Smodules and symmetric spectra
Stefan Schwede 1
Abstract: We study a symmetric monoidal adjoint functor pair between t*
*he category
of Smodules and the category of symmetric spectra. The functors induc*
*e equivalences
between the respective homotopy categories of spectra, module spectra *
*and ring spectra.
1 Introduction
Stable homotopy theory studies the homotopy category of spectra. This category *
*has a symmetric monoidal
smash product which allows the definition of ring spectra `up to homotopy'. In *
*recent years there was an
increasing interest in more refined notions of ring spectra which are associati*
*ve (and possibly commutative)
up to coherent homotopy, and a complex machinery developed around this issue. T*
*he coherence questions
can be avoided if there is a model for the category of spectra (not just its ho*
*motopy category) which admits
a symmetric monoidal smash product. For a long time no such category was known,*
* and there was even
evidence that it might not exist [L].
Then at approximately the same time, two categories of spectra with nice smash *
*products were discov
ered. Elmendorf, Kriz, Mandell and May constructed the category of Smodules [E*
*KMM ], and Jeff Smith
introduced symmetric spectra [HSS]. Both categories are Quillen model categori*
*es and have associated
notions of ring and module spectra. However these two categories arise in compl*
*etely different ways. And
even though the homotopy categories are equivalent, it is not a priori clear if*
* both frameworks give rise
to the same homotopy theory of rings and modules. Both categories have their me*
*rits, described in detail
in the introductions of [EKMM ] and [HSS], and it is desirable to be able to t*
*ranslate results obtained in
one category into conclusions valid in the other. The present paper describes a*
*n easy mechanism which
facilitates such comparisons.
Below we define a lax symmetric monoidal functor : MS ! Sp from the category*
* of Smodules to
the category of symmetric spectra. The functor preserves homotopy groups and h*
*as a strong symmetric
monoidal left adjoint. We show that the two functors induce inverse equivalence*
*s of the homotopy categories
of spectra, ring spectra, commutative ring spectra and module spectra:
Main Theorem. The functor from the category MS of Smodules to the category Sp*
* of symmetric
spectra passes to a symmetric monoidal equivalence of homotopy categories
~=
Ho (MS) ! Ho (Sp ) :
Furthermore, induces equivalences of homotopy categories
~=
Ho(Salgebras)!Ho (symmetric ring spectra) ;
~=
Ho(com. Salgebras)!Ho(com. symmetric ring spectra) ;
~=
and Ho (MR) ! Ho ((R)mod)
for any Salgebra R.
The functor and its left adjoint are not quite a Quillen equivalence with resp*
*ect to the standard stable
model category structure of symmetric spectra [HSS, 3.4.4]. However, there is a*
* slight variation which leads
to a Quillen equivalence, see Remark 2.2.
________________________________1
Research supported by BASFForschungsstipendium der Studienstiftung des deut*
*schen Volkes
May 7, 1998; 1991 AMS Math. Subj. Class.: 55U35
1
We assume a certain familiarity with the category of symmetric spectra, but for*
* most of this paper Sections
13 of [HSS] suffice. On the other hand, no deep knowledge about the internal s*
*tructure of the category of
Smodules is required, and we will mainly rely on the following formal properti*
*es: (i) the category MS is a
simplicial and monoidal model category [HSS, 5.3.2]; (ii) the model category MS*
* is stable in the sense that
homotopy fibre sequences and homotopy cofibre sequences coincide; (iii) for any*
* cofibrant approximation
Sc ! S of the sphere Smodule and any Smodule M the induced map Sc^M ! S ^M *
*~=M is a
weak equivalence (this follows from [EKMM , III 3.8] since every cofibrant Sm*
*odule is a retract of a cell
Smodule). In fact our comparison method generalizes from Smodules to model ca*
*tegories sharing these
formal properties, see Remark 2.3. When using the terms `fibrations' or `cofibr*
*ations' in the context of
Smodules, we refer the model category structure of [EKMM , VII 4.6], where th*
*ose notions are called
`qfibrations' and `qcofibrations'. In the context of symmetric spectra, we w*
*ork with the stable model
category structure of [HSS, 3.4.4], unless otherwise stated. The unit of the sm*
*ash product (i.e., the sphere
spectrum) is denoted `S' in both [EKMM ] and [HSS]. In order to distiguish bet*
*ween these two objects we
use the notation S for the symmetric sphere spectrum.
