SPECTRAL ENRICHMENTS OF MODEL CATEGORIES DANIEL DUGGER Abstract.We prove that every stable, combinatorial model category can be enriched in a natural way over symmetric spectra. As a consequence of the general theory, every object in such a model category has an associa* *ted homotopy endomorphism ring spectrum. Basic properties of these invariants are established. Contents 1. Introduction 1 2. Enrichments in category theory 5 3. Enrichments for model categories 7 4. Universal pointed model categories 11 5. The main results 15 6. A leftover proof 17 7. The additive case: Homotopy enrichments over Sp (sAb) 19 Appendix A. D-model categories 24 Appendix B. Stabilization and localization 27 References 28 1.Introduction If X and Y are two objects in a model category M, it is well-known that there is an associated `homotopy function complex' Map (X, Y ) (cf. [H , Chap. 17] * *or [Ho2 , Sec. 5.4]). This is a simplicial set, well-defined up to weak equivalenc* *e, and it is an invariant of the homotopy types of X and Y . Following [DK ] one can actually construct these function complexes so that they come with composition maps Map (Y, Z) x Map (X, Y ) ! Map (X, Z), thereby giving an enrichment of M over simplicial sets. This enrichment is an invariant (in an appropriate sense)* * of the model category M. This paper concerns analagous results for stable model categories, with the r* *ole of simplicial sets being replaced by symmetric spectra [HSS , Th. 3.4.4]. We sh* *ow that if M is a stable, combinatorial model category then any two objects can be assigned a symmetric spectrum function complex. More importantly, one can give composition maps leading to an enrichment of M over the symmetric monoidal category of symmetric spectra. One application is that any object X 2 M has an associated `homotopy endomorphism ring spectrum' hEnd (X) (where by ring spectrum we mean essentially what used to be called an A1 -ring spectrum). These ring spectra, as well as the overall enrichment by symmetric spectra, are homot* *opy invariants of the model category M. 1 2 DANIEL DUGGER 1.1. An application. Before describing the results in more detail, here is the * *mo- tivation for this paper. If R is a differential graded algebra, there is a stab* *le model category structure on (differential graded) R-modules where the weak equivalenc* *es are quasi-isomorphisms and the fibrations are surjections. Given two dgas R and* * S, when are the model categories of R- and S-modules Quillen equivalent? A complete answer to this question is given in [DS ]. The problem is subtle: even though t* *he categories of R- and S-modules are additive, examples show that it's possible f* *or them to be Quillen equivalent only through a zig-zag involving non-additive mod* *el categories. To deal with this, the arguments in [DS ] depend on using homotopy endomorphism ring spectra as invariants of stable model categories. The present paper develops some of the tools necessary for those arguments. 1.2. Statement of results. A category is locally presentable if it is cocomplete and all objects are small in a certain sense; see [AR ]. A model category is ca* *lled combinatorial if it is cofibrantly-generated and the underlying category is loc* *ally presentable. This class was introduced by Jeff Smith, and the examples are ubiq- uitous (the class even includes model categories made from topological spaces, * *if one uses -generated spaces). Background information on combinatorial model categories can be found in [D2 ]. A model category is called stable if the initial and terminal objects coincide (that is, it is a pointed category) and if the induced suspension functor is in* *vertible on the homotopy category. Our results concern enrichments of stable, combinatorial model categories. Un- fortunately we do not know how to give a canonical spectral enrichment for our model categories, however. Instead there are many such enrichments, involving choices, but the choices yield enrichments which are homotopy equivalent in a c* *er- tain sense. The machinery needed to handle this is developed in Section 3. There we define a model enrichment of one model category by another, and give a notion of two model enrichments being quasi-equivalent. A crude version of our main theorem can be stated as follows: Theorem 1.3. Every stable, combinatorial model category has a canonical quasi- equivalence class of model enrichments by Sp . Here Sp denotes the model category of symmetric spectra from [HSS ], with its symmetric monoidal smash product. `Canonical' means the enrichment has good functoriality