CLASSIFICATION OF STABLE MODEL CATEGORIES
STEFAN SCHWEDE AND BROOKE SHIPLEY
Abstract: A stable model category is a setting for homotopy theory where the *
*suspension functor
is invertible. The prototypical examples are the category of spectra in the sen*
*se of stable homotopy
theory and the category of unbounded chain complexes of modules over a ring. In*
* this paper we develop
methods for deciding when two stable model categories represent `the same homot*
*opy theory'. We
show that stable model categories with a single compact generator are equivalen*
*t to modules over a
ring spectrum. More generally stable model categories with a set of generators *
*are characterized as
modules over a `ring spectrum with several objects', i.e., as spectrum valued d*
*iagram categories. We
also prove a Morita theorem which shows how equivalences between module categor*
*ies over ring spectra
can be realized by smashing with a pair of bimodules. Finally, we characterize *
*stable model categories
which represent the derived category of a ring. This is a slight generalizatio*
*n of Rickard's work on
derived equivalent rings. We also include a proof of the model category equival*
*ence of modules over the
Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for*
* a ring R.
1. Introduction
The recent discovery of highly structured categories of spectra has opened th*
*e way for a new
wholesale use of algebra in stable homotopy theory. In this paper we use this n*
*ew algebra of
spectra to characterize stable model categories, the settings for doing stable *
*homotopy theory,
as categories of highly structured modules. This characterization also leads to*
* a Morita theory
for equivalences between categories of highly structured modules.
The motivation and techniques for this paper come from two directions, namely*
* stable homo-
topy theory and homological algebra. Specifically, stable homotopy theory studi*
*es the classical
stable homotopy category which is the category of spectra up to homotopy. For o*
*ur purposes
though, the homotopy category is inadequate because too much information is los*
*t, for example
the homotopy type of mapping spaces. Instead, we study the model category of sp*
*ectra which
captures the whole stable homotopy theory. More generally we study stable model*
* categories,
those model categories which share the main formal property of spectra, namely *
*that the sus-
pension functor is invertible up to homotopy. We list examples of stable model*
* categories in
Section 2.
The algebraic part of the motivation arises as follows. A classical theorem, *
*due to Gabriel
[Gab62], characterizes categories of modules as the cocomplete abelian categori*
*es with a sin-
gle small projective generator; the classical Morita theory for equivalences be*
*tween module
categories (see for example [AF92 , x21, 22]) follows from this. Later Rickard*
* [Ric89, Ric91]
developed a Morita theory for derived categories based on the notion of a tilti*
*ng complex. In
this paper we carry this line of thought one step further. Spectra are the homo*
*topy theoretical
generalization of abelian groups and stable model categories are the homotopy t*
*heoretic ana-
logue of abelian categories (or rather their categories of chain complexes). Ou*
*r generalization of
Gabriel's theorem develops a Morita theory for stable model categories. Here th*
*e appropriate
___________
Date: August 2, 2001; 1991 AMS Math. Subj. Class.: 55U35, 55P42.
Research supported by a BASF-Forschungsstipendium der Studienstiftung des deu*
*tschen Volkes.
Research partially supported by an NSF Postdoctoral Fellowship.
1
2 STEFAN SCHWEDE AND BROOKE SHIPLEY
notion of a model category equivalence is a Quillen equivalence since these equ*
*ivalences preserve
the homotopy theory, not just the homotopy category, see 2.5.
We have organized our results into three groups:
Characterization of module categories. The model category of modules over a *
*ring
spectrum has a single compact generator, namely the free module of rank one. Bu*
*t module cate-
gories are actually characterized by this property. To every object in a stable*
* model category we
associate an endomorphism ring spectrum, see Definition 3.7.5. We show that if *
*there is a single
compact generator, then the given stable model category has the same homotopy t*
*heory as the
modules over the endomorphism ring spectrum of the generator (Theorem 3.1.1). M*
*ore generally
in Theorem 3.3.3, stable model categories with a set of compact generators are *
*characterized
as modules over a `ring spectrum with many objects,' or spectral category, see *
*Definition 3.3.1.
This is analogous to Freyd's generalization of Gabriel's theorem [Fre64, 5.3H p*
*.120]. Examples
of these characterizations are given in 3.2 and 3.4.
Morita theory for ring spectra. In the classical algebraic context Morita the*
*ory describes
equivalences between module categories in terms of bimodules, see e.g. [AF92 , *
*Thm. 22.1]. In
Theorem 4.1.2 we present an analogous result which explains how a chain of Quil*
*len equivalences
between module categories over ring spectra can be replaced by a single Quillen*
* equivalence given
by smashing with a pair of bimodules.
Generalized tilting theory. In [Ric89, Ric91], Rickard answered the question*
* of when
two rings are derived equivalent, i.e., when various derived module categories *
*are equivalent as
triangulated categories. Basically, a derived equivalence exists if and only if*
* a so-called tilting
complex exists. From our point of view, a tilting complex is a particular compa*
*ct generator for
the derived category of a ring. In Theorem 5.1.1 we obtain a generalized tiltin*
*g theorem which
characterizes stable model categories which are Quillen (or derived) equivalent*
* to the derived
category of a ring.
Another result which is very closely related to this characterization of stab*
*le model categories
can be found in [SS1] where we give necessary and sufficient conditions for whe*
*n a stable model
category is Quillen equivalent to spectra, see also Example 3.2 (i). These uni*
*queness results
are then developed further in [Sh2, Sch]. Moreover, the results in this paper f*
*orm a basis for
developing an algebraic model for any rational stable model category. This is c*
*arried out in [SS2]
and applied in [Sh1, GS].
In order to carry out our program it is essential to have available a highly *
*structured model
for the category of spectra which admits a symmetric monoidal and homotopically*
* well behaved
smash product before passing to the homotopy category. The first examples of su*
*ch categories
were the S-modules of [EKMM ] and the symmetric spectra of Jeff Smith [HSS ]; *
*by now several
more such categories have been constructed [Lyd98, MMSS ]. We work with symmetr*
*ic spectra
because we can replace stable model categories by Quillen equivalent ones which*
* are enriched
over symmetric spectra (Section 3.6). Also, symmetric spectra are reasonably e*
*asy to define
and understand and several other model categories in the literature are already*
* enriched over
symmetric spectra. The full strength of our viewpoint comes from combining enr*
*iched (over
symmetric spectra) category theory with the language of closed model categories*
*. We give
specific references throughout; for general background on model categories see *
*Quillen's original
article [Qui67], a modern introduction [DS95], or [Hov99] for a more complete o*
*verview.
We want to point out the conceptual similarities between the present paper an*
*d the work
of Keller [Kel94]. Keller uses differential graded categories to give an elega*
*nt reformulation
(and generalization) of Rickard's results on derived equivalences for rings. O*
*ur approach is
similar to Keller's, but where he considers categories whose hom-objects are ch*
*ain complexes
STABLE MODEL CATEGORIES 3
of abelian groups, our categories have hom-objects which are spectra. Keller do*
*es not use the
language of model categories, but the `P-resolutions' of [Kel94, 3.1] are basic*
*ally cofibrant-fibrant
replacements.
Notation and conventions: We use the symbol S* to denote the category of poin*
*ted sim-
plicial sets, and we use Sp for the category of symmetric spectra [HSS ]. The*
* letters C and
D usually denote model categories, most of the time assumed to be simplicial an*
*d stable. For
cofibrant or fibrant approximations of objects in a model category we use super*
*scripts (-)c and
(-)f. For an object X in a pointed simplicial model category we use the notatio*
*n X and X for
the simplicial suspension and loop functors (i.e., the pointed tensor and coten*
*sor of an object X
with the pointed simplicial circle S1 = [1]=@ [1]); one should keep in mind th*
*at these objects
may have the `wrong' homotopy type if X is not cofibrant or fibrant respectivel*
*y. Our notation
for various kinds of morphism objects is as follows: the set of morphisms in a *
*category C is
denoted `hom C'; the simplicial set of morphisms in a simplicial category is de*
*noted `map '; we
use `Hom ' for the symmetric function spectrum in a spectral model category (De*
*finition 3.5.1);
square brackets `[X, Y ]Ho(C)' denote the abelian group of morphisms in the hom*
*otopy category
of a stable model category C; and for objects X and Y in any triangulated categ*
*ory T we use the
notation `[X, Y ]T*' to denote the graded abelian group of morphisms, i.e., [X,*
* Y ]Tn= [X[n], Y ]T
for n 2 Z and where X[n] is the n-fold shift of X.
We want to write the evaluation of a morphism f on an element x as f(x). This*
* determines
the following conventions about actions of rings and ring spectra: the endomorp*
*hism monoid,
ring or ring spectrum End(X) acts on the object X from the left, and it acts on*
* the set (group,
spectrum) Hom (X, Y ) from the right. A module will always be a right module; t*
*his way the left
multiplication map establishes an isomorphism between a ring and the endomorphi*
*sm ring of
the free module of rank one. A T -R-bimodule is a (T op R)-module (or a (T op^*
* R)-module in
the context of ring spectra).
Organization: In Section 2 we recall stable model categories and some of thei*
*r properties,
as well as the notions of compactness and generators, and we give an extensive *
*list of examples.
In Section 3 we prove the classification theorems (Theorems 3.1.1 and 3.3.3). I*
*n Section 3.6 we
introduce the category Sp(C) of symmetric spectra over a simplicial model categ*
*ory C. Under
certain technical assumptions we show in Theorem 3.8.2 that it is a stable mode*
*l category
with composable and homotopically well-behaved function symmetric spectra which*
* is Quillen
equivalent to the original stable model category C. In Definition 3.7.5 we asso*
*ciate to an object
P of a simplicial stable model category a symmetric endomorphism ring spectrum *
*End(P ). In
Theorem 3.9.3 we then prove Theorem 3.3.3 for spectral model categories (Defini*
*tion 3.5.1), such
as for example Sp(C). This will complete the classification results. In Section*
* 4 we prove the
Morita context (Theorem 4.1.2) and in Section 5 we prove the tilting theorem (T*
*heorem 5.1.1).
In two appendices we consider modules over spectral categories, the homotopy in*
*variance of
endomorphism ring spectra and the characterization of Eilenberg-Mac Lane spectr*
*al categories.
Acknowledgments: We would like to thank Greg Arone, Dan Dugger, Bill Dwyer, M*
*ark
Hovey, Jeff Smith and Charles Rezk for inspiration and for many helpful discuss*
*ions on the
subjects of this paper.
2.Stable model categories
In this section we recall stable model categories and some of their propertie*
*s, as well as the
notions of compactness and generators, and we give a list of examples.
2.1. Structure on the homotopy category. Recall from [Qui67, I.2] of [Hov99, 6.*
*1] that the
homotopy category of a pointed model category supports a suspension functor w*
*ith a right
adjoint loop functor .
4 STEFAN SCHWEDE AND BROOKE SHIPLEY
Definition 2.1.1.A stable model category is a pointed closed model category for*
* which the
functors and on the homotopy category are inverse equivalences.
The homotopy category of a stable model category has a large amount of extra *
*structure, some
of which plays a role in this paper. First of all, it is naturally a triangulat*
*ed category (cf. [Ver96]
or [HPS97 , A.1]). A complete reference for this fact can be found in [Hov99, 7*
*.1.6]; we sketch
the constructions: by definition of `stable' the suspension functor is a self-e*
*quivalence of the
homotopy category and it defines the shift functor. Since every object is a two*
*-fold suspension,
hence an abelian co-group object, the homotopy category of a stable model categ*
*ory is additive.
Furthermore, by [Hov99, 7.1.11] the cofiber sequences and fiber sequences of [Q*
*ui67, 1.3] coincide
up to sign in the stable case, and they define the distinguished triangles. Sin*
*ce we required a
stable model category to have all limits and colimits, its homotopy category ha*
*s infinite sums
and products. So such a homotopy category behaves like the unbounded derived ca*
*tegory of an
abelian category. This motivates thinking of a stable model category as a homot*
*opy theoretic
analog of an abelian category.
We recall the notions of compactness and generators in the context of triangu*
*lated categories:
Definition 2.1.2.Let T be a triangulated category with infinite coproducts. A f*
*ull triangulated
subcategory of T (with shift and triangles induced from T ) is called localizin*
*g if it is closed
under coproducts in T . A set P of objects of T is called a set of generators i*
*f the only localizing
subcategory which contains the objects of P is T itself. An object X of T is co*
*mpact (also called
small of finite) if for any family of objects {Ai}i2Ithe canonical map
M a
[X, Ai]T ----! [X, Ai]T
i2I i2I
is an isomorphism. Objects of a stable model category are called `generators' o*
*r `compact' if
they are so when considered as objects of the triangulated homotopy category.
A triangulated category with infinite coproducts and a set of compact generat*
*ors is often
called compactly generated. We avoid this terminology because of the danger of *
*confusing it with
the terms `cofibrantly generated' and `compactly generated' in the context of m*
*odel categories.
2.2. Remarks.
(i)There is a convenient criterion for when a set of compact objects generate*
*s a triangulated
category. This characterization is well known, but we have been unable to f*
*ind a reference
which proves it in the form we need.
Lemma 2.2.1. Let T be a triangulated category with infinite coproducts and*
* let P be a
set of compact objects. Then the following are equivalent:
(i) The set P generates T in the sense of Definition 2.1.2.
(ii) An object X of T is trivial if and only if there are no graded maps fr*
*om objects of P
to X, i.e. [P, X]* = 0 for all P 2 P.
Proof.Suppose the set P generates T and let X be an object with the proper*
*ty that
[P, X]* = 0 for all P 2 P. The full subcategory of T of objects Y satisfyin*
*g [Y, X]* = 0 is
localizing. Since it contains the set P, it contains all of T . Taking Y = *
*X we see that the
identity map of X is trivial, so X is trivial.
The other implication uses the existence of Bousfield localization funct*
*ors, which in this
case is a finite localization first considered by Miller in the context of *
*the stable homotopy
category [Mil92]. For every set P of compact objects in a triangulated cat*
*egory with
infinite coproducts there exist functors LP (localization) and CP (colocali*
*zation) and a
STABLE MODEL CATEGORIES 5
natural distinguished triangle
CPX ----! X ----! LPX ----! CPX[1]
such that CPX lies in the localizing subcategory generated by P, and such t*
*hat [P, LPX]* =
0 for all P 2 P and X 2 T ; one reference for this construction is in the p*
*roof of [HPS97 ,
Prop. 2.3.17] see also 3.9.4. So if we assume condition (ii) then for all X*
* the localization
LPX is trivial, hence X is isomorphic to the colocalization CPX and thus co*
*ntained_in
the localizing subcategory generated by P. *
* |__|
(ii)Our terminology for `generators' is different from the use of the term in *
*category theory;
generators in our sense are sometimes called weak generators elsewhere. By *
*Lemma 2.2.1,
a set of generators detects if objects are trivial (or equivalently if maps*
* in T are isomor-
phisms). This notion has to be distinguished from that of a categorical gen*
*erator which
detects if maps are trivial. For example, the sphere spectra are a set of g*
*enerators (in the
sense of Definition 2.1.2) for the stable homotopy category of spectra. Fre*
*yd's generating
hypothesis conjectures that the spheres are a set of categorical generators*
* for the stable
homotopy category of finite spectra. It is unknown to this day whether the*
* generating
hypothesis is true or false.
(iii)An object of a triangulated category is compact if and only if its shifts*
* (suspension and loop
objects) are. Any finite coproduct or direct summand of compact objects is *
*again compact.
Compact objects are closed under extensions: if two objects in a distinguis*
*hed triangle are
compact, then so is the third one. In other words, the full subcategory of *
*compact objects
in a triangulated category is thick. There are non-trivial triangulated cat*
*egories in which
only the zero object is compact. Examples with underlying stable model cate*
*gories arise
for example as suitable Bousfield localizations of the category of spectra,*
* see [HS99, Cor.
B.13].
(iv)If a triangulated category has a set of generators, then the coproduct of *
*all of them is
a single generator. However, infinite coproducts of compact objects are in*
* general not
compact. So the property of having a single compact generator is something *
*special. In
fact we see in Theorem 3.1.1 below that this condition characterizes the mo*
*dule categories
over ring spectra among the stable model categories. If generators exist, t*
*hey are far from
being unique.
(v)In the following we often consider stable model categories which are cofibr*
*antly generated.
Hovey has shown [Hov99, Thm. 7.3.1] that a cofibrantly generated model cate*
*gory always
has a set of generators in the sense of Definition 2.1.2 (the cofibers of a*
*ny set of generating
cofibrations will do). So having generators is not an extra condition in th*
*e situation we
consider, although these generators may not be compact. See [Hov99, Cor. 7*
*.4.4] for
conditions that guarantee a set of compact generators.
2.3. Examples.
(i)Spectra. As we mentioned in the introduction, one of our main motivating e*
*xamples is
the category of spectra in the sense of stable homotopy theory. The sphere *
*spectrum is a
compact generator. Many model categories of spectra have been constructed, *
*for example
by Bousfield and Friedlander [BF78 ]; Robinson [Rob87a, `spectral sheaves']*
*; Jardine [Jar97,
`n-fold spectra']; Elmendorf, Kriz, Mandell and May [EKMM , `coordinate fr*
*ee spectra',
`L-spectra', `S-modules']; Hovey, Shipley and Smith [HSS , `symmetric spect*
*ra']; Lydakis
[Lyd98, `simplicial functors']; Mandell, May, Schwede and Shipley [MMSS , *
*`orthogonal
spectra', `W-spaces'].
(ii)Modules over ring spectra. Modules over an S-algebra [EKMM , VII.1] or m*
*odules
over a symmetric ring spectrum [HSS , 5.4.2] form proper, cofibrantly gener*
*ated, simplicial,
6 STEFAN SCHWEDE AND BROOKE SHIPLEY
stable model categories, see also [MMSS , Sec. 12]. In each case a module *
*is compact if and
only if it is weakly equivalent to a retract of a finite cell module. The f*
*ree module of rank
one is a compact generator. More generally there are stable model categorie*
*s of modules
over `symmetric ring spectra with several objects', or spectral categories,*
* see Definition
3.3.1 and Theorem A.1.1.
(iii)Equivariant stable homotopy theory. If G is a compact Lie group, there is*
* a category
of G-equivariant coordinate free spectra [LMS86 ] which is a stable model c*
*ategory. Modern
versions of this model category are the G-equivariant orthogonal spectra of*
* [MM ] and G-
equivariant S-modules of [EKMM ]. In this case the equivariant suspension *
*spectra of the
coset spaces G=H+ for all closed subgroups H G form a set of compact gene*
*rators. This
equivariant model category is taken up again in Examples 3.4 (i) and 5.1.2.
(iv)Presheaves of spectra. For every Grothendieck site Jardine [Jar87] constru*
*cts a proper,
simplicial, stable model category of presheaves of Bousfield-Friedlander ty*
*pe spectra; the
weak equivalences are the maps which induce isomorphisms of the associated *
*sheaves of
stable homotopy groups. For a general site these stable model categories do*
* not seem to
have a set of compact generators.
(v)The stabilization of a model category. In principle every pointed model ca*
*tegory
should give rise to an associated stable model category by `inverting' the *
*suspension functor,
i.e., by passage to internal spectra. This has been carried out for certain*
* simplicial model
categories in [Sch97] and [Hov]. The construction of symmetric spectra ove*
*r a model
category (see Section 3.6) is another approach to stabilization.