The present paper could not have been written without an ongoing collaboration *
*with Mike Mandell, Peter
May and Brooke Shipley. It is a spinoff of our joint effort to understand the *
*relationship between various
spectra categories. I learned a lot of what I know about Smodules and symmetri*
*c spectra through our
extensive discussions. I would also like to thank Jeff Smith for always keepin*
*g me informed about the
progress of his work on commutative symmetric ring spectra, and Mike Hopkins an*
*d Charles Rezk for
explaining the operad trick of Lemma 3.1 to me.
2 Formal and homotopical properties
Since we are mainly working simplicially, we briefly recall how the topological*
* model category of Smodules
[EKMM , VII 4.2] can be regarded as a simplicial model category [Q , II.2 Def.*
* 2]. For every pointed simplicial
set K and every Smodule M we can use the topological enrichment and define the*
* smash product K ^M
to be the smash product K ^M of the geometric realization of K with M. Simila*
*rly we can define the
function Smodule (or cotensor) MK of K with M to be the function Smodule FMS *
*(K; M). Finally,
for two Smodules M and N we define the homomorphism simplicial set homMS (M; N*
*) to be the singular
complex of the topological mapping space MS(M; N). An application of the singul*
*ar complex functor to
the topological adjunction isomorphisms [EKMM , VII 2.2] gives adjunction isom*
*orphism of simplicial hom
sets
homMS (K ^M; N) ~= homMS(M; NK ) ~= homS*(K; homMS(M; N)) :
Here homS*denotes the simplicial hom set in the category S* of pointed simplici*
*al sets. Since the singular
complex functor takes Serre fibrations of spaces to Kan fibrations of simplicia*
*l sets, Quillen's compatibility
axiom SM7 of [Q , II.2 Def. 2] follows from the corresponding topological axiom*
* [EKMM , VII 4.3]. For the
purpose of this paper we can now forget about the topological enrichment of the*
* category of Smodules
and simply treat it as a simplicial model category.
In order to define the functor from the category MS of Smodules to the catego*
*ry Sp of symmetric
spectra we start by choosing a desuspension of the sphere Smodule. By definiti*
*on, such a desuspension con
sists of a cofibrant Smodule S1ctogether with a weak equivalence S1^S1c! S,*
* where S1 = [1]=@[1]
denotes the simplicial circle. The functor is then given by
(M)n = homMS (S1c^:^:S:1c_____z____"; M) :
n
2
The symmetric group acts on the mapping space by permuting the smash factors of*
* the source. The 0th
smash power is taken to be S, the unit of the smash product. The mfold smash p*
*ower of the desuspension
map S1^S1c! S induces a map
hom MS((S1c)^n; M) ! homMS (Sm ^(S1c)^(m+n); M) ~=homS*(Sm ; homMS((S1c*
*)^(m+n); M))
whose adjoint
Sm ^hom MS((S1c)^n; M) ! homMS((S1c)^(m+n); M)
makes (M) into a symmetric spectrum. For n 1, the Smodule (S1c)^n is a cofib*
*rant model of the (n)
sphere. So the functor takes weak equivalences of Smodules to maps which are *
*level equivalences above
level 0, and the ith homotopy group of the space (M)n is isomorphic to the (i*
*n)th homotopy group of
the Smodule M. In particular there is a natural isomorphism of stable homotopy*
* groups ss*(M) ~=ss*M,
and takes equivalences of Smodules to stable homotopy equivalences of symmetr*
*ic spectra.
The functor is lax symmetric monoidal: smashing maps induces
hom MS((S1c)^m; M) ^homMS ((S1c)^n; N) ! homMS ((S1c)^(m+n); M ^*
*N)
which assemble into a natural map (M) ^(N) ! (M ^N). The unit map S ! (S) co*
*mes from
the identity map of S which is a point in (S)0. (We remind the reader that we u*
*se the notation S for
the symmetric sphere spectrum).
