and fl.
The usual definition of a lax, strong, or strict unital symmetric monoidal f*
*unctor
between unital symmetric monoidal categories imposes the additional structure o*
*f a
morphism 0 ! F 0 (respectively, an isomorphism, an identity) subject to a commu*
*tative
diagram involving the unitaricity isomorphisms. See e.g. [Th2] (1.2).
1.4 Definition. A symmetric monoidal natural transformation j: F ! G between two
lax symmetric monoidal functors (F; f); (G; g): S ! T is a natural transformat*
*ion j
such that the following diagram commutes:
(1.4.2)
f
F A F B _____//_F (A B)
jj  j
fflffl fflffl
GA GB __g__//_G(A B)
A symmetric monoidal natural transformation between two strict or two strong*
* sym
metric monoidal functors is a symmetric monoidal natural transformation between*
* the
underlying lax symmetric monoidal functors.
Note that such an j is automatically compatible with the associativity and c*
*ommu
tativity isomorphisms ff and fl by naturality of j, ff, and fl. On the other ha*
*nd, the defi
nition of a unital symmetric monoidal natural transformation between unital sym*
*metric
monoidal functors imposes a new compatibility with the unitaricity isomorphisms*
*, as
in e.g. [Th2] (1.7). (Thus a symmetric monoidal natural transformation between*
* two
unital symmetrical monoidal functors need not be a unital symmetric monoidal na*
*tural
transformation!)
1.5: Catalog of variant 2categories of symmetric monoidal categories.
In order to model spectra, 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 t*
*o be able
Theory and Applications of Categories, Vol. 1, No. 5 *
* 83
to localize categories of such, I suppose Grothendieck's axiom of universes ([S*
*GA4] I
Appendice). This axiom is that each set is contained in a set U the elements of*
* which
form a model of settheory such that the internal power sets in the model U are*
* the true
power sets. For each such universe U, one has several categories of Usmall sym*
*metric
monoidal categories. A category is Usmall if the class of all its morphisms is*
* an element
of U. The class of objects is then also an element of U. Any of the variant cat*
*egories
of Usmall categories below will then itself be V small for any universe conta*
*ining U as
an element. The localization of any of the variants by inverting any set of mor*
*phisms
is then W small for any universe W containing V .
The variants, differing in unitaricity and in degree of laxity of morphisms,*
* are:
SymMon :
Objects: Usmall symmetric monoidal categories
Morphisms: lax symmetric monoidal functors
2cells: symmetric monoidal natural transformations
SymMonStrong :
Objects: Usmall symmetric monoidal categories
Morphisms: strong symmetric monoidal functors
2cells: symmetric monoidal natural transformations
SymMonStrict :
Objects: Usmall symmetric monoidal categories
Morphisms: strict symmetric monoidal functors
2cells: symmetric monoidal natural transformations
UniSymMon :
Objects: Usmall unital symmetric monoidal categories
Morphisms: lax unital symmetric monoidal functors
2cells: unital symmetric monoidal natural transformations
UniSymMonStrong :
Objects: Usmall unital symmetric monoidal categories
Morphisms: strong unital symmetric monoidal functors
2cells: unital symmetric monoidal natural transformations
UniSymMonStrict :
Objects: Usmall unital symmetric monoidal categories
Morphisms: strict unital symmetric monoidal functors
2cells: 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. 1, No. 5 *
* 84
characterizing homotopy colimits, one needs instead to consider both SymMon a*
*nd
SymMonStrict . Adding the nonunital and unital analogs of all these produces *
*the list
of six 2categories above.
One has the obvious diagram of forgetful 2functors:
(1.5.1)
UniSymMonStrict _____//_UniSymMonStrong _____//_UniSymMon
  
  
fflffl fflffl fflffl
SymMonStrict ________//_SymMonStrong ________//_SymMon
There is also a 2functor from the bottom to the top of each column in (1.5.*
*1).
On objects it sends`the nonunital symmetric monoidal category S to the coprodu*
*ct
of categories S 0_, where 0_is the category with one object 0 and only the id*
*entity
morphism. The symmetric monoidal structure on the coproduct is determined by sa*
*ying
that the inclusion of S is a strict symmetric monoidal functor and that there a*
*re natural
identities of functors on the coproduct
Id 0 = Id= 0 Id
which we take as the unitaricity isomorphisms. Thus 0 becomes a strict unit. *
*This
construction on objects extends to a 2functor in the obvious way.
1.6: The functor Spt: SymMon ! Spectra.
A slight elaboration ([Th2] Appendix) of either May's or Segal's infinite lo*
*op space
machines gives a functor into the category of spectra:
Spt: SymMon ! Spectra
Moreover, symmetric monoidal natural transformations canonically induce homotop*
*ies
of maps of spectra ([Th2] 2.9).
Let B: Cat ! Top denote the classifying space functor. By [Th2] 2.2 there *
*is a
natural transformation of functors from SymMon to Top
(1.6.1) : BS ! Spt(S)0
where Spt(S) 0is the underlying zeroth space of the spectrum Spt(S). When S adm*
*its
the structure of a unital symmetric monoidal category, this is a groupcomplet*
*ion.
Equivalently, it induces an isomorphism on homology groups after inverting the *
*action
of the monoid ss0BS:
(1.6.2) (ss0)1 H*(BS; Z) ~! H*(Spt(S)0; Z)
Theory and Applications of Categories, Vol. 1, No. 5 *
* 85
`
When S is not unital, Spt(S)0 is a groupcompletion of the unital S 0_sinc*
*e by
[Th2] 2.1 the inclusion of S into the this symmetric monoidal category induces *
*a weak
homotopy equivalence of spectra:
a
(1.6.3) Spt(S) ~! Spt(S 0_)
(This last assertion ultimately reduces to the observation`that any monoid M*
* has
groupcompletion isomorphic to that of the monoid M 0 formed by forgetting th*
*ere
was already a unit and freely adding a new one 0. For the group completion pro
cess forces the identification of the new 0 with all other idempotent elements,*
* and in
particular with the old forgotten unit. For details of the reduction see [Th2] *
*A.2.)
The proof of May's uniqueness theorem ([Ma4] Thm. 3) for functors defined fr*
*om
the category of permutative categories easily generalizes to show any two funct*
*ors from
SymMon to Spectrawhich satisfy the abovecited groupcompletion conditions of*
* [Th2]
2.1 and 2.2 are connected by a chain of natural weak homotopy equivalences. (S*
*ee
[Th2] pp. 1603, 1646.) In particular, the two functors become naturally isomo*
*rphic
after composition with the functor from Spectra into the stable homotopy catego*
*ry.
The same statements hold for any two functors with the groupcompletion propert*
*ies
defined on any of the six variant categories of symmetric monoidal categories c*
*onsidered
in 1.5.
1.6.4 Definition. A lax symmetric monoidal functor F : S ! T is said to be a s*
*table
homotopy equivalence if Spt(F ) is a weak homotopy equivalence of spectra. A mo*
*rphism
in any of the variant categories of symmetric monoidal categories listed in 1.5*
* is said to
be a stable homotopy equivalence if the underlying lax symmetric monoidal funct*
*or is
such.
1.6.5. If the lax symmetric monoidal functor F induces a homotopy equivalence *
*of
classifying spaces BS ~! BT , then F is a stable homotopy equivalence. This fo*
*llows
from the group completion property of 1.6.1. For a map of 1connective spectr*
*a 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 spectrum *
*are
0 in negative degrees by 1connectivity, and in nonnegative degrees are isomo*
*rphic
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 t*
*o the map
of zeroth spaces inducing an isomorphism on homology with Z coefficients.
More generally, this shows Spt(F ) is a stable homotopy equivalence if and o*
*nly if BF
induces an isomorphism on the localizations of the homology groups by inverting*
* the
action of the monoid ss0:
ss10H*(BF ): (ss0BS)1 H*(BS; Z) ~! (ss0BT )1 H*(BT ; Z)
Theory and Applications of Categories, Vol. 1, No. 5 *
* 86
1.7: Left oplax natural transformations.
I will need the notions of oplax functors and left oplax natural transform*
*ations
between them. Theses concepts are ultimately derived from Benabou's work on bic*
*ate
gories and Grothendieck's theory of pseudofunctors. I follow the terminology of*
* Street
[St] suitably dualized from lax to oplax and generalized from Cat to an arbitr*
*ary 2
category.
Let L_be a category and K__a 2category.
1.7.1 Definition. An oplax functor : L_! K__consists of functions assigning to*
* each
object L of L_ an object L of K__; to each morphism `: L ! L0 of L_ a morphism
0
`: L ! L0 of K__; to each composable pair of morphisms L  `! L0 `! L00in
L_ a 2cell in K__'`0;`: (`0`) ) (`0)(`); and to each object L of L_ a 2cell i*
*n K__
'L : (1L ) ) 1L . These are to satisfy the following three identities of 2cel*
*ls:
For each L0 `1!L1 `2!L2 `3!L3 in L_
(1.7.1.1)
(`3`2`1) (`3`2`1)
L0 ___________//_L3==OO L0 C___________//_L3OO
 '    CCC ' 
 (` ` )ff   CCC(`C`f)f 
`1  3 2 `3 = `1  CC2C1 `3
    CCC 
  ' ff   'ffCC 
fflffl  fflffl C!!
L1 ____`2_____//_L2 L1 ____`2_____//_L2
For each morphism `: L ! L0 in L_
(1.7.1.2)
L_2_______`_______//_L0EE L 2 ______`_______//_L0EEPP
__2_____________________________ff 2 ff_____*
*_____________________________
__22___________________ffff 22 ffff____*
*_________________
___22______________________ffffff 222 ffffff____*
*_____________________
___22122_________________________________=ffff 22= 1 ffff_______*
*__________________KK'
______________rrrru"'`ff id` 22'  ffKK_______*
*__________________!)
1 ____22__________________________'ffffff 2 ffffff_______*
*_______________________1
____22_______________________________ffff 222 ffff_________*
*__________________________
___2__________________________________''ffff ffff__________*
*______________________________
L L
A functor may be considered as an oplax functor with identity structure 2c*
*ells '. A
pseudofunctor is an oplax functor where the structure 2cells are isomorphism*
*s. The
definition of a lax functor is obtained by reversing the direction of the struc*
*ture 2cells
in 1.7.1.
1.7.2 Definition. A left oplax natural transformation between two oplax funct*
*ors
; : L_! K__is a function assigning to each object L of L_a morphism in K__, jL *
*: L !
Theory and Applications of Categories, Vol. 1, No. 5 *
* 87
L; and to each morphism `: L ! L0 a 2cell in K__, j`: ` O jL ) jL0 O ` These a*
*re to
satisfy the following two identities of 2cells:
For each L0 `1!L1 `2!L2 in L_:
(1.7.2.1)
jL0 jL0
L0 ____________//L0 L0 ________________//L0
`1 xxx   `1 xxx `1 xxx 
xx   xxx j`1ff xxx 
__xxx   = __xx jL1 __xx 
L1 F ks'____`2`1 j`2`1 `2`1 L1 F_______________//_L1Fks_____(`2`*
*1)
FFFF  ff  FFFF j FFFF 
` FF   ` FF f`2f ` FF 
2 F""fflffl fflffl 2 F"" 2 F##fflff*
*l
L2 _____jL2____//L2 L2 ______jL2_______//L2
And for each object L of L_:
(1.7.2.2)
jL jL
L______________//____________L L ____________//L__
_____________________________________________________________*
*_______________________________________
_____________________________________________________________*
*_______________________________________
______________________________________________________________*
*______________________________________
______________________________________________________________*
*__________________________________________
1_____________________________1_______________________ksj111___*
*______________________1_________________________1__ks=
__________________________________________________'ff_______*
*___________________________________________
_____________________________________________________________*
*_______________________________________
______________________________________________________fflfflf*
*flffl__________________________________________________________________
L _____jL_____//L L _____jL_____//L
Any natural transformation between two functors may be considered a left op*
*lax
natural transformation with identity structure 2cells j`. There is a notion of*
* right op
lax natural transformation obtained by reversing the direction of the structure*
* 2cells.
For a left lax natural transformation between two lax functors, the 2cells j` *
*go in the
same sense as for a left oplax natural transformation, but of course the struc*
*ture two
cells of and go in the opposite sense. The conditions to impose on the 2cell*
*s of a
left lax natural transformation are analogous to 2.7.2.1 and 2.7.2.2. (See [St]*
* x1.)
1.7.3 Definition. A modification s: j ) between two left oplax natural transf*
*or
mations is a function assigning to each object L of L_a 2cell sL : jL ) L of K*
*__. These
are to satisfy the 2cell identity that for each morphism `: L ! L0 in L_:
Theory and Applications of Categories, Vol. 1, No. 5 *
* 88
(1.7.3.1) jL
_____________________________________________________________*
*_________________________________________________________________________@
L _____________________________________________________________*
*________________44ffsLLL//_L
 __________________________________________________________*
*_____________________________________________________
    *
* L
   