(vi)Bousfield localization. Following Bousfield [Bou75], localized model struc*
*tures for mod-
ules over an S-algebra are constructed in [EKMM , VIII 1.1]. Hirschhorn [H*
*ir99] shows that
under quite general hypotheses the localization of a model category is agai*
*n a model cat-
egory. The localization of a stable model category is stable and localizat*
*ion preserves
generators. Compactness need not be preserved, see Example 3.2 (iii).
(vii)Motivic stable homotopy. In [MV , Voe98] Morel and Voevodsky introduced th*
*e A1-
local model category structure for schemes over a base. An associated stabl*
*e homotopy
category of A1-local T -spectra (where T = A1=(A1-0) is the `Tate-sphere') *
*is an important
tool in Voevodsky's proof of the Milnor conjecture [Voe]. This stable homot*
*opy category
arises from a stable model category with a set of compact generators, see E*
*xample 3.4 (ii)
for more details.
2.4. Examples: abelian stable model categories. Some examples of stable model *
*cate-
gories are `algebraic', i.e., the model category is also an abelian category. M*
*ost of the time the
objects consist of chain complexes in some abelian category and depending on th*
*e choice of weak
equivalences one gets a kind of derived category as the homotopy category. A di*
*fferent kind of
example is formed by the stable module categories of Frobenius rings.
For algebraic examples as the ones below, our results are essentially covered*
* by Keller's paper
[Kel94], although Keller does not use the language of model categories. Also th*
*ere is no need
to consider spectra when dealing with abelian model categories: the second auth*
*or shows [Sh3]
that every cofibrantly generated, proper, abelian stable model category is Quil*
*len equivalent to
a DG-model category, i.e., a model category enriched over chain complexes of ab*
*elian groups.
(i)Complexes of modules. The category of unbounded chain complexes of left mo*
*dules over
a ring supports a model category structure with weak equivalences the quasi*
*-isomorphisms
and with fibrations the epimorphisms [Hov99, Thm. 2.3.11] (this is called t*
*he projective
model structure). Hence the associated homotopy category is the unbounded *
*derived
category of the ring. A chain complex of modules over a ring is compact if*
* and only
if it is quasi-isomorphic to a bounded complex of finitely generated projec*
*tive modules
STABLE MODEL CATEGORIES 7
[BN93 , Prop. 6.4]. We show in Theorem 5.1.6 that the model category of unb*
*ounded
chain complexes of A-modules is Quillen equivalent to the category of module*
*s over the
symmetric Eilenberg-Mac Lane ring spectrum for A. This example can be genera*
*lized in
at least two directions: one can consider model categories of chain complexe*
*s in an abelian
category with enough projectives (see e.g. [CH , 2.2] for a very general con*
*struction under
mild smallness assumptions). On the other hand one can consider model categ*
*ories of
differential graded modules over a differential graded algebra, or even a `D*
*GA with many
objects', alias DG-categories [Kel94].
(ii)Relative homological algebra. In [CH ], Christensen and Hovey introduces mo*
*del cat-
egory structures for chain complexes over an abelian category based on a pro*
*jective class.
In the special case where the abelian category is modules over some ring and*
* the projective
class consists of all summands of free modules this recovers the (projective*
*) model cate-
gory structure of the previous example. Another special case of interest is *
*the pure derived
category of a ring. Here the projective class consists of all summands of (p*
*ossibly infinite)
sums of finitely generated modules, see also Example 5.1.3.
(iii)Homotopy categories of abelian categories. For any abelian category A, the*
*re is a
stable model category structure on the category of unbounded chain complexes*
* in A with
the chain homotopy equivalences as weak equivalences, see e.g., [CH , Ex. 3.*
*4]. The associ-
ated homotopy category is usually denoted K(A). Such triangulated homotopy c*
*ategories
tend not to have a set of small generators; for example, Neeman [Nee01, E.3.*
*2] shows that
the homotopy category of chain complexes of abelian groups K(Z) does not hav*
*e a set of
generators whatsoever.
(iv)Quasi-coherent sheaves. For a nice enough scheme X the derived category of *
*quasi-
coherent sheaves D(qc=X) arises from a stable model category and has a set o*
*f compact gen-
erators. More precisely, if X is quasi-compact and quasi-separated, then the*
* so-called injec-
tive model structure exists. The objects of the model category are unbounded*
* complexes of
quasi-coherent sheaves of OX -modules, the weak equivalences are the quasi-i*
*somorphisms
and the cofibrations are the injections [Hov01, Cor. 2.3 (b)]. If X is separ*
*ated, then the
compact objects of the derived category are precisely the perfect complexes,*
* i.e., the com-
plexes which are locally quasi-isomorphic to a bounded complex of vector bun*
*dles [Nee96,
2.3, 2.5]. If X also admits an ample family of line bundles {Lff}ff2A, then *
*the set of line
bundles {Lfmf| ff 2 A, m 2 Z}, considered as complexes concentrated in dimen*
*sion zero,
generates the derived category D(qc=X), see [Nee96, 1.11]. This class of exa*
*mples contains
the derived category of a ring as a special case, but the injective model st*
*ructure is different
from the one mentioned in (i). Hovey [Hov01, Thm. 2.2] generalizes the injec*
*tive model
structure to abelian Grothendieck categories.
(v)The stable module category of a Frobenius ring. A Frobenius ring is defined *
*by the
property that the classes of projective and injective modules coincide. Impo*
*rtant examples
are finite dimensional Hopf-algebras over a field and in particular group al*
*gebras of finite
groups. The stable module category is obtained by identifying two module hom*
*omorphisms
if their difference factors through a projective module. Fortunately the two*
* different mean-
ings of `stable' fit together nicely; the stable module category is the homo*
*topy category
associated to an underlying stable model category structure [Hov99, Sec. 2].*
* Every finitely
generated module is compact when considered as an object of the stable modul*
*e category.
Compare also Example 3.2 (v).
(vi)Comodules over a Hopf-algebra. Suppose B is a commutative Hopf-algebra over*
* a
field. Hovey, Palmieri and Strickland introduce the category C(B) of chain c*
*omplexes of
injective B-comodules, with morphisms the chain homotopy classes of maps [HP*
*S97 , Sec.
9.5]. Compact generators are given by injective resolutions of simple comodu*
*les (whose
8 STEFAN SCHWEDE AND BROOKE SHIPLEY
isomorphism classes form a set). In [Hov99, Thm. 2.5.17], Hovey shows that*
* there is a
cofibrantly generated model category structure on the category of all chain*
* complexes of
B-comodules whose homotopy category is the category C(B).
2.5. Quillen equivalences. The most highly structured notion to express that tw*
*o model
categories describe the same homotopy theory is that of a Quillen equivalence. *
* An adjoint
functor pair between model categories is a Quillen pair if the left adjoint L p*
*reserves cofibrations
and trivial cofibrations. An equivalent condition is to demand that the right a*
*djoint R preserve
fibrations and trivial fibrations. Under these conditions, the functors pass to*
* an adjoint functor
pair on the homotopy categories, see [Qui67, I.4 Thm. 3], [DS95, Thm. 9.7 (i)] *
*or [Hov99, 1.3.10].
A Quillen functor pair is a Quillen equivalence if it induces an equivalence on*
* the homotopy
categories. A Quillen pair is a Quillen equivalence if and only if the followin*
*g criterion holds
[Hov99, 1.3.13]: for every cofibrant object A of the source category of L and f*
*or every fibrant
object X of the source category of R, a map L(A) -! X is a weak equivalence if *
*and only if its
adjoint A -! R(X) is a weak equivalence.
As pointed out in [DS95, 9.7 (ii)] and [Qui67, I.4, Thm. 3], in addition to i*
*nducing an equiva-
lence of homotopy categories, Quillen equivalences also preserve the homotopy t*
*heory associated
to a model category, that is, the higher order structure such as mapping spaces*
*, suspension and
loop functors, and cofiber and fiber sequences. Note that the notions of compac*
*tness, generators,
and stability are invariant under Quillen equivalences of model categories.
For convenience we restrict our attention to simplicial model categories (see*
* [Qui67, II.2]).
This is not a big loss of generality; it is shown in [RSS ] that every cofibran*
*tly generated, proper,
stable model category is in fact Quillen equivalent to a simplicial model categ*
*ory. In [Dug ],
Dugger obtains the same conclusion under somewhat different hypotheses. In bot*
*h cases the
candidate is the category of simplicial objects over the given model category e*
*ndowed with a
suitable localization of the Reedy model structure.
3. Classification theorems
3.1. Monogenic stable model categories. Several of the examples of stable model*
* categories
mentioned in 2.3 already come as categories of modules over suitable rings or r*
*ing spectra. This
is no coincidence. In fact, our first classification theorem says that every st*
*able model category
with a single compact generator has the same homotopy theory as the modules ove*
*r a symmetric
ring spectrum (see [HSS , 5.4] for background on symmetric ring spectra). This *
*is analogous to the
classical fact [Gab62, V1, p. 405] that module categories are characterized as *
*those cocomplete
abelian categories which posses a single small projective generator; the classi*
*fying ring is obtained
as the endomorphism ring of the generator.
In Definition 3.7.5 we associate to every object P of a simplicial, cofibrant*
*ly generated, stable
model category C a symmetric endomorphism ring spectrum End(P ). The ring of h*
*omotopy
groups ß*End (P ) is isomorphic to the ring of graded self maps of P in the hom*
*otopy category
of C, [P, P ]Ho(C)*.
For the following theorem we have to make two technical assumptions. We need *
*the notion
of cofibrantly generated model categories from [DHK ] which is reviewed in some*
* detail in [SS00,
Sec. 2] and [Hov99, Sec. 2.1]. We also need properness (see [BF78 , Def. 1.2] o*
*r [HSS , Def. 5.5.2]).
A model category is left proper if pushouts across cofibrations preserve weak e*
*quivalences. A
model category is right proper if pullbacks over fibrations preserve weak equiv*
*alences. A proper
model category is one which is both left and right proper.
Theorem 3.1.1. (Classification of monogenic stable model categories)
Let C be a simplicial, cofibrantly generated, proper, stable model category wit*
*h a compact generator
STABLE MODEL CATEGORIES 9
P . Then there exists a chain of simplicial Quillen equivalences between C and *
*the model category
of End(P )-modules.
C 'Q mod-End (P )
This theorem is a special case of the more general classification result Theo*
*rem 3.3.3, which
applies to stable model categories with a set of compact generators and which w*
*e prove in
Section 3.6. Furthermore if in the situation of Theorem 3.1.1, P is a compact o*
*bject but not
necessarily a generator of C, then C still `contains' the homotopy theory of En*
*d(P )-modules, see
Theorem 3.9.3 (ii) for the precise statement. In the Morita context (Theorem 4*
*.1.2) we also
prove a partial converse to Theorem 3.1.1.
3.2. Examples: stable model categories with a compact generator.
(i)Uniqueness results for stable homotopy theory. The classification theorem *
*above
yields a characterization of the model category of spectra: a simplicial, c*
*ofibrantly gen-
erated, proper, stable model category is simplicially Quillen equivalent to*
* the category of
symmetric spectra if and only if it has a compact generator P for which the*
* unit map of
ring spectra S -! End(P ) is a stable equivalence. The paper [SS1] is devot*
*ed to other
necessary and sufficient conditions for when a stable model category is Qui*
*llen equivalent
to spectra - some of them in terms of the homotopy category of C and the na*
*tural action
of the stable homotopy groups of spheres. In [Sch], this result is extended*
* to a uniqueness
theorem showing that the 2-local stable homotopy category has only one unde*
*rlying model
category up to Quillen equivalence. In both of these papers, we eliminate t*
*he technical
conditions `cofibrantly generated' and `proper' by working with spectra in *
*the sense of
Bousfield and Friedlander [BF78 ], as opposed to the Quillen equivalent sym*
*metric spec-
tra and `simplicial' by working with framings [Hov99, Chpt. 5]. In another *
*direction, the
uniqueness result is extended to include the monoidal structure in [Sh2].
(ii)Chain complexes and Eilenberg-Mac Lane spectra. Let A be a ring. Theorem 5*
*.1.6
shows that the model category of chain complexes of A-modules is Quillen eq*
*uivalent to
the model category of modules over the symmetric Eilenberg-Mac Lane ring sp*
*ectrum
HA. This can be viewed as an instance of Theorem 3.1.1: the free A-module *
*of rank
one, considered as a complex concentrated in dimension zero, is a compact g*
*enerator for
the unbounded derived category of A. Since the homotopy groups of its endom*
*orphism
ring spectrum (as an object of the model category of chain complexes) are c*
*oncentrated
in dimension zero, the endomorphism ring spectrum is stably equivalent to t*
*he Eilenberg-
Mac Lane ring spectrum for A (see Proposition B.2.1). This also shows that *
*although the
model category of chain complexes of A-modules is not simplicial it is Quil*
*len equivalent to
a simplicial model category. So although our classification theorems do not*
* apply directly,
they do apply indirectly.
(iii)Smashing Bousfield localizations. Let E be a spectrum and consider the E-*
*local model
category structure on some model category of spectra (see e.g. [EKMM , VII*
*I 1.1]). This is
another stable model category in which the localization of the sphere spect*
*rum LES0 is a
generator. This localized sphere is compact if the localization is smashing*
*, i.e., if a certain
natural map X ^ LES0 -! LEX is a stable equivalence for all X. So for a sm*
*ashing
localization the E-local model category of spectra is Quillen equivalent to*
* modules over
the ring spectrum LES0 (which is the endomorphism ring spectrum of the loca*
*lized sphere
in the localized model structure).
(iv)K(n)-local spectra. Even if a Bousfield localization is not smashing, The*
*orem 3.1.1
might be applicable. As an example we consider Bousfield localization with*
* respect to
the n-th Morava K-theory K(n) at a fixed prime. The localization of the sph*
*ere is still
10 STEFAN SCHWEDE AND BROOKE SHIPLEY
a generator, but for n > 0 it is not compact in the local category, see [HP*
*S97 , 3.5.2].
However the localization of any finite type n spectrum F is a compact gener*
*ator for the
K(n)-local category [HS99, 7.3]. Hence the K(n)-local model category is Qui*
*llen equivalent
to modules over the endomorphism ring End(LK(n)F ).
(v)Frobenius rings. As in Example 2.3 (iv) we consider a Frobenius ring and as*
*sume that the
stable module category has a compact generator. Then we are in the situatio*
*n of Theorem
3.1.1; however this example is completely algebraic, and there is no need t*
*o consider ring
spectra to identify the stable module category as the derived category of a*
* suitable `ring'.
In fact Keller shows [Kel94, 4.3] that in such a situation there exists a d*
*ifferential graded
algebra (DGA) and an equivalence between the stable module category and the*
* unbounded
derived category of the DGA.
A concrete example of this situation arises for group algebras of p-grou*
*ps over a field
k of characteristic p. In this case the trivial module is the only simple *
*module, and it
is a compact generator of the stable module category. More generally a resu*
*lt of Benson
[Ben94, Thm. 1.1] says that the trivial module generates the stable module *
*category of
the principal block of a group algebra kG if and only if the centralizer of*
* every element of
order p is p-nilpotent. So in this situation Keller's theorem applies and i*
*dentifies the stable
module category as the unbounded derived category of a certain DGA. The hom*
*ology
groups of this DGA are isomorphic (by construction) to the ring of graded s*
*elf maps of
the trivial module in the stable module category, which is just the Tate-co*
*homology ring
Hb*(G; k).
(vi)Stable homotopy of algebraic theories. Another motivation for this paper a*
*nd an early
instance of Theorem 3.1.1 came from the stabilization of the model category*
* of algebras over
an algebraic theory [Sch01]. For every pointed algebraic theory T , the cat*
*egory of simplicial
T -algebras is a simplicial model category so that one has a category Sp(T *
*) of (Bousfield-
Friedlander type) spectra of T -algebras, a cofibrantly generated, simplici*
*al stable model
category [Sch01, 4.3]. The free T -algebra on one generator has an endomor*
*phism ring
spectrum which is constructed as a Gamma-ring in [Sch01, 4.5] and denoted T*
* s. Then
[Sch01, Thm. 4.4] provides a Quillen equivalence between the categories of *
*connective
spectra of T -algebras and the category of T s-modules (the connectivity co*
*ndition could be
removed by working with symmetric spectra instead of -spaces). This fits w*
*ith Theorem
3.1.1 because the suspension spectrum of the free T -algebra on one generat*
*or is a compact
generator for the category Sp(T ). See [Sch01, Sec. 7] for a list of ring s*
*pectra that arise
from algebraic theories in this fashion.
Remark 3.2.1. The notion of a compact generator and the homotopy groups of the *
*endomor-
phism ring spectrum only depend on the homotopy category, and so they are invar*
*iant under
equivalences of triangulated categories. However, the homotopy type of the endo*
*morphism ring
spectrum depends on the model category structure. The following example illustr*
*ates this point.
Consider the n-th Morava K-theory spectrum K(n) for a fixed prime and some numb*
*er n > 0.
This spectrum admits the structure of an A1 -ring spectrum [Rob89]. Hence it al*
*so has a model
as an S-algebra or a symmetric ring spectrum and the category of its module spe*
*ctra is a stable
model category. The ring of homotopy groups of K(n) is the graded field Fp[vn, *
*v-1n] with vn of
dimension 2pn - 2. Hence the homotopy group functor establishes an equivalence *
*between the
homotopy category of K(n)-module spectra and the category of graded Fp[vn, v-1n*
*]-modules.
Similarly the homology functor establishes an equivalence between the derived*
* category of dif-
ferential graded modules over the graded field Fp[vn, v-1n] and the category of*
* graded Fp[vn, v-1n]-
modules. So the two stable model categories of K(n)-module spectra and DG-modu*
*les over
Fp[vn, v-1n] have equivalent triangulated homotopy categories (including the ac*
*tion of the stable
STABLE MODEL CATEGORIES 11
homotopy groups of spheres _ all elements in positive dimension act trivially i*
*n both cases).
But the endomorphism ring spectra of the respective free rank one modules are t*
*he Morava K-
theory ring spectrum on the one side and the Eilenberg-Mac Lane ring spectrum f*
*or Fp[vn, v-1n]
on the other side, which are not stably equivalent. Similarly the two model cat*
*egories are not
Quillen equivalent since for DG-modules all function spaces are products of Eil*
*enberg-Mac Lane
spaces, but for K(n)-modules they are not.
3.3. Multiple generators. There is a generalization of Theorem 3.1.1 to the cas*
*e of a stable
model category with a set of compact generators (as opposed to a single compact*
* generator).
Let us recall the algebraic precursors of this result: a ringoid is a categor*
*y whose hom-sets are
abelian groups with bilinear composition. Ringoids are sometimes called pre-add*
*itive categories
or rings with several objects. Indeed a ring in the traditional sense is the sa*
*me as a ringoid with
one object. A (right) module over a ringoid is defined to be a contravariant ad*
*ditive functor to
the category of abelian groups. These more general module categories have been *
*identified as
the cocomplete abelian categories which have a set of small projective generato*
*rs [Fre64, 5.3H,
p. 120]. An analogous theory for derived categories of DG categories has been d*
*eveloped by
Keller [Kel94].