The functor has a left adjoint functor : Sp ! MS which is strong symmetric *
*monoidal. To construct
we first define a functor : S* ! MS from the category of symmetric sequences*
* [HSS, 2.1.1] to the
category of Smodules by the formula
_
(X) = Xn^n (S1c)^n :
n0
By inspection of definitions takes the unit of the tensor product of symmetric*
* sequences [HSS, 2.1] to the
sphere Smodule and there is a natural, unital, associative and commutative iso*
*morphism (X) ^(Y ) ~=
(X Y ). In other words, is a strong symmetric monoidal functor. Furthermore, *
* is left adjoint to
the composite of with the forgetful functor from symmetric spectra to symmetri*
*c sequences, and the
adjunction maps are monoidal transformations. Since the symmetric sphere spectr*
*um S is a commutative
monoid in the category of symmetric sequences, the Smodule (S ) is a commutati*
*ve Salgebra, and
induces a strong symmetric monoidal functor
: Sp = (S modules inS*) ! M(S )
from the category of symmetric spectra to the category of (S )modules.
The smash powers of the desuspension map S1^S1c! S assemble into a homomorphi*
*sm of commutative
Salgebras _
(S ) = Sn ^n (S1c)^n ! S :
n0
So we can define the desired functor by extension of scalars as (A) = S ^(S ) *
*(A). The resulting
functor is left adjoint to . Furthermore, is strong symmetric monoidal and t*
*he adjunction maps
are monoidal transformations, basically because the same was true for . We can *
*also read off the values
of for free symmetric spectra and for spectra induced from symmetric sequences*
*: if X is a symmetric
sequence, then (S X) is isomorphic to (X). In particular for a free symmetric*
* spectrum FnK [HSS,
2.2.5] we obtain an isomorphism (FnK) ~=K ^(S1c)^n.
3
The functor preserves and reflects weak equivalences, so it passes to a functo*
*r on homotopy categories.
By the following theorem the unit of the adjunction A ! ((A)) is a stable equi*
*valence for all cofibrant
symmetric spectra. So applying to a functorial cofibrant replacement gives a f*
*unctor from symmetric
spectra to Smodules which on the level of homotopy categories is inverse to . *
* This proves the first
statement of the main theorem.
Theorem 2.1For every cofibrant symmetric spectrum A, the unit of the adjunction*
* A ! ((A)) is a
stable equivalence of symmetric spectra.
Proof: Our first claim is that for every cofibration A ! B of symmetric spectr*
*a which is an isomorphism
at level 0, the map (A) ! (B) is a cofibration of Smodules. This is equivalen*
*t to showing that for
every acyclic fibration of Smodules X ! Y , the map (X) ! (Y ) has the right*
* lifting property with
respect to the map A ! B. In this situation the map (X) ! (Y ) is a level acy*
*clic fibration above
level 0, and since A ! B is an isomorphism at level 0 and a cofibration above,*
* any lifting problem has a
solution.
As a consequence we obtain the following twooutofthree property: suppose A *
*! B is a cofibration of
symmetric spectra which is an isomorphism at level 0, and assume further that t*
*he unit of the adjunction
is a stable equivalence for two of the three symmetric spectra A; B and B=A. Th*
*en the unit map is also
a stable equivalence for the third spectrum. Since A ! B ! B=A is a cofibre s*
*equence of symmetric
spectra and symmetric spectra are left proper [HSS, 5.4.2], it suffices to show*
* that the sequence ((A)) !
((B)) ! ((B=A)) is also a homotopy cofibre sequence. Since preserves homotopy*
* groups, this
would follow from knowing that the sequence of Smodules (A) ! (B) ! (B=A) in*
*duces a long
exact sequence of homotopy groups. This in turn is a consequence of the fact t*
*hat the category of S
modules is also left proper and stable. Unfortunately, properness is not stated*
* explicitly in [EKMM ], hence
we argue as follows: by [EKMM , I 6.4] there is a long exact sequence of homot*
*opy groups when (B=A)
is substituted by the mapping cone of the map (A) ! (B) (note that the forgetf*
*ul functor from S
modules to Lspectra preserves mapping cones). Since (A) ! (B) is a cofibratio*
*n of Smodules, the
map
((A) ^[1]+) [(A) (B) ! (B) ^[1]+
is an acyclic cofibration. So the map Cone((A)) [(A) (B) ! ((B) ^[1]+)=(A) *
*obtained by
cobase change is also an acyclic cofibration. The target of the last map is ho*
*motopy equivalent to the
cofibre (B=A).