properties with respect to Quillen pairs and Quillen equival* *ences. More precise statements are given in Section 5. We will show that the canonical enrichment by Sp is preserved, up to quasi-equivalence, when you prolong or restrict across a Quillen equivalence. It follows that the enrichment contains * *only `homotopy information' about the model category; so it can be used to decide whether or not two model categories are Quillen equivalent. One simple consequence of the above theorem is the following: Corollary 1.4. If M is a stable, combinatorial model category then Ho (M) is naturally enriched over Ho (Sp ). The above corollary is actually rather weak, and not representative of all th* *at the theorem has to offer. For instance, the corollary implies that every object* * of such a model category has an endomorphism ring object in Ho (Sp )_that is, a spectrum R together with a pairing R ^ R ! R which is associative and unital up to homotopy. The theorem, on the other hand, actually gives the following: SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 3 Corollary 1.5. Every object X of a stable, combinatorial model category has a naturally associated A1 -ring spectrum hEnd(X)_called the homotopy endomor- phism spectrum of X_well-defined in the homotopy category of A1 -ring spectra. If X ' Y then hEnd(X) ' hEnd(Y ). The main results concerning these endomorphism spectra are as follows. The first shows that they are homotopical invariants of the model category M: Theorem 1.6. Let M and N be stable, combinatorial model categories. Suppose they are Quillen equivalent, through a zig-zag where the intermediate steps are* * pos- sibly not combinatorial or pointed. Let X 2 M, and let Y 2 Ho (N) be the image of X under the derived functors of the Quillen equivalence. Then hEnd (X) and hEnd(Y ) are weakly equivalent ring spectra. A model category M is called spectral if it is enriched, tensored, and cotens* *ored over symmetric spectra in a homotopically well-behaved manner (M is also called* * an Sp -model category). See Section A.8 for a more detailed definition. The follow* *ing result says that in spectral model categories homotopy endomorphism spectra can be computed in the expected way, using the spectrum hom-object M_Sp (-, -): Proposition 1.7. Let M be a stable, combinatorial model category which is also spectral. Let X be a cofibrant-fibrant object of M. Then hEnd(X) and M_Sp (-, -) are weakly equivalent ring spectra. Enhanced results are proven in the case where M is also an additive model cat- egory (see Section 7 for the definition). In this context one obtains an enrich* *ment over the monoidal model category Sp (sAb) of symmetric spectra based on simpli- cial abelian groups. The paper [S] shows this category is monoidally equivalent* * to the model category of unbounded chain complexes of abelian groups, which perhaps is easier to think about. Any object X 2 M therefore has an additive homotopy endomorphism ring object in Sp (sAb) (or equivalently, a "homotopy endomor- phism dga"). These endomorphism dgas are invariant under Quillen equivalences between additive model categories, but not general Quillen equivalences. Detai* *ls are in Section 7. 1.8. The construction. In [DK ] Dwyer and Kan constructed model enrichments over sSet via their hammock localization. This is a very elegant construction, * *in particular not involving any choices. Unfortunately we have not been clever eno* *ugh to find a similar construction for enrichments by symmetric spectra. The methods of the present paper are more of a hack job: they get us the tools we need at a relatively cheap cost, but they are not so elegant. The idea is to make use of the `universal' constructions from [D1 , D2], toge* *ther with the general stabilization machinery provided by [Ho1 ]. By [D2 ] every co* *m- binatorial model category is Quillen equivalent to a localization of diagrams of simplicial sets. Using the simplicial structure on this diagram category, we c* *an apply the symmetric spectra construction of [Ho1 ]. This gives a new model cate- gory, Quillen equivalent to what we started with, where one has actual symmetric spectra function complexes built into the category. In more detail, given a pointed, combinatorial model category M one can choose a Quillen equivalence U+ C=S -~! M by modifying the main result of [D2 ]. Here U+ C is the universal pointed model category built from C, developed in Section* * 4; 4 DANIEL DUGGER S is a set of maps in U+ C, and U+ C=S denotes the Bousfield localization [H , * *Sec. 3.3]. The category U+ C=S is a nice