Our result is very much in the spirit of Freyd's or Keller's, with spectra su*
*bstituting for abelian
groups or chain complexes. A symmetric ring spectrum can be viewed as a categor*
*y with one
object which is enriched over symmetric spectra; the module category then becom*
*es the category
of enriched (spectral) functors to symmetric spectra. So we now look at `ring s*
*pectra with several
objects' which we call spectral categories. This is analogous to pre-additive, *
*differential graded
or simplicial categories which are enriched over abelian groups, chain complexe*
*s or simplicial
sets respectively.
Definition 3.3.1.A spectral category is a category O which is enriched over the*
* category Sp of
symmetric spectra (with respect to smash product, i.e., the monoidal closed str*
*ucture of [HSS ,
2.2.10]). In other words, for every pair of objects o, o0 in O there is a morp*
*hism symmetric
spectrum O(o, o0), for every object o of O there is a map from the sphere spect*
*rum S to O(o, o)
(the `identity element' of o), and for each triple of objects there is an assoc*
*iative and unital
composition map of symmetric spectra
O(o0, o00) ^ O(o, o0) ----! O(o, o00) .
An O-module M is a contravariant spectral functor to the category Sp of symmet*
*ric spectra,
i.e., a symmetric spectrum M(o) for each object of O together with coherently a*
*ssociative and
unital maps of symmetric spectra
M(o) ^ O(o0, o) ----! M(o0)
for pairs of objects o, o0in O. A morphism of O-modules M -! N consists of maps*
* of symmetric
spectra M(o) -! N(o) strictly compatible with the action of O. We denote the ca*
*tegory of
O-modules by mod-O. The free (or `representable') module Fo is given by Fo(o0) *
*= O(o0, o).
Remark 3.3.2. In Definition 3.3.1 we are simply spelling out what it means to d*
*o enriched
category theory over the symmetric monoidal closed category Sp of symmetric sp*
*ectra with
respect to the smash product and the internal homomorphism spectra. Kelly's boo*
*k [Kly82] is an
introduction to enriched category theory in general; the spectral categories, m*
*odules over these
(spectral functors) and morphisms of modules as defined above are the Sp -categ*
*ories, Sp -
functors and Sp -natural transformations in the sense of [Kly82, 1.2]. So the p*
*recise meaning
of the coherence and compatibility conditions in Definition 3.3.1 can be found *
*in [Kly82, 1.2].
We show in Theorem A.1.1 that for any spectral category O the category of O-m*
*odules is
a model category with objectwise stable equivalences as the weak equivalences. *
*There we also
12 STEFAN SCHWEDE AND BROOKE SHIPLEY
show that the set of free modules {Fo}o2O is a set of compact generators. If O *
*has a single object
o, then the O-modules are precisely the modules over the symmetric ring spectru*
*m O(o, o), and
the model category structure is the one defined in [HSS , 5.4.2].
In Definition 3.7.5 we associate to every set P of objects of a simplicial, c*
*ofibrantly generated
stable model category C a spectral endomorphism category E(P) whose objects are*
* the members
of the set P and such that there is a natural, associative and unital isomorphi*
*sm
ß*E(P)(P, P 0) ~= [P, P 0]Ho(C)*.
For a set with a single element this reduces to the notion of the endomorphism *
*ring spectrum.
Theorem 3.3.3. (Classification of stable model categories) Let C be a simplicia*
*l, cofi-
brantly generated, proper, stable model category with a set P of compact genera*
*tors. Then there
exists a chain of simplicial Quillen equivalences between C and the model categ*
*ory of E(P)-
modules.
C 'Q mod-E(P)
There is an even more general version of Theorem 3.3.3 which also provides in*
*formation if
the set P does not generate the whole homotopy category, see Theorem 3.9.3 (ii)*
*. This variant
implies that for any set P of compact objects in a proper, cofibrantly generate*
*d, simplicial,
stable model category the homotopy category of E(P)-modules is triangulated equ*
*ivalent to the
localizing subcategory of Ho(C)generated by the set P.
The proof of Theorem 3.3.3 breaks up into two parts. In order to mimic the cl*
*assical proof for
abelian categories we must consider a situation where the hom functor Hom C(P, *
*-) takes values
in the category of modules over a suitable endomorphism ring spectrum of P . In*
* Section 3.6 we
show how this can be arranged, given the technical conditions that C is cofibra*
*ntly generated,
proper and simplicial. We introduce the category Sp(C) of symmetric spectra ove*
*r C and show
in Theorem 3.8.2 that it is a stable model category with composable and homotop*
*ically well-
behaved function symmetric spectra which is Quillen equivalent to the original *
*stable model
category C.
In Theorem 3.9.3 we prove Theorem 3.3.3 under the assumption that C is a spec*
*tral model cat-
egory (Definition 3.5.1), i.e., a model category with composable and homotopica*
*lly well-behaved
function symmetric spectra. Since the model category Sp(C) is spectral and Quil*
*len equivalent
to C (given the technical assumptions of Theorem 3.3.3), this will complete the*
* classification
results.
3.4. Examples: stable model categories with a set of generators.
(i)Equivariant stable homotopy. Let G be a compact Lie group. As mentioned in*
* Exam-
ple 2.3(iii) there are several versions of model categories of G-equivarian*
*t spectra. In [MM ],
G-equivariant orthogonal spectra are shown to form a cofibrantly generated,*
* topological
(hence simplicial), proper model category (the monoidal structure plays no *
*role for our
present considerations). For every closed subgroup H of G the equivariant *
*suspension
spectrum of the homogeneous space G=H+ is compact and the set G of these sp*
*ectra for
all closed subgroups H generates the G-equivariant stable homotopy category*
*, see [HPS97 ,
9.4].
Recall from [LMS86 , V x9] that a Mackey functor is a module over the st*
*able homotopy
orbit category, i.e., an additive functor from the homotopy orbit category *
*ß0E(G) to the
category of abelian groups; by [LMS86 , V Prop. 9.9] this agrees with the o*
*riginal algebraic
definition of Dress [Dre73] in the case of finite groups. The spectral endo*
*morphism category
E(G) is a spectrum valued lift of the stable homotopy orbit category. Theor*
*em 3.3.3 shows
that the category of G-equivariant spectra is Quillen equivalent to a categ*
*ory of topological
STABLE MODEL CATEGORIES 13
Mackey functors, i.e., the category of modules over the stable orbit catego*
*ry E(G). Note
that the homotopy type of each morphism spectrum of E(G) depends on the uni*
*verse U.
After rationalization the Mackey functor analogy becomes even more concr*
*ete: we will
see in Example 5.1.2 that the model category of rational G-equivariant spec*
*tra is in fact
Quillen equivalent to the model category of chain complexes of rational Mac*
*key functors for
G finite. For certain non-finite compact Lie groups, our approach via `topo*
*logical Mackey
functors' is used in [Sh1] and [GS ] as an intermediate step in forming alg*
*ebraic models for
rational G-equivariant spectra.
(ii)Motivic stable homotopy of schemes. In [MV , Voe98] Morel and Voevodsky in*
*troduce
the A1-local model category structure for schemes over a base. The objects *
*of their category
are sheaves of sets in the Nisnevich topology on smooth schemes of finite t*
*ype over a fixed
base scheme. The weak equivalences are the A1-local equivalences - roughly *
*speaking they
are generated by the projection maps X x A1 -! X for smooth schemes X, wher*
*e A1
denotes the affine line.
Voevodsky [Voe98, Sec. 5] introduces an associated stable homotopy categ*
*ory by invert-
ing smashing with the `Tate-sphere' T = A1=(A1 - 0). The punch-line is tha*
*t theories
like algebraic K-theory or motivic cohomology are represented by objects in*
* this stable
homotopy category [Voe98, Sec. 6], at least when the base scheme is the spe*
*ctrum of a
field.
In [Jar00b], Jardine provides the details of the construction of model c*
*ategories of T -
spectra over the spectrum of a field k. He constructs two Quillen equivalen*
*t proper, sim-
plicial model categories of Bousfield-Friedlander type and symmetric A1-loc*
*al T -spectra
[Jar00b, 2.11, 4.18]. Since T is weakly equivalent to a suspension (of the *
*multiplicative
group scheme), this in particular yields a stable model category. A set of *
*compact gener-
ators for this homotopy category is given by the T -suspension spectra 1T(*
*SpecR)+ when
R runs over smooth k-algebras of finite type. So if k is countable then thi*
*s is a countable
set of compact generators, compare [Voe98, Prop. 5.5].
(iii)Algebraic examples. Again the classification theorem 3.3.3 has an algebra*
*ic analogue
and precursor, namely Keller's theory of derived equivalences of DG categor*
*ies [Kel94].
The bottom line is that if an example of a stable model category is algebra*
*ic (such as
derived or stable module categories in Examples 2.4), then it is not necess*
*ary to consider
spectra and modules over spectral categories, but one can work with chain c*
*omplexes and
differential graded categories instead. As an example, Theorem 4.3 of [Kel*
*94] identifies
the stable module category of a Frobenius ring with the unbounded derived c*
*ategory of a
certain differential graded category.
3.5. Prerequisites on spectral model categories. A spectral model category is a*
*nalogous
to a simplicial model category, [Qui67, II.2], but with the category of simplic*
*ial sets replaced
by symmetric spectra. Roughly speaking, a spectral model category is a pointed *
*model cate-
gory which is compatibly enriched over the stable model category of symmetric s*
*pectra. The
compatibility is expressed by the axiom (SP) below which takes the place of [Qu*
*i67, II.2 SM7].
For the precise meaning of `tensors' and `cotensors' over symmetric spectra see*
* e.g. [Kly82, 3.7].
A spectral model category is the same as a `Sp -model category' in the sense of*
* [Hov99, Def.
4.2.18], where the category of symmetric spectra is endowed with the stable mod*
*el structure
of [HSS , 3.4.4]. Condition two of [Hov99, 4.2.18] is automatic since the unit *
*S for the smash
product of symmetric spectra is cofibrant. Examples of spectral model categorie*
*s are module
categories over a symmetric ring spectrum, module categories over a spectral ca*
*tegory (Theorem
A.1.1) and the category of symmetric spectra over a suitable simplicial model c*
*ategory (Theorem
3.8.2).
14 STEFAN SCHWEDE AND BROOKE SHIPLEY
Definition 3.5.1.A spectral model category is a model category C which is tenso*
*red, cotensored
and enriched (denoted Hom C) over the category of symmetric spectra with the cl*
*osed monoidal
structure of [HSS , 2.2.10] such that the following compatibility axiom (SP) ho*
*lds:
(SP) For every cofibration A -! B and every fibration X -! Y in C the induced m*
*ap
HomC(B, X) ----! Hom C(A, X) xHomC(A,YH)omC(B, Y )
is a stable fibration of symmetric spectra. If in addition one of the maps A -!*
* B or X -! Y
is a weak equivalence, then the resulting map of symmetric spectra is also a st*
*able equivalence.
We use the notation K ^ X and XK to denote the tensors and cotensors for X 2 C *
*and K a
symmetric spectrum.
In analogy with [Qui67, II.2 Prop. 3] the compatibility axiom (SP) in Definit*
*ion 3.5.1 of a
spectral model category can be cast into two adjoint forms, one of which will b*
*e of use for us.
Given a categorical enrichment of a model category C over the category of symme*
*tric spectra,
then axiom (SP) is equivalent to (SPb) below. The equivalence of conditions (SP*
*) and (SPb)
is a consequence of the adjointness properties of the tensor and cotensor funct*
*ors, see [Hov99,
Lemma 4.2.2] for the details.
(SPb) For every cofibration A -! B in C and every stable cofibration K -! L o*
*f symmetric
spectra, the canonical map (pushout product map)
L ^ A [K^A K ^ B ----! L ^ B
is a cofibration; the pushout product map is a weak equivalence if in addition *
*A -! B is a weak
equivalence in C or K -! L is a stable equivalence of symmetric spectra.
In a spectral model category the levels of the symmetric function spectra Hom*
* C(X, Y ) can be
rewritten as follows. The adjunctions give an isomorphism of simplicial sets
*
* 0
HomC(X, Y )n ~= mapSp(FnS0, HomC(X, Y )) ~= mapC(FnS0 ^ X, Y ) ~= mapC(X, Y F*
*nS)
where FnS0 is the free symmetric spectrum generated at level n by the 0-sphere *
*(see [HSS , 2.2.5]
or Definition 3.6.5).
Lemma 3.5.2. A spectral model category is in particular a simplicial and stable*
* model cate-
gory. For X a cofibrant and Y a fibrant object of a spectral model category C t*
*here is a natural
isomorphism of graded abelian groups ßs*HomC(X, Y ) ~=[X, Y ]Ho(C)*.
Proof.The tensor and cotensor of an object of C with a pointed simplicial set K*
* is defined
by applying the tensor and cotensor with the symmetric suspension spectrum 1 K*
*. The ho-
momorphism simplicial set between two objects of C is the 0-th level of the hom*
*omorphism
symmetric spectrum. The necessary adjunction formulas and the compatibility axi*
*om [Qui67,
II.2 SM7] hold because the suspension spectrum functor 1 :S* -! Sp from the ca*
*tegory of
pointed simplicial sets to symmetric spectra is the left adjoint of a Quillen a*
*djoint functor pair
and preserves the smash product (i.e., it is strong symmetric monoidal). In ord*
*er to see that C
is stable we recall that the shift functor in the homotopy category of C is the*
* suspension functor.
For cofibrant objects suspension is represented on the model category level by *
*the smash prod-
uct with the one-dimensional sphere spectrum 1 S1. This sphere spectrum is inv*
*ertible, up
to stable equivalence of symmetric spectra, with inverse the (-1)-dimensional s*
*phere spectrum
(modeled as a symmetric spectrum by F1S0). Since the action of symmetric spect*
*ra on C is
associative up to coherent isomorphism this implies that suspension is a self-e*
*quivalence of the
homotopy category of C. This in turn implies that the right adjoint loop functo*
*r has to be an
inverse equivalence. If X is cofibrant and Y fibrant in C, then by the compatib*
*ility axiom (SP)
STABLE MODEL CATEGORIES 15
the symmetric spectrum Hom C(X, Y ) is stably fibrant, i.e., an -spectrum. So *
*for n 0 we
have isomorphisms
ßn Hom C(X, Y ) ~= ßn mapC(X, Y ) ~= ß0map C(X, nY ) ~= [X, nY ]Ho(C);
and for n 0 we have the isomorphisms
ßn HomC(X, Y ) ~= ß0Hom C(X, Y )n ~= ß0map C(FnS0 ^ X, Y ) ~= [ nX, Y ]Ho(*
*C).
|___|
3.6. Symmetric spectra over a category. Throughout this section we assume that *
*C is a
cocomplete category which is tensored and cotensored over the category S* of po*
*inted simplicial
sets, with this action denoted by and morphism simplicial sets denoted mapC. *
*We let S1 =
[1]=@ [1] be our model for the simplicial circle and we set Sn = (S1)^n for n *
*> 1; the
symmetric group on n letters acts on Sn by permuting the coordinates.
Definition 3.6.1.Let C be a category which is tensored over the category of poi*
*nted simplicial
sets. A symmetric sequence over C is a sequence of objects X = {Xn}n 0 in C tog*
*ether with a
left action of the symmetric group n on Xn for all n 0. A symmetric spectrum*
* over C is a
symmetric sequence in C with coherently associative px q-equivariant morphisms
Sp Xq ----!Xp+q
(for all p, q 0). A morphism of symmetric sequences or symmetric spectra X -!*
* Y consists
of a sequence of n-equivariant morphisms Xn -! Yn which commute with the struc*
*ture maps.
We denote the category of symmetric sequences by C and the category of symmetr*
*ic spectra
by Sp(C).
Since C is simplicial, the action of S* on C extends to an action of S* , the*
* category of
symmetric sequences over S*, on C :
Definition 3.6.2.Given X a symmetric sequence over C, and K a symmetric sequenc*
*e over
S*, we define their tensor product, K X, by the formula
`
(K X)n = +n px q(Kp Xq) .
p+q=n
The symmetric sequence S = (S0, S1, . .,.Sn, . .).of simplicial sets is a com*
*mutative monoid
in the symmetric monoidal category (S* , ). The unit map here is the identity*
* in the first
spot and the base point elsewhere, j :(S0, *, *, . .).= u -! S. In this languag*
*e a symmetric
spectrum over C can be redefined as a left S-module in the category of symmetri*
*c sequences
over C.
Definition 3.6.3.Let X be an object in Sp(C) and K a symmetric spectrum in Sp .*
* Define
their smash product, K ^ X, as the coequalizer of the two maps
K S X -!-!K X
induced by the action of S on X and K respectively. So Sp(C) is tensored over t*
*he symmet-
ric monoidal category of symmetric spectra. Dually, we define a symmetric spec*
*trum valued
morphism object Hom Sp(C)(X, Y ) 2 Sp for X, Y 2 Sp(C). As a preliminary step*
*, define a
shifting down functor, shn:Sp(C) -! Sp(C), by (shnX)m = Xn+m where m acts via *
*the in-
clusion into n+m . Note there is a leftover action of n on shnX. Define Hom *
*(X, Y ) 2 S*
for X, Y objects in C by Hom (X, Y )n = map(X, shnY ), the simplicial mapping*
* space given
by the simplicial structure on C with the n action given by the leftover acti*
*on of n on
shnY as mentioned above. Then Hom Sp(C)(X, Y ) 2 Sp is the equalizer of the *
*two maps
Hom (X, Y ) -! Hom (S X, Y ).
16 STEFAN SCHWEDE AND BROOKE SHIPLEY
Using this spectrum valued hom functor, for K in Sp and Y in Sp(C), we defin*
*e Y K 2 Sp(C)
as the adjoint of the functor K -. That is, for any X 2 Sp(C) define Y K such*
* that
(3.6.4) Hom Sp(C)(K X, Y ) ~=HomSp(C)(X, Y K) ~=HomSp (K, HomSp(C)(X, Y )) .
Definition 3.6.5.The nth evaluation functor Evn : Sp(C) -! C is given by Evn(X)*
* = Xn,
ignoring the action of the symmetric group. The functor Evn has a left adjoint *
*Fn: C -! Sp(C)
which has the form (FnX)m = +m m-n Sm-n X where Sn = * for n < 0. We use 1*
* as
another name for F0 and call it the suspension spectrum.
3.7. The level model structure on Sp(C). There are two model category structure*
*s on sym-
metric spectra over C which we consider; the level model category which we disc*
*uss in this
section, and the stable model category (see Section 3.8). The level model categ*
*ory is a stepping
stone for defining the stable model category, but it also allows us to define e*
*ndomorphism ring
spectra (Definition 3.7.5).
Definition 3.7.1.Let f : X -! Y be a map in Sp(C). The map f is a level equival*
*ence if each
fn : Xn -! Yn is a weak equivalence in C, ignoring the n action. It is a level*
* fibration if each
fn is a fibration in C. It is a cofibration if it has the left lifting property*
* with respect to all level
trivial fibrations.