Now we claim that if the unit map is a stable equivalence for every member of a*
* family {Ai}i2Iof cofibrant
symmetric spectra, then it is a stable equivalence for the coproduct of the fam*
*ily. To see this we factor the
unit map for the coproduct as the composite
_ _ _ _
Ai ! ((Ai)) ! ( (Ai)) ~= (( Ai)) :
i2I i2I i2I i2I
A wedge of stable equivalences is again a stable equivalence. Furthermore, for*
* every family {Bi}i2Iof
symmetric spectra, the natural map
!
M _
ss*Bi !ss* Bi
i2I i2I
is an isomorphism. The same is true for any family of cofibrant Smodules, so *
*the second map in the
factorization is a stable homotopy equivalence. Hence the unit map for the wedg*
*e is a stable equivalence.
4
Now we prove the statement of the theorem for suspension spectra, i.e, symmetri*
*c spectra of the form
A = 1 K = K ^S for some pointed simplicial set K. The functor takes this symm*
*etric suspension
spectrum to the Smodule suspension spectrum K ^S. We claim that for every cofi*
*brant approximation
of the sphere Smodule Sc! S the simplicial set homMS (Sc; K ^S) is weakly equ*
*ivalent to the singular
complex of the infinite loop space 1 1 K in such a way that the composite map
K ! homMS (S; K ^S) ! homMS (Sc; K ^S)
corresponds to the inclusion K ! Sing(1 1 K). It suffices to verify this for*
* a single choice of cofibrant
approximation. If we use the model S0Sof [EKMM , II 1.7], then by the various *
*defining adjunctions, the
simplicial set homMS (S0S; K ^S) is isomorphic to the singular complex of the i*
*nfinite loop space of the
underlying coordinate free spectrum of K ^S, which proves the claim. Now we exa*
*mine the adjunction
map 1 K ! (K ^S) at positive level n. We consider the composite
nK = (1 K)n ! (K ^S)n = homMS ((S1c)^n; K ^S)
~!hom 1 ^n nn
MS((Sc ) ; (K ^S))
~= 1 1 ^n n
! hom MS((S ^Sc ) ; ( K) ^S) :
The second map is a weak equivalence since for every Smodule M the map M ! nn*
*M is a weak
equivalence between fibrant objects [EKMM , I 6.3]. Since the Smodule (S1^S1*
*c)^n is a cofibrant replace
ment of the sphere Smodule, the composite map is weakly equivalent to the (2n *
* 1)connected inclusion
nK ! Sing(1 1 nK). The map of symmetric spectra 1 K ! (K ^S) is thus a stab*
*le homo
topy equivalence.
The next case is that of a free symmetric spectrum FnK for a pointed simplicial*
* set K and some n 1.
Since this symmetric spectrum is cofibrant and trivial at level 0 it suffices (*
*by the twooutofthree property)
to show that the adjunction map for the nfold suspension of FnK is a stable eq*
*uivalence. This suspension
admits a stable equivalence nFnK ! 1 K which is the identity at level n. The f*
*unctor takes this
stable equivalence to the weak equivalence of Smodules
(nFnK) ~= K ^Sn^(S1c)^n ~!K ^S ~= (1 K) :
So the adjunction map for FnK is a stable equivalence because the one for 1 K i*
*s.
Now we consider reduced cofibrant symmetric spectra, i.e., those cofibrant A fo*
*r which A0 is a one point
simplicial set. We let Ireddenote the set of maps of symmetric spectra of the f*
*orm Fn@[r]+ ! Fn[r]+
for n 1 and r 0. This is precisely the subset of the generating stable cofibr*
*ations FI@ [HSS, 3.2.3] with
reduced sources and targets. We can factor the cofibration * ! A as an Iredco*
*fibration followed by an
Iredinjective map [HSS, 3.2.11], by a countable number of attachment steps. Th*
*is yields an Iredinjective
map Z ! A, which is thus a level acyclic fibration above level 0. Since the ma*
*p is also an isomorphism
at level 0, it is a trivial fibration of symmetric spectra and A is a retract o*
*f Z. We are reduced to showing
that the unit map of Z is a stable equivalence.