simplicial model category, and we can form sym- metric spectra over it using the results of [Ho1 ]. This gives us a new model c* *ategory Sp (U+ C=S), which is enriched over Sp . If M was stable to begin with then we have a zig-zag of Quillen equivalences M -~ U+ C=S -~! Sp (U+ C=S) and can transport the enrichment of the right-most model category onto M. Final* *ly, theorems from [D1 ] allow us to check that the resulting enrichment of M doesn't depend (up to quasi-equivalence) on our chosen Quillen equivalence U+ C=S ! M. By now the main shortcoming of this paper should be obvious: all the results * *are proven only for combinatorial model categories. This is an extremely large clas* *s, but it is very plausible that the results about spectral enrichments hold in co* *mplete generality. Unfortunately we have not been able to find proofs in this setting,* * so it remains a worthwhile challenge. 1.9. Organization of the paper. Sections 2 and 3 contain the basic definitions of enrichments, model enrichments, and the corresponding notions of equivalence. Section 4 deals with the universal pointed model categories U+ C, and establish* *es their basic properties. The main part of the paper is Section 5, which gives t* *he results on spectral enrichments and homotopy endomorphism spectra. Section 6 returns to the proof of Proposition 3.5: this is a foundational result showing * *that quasi-equivalent enrichments have the properties one hopes for. Finally, in Sec* *tion 7 we present expanded results for additive model categories. This entails devel* *oping `universal additive model categories', a topic which may be of independent inte* *rest. We also give two appendices. Appendix A contains several basic results about model categories which are enriched, tensored, and cotensored over a monoidal model category (the main examples for us are simplicial and spectral model cat- egories). The reader is encouraged to familiarize himself with this section bef* *ore tackling the rest of the paper. Appendix B gives a general result about commuti* *ng localization and stabilization. 1.10. Terminology. We assume a familiarity with model categories and localiza- tion theory, for which [H ] is a good reference. Several conventions from [D1 ]* * are often used, so we'll now briefly recall these. A Quillen map L: M ! N is another name for a Quillen pair L: M AE N: R. If L1 and L2 are two such Quillen maps, a Quillen homotopy L1 ! L2 is a natural transformation between the left adjoints which is a weak equivalence on cofibrant objects. If M is a model category and * *S is a set of maps in M, then M=S denotes the left Bousfield localization [H , Sec. * *3.3]. 1.11. Acknowledgments. Readers should note that the present paper owes a great debt to both [Ho1 ] and [SS2]. SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 5 2. Enrichments in category theory In this section we review the notion of a category being enriched over a sym- metric monoidal category. Our situation is slightly more general than what usua* *lly occurs in the literature. There is a notion of equivalence which encodes when t* *wo enrichments carry the same information. 2.1. Basic definitions. Let C be a category, and let (D, , S) be a symmetric monoidal category (where S is the unit). An enrichment of C by D is a functor o :Copx C ! D together with (i)For every a, b, c 2 C a `composition map' o(b, c) o(a, b) ! o(a, c), natu* *ral in a and c; (ii)a collection of maps S ! o(c, c) for every c 2 C. This data is required to satisfy the associativity and unital rules for composi* *tion, which are so standard that we will not write them down. We also require that for any map f :a ! b in C, the square S _______//o(a, a) | | | f| fflffl|f fflffl| o(b, b)____//o(a, b) commutes. Note that if C = {*} is the trivial category and Ab is the category of abelian groups, then an enrichment of C by Ab is just another name for an associative a* *nd unital ring. If o and o0 are two enrichments of C by D, a map o ! o0 is a natural transfor- mation o(a, b) ! o0(a, b) compatible with the unit and composition maps. Remark 2.2. The above definition differs somewhat from related things in the literature. According to [B , Sec. 6.2], a D-category is a collection of obje* *cts I together with a Hom-object I(i, j) 2 D for every i, j 2 I, etc. This correspond* *s to our above definition in the case where C has only identity maps. If C is a category (i.e., a Set-category),`one can define a D-category SC wit* *h the same object set as D and SC(a, b) = C(a,b)S. To give an enrichment of C by D in the sense we defined above