`  ` ` = `  j` `
 ff   ff 
   jL0 
fflffl fflffl fflffl__________fflffl_____*
*_________________________________________________________________________@
L0 ____L0_____//_L0 L0 __________________________*
*_________________________________________________________________________@
L0
1.7.4 Notation. Let L_be a small category and K__a 2category. Denote by
Cat(L_; K__)
the 2category of functors from L_to K__. Its 1cells are natural transformatio*
*ns and its
2cells are modifications. Denote by
opLax(L_; K__)
the 2category of oplax functors from L_ to K__. Its objects are oplax funct*
*ors, the
morphisms are left oplax natural transformations, and the 2cells are modifica*
*tions.
Denote by
Fun(L_; K__)
the sub 2category whose objects are functors, whose morphisms are left oplax *
*natural
transformations, and whose 2cells are modifications.
1.8: Homotopy colimits.
I recall from [Th2] the homotopy colimit of a diagram of symmetric monoidal *
*cat
egories. This is a sort of oplax colimit which turns out to have good properti*
*es with
respect to stable homotopy theory. More precisely:
The homotopy colimit is a 2functor
(1.8.1) hocolim L: Fun(L_; SymMon ) ! SymMonStrict
which is left 2adjoint to the composite of the forgetful functor SymMonStrict *
* !
SymMon and the 2functor SymMon ! Fun (L_; SymMon ) sending a symmetric mo
noidal category S to the constant functor from L_sending each L to S. Thus ther*
*e is
a natural adjunction isomorphism of categories for each functor : L_! SymMon *
*and
each symmetric monoidal category S in SymMonStrict :
(1.8.2) (Fun (L_; SymMon ))(; S) ~=SymMonStrict (hocolimL ; S)
Theory and Applications of Categories, Vol. 1, No. 5 *
* 89
This adjunction isomorphism is equivalent to the universal mapping property *
*stated
in [Th2] Prop. 3.21, as follows by a straightforward calculation on expanding o*
*ut the
various definitions and using ([Th2] (2.9)).
I call this oplax colimit a homotopy colimit because of its relation to the*
* homotopy
colimit of diagrams of spectra. Recall ([BK] XII, [Th2] x3) the latter homotopy*
* colimit
is a functor:
hocolimL : Cat(L_; Spectra) ! Spectra
which sends natural stable homotopy equivalences to stable homotopy equivalences
equivalences. It induces a total derived functor of colimL on the homotopy cate*
*gories.
Among its other good properties is a natural spectral sequence for the stable h*
*omotopy
groups, whose E2 term is expressed in terms of homology of the category L_:
(1.8.3) E2p;q= Hp(L_; ssq) ) ssp+qhocolim L
But by [Th2] Thm. 4.1, there is a natural stable homotopy equivalence between
functors from Cat(L_; SymMon ) to the category of Spectra,
(1.8.4) hocolimL Spt() ~! Spt(hocolimL )
In particular, there is a spectral sequence natural in 2 Fun(L_; SymMon )
(1.8.5) E2p;q= Hp(L_; ssqSpt()) ) ssp+qSpt (hocolimL )
Since ssqSpt ~= 0 for q 1, the spectral sequence 1.8.5 lives in the first *
*quadrant
and converges strongly (e.g. [Th2] 3.14). Since a map of spectra is a stable ho*
*motopy
equivalence if and only if it induces an isomorphism on stable homotopy groups *
*ss*,
this spectral sequence directly gives good homotopytheoretic control of the sy*
*mmetric
monoidal category hocolimL. (The extended naturality of the spectral sequence (*
*1.8.5)
for left oplax natural transformations is proved using rectification of oplax*
* functors as
in the last paragraph of [Th2] x4, cf. [Th1] 3.3)
1.9: Comparison of variant homotopy categories.
As a first application of the homotopy colimit, I will now proceed to show a*
*ll the vari
ant categories of 1.5 all have equivalent localizations on inverting the stable*
* homotopy
equivalences.
As explained after the diagram 1.5.1, for each of the vertical forgetful fun*
*ctors in
this diagram there is a functor going in the opposite direction which freely ad*
*ds a unit
0. There are natural transformations between the identity functors and the comp*
*osites
of these vertical forgetful and freeunit functors. The components of these na*
*tural
Theory and Applications of Categories, Vol. 1, No. 5 *
* 90
`
transformations are`the inclusion : S ! S 0_for S nonunital and the map send*
*ing 0 to
the old unit ae: S 0_! S for S unital. By 1.6.3 Spt() is a stable homotopy eq*
*uivalence.
Since Spt(ae) is a right inverse to an instance of Spt(), it is also a stable h*
*omotopy
equivalence. Thus these natural transformations become natural isomorphisms in*
* the
localizations, and the localizations of the unital and nonunital variants in e*
*ach column
of 1.5.1 are equivalent.
It remains only to see that the forgetful functors between the nonunital va*
*riants
in the bottom row of 1.5.1 induce equivalences of localizations. But one has a*
*nother
functor in the opposite direction given by:
(1.9.1) SymMon ~=Cat (0_; SymMon ) ! Fun(0_; SymMon ) ! SymMonStrict
where the last functor is the hocolim0 of 1.8. Let this composite functor be d*
*enoted
by d( ). Using the universal mapping property (1.8.2) of hocolim0, one gets a *
*natural
transformation of functors on SymMonStrict with components Sb= hocolim0S ! S.
By 1.8.8 these components are stable homotopy equivalences. Moreover, this is *
*still
a natural stable homotopy equivalence after pre or postcomposing d( )with an*
*y of
the forgetful functors in the bottom row of 1.5.1. Thus these composites with *
*(d )
induce inverses to the forgetful functors after localization. This has proved t*
*he following
reassuring principle:
1.9.2 Lemma. The forgetful functors in diagram 1.5.1 all induce equivalences of*
* cate
gories between the localizations of these variant categories by inverting the s*
*table homo
topy equivalences.
2 Lax symmetric monoidal categories
In this section, I will study the category of lax algebras in Cat over the doct*
*rine of sym
metric monoidal categories, that is, the category of lax symmetric monoidal cat*
*egories.
Following Kelly, I give generators and relations for a club whose strict algebr*
*as are the
lax symmetric monoidal categories. The functor Spt( ) of 1.6 extends to a fun*
*ctor
on these. A Godement construction shows that each lax symmetric monoidal catego*
*ry
admits a simplicial resolution by (strict) symmetric monoidal categories. Using*
* this and
the homotopy colimit along op I define a functor from the category of lax symme*
*tric
monoidal categories to that of symmetric monoidal categories which commutes up *
*to
natural stable homotopy equivalence with Spt( ).
2.1: Clubs and doctrines.
2.1.1. I will suppose the reader is familiar with Kelly's theory of clubs, an e*
*fficient means
of describing algebraic structures imposed on categories. A club prescribes cer*
*tain nary
functors, which are to be operations on a category, and natural transformations*
* between
them. These may be considered as generated by iterated substitution and composi*
*tion
Theory and Applications of Categories, Vol. 1, No. 5 *
* 91
of a smaller basic set of operations and transformations. The club structure e*
*ncodes
this substitution process. I will consider only clubs over the skeletal categor*
*y of finite
sets with permutations as morphisms. That is, each operation in a club will hav*
*e a finite
arity n, and natural transformations between operations are allowed to specify *
*a required
permutation of the order of inputs between the source and the target operations*
*. The
"type" functor from the club to the category of finite sets specifies the arity*
* of the
operations and the permutations associated to the natural transformations. The *
*reader
may consult [Ke4] x10 and x1 for a quick review of club theory. See also [Ke2],*
* [Ke1].
2.1.2. Denote by oe the club for symmetric monoidal categories. The underlying *
*category
of this club has as objects a Tn of type n 2 N for each way to build up an nar*
*y operation
by iterated substitution of a binary operation into instances of itself. (This*
* includes
an empty set of substitutions, which yields the identity functor 1 as a 1ary o*
*peration.)
For each n 1 the type functor is an equivalence of categories between the subc*
*ategory
of objects of type n and the symmetric group of order n. Thus any two objects o*
*f the
same type are isomorphic, and the group of automorphisms of any object of type *
*n is
the symmetric group n. All this follows from standard coherence theory, given t*
*hat in
this paper symmetric monoidal categories are not assumed to have units.
In order to construct the Godement resolution needed to pass from lax symmet*
*ric
monoidal categories to symmetric monoidal ones, I will need to consider the doc*
*trine
associated to a club. The reader may consult [KS] x3.5 for a review of doctrine*
*s. I recall
the definitions for its convenience.
2.1.3 Definitions. A doctrine in a 2category K__is an endo 2functor D togethe*
*r with
2natural transformations j: Id! D and : DD ! D satisfying the defining identit*
*ies:
(2.1.3.1) O D = O D O Dj = id= O jD
A Dalgebra is an object A of K__together with a morphism a: DA ! A ("the ac*
*tion")
satisfying the defining identities:
(2.1.3.2) a O Da = a O A : DDA ! A idA = a O j
A lax morphism between two Dalgebras_(A; a) and (B; b) consists of a morphi*
*sm in
K__, f: A ! B together with a 2cell f:
(2.1.3.3)
Df
DA ___________//_DB
 