Proposition 3.7.2.For any simplicial, cofibrantly generated model category C, S*
*p(C) with the
level equivalences, level fibrations, and cofibrations described above forms a *
*cofibrantly generated
model category referred to as the level model category, and denoted by Sp(C)lv.*
* Furthermore the
following level analogue of the spectral axiom (SP) holds:
(SPlv) for every cofibration A -! B and every level fibration X -! Y in Sp(C) t*
*he induced
map
Hom Sp(C)(B, X) ----! Hom Sp(C)(A, X) xHomSp(C)(A,YH)omSp(C)(B, Y )
is a level fibration of symmetric spectra. If in addition one of the maps A -! *
*B or X -! Y is
a level equivalence, then the resulting map of symmetric spectra is also a leve*
*l equivalence.
There are various theorems in the model category literature which are useful *
*in establishing
model category structures. These theorems separate formal considerations that t*
*end to show
up routinely from the properties which require special arguments in each specif*
*ic case. Since we
will construct model category structures several times in this paper, we recall*
* one such result
that we apply in our cases. We work with the concept of cofibrantly generated m*
*odel categories,
introduced by Dwyer, Hirschhorn and Kan [DHK ]; see also Section 2.1 of Hovey's*
* book [Hov99]
for a detailed treatment of this concept.
We use the same terminology as [Hov99, Sec. 2.1]. Let I be a set of maps in a*
* category. A
map is a relative I-cell complex if it is a (possibly transfinite) composition *
*of cobase changes of
maps in I. An I-injective map is a map with the right lifting property with res*
*pect to every map
in I. An I-cofibration is a map with the left lifting property with respect to *
*I-injective maps.
For the definition of smallness relative to a set of maps see [Hov99, 2.1.3].
One of the main properties of cofibrantly generated model categories is that *
*they admit an
abstract version of Quillen's small object argument [Qui67, II 3.4].
Lemma 3.7.3. [DHK ], [Hov99, 2.1.14, 2.1.15] Let C be a cocomplete category and*
* I a set of
maps in C whose domains are small relative to the relative I-cell complexes. Th*
*en
othere is a functorial factorization of any map f in C as f = qi with q an I*
*-injective map
and i a relative I-cell complex, and thus
oevery I-cofibration is a retract of a relative I-cell complex.
STABLE MODEL CATEGORIES 17
Theorem 3.7.4. [DHK ], [Hov99, Thm. 2.1.19] Let C be a complete and cocomplete *
*category and
I and J two sets of maps of C such that the domains of the maps in I and J are *
*small with
respect to the relative I-cell complexes and the relative J-cell complexes resp*
*ectively. Suppose
also that a subcategory of C is specified whose morphisms are called `weak equi*
*valences'.
Then there is a cofibrantly generated model structure on C with the given cla*
*ss of weak equiv-
alences, with I a set of generating cofibrations, and with J a set of generatin*
*g trivial cofibrations
if the following conditions hold:
(1)if f and g are composable morphisms such that two of the three maps f, g an*
*d gf are weak
equivalences, then the third is also a weak equivalence.
(2)every relative J-cell complex is an I-cofibration and a weak equivalence.
(3)the I-injectives are precisely the maps which are both J-injective and weak*
* equivalences.
Proof of Proposition 3.7.2.Let IC and JC be sets of generators for the cofibrat*
*ions and trivial
cofibrations of C. We define sets of generators for the level model category by*
* F IC = {FnIC}n 0
and F JC = {FnJC}n 0, i.e., Fn applied to the generators of C for each n. Then*
* the F IC-
injectives are precisely the levelwise trivial fibrations and the F JC-injectiv*
*es are precisely the
level fibrations. We claim that every relative F IC-cell complex is levelwise a*
* cofibration in C,
and similarly every relative F JC-cell complex is levelwise a trivial cofibrati*
*on in C. We show
this for relative F JC-cell complexes; the argument for F IC is the same. Since*
* level evaluation
preserves colimits it suffices to check the claim for the generating cofibratio*
*ns, FnA -! FnB for
A -! B 2 JC. But the mth level of this map is a coproduct of m!=(m - n)! copies*
* of the map
Sm-n ^ A -! Sm-n ^ B. By the simplicial compatibility axiom [Qui67, II.2 SM7], *
*smashing
with a simplicial sphere preserves trivial cofibrations, so we are done.
Now we apply Theorem 3.7.4 to the sets F IC and F JC with the level equivalen*
*ces as weak
equivalences. Checking that the maps in F IC are small with respect to relative*
* F IC-cell com-
plexes comes down (by adjointness) to checking that the domains of the maps in *
*IC are small
with respect to the levels of relative F IC-cell complexes, i.e. the cofibratio*
*ns in C. By [Hir99,
14.2.14], since the domains in IC are small with respect to the relative IC-cel*
*l complexes they
are also small with respect to all cofibrations. The argument for F JC is the s*
*ame. Conditions
(1) and (3) of Theorem 3.7.4 hold and condition (2) follows from the above clai*
*m. So we indeed
have a cofibrantly generated level model structure.
To prove the property (SPlv) it suffices to check its adjoint pushout product*
* form, i.e., the
level analogue of condition (SPb) of Section 3.5; it is enough to show that the*
* pushout product
of two generating cofibrations is a cofibration, and similarly when one of the *
*maps is a trivial
cofibration (see [SS00, 2.3 (1)] or [Hov99, Cor. 4.2.5]). So let i 2 IS* and le*
*t j 2 IC. Then the
product Fni ^ Fm j is isomorphic to Fn+m (i ^ j) and the result follows since t*
*he free functors_
preserve cofibrations and trivial cofibrations. *
* |__|
We can now introduce endomorphism ring spectra and endomorphism categories.
Definition 3.7.5.Let C be a simplicial and cofibrantly generated model category*
* and P a
set of cofibrant objects. For every object P 2 P let 1fP be a fibrant replace*
*ment of the
symmetric suspension spectrum of P in the level model structure on Sp(C) of Pro*
*position 3.7.2.
We define the endomorphism category E(P) as the full spectral subcategory of Sp*
*(C) with objects
1fP for P 2 P. To simplify notation, we usually denote objects of E(P) by P i*
*nstead of
1fP . If P has a single object P we also refer to the symmetric ring spectrum *
*E(P)(P, P ) =
Hom Sp(C)( 1fP, 1fP ) as the endomorphism ring spectrum of the object P .
Up to stable equivalence, the definition of the endomorphism category does no*
*t depend on
the choices of fibrant replacements.
18 STEFAN SCHWEDE AND BROOKE SHIPLEY
Lemma 3.7.6. Let C be a simplicial and cofibrantly_generated model category and*
* P a set of
cofibrant objects. Suppose { 1fP }P2P and { 1fP}P2P are two sets of level fibra*
*nt replacements
of the symmetric suspension_spectra._ Then the two full spectral subcategories *
*of Sp(C) with
objects { 1fP }P2P and { 1fP}P2P respectively are stably equivalent.
Proof.The proof uses the notion of_quasi-equivalence,_see Definition_A.2.1. For*
* every P 2 P we
choose a level equivalence OEP : 1fP- ! 1fP . We define a E(P)-E(P)-bimodule *
*M by the rule
_____ 1
M(P, P 0) = Hom Sp(C)( 1fP, f P ) .
Because of the property (SPlv) of the homomorphism spectra in Sp(C) the bimodul*
*e M is a __
quasi-equivalence with respect to the maps OEP, and the result follows from Lem*
*ma A.2.3. |__|
3.8. The stable model structure on Sp(C). In this section we provide the detail*
*s of the stable
model category structure for symmetric spectra over C; the result is summarized*
* as Theorem
3.8.2. We use the level model category to define the stable model category stru*
*ctures on Sp(C).
The stable model category is more difficult to establish than the level model c*
*ategory, and we
need to assume that C is a simplicial, cofibrantly generated, proper, stable mo*
*del category. The
proof of the stable model structure for Sp(C) is similar to the proof of the st*
*able model structure
for Sp in [HSS , 3.4], except for one point in the proof of Proposition 3.8.8 *
*where we use the
stability of C instead of the fact that fiber sequences and cofiber sequences o*
*f spaces are stably
equivalent.
Categories of symmetric spectrum objects over a model category have been cons*
*idered more
generally by Hovey in [Hov]. Hovey relies on the general localization machiner*
*y of [Hir99].
Theorem 3.8.2 below should be compared to [Hov, Thms. 8.11 and 9.1] which are m*
*ore general
but have slightly different technical assumptions.
Definition 3.8.1.Let ~ : F1S1 -! F0S0 ~=S be the stable equivalence of symmetri*
*c spectra
which is adjoint to the identity map on the first level. A spectrum Z in Sp(C) *
*is an -spectrum if
Z is fibrant on each level and the map Z ~=ZF0S0-! ZF1S1induced by ~ is a level*
* equivalence.
A map g : A -! B in Sp(C) is a stable equivalence if the induced map
Hom Sp(C)(gc, Z) : Hom Sp(C)(Ac, Z) ----! Hom Sp(C)(Bc, Z)
is a level equivalence of symmetric spectra for any -spectrum Z; here (-)c den*
*otes a cofibrant
replacement functor in the level model category structure. A map is a stable co*
*fibration if it has
the left lifting property with respect to each level trivial fibration, i.e., i*
*f it is a cofibration in
the level model category structure. A map is a stable fibration if it has the r*
*ight lifting property
with respect to each map which is both a stable cofibration and a stable equiva*
*lence.
The above definition1of -spectrum is just a rewrite of the usual one since t*
*he n-th level of the
spectrum ZF1S is isomorphic to Zn+1. This form is more convenient here, though*
*. Lemma
3.8.7 below, combined with the fact that the stable fibrations are the J-inject*
*ive maps shows
that the stably fibrant objects are precisely the -spectra.
Theorem 3.8.2. Let C be a simplicial, cofibrantly generated, proper, stable mod*
*el category.
Then Sp(C) supports the structure of a spectral model category - referred to as*
* the stable model
structure - such that the adjoint functors 1 and evaluation Ev0, are a Quille*
*n equivalence
between C and Sp(C) with the stable model structure.
We deduce the theorem about the stable model structure on Sp(C) from a sequen*
*ces of lemmas
and propositions.
STABLE MODEL CATEGORIES 19
Lemma 3.8.3. Let K be a cofibrant symmetric spectrum, A a cofibrant spectrum in*
* Sp(C) and Z
an -spectrum in Sp(C). Then the symmetric function spectrum Hom Sp(C)(A, Z) is*
* a symmetric
-spectrum and the function spectrum ZK is an -spectrum in Sp(C).
Proof.By the adjunctions between smash products and function spectra (see 3.6.4*
*) we can
rewrite the symmetric function spectrum Hom Sp(C)(A, Z)F1S1as Hom Sp(C)(A, ZF1S*
*1) in such a
way that the map Hom Sp(C)(A, Z)~ is isomorphic to the map Hom Sp(C)(A, Z~); si*
*nce
0 F S1
Z~ : Z ~=ZF0S -! Z 1
is a level equivalence between level fibrant objects, the first claim follows f*
*rom property (SPlv)
of Proposition 3.7.2. 1 1
Similarly we can rewrite the spectrum (ZK )F1S as (ZF1S )K in such a way that*
* the map
(ZK )~ is isomorphic to the map (Z~)K ; since Z~ is a level equivalence between*
* level fibrant
objects, the second claim follows from the adjoint form (SPlv(a)) of property (*
*SPlv)_of Proposition
3.7.2, see [Qui67, II.2 SM7(a)]. *
* |__|
Lemma 3.8.4. Let C be a simplicial, cofibrantly generated and left proper model*
* category. Then
a cofibration A -! B is a stable equivalence if and only if for every -spectru*
*m Z the symmetric
function spectrum Hom Sp(C)(B=A, Z) is level contractible.
Proof.Choose a factorization of the functorial level cofibrant replacement Ac -*
*! Bc of the
given cofibration as a cofibration i : Ac -! ~Bfollowed by a level equivalence *
*q : ~B-! Bc.
Then q is a level equivalence between cofibrant objects, so for every -spectru*
*m Z, the induced
map Hom Sp(C)(q, Z) is a level equivalence. Hence f is a stable equivalence if *
*and only if
Hom Sp(C)(i, Z) : Hom Sp(C)(Ac, Z) ----! Hom Sp(C)(B~, Z)
is a level equivalence for every -spectrum Z.
The symmetric spectrum Hom Sp(C)(B~=Ac, Z) is the fiber of the level fibratio*
*n Hom Sp(C)(i, Z)
(by (SPlv)) between symmetric -spectra (by Lemma 3.8.3). Hence the given map i*
*s a stable
equivalence if and only if Hom Sp(C)(B~=Ac, Z) is level contractible.
Since Ac- ! A and ~B-! B are level equivalences and C is left proper, the ind*
*uced map on
the cofibers ~B=Ac -! B=A is a level equivalence between level cofibrant object*
*s. So for every
-spectrum Z the induced map HomSp(C)(B=A, Z) -! Hom Sp(C)(B~=Ac, Z) is a level*
* equivalence
(by (SPlv)) between symmetric -spectra (by Lemma 3.8.3). Hence the given map i*
*s a stable __
equivalence if and only if Hom Sp(C)(B=A, Z) is level contractible, which prove*
*s the lemma. |__|
We now show that Sp(C) satisfies (SPb) of Section 3.5, an adjoint form of (SP*
*) from Defi-
nition 3.5.1. This shows that Sp(C) is a spectral model category as soon as the*
* stable model
structure on Sp(C) is established.
Proposition 3.8.5.Let C be a simplicial, cofibrantly generated and left proper *
*model category.
Let i : A -! B be a cofibration in Sp(C) and j : K -! L a stable cofibration of*
* symmetric
spectra. Then the pushout product map
j i : L ^ A [K^A K ^ B ----! L ^ B
is a cofibration in Sp(C); the pushout product map is a stable equivalence if i*
*n addition i is a
stable equivalence in Sp(C) or j is a stable equivalence of symmetric spectra.
Proof.Since the cofibrations coincide in the level and the stable model structu*
*res for Sp(C) and
for symmetric spectra, we know by property (SPlv) of Proposition 3.7.2 that j *
* i is again a
cofibration in Sp(C). Now suppose that one of the maps is in addition a stable *
*equivalence. The
pushout product map j i is a cofibration with cofiber isomorphic to (L=K) ^ (*
*B=A). So by
20 STEFAN SCHWEDE AND BROOKE SHIPLEY
Lemma 3.8.4 it suffices to show that Hom Sp(C)((L=K) ^ (B=A), Z) is level contr*
*actible for every
-spectrum Z. If i is a stable acyclic cofibration, then we can rewrite this fu*
*nction spectrum as
Hom Sp(C)((L=K) ^ (B=A), Z) ~= Hom Sp(C)(B=A, Z(L=K)) ;
the latter spectrum is level contractible by Lemma 3.8.4 since Z(L=K)is an -sp*
*ectrum by
Lemma 3.8.3 and i is a cofibration and stable equivalence. If j is a stable acy*
*clic cofibration,
then we similarly rewrite the spectrum as
Hom Sp(C)((L=K) ^ (B=A), Z) ~= Hom Sp(L=K, HomSp(C)(B=A, Z)) ;
the latter spectrum is level contractible by [HSS , 5.3.9] since Hom Sp(C)(B=A,*
* Z) is_a symmetric
-spectrum by Lemma 3.8.3 and L=K is stably contractible. *
* |__|
We use Theorem 3.7.4 to verify the stable model category structure on Sp(C). *
*We first define
two sets I and J of maps in Sp(C) which will be generating sets for the cofibra*
*tions and stable
trivial cofibrations. Since the stable cofibrations are the same class of maps *
*as the cofibrations
in the level model structure we let I be the generating set F IC which was used*
* in Proposition
3.7.2 to construct the level model structure. With this choice the I-injectives*
* are precisely the
level trivial fibrations.
The generating set for the stable trivial cofibrations is the union J = F JC *
*[ K, where F JC is
the generating set of trivial cofibrations for the level model category (see th*
*e proof of Proposition
3.7.2) and K is defined as follows. In the category of symmetric spectra over s*
*implicial sets there
is a map ~ : F1S1 -! F0S0 = S which is adjoint to the identity map on the first*
* level; this map
was also used in defining an -spectrum in Definition 3.8.1. Let M~ be the mapp*
*ing cylinder
of this map, formed by taking the mapping cylinder of simplicial sets on each l*
*evel. So ~ = r~
with ~ : F1S1 -! M~ a stable equivalence and stable cofibration and r : M~ -! S*
* a simplicial
homotopy equivalence, see [HSS , 3.4.9]. Then K is the set of maps
K = {~ F IC} = {~ Fni|i 2 IC} ,
where for i : A -! B,
~ Fni : (F1S1 ^ FnB) [F1S1^FnA(M~ ^ FnA) -! M~ ^ FnB .
Here we only use the pushout product, , as a convenient way of naming these ma*
*ps, see
also [HSS , 5.3]. Now we can verify condition (2) of Theorem 3.7.4.
Proposition 3.8.6.Let C be a simplicial, cofibrantly generated and left proper *
*model category.
Then every relative J-cell complex is an I-cofibration and a stable equivalence.
Proof.All maps in J are cofibrations in the level model structure on Sp(C) of P*
*roposition 3.7.2,
hence the relative J-cell complexes are contained in the I-cofibrations.
We claim that for every J-cofibration A -! B and every -spectrum Z, the map
Hom Sp(C)(B, Z) ----! Hom Sp(C)(A, Z)
is a level trivial fibration of symmetric spectra. Hence the fiber, the symmet*
*ric spectrum
Hom Sp(C)(B=A, Z), is level contractible and A -! B is a stable equivalence by *
*Lemma 3.8.4.
The property of inducing a trivial fibration after applying Hom Sp(C)(-, Z) i*
*s closed under
pushout, transfinite composition and retract, so by the small object argument 3*
*.7.3 it suffices
to check this for the generating maps in J = F JC [ K. The generating cofibrati*
*ons in F JC are
level trivial cofibrations, so for these the claim holds by the compatibility a*
*xiom (SPlv). A map
in the set K is of the form ~ Fni where ~ : F1S1 -! M~ is a stable trivial co*
*fibration of
symmetric spectra and Fni is a cofibration in Sp(C); hence the map ~ Fni is a*
* stable trivial
cofibration between cofibrant objects by Proposition 3.8.5. So the induced map *
*of symmetric
spectra Hom Sp(C)(~ Fni, Z) is a level fibration (by (SPlv)) between -spectra *
*by Lemma 3.8.3.
STABLE MODEL CATEGORIES 21
In addition the fiber of the map Hom Sp(C)(~ Fni, Z) is level contractible by*
* Lemma 3.8.4,_so
the map is indeed a level trivial fibration. *
* |__|
Before turning to property (3) of Theorem 3.7.4, we need the following lemma.
Lemma 3.8.7. Let C be a simplicial, cofibrantly generated model category and X *
*a symmetric
spectrum over C. Then the map X -! * is J-injective if and only if X is an -sp*
*ectrum.
Proof.The maps in F JC generate the trivial cofibrations in the level model str*
*ucture of Propo-
sition 3.7.2, so X -! * is F JC-injective if an only if X is levelwise fibrant.*
* Now we assume
that X is levelwise fibrant and show the map X -! * is K-injective if and only *
*if X is an1
-spectrum. By adjointness X -! * is K-injective if and only if the map X~ : XM*
*~ -! XF1S
is a level trivial fibration. The projection r : M~ -! S is a simplicial homoto*
*py equivalence, so
it induces a level equivalence X -! XM~ . So X -! * is K-injective if and only *
*if the map_X~_
is a level equivalence, which precisely means that X is an -spectrum. *
* |__|
Proposition 3.8.8.Let C be a simplicial, cofibrantly generated, right proper, s*
*table model cat-
egory. Then a map is J-injective and a stable equivalence if and only if it is*
* a level trivial
fibration.