By construction the spectrum Z is the colimit Z = colimn0Zn of a countable sequ*
*ence of cofibrations
starting with the trivial symmetric spectrum. Furthermore every subquotient Zn*
*=Zn1 is a wedge of
spectra of the form Fn([r]=@[r]) for which we already proved the theorem. So w*
*ith the twooutof
three property we conclude inductively that the unit map is a stable equivalenc*
*e for every stage Zn in the
sequence. In every simplicial model category the colimit of a sequence of cofib*
*rations between cofibrant
objects admits a weak equivalence from the simplicial mapping telescope. We app*
*ly this to the sequence
{Zn}n0 of symmetric spectra and the sequence {(Zn)}n0 of Smodules. Then in *
*the commutative
5
diagram
~=
telnZn_______w((telnZn)) u____(teln(Zn))_
  
~   ~
  
u u u
Z ___________w((Z)) u____(colimn0(Zn))_~=
the left and right vertical maps are stable equivalences. The mapping telescope*
* is part of a cofibre sequence
!
_ _
Zn ! telnZn ! Zn :
n2N n2N
By the twooutofthree property the unit map is thus a stable equivalence for *
*the mapping telescope, hence
also for Z.
We have now shown that the conclusion of the theorem holds for all cofibrant re*
*duced symmetric spectra.
If A is an arbitrary cofibrant symmetric spectrum, then map 1 A0 ! A which is *
*the identity at level
0 is a cofibration [HSS, 5.2.2]. Since we know that the unit map for 1 A0 and f*
*or the reduced cofibrant
spectrum A=(1 A0) are equivalences, it is also one for A. *
* ___
Remark 2.2(Quillen equivalences) There is no strong symmetric monoidal functor *
*from the category
of symmetric spectra to the category of Smodules which is also a left Quillen *
*functor. The reason for this
is quite simple: any strong monoidal functor has to take the cofibrant symmetri*
*c sphere S to the non
cofibrant Smodule sphere. However and can be made into a Quillen equivalence*
* of model categories
by slightly restricting the class of cofibrations of symmetric spectra. One can*
* keep the stable equivalences
as weak equivalences, but take the cofibrations to be those stable cofibrations*
* which are an isomorphism at
level 0. The proof that this in fact defines another model category structure f*
*or symmetric spectra proceeds
along the lines of [HSS, Sec. 3.4]. As the proof of Theorem 2.1 shows, the func*
*tor preserves this restricted
notion of cofibration and acyclic cofibration. So and become a Quillen equiva*
*lence. The price to pay
is that the symmetric sphere spectrum is no longer cofibrant in the restricted *
*sense.
Remark 2.3For the definitions of the functors and and for the proof of Theore*
*m 2.1 we mainly
relied on formal properties of the model category of Smodules. In fact the con*
*structions make sense in
any cocomplete symmetric monoidal category C which is enriched over the categor*
*y of pointed simplicial
sets. We denote the monoidal product of C by ^ and the unit object by I. If w*
*e then fix an object
X of C and a morphism S1^X ! I, we can use X in place of S1cto define an adjo*
*int symmetric
monoidal functor pair and between C and the category of symmetric spectra. (I*
*n fact, this classifies
all strong symmetric monoidal functors L from the category of symmetric spectra*
* to C which preserve
simplicially enriched colimits: for any such L one obtains an object X = L(F1S0*
*) together with a map
S1^X ~=L(S1^F1S0) ! L(S ) ~=I.)
Now suppose C also has a compatible model category structure (`monoidal model c*
*ategory' in the sense of
[HSS, 5.3.2]) and satisfies the following additional property: for any cofibran*
*t replacement of the unit object
Ic ! I and for all objects Y the map Ic^Y ! I ^Y ~=Y is a weak equivalence. A*
*ssume further that
the unit object desuspends, i.e., we can choose a cofibrant object I1cand a we*
*ak equivalence S1^I1c~!I.
Then smashing with I1cis a functor which is inverse to suspension on the level*
* of homotopy categories.
So C is a stable model category. Using I1cwe obtain adjoint functors and whi*
*ch are a Quillen pair if
and only if the unit I is cofibrant. We refrain from axiomatizing the condition*
*s under which these functors
induce equivalences of homotopy categories.
6
3 Ring, module and algebra spectra
It is now relatively easy to lift the comparison of Smodules and symmetric spe*
*ctra to a comparison of the
module and ring categories. This discussion will prove the remaining parts of t*
*he main theorem. Since the
functors and are symmetric monoidal, and the adjunction maps are monoidal tra*
*nsformations, they
pass to adjoint functors between the categories of (commutative) Salgebras and*
* (commutative) symmetric
ring spectra, as well as to module categories.