is the same as giving a D-category with the same objects as C, together with a D-functor from SC to this D-category. Example 2.3. If M is a simplicial model category, the assignment X, Y 7! Map (X, Y ) is an enrichment of M by sSet. If M is a general model category, the hammock localization assignment X, Y 7! LH M(X, Y ) from [DK , 3.1] is also an enrichment of M by sSet. 2.4. Bimodules. Let oe and o be two enrichments of C by D. By a oe-o bimodule we mean a collection of objects M(a, b) 2 D for every a, b 2 C, together with `multiplication maps' oe(b, c) M(a, b) ! M(a, c) M(b, c) o(a, b) ! M(a, c) which are natural in a and c. We again assume associativity and unital conditio* *ns which we will not write down, as well as the property that for any a, b, c, d 2* * C the two obvious maps oe(c, d) M(b, c) o(a, b) ' M(a, d) 6 DANIEL DUGGER are equal. Note that a bimodule has a natural structure of a bifunctor Copx C ! D. For instance, if f :a ! b is a map in C then consider the composite S ! oe(a, a) ! oe(a, b). We then have S M(a0, a) ! oe(a, b) M(a0, a) ! M(a0, b), giving a * *map M(a0, a) ! M(a0, b) induced by f. Similar considerations give functoriality in * *the first variable. Remark 2.5. For a more precise version of the definition of bimodule, see Sec- tion 6.5. Earlier parts of Section 6 also define the notions of left and right * *oe-module, which we have for the moment skipped over. To understand the following definition, observe that two rings R and S are isomorphic if and only if there is an R - S bimodule M together with a chosen element m 2 M such that the induced maps r ! rm and s ! ms give isomorphisms of abelian groups R ! M S. Definition 2.6. Let oe and o be two enrichments of C by D. (a)By a pointed oe-o bimodule we mean a bimodule M together with a collection of maps S ! M(c, c) for every c 2 C, such that for any map a ! b the square S _______//M(a, a) | | | | |fflffl fflffl| M(b, b)_____//M(a, b) commutes. (b)We say that oe and o are equivalent if there is a pointed oe - o bimodule M :Copx C ! D for which the composites oe(a, b) S ! oe(a, b)M(a, a) ! M(a, b), and S o(a, b) ! M(b, b)o(a, b) ! M(a, b) are isomorphisms, for every a, b 2 C. Remark 2.7. A oe - o bimodule is, by restriction, an SC - SC bimodule. Note that SC has an obvious structure of SC - SC bimodule. The definition of pointed oe -* * o bimodule says that there is a map of SC - SC bimodules SC ! M. Lemma 2.8. Assume that D has pullbacks. Two enrichments oe and o are equiva- lent if and only if there is an isomorphism oe ~=o. Proof.If there is an isomorphism oe ~=o, then we let M = o and regard it as a oe - o bimodule. This shows oe and o are equivalent. If we instead assume that oe and o are equivalent via the pointed bimodule M, define `(a, b) to be the pullback `(a, b)____//_o(a, b) | | | | fflffl| fflffl| oe(a, b)__//_M(a, b). Here the lower horizontal map is the composite oe(a, b) ~=oe(a, b) S ! oe(a, b) M(a, a) ! M(a, b) SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 7 and the right vertical map is defined similarly. The universal property of the * *pull- back allows one to see that ` is naturally an enrichment of C by D, and that ` * *! oe and ` ! o are maps of enrichments. Now, our assumption that oe and o are equivalent via M includes the condition that the bottom and right maps in the above pullback square are isomorphisms. So all maps in the square are isomorphisms, which means we have oe ~=` ~=o. Remark 2.9. Since the notions of equivalence and isomorphism coincide, one might wonder why we bother with the former. The answer is in the next section, where the homotopical analogs of these two notions slightly diverge. 3. Enrichments for model categories We now give model category analogs for the material from the last section. Th* *ere is the notion of model enrichment, together with two notions of equivalence: these are called quasi-equivalence and direct equivalence. Direct equivalences have the property of obviously preserving the `homotopical' information in an e* *n- richment; but quasi-equivalences are what seem to arise in practice. Fortunately the two notions are closely connected_see Proposition 3.5. The material in this section is a simple extension of techniques from [SS2], * *which dealt with enrichments over symmetric spectra. 