Proof.Every level equivalence is a stable equivalence and the level trivial fib*
*rations are precisely
the I-injectives. Since the J-cofibrations are contained in the I-cofibrations,*
* these I-injectives
are also J-injective. The converse is more difficult to prove.
Since J in particular contains maps of the form FnA -! FnB where n runs over *
*the natural
numbers and A -! B runs over a set of generating trivial cofibrations for C, J-*
*injective maps are
level fibrations. So we show that a J-injective stable equivalence, E -! B, is *
*a level equivalence.
Let F denote the fiber and choose a cofibrant replacement F c-! F in the level *
*model category
structure. Then choose a factorization in the level model category structure
F cv_____Ecw_____wElv~
of the composite map F c-! E as a cofibration followed by a level equivalence. *
* Since C is
right proper, each level of F -! E -! B is a homotopy fibration sequence in C. *
*Each level of
F c-! Ec -! Ec=F cis a homotopy cofibration sequence in C (left properness is n*
*ot needed
here since each object is cofibrant). So since C is stable, we see that Ec=F c-*
*! B is a level
equivalence. Thus Ec -! Ec=F cis a stable equivalence. For any -spectrum Z the*
*re is a fiber
sequence of symmetric -spectra
Hom C(Ec=F c, Z) ----! Hom C(Ec, Z) ----! Hom C(F c, Z)
in which the left map is a level equivalence and the right map is a level fibra*
*tion. Hence the
symmetric spectrum HomC(F c, Z) is level contractible which means that F is sta*
*bly contractible.
Since F -! * is the pull back of the map E -! B, it is a J-injective map. So*
* F is an
-spectrum by Proposition 3.8.7. Since F is both stably contractible and an -s*
*pectrum, the
spectrum Hom C(F c, F ) is level contractible, so F is level equivalent to a po*
*int. But_this_means
that Ec -! Ec=F c, and thus also E -! B is a level equivalence. *
* |__|
Proof of Theorem 3.8.2.We apply Theorem 3.7.4 to show that the I-cofibrations, *
*stable equiv-
alences, and J-injectives form a cofibrantly generated model category on Sp(C).*
* Since the
I-cofibrations are exactly the stable cofibrations this implies that the J-inje*
*ctives are the maps
with the right lifting property with respect to the stable trivial cofibrations*
*, i.e., the stable
fibrations as defined before the statement of the theorem. The 2-out-of-3 cond*
*ition, part (1)
in Theorem 3.7.4, is clear from the definition of stable equivalences. Conditio*
*n (2) is verified
in Proposition 3.8.6 and condition (3) is verified in Proposition 3.8.8 (since *
*the I-injectives are
22 STEFAN SCHWEDE AND BROOKE SHIPLEY
precisely the levelwise trivial fibrations). So to conclude that the sets I and*
* J generate a model
structure with the stable equivalences as weak equivalence it is enough to veri*
*fy that the domains
of the generators are small with respect to the level cofibrations. This has al*
*ready been checked
in the proof of Proposition 3.7.2 for the generators in F IC and F JC. So only*
* the generators
in K remain. Since a pushout of objects which are small is small, we only need *
*to check that
FkA ^ M~ and FkA ^ F1S1 are small with respect to the level cofibrations. Here *
*A is small
with respect to relative IC-cell complexes and hence also the cofibrations by [*
*Hir99, 14.2.14].
FkA is small with respect to relative I-cell complexes in Sp(C) by adjointness *
*and F1S1 is small
with respect to all of Sp . So by various adjunctions F1S1 ^ FkA is small with *
*respect to level
cofibrations. Since M~ is the pushout of small objects, similar arguments show *
*that FkA ^ M~
is also small with respect to the level cofibrations in Sp(C).
The spectral compatibility axiom is verified in Proposition 3.8.5 in its adjo*
*int form (SPb).
Thus, it remains to show that the adjoint functors 1 and Ev0are a Quillen equi*
*valence. The
suspension spectrum functor 1 takes (trivial) cofibrations to (trivial) cofibr*
*ations in the level
model structure. Hence 1 also preserves (trivial) cofibrations with respect to*
* the stable model
structure. So the adjoint functors Ev0and 1 are a Quillen pair between C and S*
*p(C).
To show that the functors are a Quillen equivalence it suffices to show (see *
*[Hov99, Cor.
1.3.16]) that Ev0 reflects stable equivalences between stably fibrant objects a*
*nd that for every
cofibrant object A of C the map A -! Ev0R( 1 A) is a weak equivalence where R d*
*enotes any
stably fibrant replacement in Sp(C). So suppose that f : X -! Y is a map betwee*
*n -spectra
with the property that f0 : X0 -! Y0 is a weak equivalence in C. Since X is an *
* -spectrum,
X0 -! nXn is a weak equivalence, and similarly for Y . Hence the map nfn : n*
*Xn -! nYn
is a weak equivalence in C. Since C is stable, the loop functor is a self-Quill*
*en equivalence, so it
reflects weak equivalences between fibrant objects, and so fn : Xn -! Yn is a w*
*eak equivalence
in C. Hence f is a level, and thus a stable equivalence of spectra over C.
Since C is stable the spectrum 1fA (the fibrant replacement of the suspensio*
*n spectrum in
the level model structure) is an -spectrum, and thus stably fibrant. Hence we *
*may take 1fA
as the stably fibrant replacement R( 1 A), which proves that A -! Ev0R( 1 A) is*
*_a_weak
equivalence in C. |*
*__|
3.9. The Quillen equivalence. In this section we prove Theorem 3.3.3, i.e., we *
*show that a
suitable model category with a set of compact generators is Quillen equivalent *
*to the modules
over the spectral endomorphism category of the generators.
In Theorem 3.9.3 we first formulate the result for spectral model categories;*
* this gives a
more general result since the conditions about cofibrant generation and propern*
*ess in C are not
needed. We then combine this with the fact that every suitable stable model cat*
*egory is Quillen
equivalent to a spectral model category to prove our main classification theore*
*m.
Definition 3.9.1.Let G be a set of objects in a spectral model category D. We d*
*enote by E(G)
the full spectral subcategory of D with objects G, i.e., E(G)(G, G0) = Hom D(G,*
* G0). We let
Hom(G, -) : D ----! mod-E(G)
denote the tautological functor given by Hom (G, Y )(G) = Hom D(G, Y ).
We want to stress the reassuring fact that the stable equivalence type of the*
* spectral endo-
morphism category E(G) only depends on the weak equivalence types of the object*
*s in the set G,
as long as these are all fibrant and cofibrant, see Corollary A.2.4. This is no*
*t completely obvious
since taking endomorphisms is not a functor.
The earlier Definition 3.7.5 of the endomorphism ring spectrum and endomorphi*
*sm category
of objects in a simplicial stable model category C is a special case of Definit*
*ion 3.9.1 with
D = Sp(C) and G the level fibrant replacements of the suspension spectra of the*
* chosen objects
STABLE MODEL CATEGORIES 23
in C. Again, if G = {G} has a single element then E(G) is determined by the sin*
*gle symmetric
ring spectrum, EndD(G) = Hom D(G, G).
Definition 3.9.2.Let C and D be spectral model categories. A spectral Quillen p*
*air is a Quillen
adjoint functor pair L : C -! D and R : D -! C together with a natural isomorph*
*ism of
symmetric homomorphism spectra
Hom C(A, RX) ~= Hom D(LA, X)
which on the vertices of the 0-th level reduces to the adjunction isomorphism. *
*A spectral Quillen
pair is a spectral Quillen equivalence if the underlying Quillen functor pair i*
*s an ordinary Quillen
equivalence.
In the terminology of [Hov99, Def. 4.2.18] a spectral Quillen pair would be c*
*alled a `Sp -
Quillen functor.'
Theorem 3.9.3. Let D be a spectral model category and G a set of of cofibrant a*
*nd fibrant
objects.
(i) The tautological functor
Hom (G, -) : D ----! mod-E(G)
is the right adjoint of a spectral Quillen functor pair. The left adjoint is de*
*noted - ^E(G)G.
(ii) If all objects in G are compact, then the total derived functors of Hom (G*
*, -) and - ^E(G)G
restrict to a triangulated equivalence between the homotopy category of E(G)-mo*
*dules and the
localizing subcategory of Ho(D) generated by G.
(iii) If G is a set of compact generators for D, then the adjoint functor pair*
* Hom (G, -) and
- ^E(G)G form a spectral Quillen equivalence.
Proof.(i) For an E(G)-module M the object M ^E(G)G is given by an enriched coen*
*d [Kly82,
3.10]. This means that M ^E(G)G is the coequalizer of the two maps
` `
M(G0) ^ E(G)(G, G0) ^ G ----!----!M(G) ^ G .
G,G02G G2G
One map in the diagram is induced by the evaluation map E(G)(G, G0)^G -! G0and *
*the other is
induced by the action map M(G0)^E(G)(G, G0) -! M(G). The tautological functor H*
*om(G, -)
preserves fibrations and trivial fibrations by the compatibility axiom (SP) of *
*Definition 3.5.1,
since all objects of G are cofibrant. So together with its left adjoint it form*
*s a spectral Quillen
pair.
(ii) Since the functors Hom (G, -) and - ^E(G)G are a Quillen pair, they have*
* adjoint total
derived functors on the level of homotopy categories [Qui67, I.4]; we denote th*
*ese derived functors
by RHom (G, -) and - ^LE(G)G respectively. The functor - ^LE(G)G commutes with *
*suspension
and preserves cofiber sequences, and the functor RHom (G, -) commutes with taki*
*ng loops and
preserves fiber sequences [Qui67, I.4 Prop. 2]. In the homotopy category of a *
*stable model
category, the cofiber and fiber sequences coincide up to sign and they constitu*
*te the distinguished
triangles. So both total derived functors preserve shifts and triangles, i.e., *
*they are exact functors
of triangulated categories.
For every G 2 G the E(G)-module Hom (G, G) is isomorphic to the free module F*
*G =
E(G)(-, G) by inspection and FG ^E(G)G is isomorphic to G since they represent *
*the same
functor on D. As a left adjoint, the functor - ^LE(G)G preserves coproducts. We*
* claim that the
right adjoint RHom (G, -) also preserves coproducts. Since the free modules FG *
*form a set of
24 STEFAN SCHWEDE AND BROOKE SHIPLEY
compact generators for the category of E(G)-modules (see Theorem A.1.1), it suf*
*fices to show
that for all G 2 G and for every family {Ai}i2Iof objects of D the natural map
M Ho(mod-E(G)) a Ho(mod-E(G))
[FG, RHom (G, Ai)]* ~=[FG, RHom (G, Ai)]* ----!
i2I i2I
a Ho(mod-E(G))
[FG, RHom (G, Ai)]*
i2I
is an isomorphism. By the adjunctions and the identification FG ^LE(G)G ~= G t*
*his map is
isomorphic to the natural map
M Ho(D) a Ho(D)
[G, Ai]* ----! [G, Ai]* .
i2I i2I
But this last map is an isomorphism since G was assumed to be compact.
Both derived functors preserve shifts, triangles and coproducts; since they m*
*atch up the free
E(G)-modules FG with the objects of G, they restrict to adjoint functors betwee*
*n the localizing
subcategories generated by the free modules on the one side and the objects of *
*G on the other
side. We consider the full subcategories of those M 2 Ho(mod-E(G)) and X 2 Ho(D*
*) respectively
for which the unit of the adjunction
j : M ----! RHom (G, M ^LE(G)G)
or the counit of the adjunction
: RHom (G, X) ^LE(G)X ----! X
are isomorphisms. Since both derived functors are exact and preserve coproduct*
*s, these are
localizing subcategories. Since FG ^LE(G)G ~= G and RHom (G, G) ~= FG, the map*
* j is an
isomorphism for every free module, and the map is an isomorphism for every ob*
*ject of G.
Since the free modules FG generate the homotopy category of E(G)-modules, the c*
*laim follows.
(iii) Now the localizing subcategory generated by G is the entire homotopy ca*
*tegory of D, so
part (ii) of the theorem implies that the total derived functors of Hom (G, -) *
*and - ^E(G)Gare__
inverse equivalences of homotopy categories. Hence this pair is a Quillen equiv*
*alence. |__|
Now we can finally give the
Proof of Theorem 3.3.3:We can combine Theorem 3.8.2 and Theorem 3.9.3 (iii) to *
*obtain a
diagram of model categories and Quillen equivalences
1 -^E(G)G
C _____wu_____Sp(C)u______wmod-E(G)
Ev0 Hom(G,-)
(the left adjoints are on top). First, we may assume that each object in the se*
*t P of compact
generators for C is cofibrant. Since the left Quillen functor pair above induce*
*s an equivalence of
homotopy categories, the suspension spectra of the objects in P form a set of c*
*ompact cofibrant
generators for Sp(C).
We denote by G the set of level fibrant replacements 1fP of the given genera*
*tors in C.
The spectral endomorphism category E(G) in the sense of Definition 3.9.1 is equ*
*al to the en-
domorphism category E(P) associated to P by Definition 3.7.5. Since C is stable*
*, the spectra
1fP are -spectra, hence they are both fibrant (by Lemma 3.8.7) and cofibrant *
*in the sta-
ble model structure on Sp(C). So we can apply Theorem 3.9.3 (iii) to get the se*
*cond_Quillen
equivalence. |_*
*_|
STABLE MODEL CATEGORIES 25
Remark 3.9.4. Finite localization and E(P)-modules. Suppose P is a set of comp*
*act
objects of a triangulated category T with infinite coproducts. Then there alw*
*ays exists an
idempotent localization functor LP on T whose acyclics are precisely the object*
*s of the localizing
subcategory generated by P (compare [Mil92] or the proofs of Lemma 2.2.1 or [HP*
*S97 , Prop.
2.3.17]). These localizations are often referred to as finite Bousfield localiz*
*ations away from P.
Theorem 3.9.3 gives a lift of finite localization to the model category level*
*. Suppose C is
a stable model category with a set P of compact objects, and let LP denote the *
*associated
localization functor on the homotopy category of C. By Theorem 5.3.2 (ii) the a*
*cyclics for LP
are equivalent to the homotopy category of E(P)-modules, the equivalence arisin*
*g from a Quillen
adjoint functor pair. Furthermore the counit of the derived adjunction
Hom(P, X) ^LE(P)P ----! X
is the acyclicization map and its cofiber is a model for the localization LPX.
4.Morita context
In the classical algebraic context there is a characterization of equivalence*
*s of module cate-
gories in terms of bimodules, see for example [AF92 , x22]. We provide an analo*
*gous result for
module categories over ring spectra. As usual, here instead of actual equivale*
*nces of module
categories one obtains Quillen equivalences of model categories. We state the M*
*orita context for
symmetric ring spectra and spectral Quillen equivalences (see Definition 3.9.2).
Definition 4.1.1.If R is a symmetric ring spectrum and C a spectral model categ*
*ory, then an
R-C-bimodule is an object X of C on which R acts through C-morphisms, i.e., a h*
*omomorphism
of symmetric ring spectra from R to the endomorphism ring spectrum of X.
If C is the category of modules over another symmetric ring spectrum T , then*
* this notion
of bimodule specializes to the usual one, i.e., an R-(T -mod)-bimodule is the s*
*ame as a right
Rop^ T -module. In the following Morita theorem, the implication (3) =) (2) is *
*a special case of
the classification of monogenic stable model categories (Theorem 3.1.1); hence *
*the implication
(2) =) (3) is a partial converse to that classification result. The implication*
* (2) =) (1) says
that certain chains of Quillen equivalences can be rectified into a single Quil*
*len equivalence whose
left adjoint is given by smashing with a bimodule.
Theorem 4.1.2. (Morita context) Consider the following statements for a symmetr*
*ic ring
spectrum R and a spectral model category C.
(1)There exists an R-C-bimodule M such that smashing with M over R is the left*
* adjoint of
a Quillen equivalence between the category of R-modules and C.
(2)There exists a chain of spectral Quillen equivalences through spectral mode*
*l categories be-
tween the category of R-modules and C.
(3)The category C has a compact, cofibrant and fibrant generator M such that R*
* is stably
equivalent to the endomorphism ring spectrum of M.
Then conditions (2) and (3) are equivalent, and condition (1) implies both cond*
*itions (2) and
(3). If R is cofibrant as a symmetric ring spectrum, then all three conditions *
*are equivalent.
Furthermore, if C is the category of modules over another symmetric ring spec*
*trum T which
is cofibrant as a symmetric spectrum, then condition (1) is equivalent to the c*
*ondition
(4) There exists an R-T -bimodule M and a T -R-bimodule N which are cofibran*
*t as right
modules such that
oM ^T N is stably equivalent to R as an R-bimodule and
oN ^R M is stably equivalent to T as a T -bimodule.
26 STEFAN SCHWEDE AND BROOKE SHIPLEY
Remark 4.1.3. The cofibrancy conditions in the Morita theorem can always be arr*
*anged since
every symmetric ring spectrum has a stably equivalent cofibrant replacement in *
*the model cat-
egory of symmetric ring spectra [HSS , 5.4.3]; furthermore the underlying symme*
*tric spectrum
of a cofibrant ring spectrum is always cofibrant ([SS00, 4.1] or [HSS , 5.4.3])*
*. It should not be
surprising that cofibrancy conditions have to be imposed in the Morita theorem.*
* In the algebraic
context the analogous conditions show up in Rickard's paper [Ric91]: when tryin*
*g to realize de-
rived equivalences between k-algebras by derived tensor product with bimodule c*
*omplexes, he
has to assume that the algebras are flat over the ground ring k, see [Ric91, Se*
*c. 3].
Proof of the Morita theorem.Condition (1) is a special case of (2).
(2) =) (3): This implication follows from the homotopy invariance of endomorp*
*hism ring
spectra under spectral Quillen equivalences, see Corollary A.2.4. We choose a c*
*hain of spectral
Quillen equivalences through spectral model categories. Then we choose a trivi*
*al cofibration
R -! Rfof R-modules such that Rfis fibrant; then R is stably equivalent to the *
*endomorphism
ring spectrum of Rf. We define an object M of C by iteratively applying the fun*
*ctors in the
chain of Quillen equivalences, starting with Rf. In addition we take a fibrant*
* or cofibrant
replacement after each application depending on whether we use a left or right *
*Quillen functor.
By a repeated application of Corollary A.2.4 the endomorphism ring spectra of t*
*hese objects,
including the final one, M, are all stably equivalent to R. By construction th*
*e object M is
isomorphic in the homotopy category of C to the image of the free R-module of r*
*ank one under
the equivalence of homotopy categories induced by the Quillen equivalences. Hen*
*ce M is also a
compact generator for C and satisfies condition (3).
(3) =) (2): This is essentially a special case of Theorem 3.9.3 (iii). More *
*precisely, that
theorem constructs a spectral Quillen equivalence between C and the category of*
* modules over
the endomorphism ring spectrum of the generator M. Furthermore, restriction and*
* extension
of scalars are spectral Quillen equivalences for two stably equivalent ring spe*
*ctra [HSS , Thm.