 
a _ b
 fff 
 
fflffl fflffl
A ______f______//_B
Theory and Applications of Categories, Vol. 1, No. 5 *
* 92
which is to satisfy the following two identities of 2cells:
(2.1.3.4)
DDf DDf
DDA ____________//DDB DDA ___________//_DDB
   
   
A  B Da  _ Db
   fDff 
   
fflffl Df fflffl fflffl Df fflffl
DA _____________//_DB = DA _____________//_DB
   
   
a  _ b a _ b
 fff   fff 
   
fflffl fflffl fflffl fflffl
A _______f_______//B A _______f_______//B
(2.1.3.5)
f
A _____________//_B
 
 
  f
jA jB A ___________//_B
   
   
fflfflDf fflffl  
DA ___________//_DB = 1 1
   
   
  fflffl fflffl
a _f b A _____f_____//_B
 ff 
 
fflffl fflffl
A ______f______//_B
*
*__
A strong morphism between Dalgebras is a lax morphism such that the 2cell_*
*f is
an isomorphism. A strict morphism of Dalgebras is a lax morphism such that f i*
*s the
identity.
A D2cell between two (lax, strong, or strict) morphisms of Dalgebras f; g*
*: A ! B
is a 2cell of K__s: f ) g such that one has the identity of 2cells:
__
(2.1.3.6) __gO Ds = s O f
Denote by DAlg the 2category whose objects are Dalgebras, whose 1cells a*
*re lax
morphisms of Dalgebras, and whose 2cells are D2cells. The variant sub 2cat*
*egories
Theory and Applications of Categories, Vol. 1, No. 5 *
* 93
whose 1cells are the strong or strict morphisms of Dalgebras are denoted DAl*
*gStrong
and DAlgStrict respectively.
2.1.4. The doctrine D = O on the 2category Cat associated to a club sends a c*
*ategory
C_ to the underlying category of the free algebra on C_. From [Ke1] x2 and [Ke*
*2] x 2.3
or [Ke4] x10, one derives the following description. Denote by (n) the category*
* whose
objects are (T; o) where T is an object of type n in the club, and o 2 n. A mor*
*phism
(T; o) ! (T 0; o0) consists of a morphism u: T ! T 0in whose type satisfies o*
* = o0.
(n) is thus roughly speaking the category of operations induced by elements of *
*the
club and permutations of inputs. Then the doctrine corresponding to is:
a
(2.1.4.1) C_= (n) nn C_n
n0
Here n acts on the nfold product C_nby permuting the factors, and on (n) by*
* the
free right action (T; o) x oe 7! (T; ooe).
The natural transformation : O C_ ! C_ is given as part of the club structu*
*re
and encodes the substitution of operations into other operations. It is induce*
*d by a
collection of functors:
(2.1.4.2) (n) x (j1) x . .x.(jn) ! (j1 + . .+.jn)
The natural transformation j: C_! C_ is induced by the inclusion
(2.1.4.3) 1 2 (1)
of the distinguished object (1; 1) corresponding to the identity operation.
Assigning to each object and morphism in the club an operation and natural t*
*rans
formation of operations on C_of the appropriate arity corresponds to giving a m*
*orphism
c: C_ ! C_. If the assignment is compatible with substitution of operations an*
*d the
identity, then c is the structure of an algebra for the doctrine .
2.1.5. In the case of the club oe of 2.1.2, the structure of a oealgebra is ex*
*actly the
structure of a symmetric monoidal category. The lax, strong, and strict morphi*
*sms
of oealgebras correspond respectively to lax, strong, and strict symmetric mon*
*oidal
functors. The coherence theorem for symmetric monoidal categories noted in 2.1*
*.2 is
equivalent to the fact that for each n 1 there is a unique isomorphism between*
* any
two objects of oe(n).
2.2: Godement enriched and lax.
Recall that op, the category such that functors op ! C_are the simplicial ob*
*jects of
C_, is the opposite category of the skeletal category of finite nonempty tota*
*lly ordered
sets n_= {0 < 1 < . .<.n} and monotone increasing maps. Following [Th2] 1.1 let*
* +
Theory and Applications of Categories, Vol. 1, No. 5 *
* 94
be the category with objects n_= {1 < 0 < 1 < . .<.n} for n 1 and morphisms
the monotone increasing maps sending 1 to 1. There is a standard inclusion fu*
*nctor
from to + sending n_ to n_. A functor X: +op ! C_ is a simplicial object X:
together with an augmentation ffl = d0: X0 ! X1 and a system of extra degenera*
*cies
s1 : Xn ! Xn+1 for n 1. These must satisfy certain relations extending the u*
*sual
simplicial identities. That is, one requires on Xn that:
didj = dj1di if0 i < j n
sisj = sj+1si if 1 i j n
(2.2.1) disj = sj1di if0 i < j n
disj = id if 1 j n and 0 i 2 {j; j + 1}
disj = sjdi1 if 1 j and j + 1 < i n + 1
The extra degeneracies imply that the augmentation X: ! X1 is a simplicial hom*
*o
topy equivalence, even that there is a simplicial homotopy on X: that is a defo*
*rmation
retraction to the constant simplicial object on X1 . (Cf. e.g. [Ma1] x9)
2.2.2 Proposition. Let D be a doctrine on the 2category K__. Recall the notati*
*ons of
1.7.5 and of x2 above.
Then there is a natural 2functor, the Godement resolution,
R*: DAlg ! Fun(+op; K__)
such that Rn = Dn+1 for n 0 while R1 is the functor sending a Dalgebra to its
underlying object in K__.
After restriction of the values in Fun from +op to the standard subcategory *
*op, R*
lifts canonically to a 2functor
R*  op: DAlg ! Fun(op; DAlgStrict)
Considering R1 = Id: DAlg ! DAlg as taking values in constant simplicial D
algebras, the augmentation induces a 2natural transformation d0*+1: R*  op ) *
*R1
of functors into Fun (op; DAlg)
The restriction of R to DAlgStrict factors through the sub 2category of Fu*
*n whose
1cells are (strict) natural transformations.
R*: DAlgStrict! Cat(+op; K__)
Thus for a Dalgebra A, R*A is a simplicial free Dalgebra with a Dalgebra *
*augmen
tation to A, the augmentation being a simplicial homotopy equivalence after for*
*getting
the Dalgebra structure. The whole thing is strictly natural for strict morphis*
*ms of D
algebras, and is left oplax natural for lax morphisms of Dalgebras. If the 2*
*category
Theory and Applications of Categories, Vol. 1, No. 5 *
* 95
K__has only identity 2cells, the doctrine D is just a monad and all reduces to*
* the clas
sical Godement resolution [Go] Appendice or to May's reformulation as a twosid*
*ed bar
construction B*(D; D; Id) in [Ma1] x9.
Proof. For (A; a) a Dalgebra, define the functor R*(A; a) from +op by giving v*
*alues
on on objects and on the generating morphisms di and si of +op. On objects, RnA*
* =
Dn+1 A for n 1. For the morphisms:
aeDiDni1 : Dn+1 A ! DnA for0 i < n
(2.2.2.1) di: RnA ! Rn1 A =
Dna: Dn+1 A ! DnA fori = n
(2.2.2.2) si: RnA ! Rn+1 A = Di+1jDni: Dn+1 A ! Dn+2 A for  1 i n
Naturality and the defining identities 2.1.3 for the structure maps , j, and a *
*of doctrines
and algebras give that these di and si satisfy the extended simplicial identiti*
*es (2.2.1)
and so define a functor on +op. All di and the si other than s1 are strict ma*
*ps of
Dalgebras.
Given a morphism of Dalgebras f: A ! B, let Rnf = Dn+1 f. Note that for a s*
*trict
algebra morphism f this formula defines a natural transformation R*f: R*A ! R*B
of functors on +op. For such an f strictly commutes with , j, and a. For a D*
*2
cell between two morphisms s: f ) g let Rns = Dn+1 s: Dn+1 f ) Dn+1 g. Routine
verification now yields the results of the last paragraph of Proposition 2.2.2.
The essential point remaining is to define the structure of a left oplax_na*
*tural trans
formation (1.7.2) R*f: R*A ! R*B for each lax morphism of Dalgebras (f; f): A *
*! B.
The structure 1cells of this R*f are the Rnf = Dn+1 above. The structure 2cell
(2.2.2.3)
Dn+1f
Dn+1 A ___________//_Dn+1 B
 