3.1. Model enrichments. Let M be a model category and let V be a symmetric monoidal model category [Ho2 , Def. 4.2.6]. A model enrichment of M by V is an enrichment o with the property that whenever a ! a0is a weak equivalence between cofibrant objects, and x ! x0is a weak equivalence between fibrant objects, then the induced maps o(a0, x) ! o(a, x) and o(a, x) ! o(a, x0) are weak equivalences. A quasi-equivalence between two model enrichments oe and o consists of a pointed oe - o bimodule M such that the compositions oe(a, b) S ! oe(a, b)M(a, a) ! M(a, b) and S o(a, b) ! M(b, b)o(a, b) ! M(a, b) are weak equivalences whenever a is cofibrant and b is fibrant. Definition 3.2. Let ME0(M, V) be the collection of equivalence classes of model enrichments, where the equivalence relation is the one generated by quasi- equivalence. Example 3.3. Let M be a simplicial model category, and let o(X, Y ) be the simplicial mapping~space between X and Y . This is a model enrichment of M by sSet. Let QX - i X be a cofibrant-replacement functor for M, and define o0(X, Y ) = o(QX, QY ). This is another model enrichment of M, but note that there are no obvious maps between o and o0. There is an obvious quasi-equivalen* *ce, however: define M(X, Y ) = Map (QX, Y ). This is a o - o0 bimodule, and the maps QX ! X give the distinguished maps * ! M(X, X). This example illustrates that quasi-equivalences arise naturally, more so than the notion of `direct equivalence' we define next. 8 DANIEL DUGGER 3.4. Direct equivalences. A map of model enrichments o ! o0is a direct equiv- alence if o(a, b) ! o0(a, b) is a weak equivalence whenever a is cofibrant and * *b is fibrant. To say something about the relationship between quasi-equivalence and direct equivalence, we need a slight enhancement of our definitions. If I is a full su* *bcat- egory of M, we can talk about model enrichments defined over I: meaning that o(a, b) is defined only for a, b 2 I. In the same way we can talk about "d* *irect equivalences over I", and so on. Now we can give the following analog of Lemma 2.8. This is the most important result of this section. Proposition 3.5. Let V be a combinatorial, symmetric monoidal model category satisfying the monoid axiom [SS1, Def. 3.3]. Assume also that the unit S 2 V is cofibrant. Let oe and o be model enrichments of M by V. Let I be a small, full subcategory of M consisting of cofibrant-fibrant objects. If oe and o are * *quasi- equivalent over I, then there is a zig-zag of direct equivalences (over I) betw* *een oe and o. The assumption about the smallness of I is needed so that there is a model structure on certain categories of modules and bimodules, a key ingredient of t* *he proof. Sketch of proof.The proof can be adapted directly from [SS2, Lemma A.2.3], which dealt with the case where V is symmetric spectra and I has only identity maps. Essentially the proof is a homotopy-theoretic version of the pullback tri* *ck in Lemma 2.8. Let M be a bimodule giving an equivalence between oe and o. When the maps oe(a, b) ! M(a, b) are trivial fibrations, the pullback trick immediately gives* * a zig-zag of direct equivalences between oe and o. For the general case one uses certain model structures on module categories to reduce to the previous case. A* * full discussion requires quite a bit of machinery, so we postpone this until Section* * 6. Corollary 3.6. Let oe and o be model enrichments of M by V. Let X be a cofibran* *t- fibrant object of M. If oe and o are quasi-equivalent, then the V-monoids oe(X,* * X) and o(X, X) are weakly equivalent in V (meaning there is a zig-zag between them where all the intermediate objects are monoids in V, and all the maps are both monoid maps and weak equivalences). Proof.This is an application of Proposition 3.5, where I is the full subcategor* *y of M whose sole object is X. Corollary 3.7. Let oe be a model enrichment of M. Let I be a small category, and let G1, G2: I ! M be two functors whose images lie in the cofibrant-fibrant obj* *ects. Assume there is a natural weak equivalence G1 -~! G2. Then the enrichments on I given by oe(G1i, G1j) and oe(G2i, G2j) are connected by a zig-zag of direct equivalences. Proof.Call the two enrichments oe1 and oe2. Define a oe2-oe1 bimodule by M(i, j* *) = oe(G1i, G2j). The maps G1i -~!G2i give rise to maps S ! M(i, i), making M into a pointed bimodule. One readily checks that this is a quasi-equivalence between* * oe2 and oe1, and then applies Proposition 3.5. SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 9 3.8. Homotopy invariant enrichments. We give a few other basic results about model enrichments. ~ Proposition~3.9. Let Qa -i a be a cofibrant-replacement functor in M, and let x ae F x be a fibrant-replacement functor. If o is a model enrichment of M, then o(Qa, Qb) and o(F a, F b) give model enrichments which are quasi-equivalent to * *o. Proof.Left to the reader (see Example 3.3). A model enrichment o of M by V will be called homotopy invariant if whenever a ! a0 and x ! x0 are weak equivalences then the maps o(a0, x) ! o(a, x) ! o(a, x0) are both weak equivalences as well. Note that there is no co* *fi- brancy/fibrancy assumption on the objects. Corollary 3.10. Every model enrichment is quasi-equivalent to one which is ho- motopy invariant. Proof.Let o be a model enrichment of M by V. By Proposition 3.9 (used twice), the enrichments o(a, b), o(Qa, Qb), and o(QF a, QF b) are all quasi-equivalent.