5.4.5], see also [MMSS , Thm. 11.1] or Theorem A.1.1, which establishes condit*
*ion (2).
(3) =) (1), provided R is cofibrant as a symmetric ring spectrum: Since M is *
*fibrant and
cofibrant the endomorphism ring spectrum EndC(M) is fibrant. Since R is cofibra*
*nt as a sym-
metric ring spectrum any isomorphism between R and EndC(M) in the homotopy cate*
*gory of
symmetric ring spectra can be lifted to a stable equivalence j : R -! EndC(M). *
*In particular
the stable equivalence j makes M into an R-C-bimodule. The functor X 7! X ^R M*
* is left
adjoint to the functor Y 7! Hom C(M, Y ) from C to the category of R-modules. T*
*o show that
these adjoint functors form a Quillen equivalence we note that they factor as t*
*he composite of
two adjoint functor pairs
-^EndC(M)M _j*__
C u______wmod-EndC(M) u____w mod-R
HomC(M,-) j*
(the left adjoints are on top). Since M is a cofibrant and fibrant compact gene*
*rator for C, the left
pair is a Quillen equivalence by Theorem 3.9.3 (iii). The other adjoint functor*
* pair is restriction
and extension of scalars along the stable equivalence of ring spectra j : R -! *
*EndC(M), which
is a Quillen equivalence by [HSS , Thm. 5.4.5] or [MMSS , Thm. 11.1].
For the rest of the proof we assume that C is the category of modules over a *
*symmetric ring
spectrum T which is cofibrant as a symmetric spectrum.
(1) =) (4): Since smashing with M over R is a left Quillen functor and since t*
*he free R-
module of rank one is cofibrant, M ~=R ^R M is cofibrant as a right T -module. *
*We choose a
fibrant replacement T -! T fof T in the category of T -bimodules and we let N b*
*e a cofibrant
replacement of the T -R-bimodule Hom T(M, T f). The forgetful functor from T -R*
*-bimodules to
STABLE MODEL CATEGORIES 27
right R-modules preserves cofibrations since its right adjoint
HomSp (T, -) : mod-R ----! T -mod-R
preserves trivial fibrations (because T is cofibrant as a symmetric spectrum). *
*In particular the
bimodule N is cofibrant as a right R-module. We will exhibit two chains of stab*
*le equivalences
of bimodules
N ^R M --~--!T f--~-- T and M ^T N --~--!Hom T(M, (M ^T T f)f) --~-- R
where M ^T T f-! (M ^T T f)fis a fibrant approximation in the category of R-T -*
*bimodules.
This will establish condition (4).
Since - ^R M was assumed to be a left Quillen equivalence and the approximati*
*on map
N -! Hom T(M, T f) is a weak equivalence, so is its adjoint N ^R M -! T f; but *
*this adjoint
is even a map of T -bimodules. The equivalence T -! T fwas chosen in the beginn*
*ing. For the
next equivalence we smash the T -bimodule equivalence N ^R M -! T fwith the rig*
*ht-cofibrant
bimodule M to get a stable equivalence of R-T -bimodules
M ^T N ^R M --~--!M ^T T f.
We then compose with the approximation map and obtain a stable equivalence of R*
*-T -bimodules
M ^T N ^R M -! (M ^T T f)f. Since M and N are cofibrant as right modules, the R*
*-bimodule
M ^T N is cofibrant as a right R-module. Since -^R M is a left Quillen equivale*
*nce, the adjoint
M ^T N -! Hom T(M, (M ^T T f)f) is thus a stable equivalence of R-bimodules. Fo*
*r the same
reason the adjoint of the composite stable equivalence
R ^R M ~=M ^T T --~--!M ^T T f--~--!(M ^T T f)f
gives the final stable equivalence of R-bimodules R -! Hom T(M, (M ^T T f)f).
(4) =) (1): Let M and N be bimodules which satisfy the conditions of (4). Th*
*e functor
X 7! X ^R M is left adjoint to the functor Y 7! Hom T(M, Y ) from the category *
*of T -modules to
the category of R-modules. Since M is cofibrant as a right T -module, this righ*
*t adjoint preserves
fibrations and trivial fibrations by the spectral axiom (SP). So - ^R M and Hom*
* T(M, -) form
a Quillen functor pair. By condition (4) the left derived functor of - ^R M is *
*an equivalence of
derived categories (with inverse the left derived functor of smashing with the *
*bimodule N)._So
the functor - ^R M is indeed the left adjoint of a Quillen equivalence. *
* |__|
5. A generalized tilting theorem
In this section we state and prove a generalization of Rickard's öM rita theo*
*ry for derived
categories", [Ric89]. Rickard studies the question of when two rings are derive*
*d equivalent, i.e.,
when there exists a triangulated equivalence between various derived categories*
* of the module
categories. He shows [Ric89, Thm. 6.4] that a necessary and sufficient conditi*
*on for such a
derived equivalence is the existence of a tilting complex. A tilting complex fo*
*r a pair of rings
and is a bounded complex X of finitely generated projective -modules which g*
*enerates the
derived category and whose graded ring of self extension groups [X, X]D(*)is is*
*omorphic to ,
concentrated in dimension zero.
We generalize the result in two directions. First, we allow the input to be *
*a stable model
category (which generalizes categories of chain complexes of modules). Second,*
* we allow for
a set of special generators, as opposed to a single tilting complex. The compa*
*ct objects in
the unbounded derived category of a ring are precisely the perfect complexes, i*
*.e., those chain
complexes which are quasi-isomorphic to a bounded complex of finitely generated*
* projective
modules [BN93 , Prop. 6.4]. In our context we thus define a set of tiltors in *
*a stable model
category to be a set T of compact generators such that for any two objects T, T*
* 02 T the graded
28 STEFAN SCHWEDE AND BROOKE SHIPLEY
homomorphism group [T, T 0]Ho(C)*in the homotopy category is concentrated in di*
*mension zero.
Then Theorem 5.1.1 shows that the existence of a set of tiltors is necessary an*
*d sufficient for
a stable model category to be Quillen equivalent or derived equivalent to the c*
*ategory of chain
complexes over a ringoid (ring with several objects). Recall that a ringoid is *
*a small category
whose hom-sets carry an abelian group structure for which composition is biline*
*ar. A module
over a ringoid is a contravariant additive functor to the category of abelian g*
*roups.
Of course an interesting special case is that of a stable model category with*
* a single tiltor,
i.e., a single compact generator whose graded endomorphism ring in the homotopy*
* category is
concentrated in dimension zero. Then ringoids simplify to rings. In particular *
*when C is the
model category of chain complexes of -modules for some ring , then a single t*
*iltor is the same
(up to quasi-isomorphism) as a tilting complex, and the equivalence of conditio*
*ns (2') and (3)
below recovers Rickard`s `Morita theory for derived categories' [Ric89, Thm. 6.*
*4].
Condition (1) in the tilting theorem refers to a standard model structure on *
*the category of
chain complexes of A_-modules. The model structure we have in mind is the proje*
*ctive model
structure: the weak equivalences are the quasi-isomorphisms and the fibrations*
* are the epi-
morphisms. Every cofibrations in this model structure is in particular a monomo*
*rphism with
degreewise projective cokernel, but for unbounded complexes this condition is n*
*ot sufficient to
characterize a cofibration. In the single object case, i.e., for modules over a*
* ring, the projective
model structure on complexes is established in [Hov99, Thm. 2.3.11]. For module*
*s over a ringoid
the arguments are very similar, just that the free module of rank one has to be*
* replaced by
the set of free (or representable) modules Fa = A_(-, a) for a 2 A_. This model*
* structure can
be established using a version of Theorem A.1.1 where the enrichment over Sp i*
*s replaced by
one over chain complexes. The projective model structure for complexes of A_-mo*
*dules is also a
special case of [CH , Thm. 5.1]. Indeed, the projective (in the usual sense) A_*
*-modules together
with the epimorphisms form a projective class (in the sense of [CH , Def. 1.1])*
*, and this class is
determined (in the sense of [CH , Sec. 5.2]) by the set of small, free modules *
*{Fa}a2A_.
Theorem 5.1.1. (Tilting theorem) Let C be a simplicial, cofibrantly generated, *
*proper, stable
model category and A_a ringoid. Then the following conditions are equivalent:
(1) There is a chain of Quillen equivalences between C and the model categor*
*y of chain
complexes of A_-modules.
(2) The homotopy category of C is triangulated equivalent to D(A_), the unbo*
*unded derived
category of the ringoid A_.
(2') C has a set of compact generators and the full subcategory of compact o*
*bjects in Ho(C)
is triangulated equivalent to Kb(proj-A_), the homotopy category of bounded cha*
*in complexes of
finitely generated projective A_-modules.
(3) The model category C has a set of tiltors whose endomorphism ringoid in *
*the homotopy
category of C is isomorphic to A_.
Example 5.1.2.Let G be a finite group. As in Example 3.4 (i) the category of G-*
*equivariant
orthogonal spectra [MM ] based on a complete universe U is a simplicial, stabl*
*e model category,
and the equivariant suspension spectra of the homogeneous spaces G=H+ form a se*
*t of compact
generators as H runs through the subgroups of G. Rationalization is a smashing*
* Bousfield
localization so the rationalized suspension spectra form a set of compact gener*
*ators of the rational
G-equivariant stable homotopy category. The homotopy groups of the function spe*
*ctra between
the various generators are torsion in dimensions different from zero [GM95 , Pr*
*op. A.3], so the
rationalized suspension spectra form a set of tiltors. Modules over the associa*
*ted ringoid are
nothing but rational Mackey functors, so the tilting theorem 5.1.1 shows that t*
*he rational G-
equivariant stable homotopy category is equivalent to the derived category of r*
*ational Mackey
functors. In turn, since these rational Mackey functors are all projective and*
* injective, the
STABLE MODEL CATEGORIES 29
derived category is equivalent to the graded category. So this recovers the Th*
*eorem A.1 in
[GM95 ].
For a non-complete universe, one considers rational U-Mackey functors [Le99] *
*which are mod-
ules over the endomorphism ringoid of the rationalized suspension spectra of G=*
*H+ . For exam-
ple, for the trivial universe U, these rational U-Mackey functors are rational *
*coefficient systems.
The rational G-equivariant stable homotopy category based on a non-complete uni*
*verse U is
then equivalent to the derived category of the associated rational U-Mackey fun*
*ctors.
Example 5.1.3.Let A be a ring and consider the pure projective model category s*
*tructure
in the sense of Christensen and Hovey [CH , 5.3] on the category of chain compl*
*exes of A-
modules (see also Example 2.3 (xiii)). A map X -! Y of complexes is a weak equi*
*valence if
and only if for every finitely generated A-module M the induced map of mapping *
*complexes
Hom A(M, X) -! Hom A(M, Y ) is a quasi-isomorphism. Let G be a set of represen*
*tatives of
the isomorphism classes of indecomposable finitely generated A-modules. Then G *
*forms a set
of compact generators for the pure derived category DP(A). Since furthermore ev*
*ery finitely
generated module is pure projective, maps in the pure derived category between *
*modules in G
are concentrated in dimension zero. In other words, the indecomposable finitely*
* generated A-
modules form a set of tiltors. So Theorem 5.1.1 implies that the pure projectiv*
*e model category
of A is Quillen equivalent to the modules over the ringoid given by the full su*
*bcategory of
A-modules with objects G.
Remark 5.1.4. We want to emphasize one special feature of the tilting situation*
*. For general
stable model categories the notion of Quillen equivalence is considerably stron*
*ger than trian-
gulated equivalence of homotopy categories (see Remark 3.2.1 for an example). *
*Hence it is
somewhat remarkable that for chain complexes of modules over ringoids the two n*
*otions are
in fact equivalent. In general the homotopy category determines the homotopy gr*
*oups of the
spectral endomorphism category, but not its homotopy type. The real reason behi*
*nd the equiv-
alences of conditions (1) and (2) above is the fact that in contrast to arbitra*
*ry ring spectra or
spectral categories, Eilenberg-Mac Lane objects are determined by their homotop*
*y groups, see
Proposition B.2.1.
As a tool for proving the generalized tilting theorem we introduce the Eilenb*
*erg-Mac Lane
spectral category HA_of a ringoid A_. This is simply the many-generator version*
* of the symmetric
Eilenberg-Mac Lane ring spectrum [HSS , 1.2.5]. The key property is that module*
* spectra over
the Eilenberg-Mac Lane spectral category HA_ are Quillen equivalent to chain co*
*mplexes of
A_-modules.
Definition 5.1.5.Let A_be a ringoid. The Eilenberg-Mac Lane spectral category H*
*A_is defined
by
HA_ = A_ HZ ,
where HZ is the symmetric Eilenberg-Mac Lane ring spectrum of the integers [HSS*
* , 1.2.5]. In
more detail, HA_has the same set of objects as A_, and the morphism spectra are*
* defined by
HA_(a, b)p = A_(a, b) eZ[Sp] .
Here eZ[Sp] denotes the reduced simplicial free abelian group generated by the *
*pointed simplicial
set Sp = S1 ^ . .^.S1 (p factors), and the symmetric group permutes the factors*
*. Composition
30 STEFAN SCHWEDE AND BROOKE SHIPLEY
is given by the composite
HA_(b, c)p^ HA_(a, b)q = (A_(b, c) eZ[Sp]) ^ (A_(a, b) eZ[Sq])
- shuffle-----!A e p e q
__(b, c) A_(a, b) Z[S ] Z[S ]
- -O-~=---!A e p+q
__(a, c) Z[S ] = HA_(a, c)p+q .
The unit map
Sp ----! A_(a, a) eZ[Sp] = HA_(a, a)p
is the inclusion of generators.
We prove the following result in Appendix B.
Theorem 5.1.6. For any ringoid A_, the category of complexes of A_-modules and *
*the category
of modules over the Eilenberg-Mac Lane spectral category HA_are Quillen equival*
*ent,
mod-HA_'Q ChA_
Proof of Theorem 5.1.1.Every Quillen equivalence of stable model categories ind*
*uces an equiv-
alence of triangulated homotopy categories, so condition (1) implies condition *
*(2). Any trian-
gulated equivalence restricts to an equivalence between the respective subcateg*
*ories of compact
objects. By the same argument as [BN93 , Prop. 6.4] (which deals with the speci*
*al case of com-
plexes of modules over a ring), Kb(proj-A_) is equivalent to the full subcatego*
*ry of compact
objects in D(A_). Since the derived category of a ringoid has a set of compact *
*generators, so does
any equivalent triangulated category. Hence condition (2) implies condition (2'*
*).
Now we assume condition (2') and we choose a triangulated equivalence between*
* Kb(proj-A_)
and the full subcategory of compact objects in Ho(C). For a 2 A_we let Ta be a *
*representative in
C of the image of the representable A_-module Fa = A_(-, a), viewed as a comple*
*x concentrated
in dimension zero. Since the collection of modules {Fa}a2A_is a set of tiltors *
*for the derived
category, the set {Ta}a2A_has all the properties of a set of tiltors, except th*
*at it may not
generate the full homotopy category of C. However the localizing subcategory ge*
*nerated by the
Ta's coincides with the localizing subcategory generated by all compact objects*
* since on the
other side of the equivalence the complexes Fa generate the category Kb(proj-A_*
*). In general the
compact objects might not generate all of Ho(C)(see [HS99, Cor. B.13] for some *
*extreme cases
where the zero object is the only compact object), but here this is assumed in *
*(2'). So the Ta's
generate C, hence they are a set of tiltors, and so condition (3) holds.
If on the other hand C has a set of tiltors T, then T is in particular a set *
*of compact generators,
so by Theorem 3.1.1, C is Quillen equivalent to the category of modules over th*
*e endomorphism
category End(T). In this special case the homotopy type of the spectral catego*
*ry End(T) is
determined by its homotopy groups: since the homotopy groups of End(T) are conc*
*entrated
in dimension 0, End(T) is stably equivalent to HA_, the Eilenberg-Mac Lane spec*
*tral category
of its component ringoid A_by Proposition B.2.1. Thus the categories of End(T)-*
*modules and
HA_-modules are Quillen equivalent by Theorem A.1.1. Theorem 5.1.6 gives the fi*
*nal step in the
chain of Quillen-equivalences
C 'Q mod-End(T) 'Q mod-HA_ 'Q ChA_.
|___|
Appendix A. Spectral categories
In this appendix we develop some general theory of modules over spectral cate*
*gories (Defini-
tion 3.3.1). The arguments are very similar to the case of spectral categories *
*with one object,
i.e., symmetric ring spectra.
STABLE MODEL CATEGORIES 31
A.1. Model structures for modules over spectral categories. A morphism : O -!*
* R
of spectral categories is simply a spectral functor. The restriction of scalars
* : mod-R ----! mod-O , M 7-! M O
has a left adjoint functor *, also denoted - ^O R, which we refer to as extens*
*ion of scalars. As
usual it is given by an enriched coend, i.e., for an O-module N the R-module **
*N = N ^O R
is given by the coequalizer of the two R-module homomorphisms
` `
N(p) ^ O(o, p) ^ F (o)----!----!N(o) ^ F (o),
o,p2O o2O
where F (o)= R(-, (o)) is the free R-module associated to the object (o). We *
*call : O -!
R a stable equivalence of spectral categories if it is a bijection on objects a*
*nd if for all objects
o, o0in O the map
o,o0: O(o, o0) ----! R( (o), (o0))
is a stable equivalence of symmetric spectra.
Next we establish the model category structure for O-modules, show its invari*
*ance under
restriction of scalars along a stable equivalence of spectral categories and ex*
*hibit a set of compact
generators.
Theorem A.1.1. (i)Let O be a spectral category. Then the category of O-module*
*s with the
objectwise stable equivalences, objectwise stable fibrations, and cofibrati*
*ons is a cofibrantly
generated spectral model category.
(ii)The free modules {Fo}o2O given by Fo = O(-, o) form a set of compact gener*
*ators for the
homotopy category of O-modules.
(iii)Assume : O -! R is a stable equivalence of spectral categories. Then re*
*striction and
extension of scalars along form a spectral Quillen equivalence of the mod*
*ule categories.
Proof.We use [SS00, 2.3] to lift the stable model structure from (families of) *
*symmetric spectra
to O-modules. Let SO denote the spectral category with the same set of objects *
*as O, but with
morphism spectra given by
æ 0
SO (o, o0) = S* ifeol=soe,.
An SO -module is just a family of symmetric spectra indexed by the objects of O*
*. Hence it
has a cofibrantly generated model category structure in which the cofibrations,*
* fibrations and
weak equivalences are defined objectwise on underlying symmetric spectra. Here *
*the generating
trivial cofibrations are maps between modules concentrated at one object, i.e. *
*of the form Ao
with Ao(o) = A and Ao(o0) = * if o 6= o0.
The unit maps give a morphism of spectral categories SO - ! O, which in turn *
*induces
adjoint functors of restriction and extension of scalars between the module cat*
*egories. This
produces a triple - ^SO O on SO -modules with the algebras over this triple the*
* O-modules.
Then the generating trivial cofibrations for O-modules are maps between modules*
* of the form
Ao^SO O = A ^ O(-, o) = A ^ Fo. Hence the monoid axiom [HSS , 5.4.1] applies to*
* show that
the new generating trivial cofibrations and their relative cell morphisms are w*
*eak equivalences.