 
'*  ___Rf '*
 ff' 
 
fflffl fflffl
Dq+1A ____Dq+1f___//Dq+1B
associated to a morphism ': q_! n_in + is derived from the structure 2cells of*
* the
lax algebra map as follows:
If ': {1 < 0 < . . .< q} ! {1 < . . .< n} preserves the maximal element,
'(q) = n, then ' in +op can be written as a composite of si and those di for wh*
*ich i
is not maximal. Thus '* can be written as a composite of DiDk and DijDk, without
using any action_maps Dia. In this case the diagram of 1cells in (2.2.2.3) co*
*mmutes
and one takes Rf ' to be the identity 2cell. In the other case where ' does no*
*t preserve
the maximal element, it factors uniquely in + as:
' = {1 < . .<.q} ! {1 < . .<.q < q + 1} ! {1 < . .<.p}
Theory and Applications of Categories, Vol. 1, No. 5 *
* 96
where is the inclusion of the initial segment and is determined by = ' with
(q + 1) = n. Thus preserves_the_maximal element. As for , * = d*q= Dq+1a.
One sets the structure 2cell Rf ' to be that induced by the structure 2cell o*
*f the lax
algebra morphism:
(2.2.2.4)
Dn+1f
Dn+1 A ___________//_Dn+1 B
 