Ring spectra. The categories of Salgebras and symmetric ring spectra are model*
* categories with the
weak equivalences defined on underlying spectra ([EKMM , VII 4.8] and [HSS, 5.*
*5.3]). So preserves and
reflects weak equivalences between ring spectra. Furthermore, every cofibrant s*
*ymmetric ring spectrum
is also cofibrant as a symmetric spectrum [HSS, 5.5.3]. So Lemma 2.1 implies t*
*hat for every cofibrant
symmetric ring spectrum F, the unit of the adjunction F ! ((F)) is a stable eq*
*uivalence of symmetric
ring spectra. Since symmetric ring spectra can functorially be replaced by cofi*
*brant ones, the functors
and induce inverse equivalences of homotopy categories
Ho(Salgebras) ~= Ho(symmetric ring spectra) :
Algebras over simplicial operads. It will be useful to extend our discussion fr*
*om the case of associative
or commutative algebras to algebras over operads [May]. By definition, a simpli*
*cial operad is an operad in
the category of simplicial sets, with cartesian product as the monoidal product*
*. We obtain strong symmetric
monoidal functors from the category of simplicial sets to either the category o*
*f Smodules or the category
of symmetric spectra by forming the suspension spectrum of a simplicial set wit*
*h an extra basepoint. This
allows us to view all simplicial operads as operads in the category of Smodule*
*s or symmetric spectra. If
A is a simplicial operad we denote by MS[A] and Sp [A] the categories of Aalge*
*bras in the category of
Smodules and symmetric spectra respectively. In the context of Smodules this *
*means that the simplicial
operad A acts via its geometric realization which is a topological operad. Sinc*
*e the functors and are
symmetric monoidal and simplicial, they pass to an adjoint functor pair
_____
MS[A] u____w_Sp [A]
for every simplicial operad A. This includes the case of ring spectra and commu*
*tative ring spectra which
correspond to the associative and commutative operad. The comparison of ring sp*
*ectra indicates what the
crucial condition on an operad A is so that the functor passes to an equivalen*
*ces of homotopy categories
of the Aalgebras: if we knew that every algebra in Sp [A] can be replaced by a*
* stably equivalent algebra
whose underlying symmetric spectrum is cofibrant, then Theorem 2.1 would imply *
*the equivalence of
homotopy categories. This can in fact be arranged for cofibrant operads.
We call a morphism of simplicial operads a fibration (resp. weak equivalence) i*
*f it is termwise a fibration (resp.
weak equivalence) of simplicial sets. A simplicial operad is the same as a simp*
*licial object of operads over
the category of sets. Since free operads exist, this latter category has a set *
*of small projective generators.
Quillen's Theorem [Q , II.4 Thm. 4] shows that the termwise fibrations and weak*
* equivalences are part of a
simplicial model category structure for simplicial operads.
Lemma 3.1 (C. Rezk [R]) Let A be a cofibrant simplicial operad. Then there exi*
*sts a functor c :
Sp [A] ! Sp [A] and a natural stable equivalence cX ~! X of Aalgebras such t*
*hat the underlying
symmetric spectrum of cX is cofibrant.
7
Proof: Morally the functor c is a cofibrant approximation for Aalgebras. Howev*
*er we are not assuming
that the category Sp [A] has a model category structure, so we manufacture the *
*desired object directly
using the small object argument. We denote by IA the set of Aalgebra maps fre*
*ely generated by the
generating stable cofibrations of symmetric spectra FI@ of [HSS, 3.3.2]. Then c*
*X is defined by applying
the Aalgebra analog of the factorization lemma [HSS, 3.2.11], with respect to *
*the set IA, to the map from
the initial Aalgebra to X. This yields an IAinjective Aalgebra map cX ! X, *
*which in particular is an
acyclic fibration of underlying symmetric spectra.
To show that the underlying spectrum of cX is cofibrant we choose an acyclic fi*
*bration f : Y ! cX in
the category of symmetric spectra such that Y is a cofibrant symmetric spectrum*
*. We claim that Y can
be given an Aalgebra structure in such a way that f becomes a morphism of Aal*
*gebras. This will finish
the proof: by construction cX has the left lifting property for all Aalgebra m*
*aps which are also acyclic
fibrations of underlying spectra. So cX is a retract of Y , hence cofibrant as *
*a symmetric spectrum.