Thus, since all symmetric spectra, hence all SO -modules are small, the model c*
*ategory structure
follows by criterion (1) of [SS00, 2.3]. We omit the verification of the spectr*
*al axiom (SP), which
then implies stability by Lemma 3.5.2.
The proof of (ii) uses the adjunction defined above between SO -modules and O*
*-modules.
Since Fo = So^SO O,
)
[Fo, M]Ho(mod-O)*~=[So, M]Ho(mod-SO)*~=[S, M(o)]Ho(Sp*.
32 STEFAN SCHWEDE AND BROOKE SHIPLEY
Thus, since S is a generator for Sp and an O-module is trivial if and only if *
*it is objectwise
trivial, the set of free O-modules is a set of generators. The argument that Fo*
* is compact is
similar because the map
M a
[Fo, Mi]Ho(mod-O)----![Fo, Mi]Ho(mod-O)
i2I i2I
is isomorphic to the map
M a
[S, Mi(o)]Ho(Sp )----![S, Mi(o)]Ho(Sp )
i2I i2I
and S is compact.
The proof of (iii) follows as in [SS00, 4.3]. The restriction functor * pres*
*erves objectwise
fibrations and objectwise equivalences, so restriction and extension of scalars*
* form a Quillen
adjoint pair. For every cofibrant right O-module N, the induced map
N ~=N ^O O - ! N ^O R
is an objectwise stable equivalence, by a similar `cell induction' argument as *
*for ring spectra [HSS ,
5.4.4] or [MMSS , 12.7]. Thus if M is any right R-module, an O-module map N -!*
* *M is an
objectwise stable equivalence if and only if the adjoint R-module map *N = N ^*
*O R -!_M
is an objectwise stable equivalence. *
* |__|
A.2. Quasi-equivalences. This section introduces quasi-equivalences which are a*
* bookkeeping
device for producing stable equivalences between symmetric ring spectra or spec*
*tral categories,
see Lemma A.2.3 below. The name is taken from [Kel94, Summary, p. 64], where th*
*is notion
is discussed in the context of differential graded algebras. Every (stable) eq*
*uivalence of ring
spectra gives rise to a quasi-equivalence; conversely the proof of Lemma A.2.3 *
*shows that a
single quasi-equivalence encodes a zig-zag of four stable equivalences relating*
* two ring spectra
or spectral categories. One place where quasi-equivalences arise `in nature' i*
*s the proof that
weakly equivalent objects in a model category have weakly equivalent endomorphi*
*sm monoids,
see Corollary A.2.4.
If R and O are spectral categories, their smash product R ^ O is the spectral*
* category whose
set of objects is the cartesian product of the objects of R and O and whose mor*
*phism objects
are defined by the rule
R ^ O((r, o), (r0, o0)) = R(r, r0) ^ O(o, o0) .
An R-O-bimodule is by definition an Rop^ O-module. Since modules for us are alw*
*ays con-
travariant functors, an R-O-bimodule translates to a covariant spectral functor*
* from Oop^ R
to Sp .
Definition A.2.1.Let R and O be two spectral categories with the same set I of *
*objects.
Then a quasi-equivalence between R and O is an R-O-bimodule M together with a c*
*ollection
of `elements' 'i 2 M(i, i) (i.e., morphisms S -! M(i, i)) for all i 2 I such th*
*at the following
holds: for all pairs i and j of objects the right multiplication with 'iand the*
* left multiplication
with 'j,
R(i, j) --.'i--!M(i, j) --'j.--O(i, j)
are stable equivalences.
Remark A.2.2. In the important special case of spectral categories with a singl*
*e object, i.e.,
for two symmetric ring spectra R and T , a quasi-equivalence is an R-T -bimodul*
*e M together
STABLE MODEL CATEGORIES 33
with an element ' 2 M (i.e., a vertex of the 0-th level of M or equivalently a *
*map S -! M of
symmetric spectra) such that the left and right multiplication maps with ',
R --.'--!M -'.---T
are stable equivalences of symmetric spectra.
If : O -! R is a stable equivalence of spectral categories, then the target*
* R becomes an
R-O-bimodule if O acts on the right via . Furthermore the identity elements in*
* R(i, i) for all
objects i of R make the bimodule R into a quasi-equivalence between R and O. Th*
*e following
lemma shows conversely that quasi-equivalent spectral categories are related by*
* a chain of weak
equivalences:
Lemma A.2.3. Let R and O be two spectral categories with the same set of objec*
*ts. If a quasi-
equivalence exists between R and O, then there is a chain of stable equivalence*
*s between R
and O.
Proof.i) Special case: suppose there exists a quasi-equivalence (M, {'i}i2I) fo*
*r which all of the
right multiplication maps .'i: R(i, j) -! M(i, j) are trivial fibrations. In th*
*is case we define a
new spectral category E(M, ') with objects I as the pullback of R and O over M.*
* More precisely
for every pair i, j 2 I the homomorphism object E(M, ')(i, j) is defined as the*
* pullback in Sp
of the diagram
R(i, j) --.'i--!M(i, j) --'j.--O(i, j) .
Using the universal property of the pullback there is a unique way to define co*
*mposition and
identity morphisms in E(M, ') in such a way that the maps E(M, ') -! O and E(M,*
* ') -! R
are homomorphisms of spectral categories.
Since M is a quasi-equivalence, all the maps in the defining pullback diagram*
*s are weak
equivalences. By assumption the horizontal ones are even trivial fibrations, so*
* both base change
maps E(M, ') -! O and E(M, ') -! R are pointwise equivalences of spectral categ*
*ories. The
same argument works if instead of the right multiplication maps all the left mu*
*ltiplication maps
'j. : O(i, j) -! M(i, j) are trivial fibrations.
ii) General case: taking fibrant replacement if necessary we can assume that *
*the bimodule M is
objectwise fibrant. The `element' 'j of M(j, j) corresponds to a map Fj = O(-, *
*j) -! M(-, j)
from the free O-module; the map is left multiplication by 'j and is an objectwi*
*se equivalence
since M is a quasi-equivalence. We factor this O-module equivalence as a trivi*
*al cofibration
ffj : Fj -! Nj followed by a trivial fibration _j : Nj -! M(-, j); in particula*
*r, the objects Nj
so obtained are cofibrant and fibrant. We let E(N) denote the endomorphism spec*
*tral category
of the cofibrant-fibrant replacements, i.e., the full spectral subcategory of t*
*he category of O-
modules with objects Nj for j 2 I. Now we appeal twice to the special case that*
* we already
proved, obtaining a chain of four stable equivalences of spectral categories
O --~-- E(W, ff) --~--! E(N) --~-- E(V, _) --~--! R .
In more detail, we define a E(N)-O-bimodule W by the rule
W (i, j) = Hom mod-O(Fi, Nj) ~= Nj(i) .
The bimodule W is a quasi-equivalence with respect to the maps ffj. Moreover, *
*the right
multiplication map .ffiis the restriction map
ff*i: Hom mod-O(Ni, Nj) ----! Hom mod-O(Fi, Nj) .
So ff*iis a trivial fibration since ffiis a trivial cofibration of O-modules an*
*d Njis a fibrant module.
Case i) above then provides a chain of stable equivalences between O and E(N), *
*passing through
E(W, ff).
34 STEFAN SCHWEDE AND BROOKE SHIPLEY
Now we define an R-E(N)-bimodule V by the rule
V (i, j) = Hom mod-O(Ni, M(-, j)).
The bimodule V is a quasi-equivalence with respect to the maps _j. The left mul*
*tiplication map
_j. is the composition
(_j)* : Hom mod-O(Ni, Nj) ----! Hom mod-O(Ni, M(-, j)) .
This time (_j)* is a trivial fibration since _j is a trivial fibration of O-mod*
*ules and Ni is a
cofibrant module. Furthermore the right multiplication map
._i: R(i, j) ----! Hom mod-O(Ni, M(-, j))
is an equivalence because its composite with the map
ff*i: Hom mod-O(Ni, M(-, j)) ----! Hom mod-O(Fi, M(-, j)) ~= M(i, j)
is right multiplication by _i, an equivalence by assumption. Recall M is object*
*wise fibrant, so
ff*iis a weak equivalence. So case i) gives a chain of pointwise equivalences b*
*etween_R and E(N),
passing through E(V, _). *
*|__|
As a corollary we obtain the homotopy invariance of endomorphism spectral cat*
*egories under
spectral Quillen equivalences.
Corollary A.2.4.Suppose C and D are spectral model categories and L : C -! D is*
* the left
adjoint of a spectral Quillen equivalence. Suppose I is a set, {Pi}i2Iand {Qi}i*
*2Iare sets of
cofibrant-fibrant objects of C and D respectively, and that for all i 2 I, LPii*
*s weakly equivalent to
Qiin D. Then the spectral endomorphism categories of {Pi}i2Iand {Qi}i2Iare stab*
*ly equivalent.
In particular the spectral endomorphism category of {Pi}i2Idepends up to pointw*
*ise equivalence
only on the weak equivalence type of the objects Pi.
Proof.Since the object LPi is cofibrant and weakly equivalent to the fibrant ob*
*ject Qi, we
can choose a weak equivalence 'i: LPi- ! Qifor every i 2 I. We claim that the c*
*ollection of
homomorphism objects HomD (LPi, Qj) forms a quasi-equivalence for the endomorph*
*ism spectral
categories of {Pi}i2Iand {Qi}i2Iwith respect to the equivalences 'i. Indeed the*
* endomorphism
category of {Qi}i2Iacts on the left by composition; also right multiplication b*
*y 'j is a stable
equivalence since Qj is fibrant and 'j is a weak equivalence between cofibrant *
*objects. If R
denotes the right adjoint of L, then Hom D(LPi, Qj) is isomorphic to Hom C(Pi, *
*RQj), so the
endomorphism category of {Pi}i2Iacts on the right by composition. Since R and *
*L form
a spectral Quillen equivalence, the adjoints b'i: Pi -! RQi are weak equivalenc*
*es between
fibrant objects; so left multiplication by 'iis a stable equivalence since Piis*
* cofibrant._The_last
statement is the special case where D = C and L is the identity functor. *
* |__|
Appendix B. Eilenberg-Mac Lane spectra and chain complexes
The proof of the generalized tilting theorem in Section 5 uses the Eilenberg-*
*Mac Lane spectral
category HA_of a ringoid A_. Recall that a ringoid is a small category whose ho*
*m-sets carry an
abelian group structure for which composition is bilinear. A right module over *
*a ringoid is a
contravariant additive functor to the category of abelian groups. The Eilenberg*
*-Mac Lane spec-
tral category HA_of a ringoid A_is defined in 5.1.5. In this appendix we provid*
*e some general
facts about Eilenberg-Mac Lane spectral categories. The main results are that m*
*odule spectra
over the Eilenberg-Mac Lane spectral category HA_are Quillen equivalent to chai*
*n complexes
of A_-modules (Theorem 5.1.6) and that Eilenberg-Mac Lane spectral categories a*
*re determined
up to stable equivalence by their coefficient ringoid (Theorem B.2.1). These pr*
*operties are not
unexpected, and variations have been proved for the special case of ring spectr*
*a in different
STABLE MODEL CATEGORIES 35
frameworks. Indeed the Quillen equivalence of Theorem 5.1.6 is a generalization*
* and strength-
ening of the fact first proved in [Rob87b] that the unbounded derived category *
*of modules over
a ring R is equivalent to the homotopy category of HR-modules, see also [EKMM *
*, IV Thm. 2.4]
in the context of S-algebras.
B.1. Chain complexes and module spectra. Throughout this section we fix a ringo*
*id A_,
and we want to prove Theorem 5.1.6 relating the modules over the Eilenberg-Mac *
*Lane spectral
category HA_to complexes of A_-modules by a chain of Quillen equivalences. We d*
*o not know
of a Quillen functor pair which does the job in a single step. Instead, we com*
*pare the two
categories through the intermediate model category of naive HA_-modules, obtain*
*ing a chain of
Quillen equivalences
mod-HA_ __U__wu_____Nvmod-HA_Hu______wCh A_
L
(the right adjoints are on top), see Corollary B.1.8 and Theorem B.1.11. We me*
*ntion here
that an analogous statement holds for differential graded modules over a differ*
*ential graded ring
and modules over the associated Eilenberg-Mac Lane spectrum, but the proof beco*
*mes more
complicated; see Remark B.1.10 and [SS2].
Definition B.1.1.Let O be a spectral category. A naive O-module M consists of a*
* collection
{M(o)}o2O of N-graded, pointed simplicial sets together with associative and un*
*ital action maps
M(o)p ^ O(o0, o)q ----!M(o0)p+q
for pairs of objects o, o0in O and for natural numbers p, q 0. A morphism of *
*naive O-modules
M -! N consists of maps of graded spaces M(o) -! N(o) strictly compatible with *
*the action
of O. We denote the category of naive O-modules by Nvmod-O.
Note that a naive module M has no symmetric group action on M(o)n, and hence *
*there is no
equivariance condition for the action maps. A naive O-module has strictly less *
*structure than a
genuine O-module, so there is a forgetful functor
U : mod-O ----! Nvmod-O .
The free naive O-module Fo at an object o 2 O is given by the graded spaces F*
*o(o0) = O(o0, o)
with action maps
Fo(o0)p ^ O(o00, o0)q = O(o0, o)p ^ O(o00, o0)q ----!O(o00, o)p+q = Fo(o00*
*)p+q
given by composition in O. In other words, the forgetful functor takes the fre*
*e, genuine O-
module to the free, naive O-module. The free naive module Fo represents evalua*
*tion at the
object o 2 O, i.e., there is an isomorphism of simplicial sets
(B.1.2) map Nvmod-O(Fo, M) ~= M(o)0
which is natural for naive O-modules M.
If M is a naive O-module, then at every object o 2 O, M(o) has an underlying *
*spectrum in
the sense of Bousfield-Friedlander [BF78 , x2] (except that in [BF78 ], the sus*
*pension coordinates
appear on the left, whereas we get suspension coordinates acting from the right*
*). Indeed, using
the unital structure map S1 -! O(o, o)1 of the spectral category O, the graded *
*space M(o) gets
suspension maps via the composite
M(o)p ^ S1 ----! M(o)p ^ O(o, o)1 ----!M(o)p+1 .
A morphism of naive O-modules f : M -! N is an objectwise ß*-isomorphism if for*
* all o 2 O
the map f(o) : M(o) -! N(o) induces an isomorphism of stable homotopy groups. T*
*he map f
36 STEFAN SCHWEDE AND BROOKE SHIPLEY
is an objectwise stable fibration if each f(o) is a stable fibration of spectra*
* in the sense of [BF78 ,
Thm. 2.3]). A morphism of naive O-modules is a cofibration if it has the left l*
*ifting properties
for maps which are objectwise ß*-isomorphisms and objectwise stable fibrations.
Theorem B.1.3. Let A_be a ringoid.
(i)The category of naive HA_-modules with the objectwise ß*-isomorphisms, obj*
*ectwise stable
fibrations, and cofibrations is a cofibrantly generated, simplicial, stable*
* model category.
(ii)The collection of free HA_-modules {Fa}a2A_forms a set of compact generato*
*rs for the
homotopy category of naive HA_-modules.
(iii)Let C be a stable model category and consider a Quillen adjoint functor p*
*air
C __æ__wu_____Nvmod-HA_
~
where æ is the right adjoint. Then (~, æ) is a Quillen equivalence, provide*
*d that
(a)for every object a 2 A_, the object ~(Fa) is fibrant in C
(b)for every object a 2 A_, the unit of the adjunction Fa -! æ~(Fa) is an *
*objectwise
ß*-isomorphism, and
(c)the objects {~(Fa)}a2A_form a set of compact generators for the homotop*
*y category
of C.
Proof.(i) We use Theorem 3.7.4 to establish the model category structure. The c*
*ategory of naive
HA_-modules is complete and cocomplete and every naive HA_-module is small. The*
* objectwise
ß*-isomorphisms are closed under the 2-out-of-3 condition (Theorem 3.7.4 (1)).
As generating cofibrations I we use the collection of maps
(@ i)+ ^ Fa[n] ----! ( i)+ ^ Fa[n]
for all i, n 0 and a 2 A_. Here i denotes the simplicial i-simplex and @ i i*
*s its boundary;
the square bracket [n] means shifting (reindexing) of a naive HA_-modules and s*
*mashing of a
module and a pointed simplicial set is levelwise. Since the free modules repres*
*ent evaluation at
an object (see (B.1.2)above), the I-injectives are precisely the maps which are*
* objectwise level
acyclic fibrations.
As generating acyclic cofibrations J we use the union J = Jlv[ Jst. Here Jlvi*
*s the set of
maps
( ik)+ ^ Fa[n] --~--! ( i)+ ^ Fa[n]
for i, n 0, 0 k i and a 2 A_, where i,kis the k-th horn of the i-simplex*
*. The Jlv-injectives
are the objectwise level fibrations. Finally, Jstconsists of the mapping cylind*
*er inclusions of the
maps
(B.1.4)
S1 ^ ( i)+ ^ Fa[n+ 1] [S1^(@ i)+^Fa[n+1](@ i)+ ^ Fa[n] ----! ( i)+ ^ Fa[*
*n] .
Here the mapping cylinders are formed on each simplicial level, just as in [HSS*
* , 3.1.7]. Every
map in J is an I-cofibration, hence every relative J-cell complex is too; we cl*
*aim that in addition,
every map in J is an objectwise injective ß*-isomorphism. Since this property i*
*s closed under
infinite wedges, pushout, sequential colimit and retracts this implies that eve*
*ry relative J-cell
complex is an objectwise injective ß*-isomorphism and so condition (2) of Theor*
*em 3.7.4 holds.
The maps in Jlvare even objectwise injective level-equivalences, so it remain*
*s to check the
maps in Jst. These maps are defined as mapping cylinder inclusions, so they are*
* injective, and
STABLE MODEL CATEGORIES 37
we need only check that the maps in (B.1.4)above are objectwise ß*-isomorphisms*
*. This in turn
follows once we know that the maps
(B.1.5) S1 ^ Fa[n+ 1] ----! Fa[n]
are objectwise ß*-isomorphisms. At level p n + 1 and an object b 2 A_, this m*
*ap is given by
the inclusion
S1 ^ (A_(b, a) eZ[Sp-n-1]) ----!A_(b, a) eZ[Sp-n]
whose adjoint is a weak equivalence. This map is roughly 2(p - n)-connected, so*
* in the limit we
indeed obtain a ß*-isomorphism.
It remains to check condition (3) of Theorem 3.7.4, namely that the I-injecti*
*ves coincide
with the maps which are both J-injective and objectwise ß*-isomorphisms. Every *
*map in J is
an I-cofibration, so I-injectives are J-injective. Since I-injectives are leve*
*l acyclic fibrations,
they are also objectwise ß*-isomorphisms. Conversely, suppose f : M -! N is an *
*objectwise
ß*-isomorphism of naive HA_-modules which is also J-injective. Since f is Jlv-i*
*njective, it is an
objectwise level fibration. Since f is Jlv-injective, at every object a 2 A_, t*
*he underlying map of
spectra f(a) : M(a) -! N(a) has the right lifting property for the maps
S1 ^ ( i)+ ^ S[n+ 1] [S1^(@ i)+^S[n+1](@ i)+ ^ S[n] ----! ( i)+ ^ S[n] ,
where S is the sphere spectrum. But then f(a) is a stable fibration of spectra *
*[Sch01, A.3], so f
is an objectwise stable fibration and ß*-isomorphism. By [BF78 , A.8 (ii)], f i*
*s then an objectwise
level fibration, so it is I-injective. So conditions (1)-(3) of Theorem 3.7.4 a*
*re satisfied and this
theorem provides the model structure. We omit the verification that the model s*
*tructure for
naive HA_-modules is simplicial and stable; the latter is a consequence of the *
*fact that stable
equivalences of HA_-modules are defined objectwise and spectra form a stable mo*
*del category.