 
*  *
 
 
fflfflDq+2f fflffl
Dq+2A ___________//_Dq+2B
 
 
Dq+1a  Dq+1_f Dq+1b
 ff 
 
fflffl fflffl
Dq+1A ___Dq+1f___//_Dq+1B
___
That these Rf 'satisfy the defining identities 1.7.2.1 and 1.7.2.2 for an op*
*lax natural
transformation follows by routine case by case analysis of these specifications*
* and the
identities 2.1.3.5 and 2.1.3.4 of lax morphisms of algebras.
Finally, for s a D2cell, R*s is a modification of left oplax natural tran*
*sformations.
The required identity 1.7.3.1 results from 2.1.3.6. *
* 
2.2.3 Addendum. Preserving the hypotheses and notations of 2.2.2, suppose that
E: K__! Q__is a 2functor which is a right Dalgebra. That is, suppose there i*
*s a 2
natural transformation e: ED ! E such that
e = e O eD : EDD ! E
and
1 = e O Ej: E ! ED ! E
.
Then there is a 2functor
ED*: DAlg ! Fun(op; Q__)
specified as follows:
Given (A; a) a Dalgebra, the simplicial object ED*A: op ! Q__is defined by:
n 7! EDnA
Theory and Applications of Categories, Vol. 1, No. 5 *
* 97
si = EDijDni: EDnA ! EDn+1 A
8
> EDi1Dni1 for 0 < i < n
: EDn1 a for i = n
__
Given (f; f): (A; a) ! (B; b) a lax morphism of Dalgebras, the associated l*
*eft oplax
natural transformation ED*A ! ED*B has structure 1cells EDnf: EDnA ! EDnB
and structure 2cells given by the obvious generalization of 2.2.2.4 on replaci*
*ng all
Dn+1 A by EDnA, etc.
Given s: f ) g a modification of lax morphisms of Dalgebras, the associated*
* modi
fication of left oplax natural transformations has component 2cells
EDns: EDnf ) EDng
Verification of all these points results from calculations which are trivial*
* modifications
of those required for the verification of 2.2.2.
2.3: LaxSymMon .
2.3.1 Definition. The club "oefor lax symmetric monoidal categories is the club*
* over
the category of finite ordinals and permutations with the following_presentatio*
*n:
The objects of "oeare generated under substitution by a T for each_object_T *
*of the
club for symmetric monoidal categories oe. The type of the generator Tn equals *
*the type
of Tn. No relations are imposed on the objects. (In particular, if 1 denotes_th*
*e objects
in oe and "oecorresponding to the identity operation, one does not have 1 = 1.)
The morphisms of "oeare generated by:
i) A morphism __ __ __
u: T ! R
for each morphism u: T ! S in oe. The type of __uis the permutation which is th*
*e type
of u.
ii) A morphism
____ __ _____________
"aT[S1;S2;...;Sn]: T(S 1; . .;.Sn) ! T (S1; . .;.Sn)
of type the identity permutation for each object T of oe and each ntuple of ob*
*jects
S1; . .;.Sn of oe. Here n 2 N is the type of T .
iii) A morphism __
^a: 1 ! 1
of type 1.
The relations imposed on these generators are a)e) below:
Theory and Applications of Categories, Vol. 1, No. 5 *
* 98
(2.3.1.a)
__uO __v= ___uv
____
idT = id__T
(2.3.1.b)
__ __T(^a;...;^a)__ __
T(1; . .;.1)____//_T(1; . .;.1)
OO
OOOO "aT[1;...;1]
idOOOOOO 
O''fflffl_
T
(2.3.1.c)
__ ^a[__S]_ __
1[S ]____//_D1[S ]
DDD "a
DDD  1[S]
id DD!!fflffl_
S
(2.3.1.d)
__ __ __ __ __ "a _____________ __ __
T (S 1(R 11; . .).; . .;.Sn(R n1;_._.).)//_T (S1; . .;.Sn)(R 11; . .;.Rn*
*q)
__T("a;...;"a) 
 "a
______________ fflffl____________ _______________fflffl_________
T(S1(R11; . .).; . .;.Sn(Rn1; ._.).)//"a_T (S1(R11; . .).; . .;.Sn(. .;*
*.Rnq))
(2.3.1.e)
____ __ "aT[S1;...;Sn]___________
T(S 1; . .;.Sn)___________//T (S1; . .;.Sn)
_u(_v1;...;_vn) _________
 u(v1;...;vn)
___ ___fflffl___ _______fflffl_
T 0(S01; . .;.S0n)"aT0[S0__//T 0(S01; . .;.S0n)
1;...;S0n]
for each ordered set of morphisms in oe consisting of a u: T ! T 0and of vi: Si*
* ! S0'(i)
where ' is the type of u.
2.3.2. Comparison with [Ke3] 4.10, 4.14.2 shows that this club "oeis indeed th*
*e club
whose doctrine has as strict algebras the lax algebras over the doctrine of sym*
*metric
monoidal categories.
There is a map of clubs
(2.3.2.1) s: "oe! oe
Theory and Applications of Categories, Vol. 1, No. 5 *
* 99
__ __
sending the object T to T , the morphism u to u, and the morphisms ^aand "aT[S1*
*;...;Sn]
to identity morphisms. This map of clubs induces a morphism of the correspondi*
*ng
doctrines.
By [Ke3] 4.8 Thm. 4.1 there is also a lax doctrine map going in the opposite*
* direction:
(2.3.2.2) (h; "h; ^h): oe ! "oe
As in [Ke3] 4.10 h is induced by_the functor between the underlying categori*
*es of the
clubs which sends the object T to T and the morphism u: T ! S to __u. This fun*
*ctor
is not a (strict) morphism of clubs since it does not strictly preserve the ope*
*rational
substitution which is part of the club structure.
The following facts are immediately deduced from the "cheap" coherence resul*
*t of
[Ke3] 1.4 and the description of oe_above in 2.1. For each object U of "oethere*
* is an object
V in oe, and a_morphism in "oeU ! V = hV . Given U, any V for_which there exist*
*s a
morphism U ! V is isomorphic to sU in oe. Two morphisms U ! V in "oeare equal if
and only if their images under s are equal, that is to say, if and only if they*
* have the
same permutation as type.
Rephrasing this, we get that there is a 2cell of lax maps of doctrines o: 1*
* ) hs such
that so = 1 and oh = 1. In particular for any category C_the functor h: oeC_ ! *
*"oeC_is
right adjoint to s: "oeC_! oeC_.
Note that these results of coherence theory together with 2.1.4, 2.1.5, and *
*the ele
ments of the homotopy theory of categories ([Qu] x1) imply that one has the fol*
*lowing
homotopy equivalences of classifying spaces, naturally in any category C_:
~ a1 n
(2.3.2.3) B("oeC_) ~!sB(oeC_)!typeEn xn B(C_)
n=1
2.3.3 Definition. The 2category LaxSymMonStrict has as objects the Usmall al*
*ge
bras for the club "oe, i.e. the (Usmall) lax symmetrical monoidal categories. *
*The 1cells
are the_strict_morphisms of "oealgebras, those functors which strictly preserv*
*e the oper
ations T and the natural transformations __u, "a, and ^a. The 2cells are the m*
*odifications
of 1cells (2.1.3).
There are variant 2categories LaxSymMonStrong and LaxSymMon whose 1cel*
*ls are
respectively the strong and the lax morphisms of "oealgebras (2.1.3) and whose*
* 2cells
are the modifications of such.
To preserve compatibility with the more common terminology of x1, I will use*
* the
term "lax symmetric monoidal functor" only for a 1cell in the category SymMon *
* of
1.5.1. A 1cell in LaxSymMonStrict will be a "strict functor between lax sym*
*metric
monoidal categories".
There is a forgetful 2functor
Theory and Applications of Categories, Vol. 1, No. 5 *
* 100
(2.3.4) SymMon ! LaxSymMon
induced by the map of clubs s: "oe! oe.
2.4: Spt( ) on LaxSymMon .
The functor Spt: SymMon ! Spectra of 1.6 extends to a functor *
*on
LaxSymMon . Indeed the construction in [Th2] Appendix generalizes easily. The*
* only
differences arise because now for A a lax symmetric monoidal category which has*
* a strict
unit 0, one step of the construction gives a lax functor (1.7.1, or [St]) A: op*
* ! Cat
instead of a pseudofunctor. To construct this, choose for each finite ordinal n*
* 1 one
of the isomorphic objects Tn of typeQn in the club oe. Let T0 denote the strict*
* unit of
A. The lax functor A sends p to Q pA. ForQeach morphism ': p ! q in the categor*
*y of
finite based sets op, let A('): pA ! qA be the functor sending (A1; . .;.Ap*
*) to
__ 1 __ 1
(T '1(1)(Ai; i 2 ' (1)); . .;.T'1(q)(Ai; i 2 ' (q)))
__
Here in each T'1(j)(Ai; i 2 '1 (j)) one orders the Aiby increasing order of t*
*he indexes
i. The structure natural transformations of the lax functor A(')A( ) ) A(' ) a*
*nd
id ) A(id) have components induced by the unique tuple of morphisms in the club
"oethat have the appropriate type to universally define such a natural transfor*
*mation.
The essential ingredients are the "aand ^aof 2.3.1. The identities required by*
* the lax
functor axioms hold by the coherence result mentioned in 2.3.2. In the case wh*
*ere A
is a strict symmetric monoidal category, this lax functor is in fact a pseudofu*
*nctor,
that associated to A in [Th2] Appendix. A lax morphism of lax symmetric monoid*
*al
categories induces a left lax natural transformation of lax functors op ! Cat b*
*y trivial
generalization of the formulae in [Th2]. To the lax functor A one now applies S*
*treet's
"second construction" of [St] to convert this into a strict functor from op to *
*Cat which
sends p to a category related to Ap by an adjoint pairQof functors. From this i*
*t follows
that its classifying space is homotopy equivalent to pB(A) ([Qu] x1 Prop.2 Co*
*r.1).
Thus on applying the classifying space functor B: Cat ! Top to this, one has a *
*"special
space " in the sense of Segal [Se], to which his infinite loop space machine a*
*ssociates
a spectrum. By [St] Thm. 2, Street's second construction is in fact a functor
Lax (op; Cat) ! Cat(op; Cat)
and thus a left lax natural transformation yields a map of "special spaces".
The rest of the argument, including generalization to the nonunital case, n*
*ow pro
ceeds exactly as in [Th2].
In the case where A is a strict symmetric monoidal category the construction*
* coincides
with that given in [Th2], except that instead of considering a pseudofunctor on*
* op as an
oplax functor and applying Street's first construction for oplax functors, on*
*e considers
Theory and Applications of Categories, Vol. 1, No. 5 *
* 101
it as a laxfunctor and applies the second construction. The proof given in [Th*
*2] shows
that this variant also yields a functor Spt: SymMon ! Spectra satisfying cond*
*itions
2.1 and 2.2 of [Th2], and thus ([Th2] p. 1603) is linked by a chain of natural *
*stable
homotopy equivalences to the original version Spt( ).
2.4.1. By an argument identical to that in [Th2], one obtains the analogs of th*
*e group
completion results (1.6.1), (1.6.2). Similarly as in 1.6.5, these imply that if*
* F : S ! T is
a 1cell in LaxSymMon such that BF is a homotopy equivalence of spaces, then *
*Spt(F )
is a stable homotopy equivalence of spectra.
2.5 Proposition. There is a 2functor:
(2.5.1) S: LaxSymMon ! SymMon
and a chain of natural stable homotopy equivalences of spectra:
(2.5.2) Spt(SS) ~ hocolimop Spt(R"*S) ~! Spt(S)
It follows that S preserves stable homotopy equivalences between lax symmetric *
*monoidal
categories.
Denote by I: SymMon ! LaxSymMon the inclusion functor. There is a 2nat*
*ural
transformation to the identity functor:
(2.5.3) j: S O I ! 1SymMon
such that Spt(j) is a stable homotopy equivalence of spectra.
If U is a Grothendieck universe inspection of the construction in 2.6 shows *
*that the
functor S preserves Usmallness.
Proof. Recall that LaxSymMon is the 2category of algebras for the doctrine "*
*oe.
The strict map of doctrines s: "oe! oe of 2.3.2 induces the structure of a r*
*ight "oe
algebra on the doctrine for symmetric monoidal categories oe. The action map is*
* given
by: O oes: oe"oe! oeoe ! oe. Thus the addendum 2.2.3 to the Godement resolut*
*ion
yields a 2functor oe("oe)* from LaxSymMon to Fun (op; SymMonStrict ) sendin*
*g S to
the simplicial symmetric monoidal category n 7! oe"oenS
Let S: LaxSymMon ! SymMon be the composite of this 2functor with the ho*
*motopy
colimit functor of 1.8
hocolimop : Fun(op; SymMon ) ! SymMonStrict SymMon
To construct the chain of stable homotopy equivalences of spectra 2.5.2, let*
* "R*be the
Godement resolution functor 2.2.2 n 7! "oe"oen. Then s"oen: "oe"oen! oe"oengive*
*s a simplicial
Theory and Applications of Categories, Vol. 1, No. 5 *
* 102
map R" ! oe"oe*which is a stable homotopy equivalence in each degree n 2 op by
(2.3.2.3). By 1.8.4 and 1.8.3 this map induces a natural stable homotopy equiva*
*lence:
(2.5.4)
hocolimop Spt"R*S ~! hocolimop Sptoe"oe*S ~! Spt(hocolimop oe"oe*S) = Sp*
*tSS
On the other hand, since "R*S is a resolution of S, the augmentation map "R**
*S ! S
induces a stable homotopy equivalence hocolimop Spt"R*S  ~! SptS. Indeed, by*
* a
standard reasoning on 1connective spectra using the group completion property*
* 2.4.1
(cf. [Th2] 4.5 ) it suffices to show that the map induced by the augmentation f*
*rom the
geometric realization of the simplicial classifying space kn 7! B"oen+1Sk ! BS *
*is a weak
homotopy equivalence of spaces. But this map is such since the resolution simpl*
*icially
deformation retracts to S because of the extra degeneracies s1 (2.2). (cf. e.*
*g. [Ma1]
9.8). This completes the construction of the chain of stable homotopy equivale*
*nces
(2.5.2).
It remains to construct the natural stable homotopy equivalence j: S O I ! I*
*d. But
there is a 2natural transformation
(2.5.5) oe("oe)* O I ) R*: SymMon ! Fun(op; SymMon )
to the Godement resolution 2.2.2 for oealgebras. The component at S is the na*
*tural
transformation of functors on op whose component at n 2 op is oesnS: oe"oenS !
oen+1 S. The 2natural transformation (2.5.5) induces a map of homotopy colimits
(2.5.6) SI = hocolimop (n 7! oe"oenS) ! hocolimop (n 7! oen+1 S)
By the universal mapping property of homotopy colimits 1.8.2, the augmentati*
*on of
the Godement resolution for oealgebras oe*S ! S yields a natural 1cell
(2.5.7) hocolimop (n 7! oen+1 S) ! S
The 2natural transformation j: SI(S) ! S is the composite of 2.5.6 and 2.5.*
*7. To
show it is a stable homotopy equivalence, it suffices to show 2.5.6 and 2.5.7 a*
*re such.
 From the homotopy equivalences (2.3.2.3) it follows that s: "oeC_! oeC_ in*
*duces a
homotopy equivalence on classifying spaces of categories, and that both functor*
*s "oe
and oe preserve homotopy equivalences of categories. Thus in each degree n 2 op
the component of (2.5.5) is a homotopy equivalence of categories. Then it is a*
* stable
homotopy equivalence for each n and so by 1.8 the induced map 2.5.6 of homotopy
colimits is a stable homotopy equivalence.
As for the map 2.5.7, to show it is a stable homotopy equivalence it suffice*
*s by 1.8.4
to show that the augmentation induces a stable homtopy equivalence
hocolimop SptR*S ~! SptS
But as above this holds since R*S is a resolution of S. *
* 
Theory and Applications of Categories, Vol. 1, No. 5 *
* 103
3 From Spectra to E1 spaces
In this section, I will recall May's notion of E1 operad and his result on the*
* equivalence
of the homotopy category of 1connective spectra with a localization of the ca*
*tegory of
E1 spaces. This is an essential link in the chain of equivalences connecting t*
*he former
with the localization of SymMon . I begin by recalling some definitions from [*
*Ma1], since
one will need to have the details available for x4.
3.1 Definition. An operad in the category of compactly generated Hausdorff spac*
*es
consists of the following data:
i)For each integer n 0, a space E(n) and a right action of the symmetric *
*group
n on E(n).
ii)A point 1 2 E(1)
iii)For each n 1 and each sequence j1; j2; . .;.jn of nonnegative integer*
*s, a
continuous function:
(3.1.1) fl: E(n) x E(j1) x E(j2) x . .x.E(jn) ! E(j1 + j2 + . .+.jn)
These are to satisfy the following conditions:
a) The space E(0) = * is a single point.
b) The distinguished point 1 2 E(1) is an identity for the composition law *
*fl. That
is, for any f 2 E(n) and g 2 E(j) one has:
fl(f; 1; . .;.1)= f
fl(1; g)= g
c) The composition law is compatible with the action of the symmetric group.
That is, for any oe 2 n, any sequence of oek 2 jk for k = 1; . .;.n and *
*any
f 2 E(n); gk 2 E(jk) one has:
fl(foe; g1; . .;.gn)= fl(f; goe1(1); . .g.oe1(n))oe(j1; . .;.*
*jn)
fl(f; g1oe1; . .;.gnoen)= fl(f; g1; . .;.gn)(oe1 q . .q.oen)
Here oe(j1; . .;.jn) denotes the permutation in j1+...+jnthat permutes t*
*he n
blocks of jk successive integers according to oe 2 n, leaving the order *
*within
each block fixed. oe1 q . .q.oen is the permutation leaving the n blocks*
* invariant
and which restricts to oek on the kth block.
d) The composition law is associative. That is, given f 2 E(n); gi 2 E(ji)*
* for
i = 1; 2; . .;.n; and hik2 E(`k) for k = 1; 2; . .;.ji; one has:
fl(f; fl(gi; hik)) = fl(fl(f; gi); hik)
Theory and Applications of Categories, Vol. 1, No. 5 *
* 104
3.2 Definition. An E1 operad is an operad such that for each n the space E(n) *
*is
homotopy equivalent to a point and n acts freely on E(n).
3.3 Definition. An Espace for E an operad is a based space X together with con
tinuous functions for each nonnegative integer n:
Yn
ffn: E(n) x X ! X
such that this action ff is based, unital, and respects the permutations and th*
*e compo
sition law of the operad. More precisely, one requires that:
a) ff0(*) is the basepoint of X.
b) ff1(1; x) = x for the distinguished point 1 2 E(1).
c)
ffn(foe; x1; x2; . .;.xn) = ffn(f; xoe1(1); . .;.xoe1(n))
for any f 2 E(n) and oe 2 n.
d)
ffj1+...+jn(fl(f; g1; . .;.gn); x11; . .;.x1j1; x21; . .;.xnjn) =
ffn(f; ffj1(g1; x11; . .;.x1j1); . .;.ffjn(gn; xn1; *
*. .;.xnjn))
for any f 2 E(n), and any sequence gi 2 E(ji) for i = 1; : :;:n
A morphism of Espaces is a continuous function ': X ! Y such that for all n*
* the
following diagram commutes:
(3.4)
Q n idxQ n' Q n
E(n) x X ___________//_E(n) x Y
ffn ffn
fflffl fflffl
X __________'___________//_Y
A morphism of Espaces is a weak homotopy equivalence if it is a weak homoto*
*py
equivalence on the underlying spaces.
These definitions are due to May ([Ma1] 1.1, 3.5, 1.4) inspired by earlier w*
*ork of
Adams, Beck, Boardman, MacLane, Stasheff, and Vogt.
One fixes an E1 operad E in a Grothendieck universe U and considers the cat*
*egory
of Usmall Espaces. For any two choices of such E, there are functors between*
* the
two categories such that the composites each way are linked to the identity by *
*a chain
of natural homotopy equivalences of Espaces ([MaT] 1.8, x4, x5; cf. [Ma1] x13 *
*up to
13.1). Thus these categories are essentially interchangeable. I will therefore *
*conform to
the standard abuse and speak of any of them as the category of E1 spaces.
Theory and Applications of Categories, Vol. 1, No. 5 *
* 105
May's approach to infinite loop space theory produces an functor Spt' from t*
*he
category of E1 spaces to the category of 1connective spectra ([Ma2]). In fa*
*ct, he
shows this functor induces an equivalence of a localization of the category of *
*E1 spaces
and the stable homotopy category of  1connective spectra. The inverse funct*
*or is
given by imposing a natural action of an E1 operad on the zeroth space of a sp*
*ectrum.
3.5 Proposition (May). The functor Spt': E1 spaces! Spectrainduces an equiva
lence from the localization of E1 spacesby inverting all maps that Spt'sends t*
*o stable
homotopy equivalences, to the full subcategory of the stable homotopy category *
*consisting
of 1connective spectra. The inverse equivalence is induced by the zeroth spac*
*e functor
1 .
This is just a combination and slight reinterpretation of parts of the state*
*ments
([Ma2] Thm. 2.3, Cor. 2.4, Thm. 3.2), to which I refer the reader for the proof.
3.5.1. The morphisms of E1 spaces ': X ! Y that Spt' sends to stable homotopy
equivalences are precisely those which induce isomorphisms on homology after lo*
*calizing
by inverting the action of the abelian monoid ss0 of the E1 spaces:
'*: ss0(X)1 H*(X; Z) ~! ss0(Y )1 H*(Y ; Z)
This follows from the groupcompletion theorem ([Ma2] 2.3iv, 1.3, [Se] x4).
3.6. Let be a club such that the categories (n) of 2.1.4 have contractible cla*
*ssifying
spaces B(n). Suppose further that (0) is isomorphic to the category with one mo*
*r
phism. By coherence theory the club for strict unital symmetric monoidal categ*
*ories
is such a club. Then setting E(n) = B(n) for n 0 yields an E1 operad. For t*
*he
free action of n on (n) induces a free right n action on B(n). The distinguished
object 1 2 (1) induces a distinguished point 1 2 B(1). Since the classifying s*
*pace
functor B preserves finite products, the composition law of the club expressed *
*in form
(2.1.4.2) induces a composition law fl on the B(n) for n 1. The conditions 3.1*
*.ad
for an operad hold as a consequence of the similar conditions satisfied by a cl*
*ub.
Moreover, an action of such a club on a category C_induces an action of the*
* operad
B on BC_. For the action map of the club
a
(3.6.1) (n) nn C_n~= C_! C_
n0
induces an action of the operad on applying B.
This procedure is natural for strict morphisms of actions. In particular, t*
*aking
to be the club for unital symmetric monoidal categories, this procedure gives a*
* functor
(3.6.2) B: UniSymMonStrict ! E1 spaces
The May functor [Ma2] from UniSymMonStrict to Spectra is the composition of
this functor B and the May machine Spt'. As noted in 1.6, this functor is link*
*ed by
a chain of natural stable homotopy equivalences to the restriction from SymMon *
* to
UniSymMonStrict of our functor Spt.
Theory and Applications of Categories, Vol. 1, No. 5 *
* 106
4 From E1 spaces to lax symmetric monoidal categories
In this section, I construct a functor from E1 spaces to LaxSymMonStrict that*
* will be
the essential constituent of an inverse up to natural stable homotopy equivalen*
*ce to the
functor Spt of x2. The strategy of the construction is first to show that the *
*category
of contractible spaces over a space X has a classifying space weak homotopy equ*
*ivalent
to X, and then to use the action of the E1 operad on X to produce a lax symmet*
*ric
monoidal structure on this category.
4.1. I begin by recalling some wellknown facts from the theory of simplicial s*
*ets. Denote
by [ ]: ! Top the functor sending p to the standard topological psimplex. Re*
*call
that the singular functor
Sing( ): Top ! opSets
sends a space X to the simplicial set which in degree p is the set Top ([p]; X)*
*. There
is a natural weak homotopy equivalence from the geometric realization of the si*
*ngular
complex
(4.1.1) Sing(X)~! X
Denote by
(4.1.2) =X
the category whose objects are the singular simplices of X, c: [p] ! X and whose
morphisms from c to c0: [q] ! X are morphisms ': p ! q in such that the follow*
*ing
diagram commutes in Top :
(4.1.3)
[p] C
 CCCc
 CCC
 C!!
[']  =X=
 