It remains to put an Aalgebra structure on Y , and here we will use that A is *
*cofibrant as an operad. We
denote by E(cX) the endomorphism operad of cX [May, Ex. 5]. This is a simplicia*
*l operad with nth space
equal to
E(cX)n = homSp ((cX)^n; cX) :
The map f : Y ! cX also has an endomorphism operad E(f) with nth space
E(f)n = homSp (Y ^n; Y ) xhomSp (Y ^n;cX)homSp((cX)^n; cX) :
The pullback is formed using the composition with f on the left and composition*
* with f^n on the right.
The given data correspond to a chain of operad morphisms
A ______E(cX)wu~u__E(f)_____E(Yw)
The left morphism is given by the Aalgebra structure on cX, and the other two *
*morphisms are projections.
Since Y is cofibrant as a symmetric spectrum, so is its nfold smash power for *
*all n 0. Since in addition the
map f is an acyclic fibration, the projection E(f)n ! E(cX)n is an acyclic fib*
*ration. So E(f) ! E(cX) is
an acyclic fibration of simplicial operads. Since A is cofibrant, there exists *
*an operad morphism A ! E(f)
covering the Aalgebra structure on cX. Such a lift corresponds precisely to an*
* Aalgebra structure on Y
for which the map f is a homomorphism. *
* ___
As a combination of Theorem 2.1 and Lemma 3.1 we obtain
Corollary 3.2Let A be a cofibrant simplicial operad and c the functor of Lemma *
*3.1. Then the functors
: MS[A] ! Sp [A] and the composite functor c : Sp [A] ! MS[A] pass to inver*
*se equivalences of
homotopy categories.
Commutative ring spectra. Since the functors and are symmetric monoidal, they*
* pass to a point set
level adjoint functor pair between the category of commutative Salgebras and t*
*he category of commutative
symmetric ring spectra. The right adjoint functor still preserves and reflects*
* weak equivalences and so
it passes to a functor on homotopy categories. However, the commutative operad *
*is not cofibrant and it is
not clear if every commutative symmetric ring spectrum can be replaced by a sta*
*bly equivalent one with
cofibrant underlying spectrum. The proof that commutative Salgebras and commut*
*ative symmetric ring
spectra have equivalent homotopy categories thus passes through E1 ring spectr*
*a.
We let E be a cofibrant replacement of the commutative operad in the model cate*
*gory of simplicial operads.
Any such E will be an E1 operad in the sense that its nth space is a free and*
* (nonequivariantly) weakly
8
contractible nsimplicial set. By a theorem of M. Mandell [Man , 12.2] the incl*
*usion (pullback) functor
from the category of commutative Salgebras to the category MS[E] of Ealgebras*
* induces an equivalence
of homotopy categories. By an unpublished theorem of J. Smith the same is true *
*for symmetric spectra:
the inclusion of the category of commutative symmetric ring spectra into the ca*
*tegory of Ealgebras induces
an equivalence of homotopy categories.
So in the commutative diagram
(com. Salgebras)__(com.wsymmetric ring spectra)

incl. incl.
 
u u
MS[E] _________________wSp [E]
the vertical and the lower horizontal functors pass to equivalences of homotopy*
* categories (see Lemma
3.1). Consequently, the functor identifies the homotopy category of commutativ*
*e Salgebras with the
homotopy category of commutative symmetric ring spectra
Ho(com. Salgebras) ~= Ho(com. symmetric ring spectra) :
Remark 3.3According to J. Smith there is a model category structure for commuta*
*tive symmetric ring
spectra in which the weak equivalences are the stable equivalences of underlyin*
*g symmetric spectra. Fur
thermore, in this model category the fibrant objects are those commutative ring*
* spectra whose underlying
symmetric spectra are spectra above level 0. Similarly any map between fibrant*
* objects which is a level
fibration (resp. level acyclic fibration) above level 0 is a model category fib*
*ration (resp. acyclic fibration).
This means that with respect to this model structure is a right Quillen functo*
*r from the category of
commutative Salgebras to the category of commutative symmetric ring spectra. I*
*t is quite conceivable
that there is a more direct proof, avoiding E1 operads, that and are in fact*
* a Quillen equivalence.