(ii) The stable model structure for naive HA_-modules is defined so that eval*
*uation at a 2 A_
is a right Quillen functor to the stable model category of Bousfield-Friedlande*
*r type spectra.
Moreover, evaluation at a 2 A_has a left adjoint which takes the sphere spectru*
*m S to the free
module Fa. So the derived adjunction provides an isomorphism of graded abelian *
*groups
[Fa, M]Ho(Nvmod-HA_)*~=[S, M(a)]Ho(Sp)*~=ß*M(a) .
This implies that in the homotopy category the free modules detect objectwise ß*
**-isomorphisms,
so they form a set of generators. It also implies that the representable module*
*s are compact,
because evaluation at a 2 A_and homotopy groups commute with infinite sums.
(iii) We have to show that the derived adjunction on the level of homotopy ca*
*tegories
Ho(C) __Ræ_wu_____Ho(Nvmod-HA_)
L~
yields equivalences of (homotopy) categories. The right adjoint Ræ detects iso*
*morphisms: if
f : X -! Y is a morphism in Ho(C) such that Ræ(f) is an isomorphism in the homo*
*topy
category of naive HA_-modules, then for every a 2 A_, the map f induces an isom*
*orphism on
[L~(Fa), -] by adjointness. Since the objects L~(Fa) generate the homotopy cate*
*gory of C, f is an
isomorphism. It remains to show that the unit of the derived adjunction jM : M *
*-! Ræ(L~(M))
on the level of homotopy categories is an isomorphism for every HA_-module M. F*
*or the free
HA_-modules Fa this follows from assumptions (a) and (b): by (a), ~(Fa) is fibr*
*ant in C, so
the point set level adjunction unit Fa -! æ~(Fa) models the derived adjunction *
*unit, then by
(b) jM is an isomorphism. The composite derived functor Ræ O L~ is exact; the*
* functor Ræ
commutes with coproducts (a formal consequence of (ii)), hence so does Ræ O L~ *
*since L~ is a
38 STEFAN SCHWEDE AND BROOKE SHIPLEY
left adjoint. Hence the full subcategory of those HA_-modules M for which the d*
*erived unit jM
is an isomorphism is a localizing subcategory. Since it also contains the gener*
*ating representable_
modules, it coincides with the full homotopy category of naive HA_-modules. *
* |__|
Remark B.1.6. The reader may wonder why we do not state Theorem B.1.3 for a gen*
*eral
spectral category O. The reason is that already the analog of part (i), the exi*
*stence of the stable
model structure for naive O-modules, can fail without some hypothesis on O. The*
* problem can
be located: one needs that the analog of the map (B.1.5),
S1 ^ Fo[n + 1] ----! Fo[n]
which is given by the action of the suspension coordinates from the left, induc*
*es an isomorphism
of homotopy groups, taken with respect to suspension on the right. But in gener*
*al, the effects
of left and right suspension on homotopy groups can be related in a complicated*
* way. We hope
to return to these questions elsewhere.
As a corollary, we use the criteria in part (iii) of the previous theorem to *
*establish the Quillen
equivalence between the model category of (right) HA_-modules of symmetric spec*
*tra and the
model category of (right) naive HA_-modules. These criteria are also used to es*
*tablish the Quillen
equivalence between naive HA_-modules and chain complexes of A_-modules, see Th*
*eorem B.1.11
below.
First we recall a general categorical criterion for the existence of left adj*
*oints. Recall from
[AR94 , Def. 1.1, 1.17] that an object K of a category C is finitely presentabl*
*e if the hom functor
Hom C(K, -) preserves filtered colimits. A category C is called locally finitel*
*y presentable if it is
cocomplete and there exists a set A of finitely presentable objects such that e*
*very object of C is
a filtered colimit of objects in A. The condition `locally finitely presentable*
*' implies that every
object is small in the sense of [Hov99, 2.1.3]. For us the point of this defini*
*tion is that every
functor between locally finitely presentable categories which commutes with lim*
*its and filtered
colimits has a left adjoint (this is a special case of [AR94 , 1.66]). We omit*
* the proof of the
following lemma.
Lemma B.1.7. Let A_be a ringoid. Then the categories of complexes of A_-module*
*s, of (genuine)
HA_-modules and of naive HA_-modules are locally finitely presentable.
Corollary B.1.8.The forgetful functor from HA_-modules to naive HA_-modules is *
*the right
adjoint of a Quillen equivalence.
Proof.The forgetful functor U from HA_-modules to naive HA_-modules preserves l*
*imits and
filtered colimits. Since source and target category are locally finitely presen*
*table, U has a left
adjoint L by [AR94 , 1.66]. The forgetful functor from symmetric spectra to (no*
*n-symmetric)
spectra is the right adjoint of a Quillen functor pair, see [HSS , 4.2.4]. So t*
*he forgetful functor U
from HA_-modules to naive HA_-modules preserves objectwise stable equivalences *
*and objectwise
stable fibrations. Thus U and L form a Quillen pair, and we can apply part (iii*
*) of Theorem
B.1.3. The left adjoint L sends the naive free modules Fa to the genuine free m*
*odules, so the
relevant adjunction unit in condition (b) is even an isomorphism. For every pa*
*ir of objects
a, b 2 A_, the symmetric spectrum (LFa)(b) = HA_(b, a) is a symmetric -spectru*
*m, hence stably
fibrant, which gives condition (a). The free modules form a set of compact gene*
*rators for the__
homotopy category of genuine HA_-modules, by Theorem A.1.1, so condition (c) is*
* satisfied. |__|
To finish the proof of Theorem 5.1.6 we now construct a Quillen-equivalence b*
*etween naive
HA_-modules and complexes of A_-modules. We define another Eilenberg-Mac Lane f*
*unctor
H : Ch A_----!Nvmod- HA_
STABLE MODEL CATEGORIES 39
from the category of chain complexes of A_-modules to the category of naive mod*
*ules over HA_.
For any simplicial set K we denote by NK the normalized chain complex of the *
*free simplicial
abelian group generated by K. So NK is a non-negative dimensional chain complex*
* which in
dimension n is isomorphic to the free abelian group on the non-degenerate n-sim*
*plices of K. A
functor W from the category of chain complexes ChZ to the category of simplicia*
*l abelian groups
is defined by
(W C)k = hom ChZ(N [k], C) .
For non-negative dimensional complexes, W is just the inverse to the normalized*
* chain functor in
the Dold-Kan equivalence between simplicial abelian groups and non-negative dim*
*ensional chain
complexes [Do58, 1.9]. For an arbitrary complex C there is a natural chain map *
*NW C -! C
which is an isomorphism in positive dimensions and which expresses NW C as the *
*(-1)-connected
cover of C.
For a chain complex of abelian groups C we define a graded space by the formu*
*la
(HC)n = W (C[n])
where C[n] denotes the n-fold shift suspension of the complex C. To define the *
*module structure
maps we use the Alexander-Whitney map, see [EM54 , 2.9] or [May67 , 29.7]. Thi*
*s map is a
natural, associative and unital transformation of simplicial abelian groups
AW : W (C) W (D) ----! W (C D) .
Here the left tensor product is the dimensionwise tensor product of simplicial *
*abelian groups,
whereas the right one is the tensor product of chain complexes. The Alexander-W*
*hitney map
is neither commutative, nor an isomorphism. By our conventions the p-sphere Sp*
* is the p-
fold smash product of the simplicial circle S1 = [1]=@ [1], so the reduced fre*
*e abelian group
generated by Sp is the p-fold tensor product of the simplicial abelian group eZ*
*[S1] = W (Z[1])
(where Z[1] is the chain complex which contains a single copy of the group Z in*
* dimension 1).
Since the p-th space in the Eilenberg-Mac Lane spectrum HA_(a, b) is given by H*
*A_(a, b)p =
A_(a, b) eZ[Sp], for every chain complex D of A_-modules the Alexander-Whitne*
*y map gives a
map
H(D(b))p^ HA_(a, b)q----! H(D(b))p HA_(a, b)q
~=
----! W (D(b)[p]) A_(a, b) W_(Z[1])___.-.z.W_(Z[1]*
*)_")
0 q 1
--AW--!W B@D(b)[p] A C
__(a, b) Z[1]___._._.Z[1]-z_____*
*"A
q
----! W (D(a)[p + q]) = H(D(a))p+q .
These maps make HD into a naive HA_-module. The spectra underlying HD(a) are a*
*lways
-spectra and the stable homotopy groups of HD(a) are naturally isomorphic to t*
*he homology
groups of the chain complex D(a),
(B.1.9) ß*HD ~= H*D
as graded A_-modules.
Remark B.1.10. The functor H should not be confused with the Eilenberg-Mac Lane*
* functor
H of Definition 5.1.5. The functor H takes values in symmetric spectra, but it*
* cannot be
extended in a reasonable way to chain complexes; the functor H is defined for c*
*omplexes, but it
only takes values in naive HA_-modules.
40 STEFAN SCHWEDE AND BROOKE SHIPLEY
The essential difference between the two functors can already be seen for an *
*abelian group A.
The simplicial abelian group (HA)n = W (A[n]) is the minimal model of an Eilenb*
*erg-Mac Lane
space of type K(A, n) and it is determined by the property that its normalized *
*chain complex
consists only of one copy of A in dimension n. The simplicial abelian group (HA*
*)n = A eZ[Sn]
is another Eilenberg-Mac Lane space of type K(A, n), but it has non-degenerate *
*simplices in
dimensions smaller than n. The Alexander-Whitney map gives a weak equivalence o*
*f simplicial
abelian groups A eZ[Sn] -! W (A[n]).
However, the Alexander-Whitney map is not commutative, and for n 2 there is*
* no n-action
on the minimal model W (A[n]) which admits an equivariant weak equivalence from*
* eZ[Sn] A.
More generally, the graded space HA cannot be made into a symmetric spectrum wh*
*ich is
level equivalent to the symmetric spectrum HA. This explains why the compariso*
*n between
HA-modules and complexes of A-modules has to go through the category of naive H*
*A-modules.
Theorem B.1.11. Let A_be a ringoid. Then the Eilenberg-Mac Lane functor H is t*
*he right
adjoint of a Quillen equivalence between chain complexes of A_-modules and naiv*
*e HA_-modules.
Proof.The functor H commutes with limits and filtered colimits, and since sourc*
*e and target
category of H are locally finitely presentable, a left adjoint exists by [AR9*
*4 , 1.66]. The
Eilenberg-Mac Lane functor takes values in the category of -spectra, which are*
* the (stably)
fibrant objects in the category of naive HA_-modules. Moreover, it takes object*
*wise fibrations of
chain complexes (i.e., epimorphisms) to objectwise level fibrations. Since leve*
*l fibrations between
-spectra are stable fibrations, H preserves fibrations. Because of the isomor*
*phism labeled
(B.1.9), H takes objectwise quasi-isomorphisms of A_-modules to objectwise stab*
*le equivalences
of HA_-modules, so it also preserves acyclic fibrations. Thus H and form a Qu*
*illen adjoint
functor pair.
Now we apply criterion (iii) of Theorem B.1.3. Every chain complex of A_-modu*
*les is fibrant in
the projective model structure, so condition (a) holds. If we consider the free*
* A_-module A_(-, a),
as a complex in dimension 0, then the identity element in A_(a, a) ~=HA_(-, a)(*
*a)0 is represented
by a map of naive HA_-modules ~ : Fa -! H(A_(-, a)). By the adjunction and repr*
*esentability
isomorphisms
hom ChA_( (Fa), D) ~= homNvmod-HA_(Fa, HD) ~= (HD(a))0 ~= homChA_(A_(-, a), *
*D) ,
so the complexes (Fa) and A_(-, a) represent the same functor. Thus, the adjoi*
*nt of ~ is an
isomorphism from (Fa) to A_(-, a). The adjunction unit relevant for condition *
*(b) is the map
~ : Fa -! H(A_(-, a)) ~=H (Fa). At an object b 2 A_and in dimension p, the map *
*~ specializes
to the Alexander-Whitney map
Fa(b)p = A_(b, a) eZ[Sp] ----! W (A_(b, a)[p]) = H(A_(b, a))p .
Both sides of this map are Eilenberg-Mac Lane spaces of type K(A_(b, a), p), th*
*e target being
the minimal model. The map is a weak equivalence, so condition (b) of Theorem *
*B.1.3 (iii)
holds. The free modules A_(-, a) (viewed as a complexes in dimension 0) form a *
*set of compact __
generators for the derived category of A_-modules, so condition (c) is satisfie*
*d. |__|
B.2. Characterization of Eilenberg-Mac Lane spectra. In this section we show th*
*at
Eilenberg-Mac Lane spectral categories are determined up to stable equivalence *
*by the prop-
erty that their homotopy groups are concentrated in dimension zero.
Proposition B.2.1.Let R be a spectral category all of whose morphism spectra ar*
*e stably fibrant
and have homotopy groups concentrated in dimension zero. Then there exists a na*
*tural chain of
stable equivalences of spectral categories between R and Hß0R_.
STABLE MODEL CATEGORIES 41
The proposition is a special case of the following statement. Here we call a*
* stably fibrant
spectrum connective if the negative dimensional stable homotopy groups vanish.
Lemma B.2.2. Let I be any set. There are functors M and E from the category of*
* spectral
categories with object set I to itself and natural transformations
Id --ff--!M --fi--E --fl--!Hß0_
with the following properties: for every spectral category R with connective st*
*ably fibrant mor-
phism spectra the maps ffR and fiR are stable equivalences and the map flR indu*
*ces the canonical
isomorphism on component ringoids.
Proof.The strategy of proof is to transfer the corresponding statement from the*
* category of
Gamma-rings (where it is easy to prove) to the category of symmetric ring spect*
*ra and extend
it to the `multiple object case'. We use in a crucial way Bökstedt's hocolimI c*
*onstruction [Bök].
The functors M, E as well as an intermediate functor D all arise as lax monoida*
*l functors from
the category of symmetric spectra to itself, and the natural maps between them *
*are monoidal
transformations. This implies that when we apply them to the morphism spectra o*
*f a spectral
category, then the outcome is again a spectral category in a natural way, and t*
*he transformations
assemble into spectral functors.
The two functors M and D from the category of symmetric spectra to itself are*
* defined in
[Sh00, Sec. 3]. The n-th space of the symmetric spectrum MX is defined as the *
*homotopy
colimit
(MX)n = hocolimk2I kSing|Xk+n| .
Here I is a skeleton of the category of finite sets and injections with objects*
* k = {0, 1, . .,.k};
for the precise definition and the structure maps making this a symmetric spect*
*rum see [Sh00,
Sec. 3]. The map ff : X -! MX is induced by the inclusion of X into the colimit*
* diagram at
k = 0. In the proof of [Sh00, Prop. 3.1.9] it is shown that the map ff is a sta*
*ble equivalence
(even a level equivalence) for every stably fibrant symmetric spectrum X.
The n-th level of the functor D (for `detection' - it detects stable equivale*
*nces of symmetric
spectra) is defined as
(DX)n = hocolimk2I kSing|Xk ^ Sn| ,
see [Sh00, Def. 3.1.1]. Also in the proof of [Sh00, Prop. 3.1.9] a natural map *
*DX -! MX is
constructed which we denote fi1Xand which is a stable equivalence (even a level*
* equivalence) for
every stably fibrant spectrum X.
The symmetric spectrum DX in fact arises from a simplicial functor QX. The va*
*lue of QX
at a pointed simplicial set K is given by
QX(K) = hocolimk2I kSing|Xk ^ K| .
A simplicial functor F can be evaluated on the simplicial spheres to give a sym*
*metric spectrum,
which we denote F (S). In the situation at hand we thus have DX = QX(S). If we *
*restrict the
simplicial functor QX to the category opof finite pointed sets we obtain a -s*
*pace [Seg74, BF78]
denoted æQX. Every -space can be prolonged to a simplicial functor defined on *
*the category of
pointed simplicial sets [BF78 , x4]. Prolongation is left adjoint to the restri*
*ction functor æ, and
we denote it by P . We then set EX = (P æQX)(S). The unit P æQX -! QX of the ad*
*junction
between restriction and prolongation, evaluated at the spheres, gives a map of *
*symmetric spectra
fi2X: EX = (P æQX)(S) ----!QX(S) = DX .
We claim that if X is a connective symmetric -spectrum, then fi2Xis a level eq*
*uivalence between
connective symmetric -spectra. Indeed, if X is a symmetric -spectrum, it is *
*in particular
42 STEFAN SCHWEDE AND BROOKE SHIPLEY
convergent in the sense of [MS , 2.1]. By [MS , Thm. 2.3] and the remark therea*
*fter, the natural
map QX(K) -! (QX( K)) is then a weak equivalence for all pointed simplicial se*
*ts K.
This implies (see e.g. [MMSS , Lemma 17.9]) that QX is a linear functor, i.e.,*
* that it takes
homotopy cocartesian squares to homotopy cartesian squares. In particular, QX c*
*onverts wedges
to products, up to weak equivalence, and takes values in infinite loop spaces, *
*so the restricted
-space æQX is very special [BF78 , p. 97]. By [BF78 , Thm. 4.2], EX = (P æQX)*
*(S) is a
connected -spectrum. Since both EX and DX are connected -spectra and the map *
*fi2Xis an
isomorphism at level 0, fi2Xis in fact a level equivalence. The map fiX : EX -!*
* MX is defined
as the composite of the maps fi2X: EX -! DX and fi1X: DX -! MX; if X is stably *
*fibrant
and connective, then both of these are level equivalences, hence so is fiX .
Every -space Y has a natural monoidal map Y -! Hß0Y to the Eilenberg-Mac Lan*
*e -space
([Seg74, x0], [Sch99, Sec. 1]) of its component group which induces the canonic*
*al isomorphism
on ß0, see [Sch99, Lemma 1.2]. (This map is in fact the unit of another monoida*
*l adjunction,
namely, between the Eilenberg-Mac Lane -space functor and the ß0-functor.) In*
* particular
there is such a map of -spaces æQX -! Hß0(æQX). From this we get the map
flX : EX = (P æQX)(S) ----! (P Hß0(æQX))(S) = Hß0X
by prolongation and evaluation of the adjunction unit on spheres. Whenever X i*
*s a stably
fibrant, the component groups ß0X and ß0(æQX) are isomorphic. The symmetric sp*
*ectrum
associated to the Eilenberg-Mac Lane -space by prolongation and then restricti*
*on to spheres_is
the Eilenberg-Mac Lane model of Definition 5.1.5. *
* |__|
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Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
E-mail address: schwede@mathematik.uni-bielefeld.de
Department of Mathematics, Purdue University, W. Lafayette, IN 47907, USA
E-mail address: bshipley@math.purdue.edu