  0
fflfflc
[q]
This construction yields a functor from Top to Cat.
4.2 Proposition (Quillen). There is a natural weak homotopy equivalence from the
classifying space of the category =X to X:
(4.2.1) B=X ~! X
Proof. Given the weak homotopy equivalence of 4.1.1, it suffices to find a natu*
*ral weak
equivalence of simplicial sets from the nerve of =X to Sing(X). For the geomet*
*ric
Theory and Applications of Categories, Vol. 1, No. 5 *
* 107
realization of this weak equivalence is then a weak equivalence of spaces and c*
*an then
be composed with the map of 4.1.1 to yield the equivalence of 4.2.1.
By definition, a psimplex of the nerve N=X is a sequence of p composable mo*
*r
phisms in =X:
(4.2.2)
['1] ['2] ['p]
[k0] _____//_WWW[k1]_//_R._._.//_[kp]
WWWW RRRR
WWWWWW RRRR cp
c0WWWWWWWWWWRRRRR
WWWW++))RRfflfflX
One sends this to the psimplex of Sing(X) specified as follows. The sequen*
*ce of
morphisms 'i determine a map in : {0; 1; . .;.p} ! {0; 1; . .;.kp} in the cat*
*egory of
finite ordinals and monotone maps by (i) = 'p'p1 . .'.i+1(ki) for i = 0; 1; *
*. .;.p.
(For i = p one considers the composition of an empty set of ' to be the identit*
*y, so
(p) = kp.) The desired psimplex of Sing(X) is the composite:
cp[ ]: [p] ! [kp] ! X
One easily checks this defines a map of simplicial sets:
(4.2.3) N=X ! Sing(X)
It remains to show this map is a weak homotopy equivalence.
This could be deduced from trivial modifications to the argument given in [I*
*l] VII x3
for the weaker formulation of the proposition given there. I prefer an alternat*
*e proof:
=X isRisomorphic to the opposite category of the Grothendieck construction (*
*[Th1]
1.1) op Sing(X) on Sing(X) considered as a functor op ! Sets Cat. Thus one
may conclude by dualizing the homotopy colimit theorem of [Th1] 1.2, that there*
* is a
natural weak homotopy equivalence of simplicial sets:
opZ
(4.2.4) hocolimop (p 7! Singp(X)) ~! I (p 7! Singp(X)) ~=N(=X)
Here I is the version of the nerve functor used in [BK] XI 2.1 and [Th1], wh*
*ich has
the opposite orientation to the nerve functor N of [Qu] x1 used in this paper (*
*cf. 4.2.2).
The two are related by:
NC = I Cop
By [BK] XII 3.4, there is a natural weak homotopy equivalence from the homot*
*opy
colimit of a F : op ! opSets to the diagonal simplicial set of F considered as*
* a
bisimplicial set. Applied to the simplicial set Sing(X) considered as a bisimpl*
*icial set
constant in one direction, this gives a weak equivalence:
Theory and Applications of Categories, Vol. 1, No. 5 *
* 108
(4.2.5) hocolimop Sing(X) ~! Diag p; q 7! Singp(X) ~=Sing(X)
Explicit formulas for the equivalences 4.2.4 and 4.2.5 may be deduced from [*
*Th1]
1.2.1 and [BK] XI 2.6, XII 3.4. A routine calculation then yields that these eq*
*uivalences
fit into a commutative triangle (4.2.6) with the map (4.2.3), which is therefor*
*e also an
equivalence as required.
(4.2.6)
hocolimop NSing(X)G
wwww GGG
ww GGG
~wwww GG~GG
www GGG
www GGG
ww GG##
N=X ________________________________//Sing(X)
(In verifying the commutativity it is important to note that [BK] XII 3.4 is*
* incorrect
in stating that "obviously" op\p = =p. What is correct is that op\p = (=p)op.
When this correction is fed through [BK] XII 3.4, the result is that the descri*
*ption given
above of 4.2.2 is right, whereas the erroneous formulae in [BK] XII 3.4 and XI *
*2.6 would
lead one to expect a description using minima 0 instead of the maxima ki) *
* 
4.3 Notation. For X a topological space, let Null=X be the category whose objec*
*ts
are maps of spaces c: C ! X where C is weak homotopy equivalent to a point. A
morphism (C; c) ! (C0; c0) is a map of spaces fl: C ! C0 such that c = c0fl.
4.4 Lemma. The obvious inclusion of categories : =X ! Null=X induces a natural
weak homotopy equivalence of classifying spaces:
(4.4.1) B=X ~! BNull=X
Proof. By Quillen's Thm. A ([Qu] x1), it suffices to show for each object c: C *
*! X
that the comma category =(C; c) is contractible. But this comma category is ju*
*st
=C. Thus Prop. 4.2 gives that its classifying space is weakly equivalent to C, *
*hence
contractible as required. *
* 
It follows immediately from 4.4 and 4.2 that the functors B: Cat ! Top and N*
*ull=( ):
Top ! Cat induce inverse equivalences of localized categories on inverting the*
* weak
homotopy equivalences. (Modulo the usual precautions to take the categories of*
* U
small objects.)
Theory and Applications of Categories, Vol. 1, No. 5 *
* 109
4.5 Proposition. The functor Null=( ): Top ! Cat lifts to a functor E1 spaces*
* !
LaxSymMonStrict .
Proof. This claim is that one can construct a natural lax symmetric monoidal st*
*ruc
ture on Null=X from an action of an E1 operad {E(n)} on X. By the presentation*
* of
the club "oefor lax symmetric monoidal categories given in 2.3.1, and the descr*
*iption of
a club action in terms of its presentation ([Ke4] 10.210.7), this amounts to g*
*iving the
following data:
4.5.1
i)For each object T of type n in the club oe for symmetric monoidal catego*
*ries, i.e.
for each nary operation built up by iterated substitution of a binary o*
*peration
into itself, a functor:
__ nY
T: Null=X ! Null=X
ii)For each morphism u: T ! S in the club oe, a natural transformation with
components:
__u: __T(c __
1; . .c.n) ! S(cAE1(1); . .;.cAE1(n))
Here AE is the permutation which is the type of u.
iii)A natural transformation __
^a: 1 ! 1
iv)For each T in oe of type n, and each ntuple Si of objects in oe, a nat*
*ural
transformation of type the identity
____ __ _____________
"aT[S1;S2;...;Sn]: T(S 1; . .;.Sn) ! T (S1; . .;.Sn)
such that the diagrams 2.3.1.a,b,c,d, and e commute.
I construct all this as follows. Note that the category Top is a symmetric m*
*onoidal
category under product of spaces. Thus for each T of type n in the club oe, and*
* each
ntuple of spaces Y1; . .;.Yn, one has the product space T (Y1; . .;.Yn), usual*
*ly denoted
Y1 x_. .x.Yn with the choice of parentheses censored.
T will be the functor sending an ntuple of objects ci: Ci ! X in Null=X to *
*the
contractible space over X given by the product of the Ci and E(n):
(4.5.2)
1xT(c1;...;cn) *
* ffn
E(n) x T (C1; . .;.Cn)!E(n) x T (X; . .;.X*
*) ! X
__ __
The natural transformations of ii) __u: T ! S for u: T ! S in oe will have a*
*s components
the maps induced by the symmetric monoidal structure of Top :
Theory and Applications of Categories, Vol. 1, No. 5 *
* 110
(4.5.3)
1 x u: E(n) x T (C1; . .;.Cn) ! E(n) x S(CAE1(1); . .;.CAE1(n))
This map is compatible with the structure maps to X by naturality of u and t*
*he
compatibility (3.3.c) of the operad action ff with the symmetric group action o*
*n the
E(n). Thus it is a map in Null=X. __
The natural transformation of iii), ^a1: ! 1 will have components:
(4.5.4) __
C = 1 x C ! E(1) x C = 1(C)
induced by the inclusion of the operad's distinguished point 1 2 E(1).
This is a map over X by 3.3.b.
The natural transformation of iv), "aT[S1;...;Sn]will have as components the*
* maps
induced by the composition law of the operad fl:
(4.5.5)
E(n) x T E(k1) x S1(C11; . .;.C1k1); . .;.E(kn) x Sn(Cn1; . .;.Cnkn)
~=
fflffl
E(n) x E(k1) x . .x.E(kn) x T (S1; . .;.Sn) C11; . .;.Cnkn
flxid
fflffl
E(k1 + . .+.kn) x T (S1; . .;.Sn) C11; . .;.Cnkn
This is a map over X by 3.3.d.
These data satisfy the conditions imposed because of the conditions satisfie*
*d by
the structure of an operad and of and operad action. Thus condition 2.3.1.a ho*
*lds
by naturality, conditions 2.3.1.b and 2.3.1.c by 3.1.b, 2.3.1.d by 3.1.d, and 2*
*.3.1.e by
naturality with 3.1.c and 3.3.c.
Finally, the construction is functorial, that is, it takes maps of operad ac*
*tions to
strict morphisms between lax symmetric monoidal categories. *
* 
4.6 Lemma. The functor Null=( ): E1 spaces! LaxSymMonStrict of 4.5 preserves
stable homotopy equivalences.
Proof. By group completion (1.6.5, 2.4.1, 3.5.1) it suffices to show that if f:*
* X ! Y is
a map of E1 spaces which induces an isomorphism on homology localized by the a*
*ction
of the monoid ss0:
(4.6.1) f*: ss0(X)1 H*(X; Z) ~! ss0(Y )1 H*(Y ; Z)
then is also an isomorphism the map:
B(Null=f)*: ss0(BNull=X)1 H*(BNull=X; Z) ~! ss0(BNull=Y )1 H*(BNull=Y ; Z)
Theory and Applications of Categories, Vol. 1, No. 5 *
* 111
But by 4.4 and 4.2, there is a chain of natural homotopy equivalences between t*
*he E1 
space Z and BNull=Z, inducing an isomorphism on H*( ; Z) and on the set ss0( *
*).
Thus it suffices to show this isomorphism respects the actions of of ss0 on the*
* homology
groups, and so induces an isomorphism of the localizations. For z 2 Z consider*
*ed as
a representative of a class in ss0Z, and for any choice of m in the contractibl*
*e E(2),
the action on H*(Z; Z) is the map on homology induced by the homotopy class of *
*the
endomorphism Z ! Z given by
(4.6.2) Z ~=m x z x Z E(2) x Z x Z ff2!Z
By naturality of the chain of homotopy equivalences, this map on homology agree*
*s with
the endomorphism of H*(BNull=Z; Z) induced by Z ! Z. This is the map on homology
induced by the endofunctor of Null=Z sending C ! Z to
(4.6.3) C ~=m x z x C E(2) x Z x Z ff2!Z
On the other hand, the action of z 2 ss0Z ~=ss0BNull=Z for the lax symmetric mo*
*noidal
structure is the map on homology induced by the endofunctor sending C ! Z to
__ ff2
(4.6.4) T (z ! Z; C ! Z) = E(2) x z x C ! E(2) x Z x Z ! Z
for any choice of T in the contractible oe(2). But the two endofunctors are hom*
*otopic,
since they are linked by the natural transformation with components
(4.6.5) C = m x z x C ,! E(2) x z x C
This shows the actions on homology by an element of ss0 are compatible under the
isomorphism induced by the chain of 4.4 and 4.2. Similarly, the isomorphism of*
* ss0Z
with ss0BNull=Z respects the translation action of ss0 on itself, and so is an *
*isomorphism
of monoids. Thus the two ss0 actions are isomorphic.