Remark 3.4The argument just used to compare the commutative ring spectra should*
* generalize to arbi
trary simplicial operads as follows. For every simplicial operad A, the categor*
*y MS[A] is a model category
with weak equivalences and fibrations defined on underlying Smodules ([Man , 1*
*2.1], [EKMM , VII 4.7]). If
A ! B is a (termwise and nonequivariant) weak equivalence of simplicial opera*
*ds, then by [Man , 12.2]
the pullback functor MS[B] ! MS[A] is the right adjoint of a Quillen equivalen*
*ce of model categories. In
particular it induces an equivalence of homotopy categories. We also expect tha*
*t the homotopy categories
of Aalgebras and Balgebras over symmetric spectra are equivalent. Every simpl*
*icial operad is equivalent
to a cofibrant one, so together with Lemma 3.1 this would imply that the functo*
*r induces an equivalence
of homotopy categories of algebras over any simplicial operad.
Module spectra. Modules over an Salgebra or a symmetric ring spectrum are in g*
*eneral not algebras
over a simplicial operad, but they behave quite similarly. We first consider a*
* cofibrant symmetric ring
spectrum F and look at the category of left Fmodules. For every Fmodule M the*
* Smodule (M) is
naturally a module over the Salgebra (F). Conversely, for every (F)module N, *
*the symmetric spectrum
(N) is naturally a ((F))module, which we regard as an Fmodule by restriction *
*of scalars along the
morphism F ! ((F)) of symmetric ring spectra. Moreover, the functors and are*
* again adjoint
when considered as functors between the categories of Fmodules and the categor*
*y of (F)modules. The
categories of Fmodules and (F)modules are model categories with weak equivale*
*nces and fibrations
defined on underlying spectra. Since F is a cofibrant symmetric ring spectrum, *
*every cofibrant Fmodule
will also be cofibrant as a symmetric spectrum (by an argument as in Lemma 3.1)*
*. So by Theorem 2.1 the
9
unit map M ! ((M)) is a stable equivalence for every cofibrant Fmodule M. So *
* and pass to
inverse equivalence of homotopy categories
Ho(M(F)) ~= Ho(Fmod) :
To prove the last statement in the main theorem, we start with an Salgebra R a*
*nd choose a cofibrant re
placement (R)c! (R) in the model category of symmetric ring spectra. We consid*
*er the commutative
diagram of module categories and functors
MR ________(R)modw
 
 
restr. restr.
 
u u
M((R)c) _____(R)cmod.w
A left adjoint to the upper horizontal functor is given by M 7! R ^((R))(M). Th*
*e vertical functors are
restriction of scalars along the weak equivalences of ring spectra ((R)c) ! R *
*and (R)c ! (R),
so they pass to equivalences of homotopy categories (compare [HSS, Thm. 5.5.9])*
*. The lower horizontal
functor induces an equivalence of homotopy categories by the previous paragraph*
*, hence so does the upper
functor.
References
[EKMM] A. D. Elmendorf, I. Kriz, M. A. Mandell J. P. May. Rings, modules, and a*
*lgebras in stable homotopy
theory. Mathematical Surveys and Monographs 47. Amer. Math. Soc., Provid*
*ence, RI, 1997. With an
appendix by M. Cole.
[HSS] M. Hovey, B. E. Shipley, J. H. Smith. Symmetric spectra. Preprint, 1998.
[L] L. G. Lewis, Jr. Is there a convenient category of spectra ? J. Pure App*
*l. Algebra 73 (1991), 233246.
[Man] M. A. Mandell. Algebraization of E1 ring spectra. Preprint, 1998.
[May] J. P. May. Definitions: operads, algebras and modules. Operads: Proceedi*
*ngs of Renaissance Conferences.
Contemp. Math. 202, Amer. Math. Soc., Providence, RI, 1997. 17.
[Q] D. G. Quillen. Homotopical Algebra. Lecture Notes in Math. 43, Springer,*
* 1967.
[R] C. W. Rezk. Spaces of Algebra Structures and Cohomology of Operads. PhD *
*thesis, Massachusetts Insti
tute of Technology, Cambridge, 1996.
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
schwede@math.mit.edu
10