5 The main theorem
Recall that one has fixed a Grothendieck universe U and by convention considers*
* only
symmetric monoidal categories and spectra which are Usmall.
5.1 Theorem. Let Spectra0 be the category of 1connective spectra, and SymMon *
* the
category of symmetric monoidal categories (1.5). Then the functor (1.6)
Spt: SymMon ! Spectra0
induces an equivalence between their homotopy categories formed by inverting th*
*e stable
homotopy equivalences.
Theory and Applications of Categories, Vol. 1, No. 5 *
* 112
The inverse equivalence is induced by the functor:
(5.1.1) Spectra0 ! E1 spaces! LaxSymMon ! SymMon
which is the composite of the zeroth space functor 1 : Spectra0 ! E1 spacesof
3.5, the functor Null=( ): E1 spaces ! LaxSymMonStrict of 4.5, and the func*
*tor
S: LaxSymMon ! SymMon of 2.5.
Proof. The functors of 2.5, 3.5, and 4.5 all preserve stable homotopy equivalen*
*ces.
Thus the composite functor does induce a functor on the homotopy categories. It
remains to show the two functors are inverse on the homotopy categories.
One already knows by 4.5 that Spectra0 and E1 spaces are linked by functo*
*rs
1 and Spt'inducing inverse equivalences of homotopy categories. Similarly, by *
*1.9.2,
the inclusions between all the variants of SymMon listed in 1.5.1 induce equi*
*valences
of homotopy categories. Moreover one knows (3.6.2) that the restriction of Spt*
* to
UniSymMonStrict is linked by a chain of natural stable homotopy equivalences *
*to the
composite of a lift of the classifying space functor B: UniSymMonStrict ! E1 *
*spaces
and the May machine functor Spt': E1 spaces! Spectra0 .
In light of this, to prove the two functors of the theorem are inverse to ea*
*ch other on
the homotopy category, it suffices to show:
a) The composite functor
SNull=( )
UniSymMonStrict B! E1 spaces!SymMon
is linked to the inclusion functor by natural stable homotopy equivalenc*
*es.
b) The composite functor
Spectra0 ! SymMon ! Spectra0
is linked to the identity functor by natural stable homotopy equivalence*
*s.
These will be proved in the course of this section. The statement a) will be*
* proven
by direct construction of the link, and b) will result from the uniqueness theo*
*rem for
infinite loop space machines of [MaT]. *
* 
5.2: proof of 5.1.a).
In order to prove 5.1.a) it suffices to construct a chain of functors and na*
*tural stable
homotopy equivalences linking the composite
Null=( )
(5.2.1) UniSymMonStrict B! E1 spaces!LaxSymMon
to the inclusion I of UniSymMonStrict into LaxSymMon . For then on applyin*
*g the
functor S: LaxSymMon ! SymMon of 2.5 to this link one obtains a link of SNu*
*ll=B( )
Theory and Applications of Categories, Vol. 1, No. 5 *
* 113
to SI, which in turn is linked to the inclusion of UniSymMonStrict in SymMon *
* by the
natural stable homotopy equivalence j of 2.5.
I will need to define a functor
(5.2.2) Null_=( ): UniSymMonStrict ! LaxSymMon
which will be the analog for categories of Null=( ) for spaces. First, I will*
* construct
the underlying endofunctor of Cat and show there is a natural homotopy equivale*
*nce
j from the identity to this endofunctor. For A a category, let Null_=A be the c*
*ategory
whose objects are (C; c) where C is a Usmall category such that BC is contract*
*ible and
c: C ! A is a functor. A morphism in Null_=A from (C; c) to (C0; c0) is a funct*
*or fl: C !
C0 such that c = c0fl. Let Term_=A be the full subcategory of those (C; c) such*
* that C has
a terminal object. For each such C, chose a terminal tC out of the isomorphism *
*class of
terminal objects. Then one may construct a functor ae0: Term_=A ! A sending the*
* object
(C; c) to the image of the terminal object, c(tC ). ae0sends the morphism (C; c*
*) ! (C0; c0)
to c0of the unique morphism in C0 from the image of tC to the terminal object t*
*C0. This
functor ae0is not strictly natural in A because of the choice of terminal objec*
*t. Consider
now the full subcategory __=A whose objects are those (C; c) for which C is the*
* total
order {0 < 1 < . .<.n} for some n. Since here the terminal object n is unique,*
* the
restriction of ae0 gives a functor ae: __=A ! A, is strictly natural in A. More*
*over, given
any object A 2 A, the comma category ae=A has a terminal object (({0}; c)1: c(0*
*) = A),
and so is contractible. By Quillen's Theorem A ([Qu] x1) it follows that ae is *
*a homotopy
equivalence of categories. This statement is the analog of Prop. 4.2. As in t*
*he proof
of Lemma 4.4, the comma categories =(C; c) of the inclusion : __=A ! Null_=A are
isomorphic to __=C and hence are homotopy equivalent to the contractible C. The*
*n by
Quillen's Theorem A the inclusion is a homotopy equivalence. Similarly, the in*
*clusion
of __=A into Term_=A is a homotopy equivalence. Thus Null_=A is linked by a cha*
*in of
homotopy equivalences of categories to A. In fact the following natural functor
(5.2.3) j: A ! Null_=A
is a homotopy equivalence. This j sends an object A to the canonical forgetful *
*functor
from the comma category C = A=A ! A. j sends a morphism A= ! A0to the canonical
map of comma categories A=A ! A=A0. As each A=A has a terminal object, j is the
composite of a functor j0 into Term_=A and the homotopy equivalence given by the
inclusion of the latter in Null=A. But ae0j0 = 1, and since ae0 is a homotopy e*
*quivalence,
so is j0 and j. This completes the proof of the chain of homotopy equivalences*
* of
categories for a general category A.
Now return to the case where A runs over UniSymMonStrict . Let be the club*
* for
strict unital symmetric monoidal categories. Then as in 3.6, the natural action*
* of on
the unital symmetric monoidal A is expressed by action maps (3.6.1). The actio*
*n of
the E1 operad E(n) = B(n) on BA is given by applying the classifying space fun*
*ctor
B to the categorical action (3.6.1). Then replacing everywhere in the construct*
*ion 4.5
Theory and Applications of Categories, Vol. 1, No. 5 *
* 114
of the lax symmetric monoidal structure on Null=X the operad E(n) = B(n) by the
categories (n) of 2.1.4, and Top by Cat, one finds a construction of a lax symm*
*etric
monoidal category structure on Null_=A strictly natural for A in UniSymMonStric*
*t .
There is a natural strict morphism of lax symmetric monoidal categories:
(5.2.4) B: Null_=A ! Null=BA
This B sends the object (C; c: C ! A) to (BC; Bc: BC ! BA). Moreover, this B
induces a homotopy equivalence on the underlying categories. For since B({0 < .*
* .<.
n}) = [n], B restricts to a functor __=A ! =BA, and one has a commutative ladder
of classifying spaces whose sides are given by the homotopy equivalences of 4.2*
* and 4.4,
and the maps induced by their above categorical analogs:
(5.2.5)
BNull_=A oo_~___ B__=A _~____//BA
BB   1
fflffl ~ fflffl~ fflffl
BNull=BA oo____B=BA ______//BA
This diagram shows that up to homotopy equivalence, the B of (5.2.4) is inde*
*ntified
to the identity map of A. A fortiori is a fortiori a natural stable homotopy eq*
*uivalence.
For A in UniSymMonStrict the natural homotopy equivalence j: A ! Null_=A of*
* 5.2.3
is a lax morphism of lax symmetric monoidal categories. To give j such a struct*
*ure, it
suffices considering the presentation of the club for lax symmetric monoidal ca*
*tegories
2.3.1 and the description of lax morphisms in terms of presentations ([Ke3] 4.3*
* [Ke4]
10.210.7) to give for each object T of type n in the club oe for symmetric mon*
*oidal
categories, a natural transformation
__
(5.2.6) jT : T(jA1 ; . .j.An) ! j(T (A1; . .A.n))
in Null_=A satisfying the following compatibilities with the natural transforma*
*tions which
are part of the lax symmetric monoidal structure.
(5.2.7.a) For each morphism u: T ! R in oe of type 2 n, the following diagram
commutes:
__ jT
T(jA1 ; . .;.jAn_)_________//j(T (A1; . .;.An))
_u(jA) j(u(A))

__ fflffl fflffl
R (jA11 ; . .;.jA1n__)jR//_j(R(A11 ; . .;.A1n )
Theory and Applications of Categories, Vol. 1, No. 5 *
* 115
(5.2.7.b) For each natural transformation "aT[R1;...Rn]as in 4.5.1.iv), the fol*
*lowing dia
gram commutes:
____ __T(jRi(Aij))_ jT(Ri(Aij)
T(R i)(jAj)___________//_T(j(Ri(Aij))__________//_j(T (Ri(Aij)))
"aT[Ri](Aj) =
_____fflffl_ fflffl
T (Ri)(jAj)______________jT(R__________________//_j(T (Ri(Aij)))
i)(Aj)
__
(5.2.7.c) For the natural transformation ^a: 1 ! 1 of 4.5.1.iii), the following*
* diagram
commutes:
1[j(A)]__=__//_j(1[A])
^a j(1A)
__ fflffl fflffl_
1[j(A)]__j1_//_j(1(A))
__
But for T of type n, T(jA1 ; . .j.An) is the contractible category over A
oe(n) x A=A1 x . .A.=An ! oe(n) x A x . .A.! A
Since T is a terminal object in oe(n) and Ai is terminal in A=Ai, the above *
*functor
factors canonically through the comma category A=T (A1; . .;.An) = j(T (A1; . .*
*;.An)).
Then letting jT be this factorization yields the required natural transformatio*
*n. It is
routine to check the properties (5.2.7) hold.
Thus the j of 5.2.3 is a natural stable homotopy equivalence between functors
SymMonStrict ! LaxSymMon . Composing this with the natural stable homotopy
equivalence of 5.2.4 yields a natural stable homotopy equivalence from the incl*
*usion
functor I to Null=B( ), as required to complete the proof of 5.1.a.
5.3: proof of 5.1.b.
It remains to link the functor Spectra0 ! SymMon ! Spectra0 to the ident*
*ity
functor by a chain of natural stable homotopy equivalences.
After composing with the zeroth space functor ( )0: Spectra! Top , one has *
*the
chain of homotopy equivalences of spaces, all naturally in X 2 Spectra:
(5.3.1) Spt (Null=X0)0 ~ B(Null=X0) ~ B(=X0) ~! X0
The second and third maps of 5.3.1 are the natural homotopy equivalences of *
*4.2 and
4.4. The first map is the canonical group completion map 1.6.1, which is a homo*
*topy
Theory and Applications of Categories, Vol. 1, No. 5 *
* 116
equivalence in this case. For as in the proof of 4.6, ss0B(Null=X0) is isomorp*
*hic as a
monoid to ss0X0 and so is already a group.
It would be quite difficult to specify by hand a chain of E1 homotopy equiv*
*alences
between the two ends of 5.3.1. Instead, I will circumvent the need to do this *
*by the
following Lemma. Its proof will complete the proof of 5.1.b and hence of the Th*
*eorem
5.1. It is essentially an avatar of the MayThomason uniqueness theorem for in*
*finite
loop space machines [MaT].
5.3.2 Lemma. Let F : Spectra0 ! Spectra0 be an endofunctor of the category of 
1connective spectra. Suppose there exists a chain of functors Gi: Spectra0 ! *
*Top and
of natural homotopy equivalences of spaces:
(5.3.2.1) (F )0 = G0 ~ G1 ~! . ..~ Gn1 ~! Gn = ( )0
Then there is a chain of endofunctors of Spectra0 and natural stable homoto*
*py
equivalences linking F to Id.
Proof. Recall (1.6.5) that a map of 1connective spectra is a stable homotopy *
*equiva
lence if and only if it induces a homotopy equivalence on the zeroth space. Fro*
*m this and
the chain of homotopy equivalences of zeroth spaces . .G.i. .o.ne sees that F p*
*reserves
stable homotopy equivalences. Also all the Gi send products of spectra to produ*
*cts of
spaces, at least up to homotopy.
Consider the category of "special spaces " of Segal [Se], the category of f*
*unctors
from the category of finite based sets op to Top that take wedges to products u*
*p to
homotopy. Segal's infinite loop space machine Sg is a functor from the categor*
*y of
special spaces to Spectra0 . By [Se] x3 there is a functor is the opposite di*
*rection,
with the two composites linked to the identity by stable homotopy equivalences.*
*  From
this it follows that it suffices to link F Sg to Sg by a chain of natural stabl*
*e homotopy
equivalences of functors from special spaces to Spectra0 .
But this can be shown to hold by a slight modification of the proof of [MaT]*
* 2.5. In
more detail, one has a functor given by smash product of finite based sets:
op x op ! op
If A: op ! Top is a special space, then the induced A: opx op ! Top is such th*
*at
for each p q 7! A(pq) is a special space. Indeed, as p varies it is a specia*
*l (special
space), so op ! Spectra0 sending p to Sg(q 7! A(pq)) is homotopy equivalent t*
*o the
pfold product of the spectrum Sg(q 7! A(1q)). Applying the chain (5.3.2.1) of *
*product
preserving natural homotopy equivalences of spaces to the spectra Sg(q 7! A(pq)*
*) yields
a chain of homotopy equivalences of special spaces linking p 7! (F Sg(q 7! A(p*
*q)))0 to
p 7! (Sg(q 7! A(pq)))0. The Segal machine gives a group completion map natural *
*in the
space A(p*): A(p1) ! (Sg(q 7! A(pq)))0. Thus it induces a homotopy equivalence*
* of
associated spectra Sg(p 7! A(p)) ~! Sg(p 7! (Sg(q 7! A(pq)))0). Applying Sg(p *
*7! )
Theory and Applications of Categories, Vol. 1, No. 5 *
* 117
to the chain of homotopy equivalences of special spaces gives a chain of natur*
*al stable
homotopy equivalences from the latter spectrum to Sg(p 7! (F Sg(q 7! A(pq)))0).*
* By
the "over and across theorem" [MaT] 3.9 for bispectra associated to special sp*
*ectra,
there is a natural chain of stable homotopy equivalences linking Sg(p 7! (F Sg(*
*q 7!
A(pq)))0) to F Sg(q 7! A(1q)) = F Sg(A). Combining the above chains give the re*
*quired
chain of stable homotopy equivalences linking Sg to F Sg. *
* 
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Institut Mathematique de Jussieu (CNRS),
Universite Paris VII, Case Postale 7012,
75251 Paris CEDEX 05, FRANCE
Email: thomason@frmap711.mathp7.jussieu.fr
This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anonym*
*ous ftp
at ftp://ftp.tac.mta.ca/pub/tac/volumes/1995/n5/v1n5.{dvi,ps}.
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