* * The last of these is homotopy invariant. Recall that the monoidal product on Ho (V) is defined by v1 L v2 = Cv1 Cv2, where C is some chosen cofibrant-replacement functor in V. It is easy to check * *that a homotopy invariant enrichment o induces an enrichment of Ho (M) by Ho (V), where the composition maps are the composites o(b, c) L o(a, b) ! o(b, c) o(a, b) ! o(a, c). We note the following: Corollary 3.11. If two homotopy invariant enrichments oe and o are quasi- equivalent, then the induced enrichments of Ho (M) by Ho (V) are equivalent. Proof.First note that if M is a quasi-equivalence between oe and o then M is automatically homotopy invariant itself (in the obvious sense)_this follows from the two-out-of-three property for weak equivalences. Therefore M may be extended to a functor on the homotopy category, where it clearly gives an equivalence be* *tween the enrichments induced by oe and o. To say that oe and o are quasi-equivalent, though, does not say that such an M necessary exists_it only says that there is a chain of such M's. Note that the intermediate model enrichments in the chain need not be homotopy invariant. To get around this, we do the following. If ~ is a model enrichment of M by V, let ~h be the model enrichment ~h(a, b) = ~(QF a, QF b). We have seen that this is homotopy invariant and quasi-equivalent to ~. If M is a quasi-equivalence betwe* *en ~1 and ~2, note that Mh (with the obvious definition) is a quasi-equivalence be* *tween ~h1and ~h2. It follows readily that if our oe and o are quasi-equivalent then * *they are actually quasi-equivalent through a chain where all the intermediate steps * *are homotopy invariant. Now one applies the first paragraph to all the links in th* *is chain. 3.12. Transporting enrichments. Let G: M ! N be a functor, and suppose o is an enrichment of N. Define an enrichment G*o of M by the formula G*o(m1, m2) = o(Gm1, Gm2). Call this the pullback of o along G. 10 DANIEL DUGGER Lemma 3.13. Let M and N be model categories, and let G: M ! N be a functor which preserves weak equivalences and has its image in the cofibrant-fibrant ob* *jects of N. If o is a model enrichment of N, then G*o is a model enrichment of M. Mor* *e- over, G* preserves quasi-equivalence: it induces G*: ME0(N, V) ! ME0(M, V). Proof.Routine. Lemma 3.14. Let M and N be model categories, and let o be a homotopy invariant enrichment of N. Suppose G1, G2: M ! N are two functors which preserve weak equivalences, and assume there is a natural weak equivalence G1 -~!G2. Then G*1o and G*2o are model enrichments of M, and they are quasi-equivalent. Proof.The quasi-equivalence is given by M(a, b) = o(G1a, G2b). The weak equiv- alences G1a ! G2a give the necessary maps S ! M(a, a). Details are left to the reader. Recall that a Quillen map L: M ! N is an adjoint pair L: M AE N: R in which L preserves cofibrations and trivial cofibrations (and R preserves fibrat* *ions ~ and trivial fibrations). Choose cofibrant-replacement functors QM X -i X and ~ ~ ~ QN Z -i Z as well as fibrant-replacement functors A ae FM A and B ae FN B. If o is a model enrichment of N by V, we can define a model enrichment on M by the formula L*o(a, x) = o(FN LQM a, FN LQM x). Similarly, if oe is a model enrichment of M by V we get a model enrichment on N by the formula L*oe(c, w) = oe(QM RFN c, QM RFN w). Proposition 3.15. (a)The constructions L* and L* induce maps L*: ME0(N, V) ! ME0(M, V) and L*: ME0(M, V) ! ME0(N, V). (b)The maps in (a) do not depend on the choice of cofibrant- and fibrant- repla* *ce- ment functors. (c)If L, L0:M ! N are two maps which are Quillen-homotopic, then L* = L0*and L* = (L0)* as maps on ME0(-, V). (d)If L: M ! N is a Quillen equivalence, then the functors L* and L* are inverse isomorphisms ME0(M, V) ~=ME0(N, V). (e)Suppose M and N are V-model categories, with the associated V-enrichments denoted oeM and oeN . If L: M ! N is a V-Quillen equivalence, then L*(oeM )* * = oeN and L*(oeN ) = oeM (as elements of ME0(-, V)). For the notion of `V-Quillen equivalence' used in part (e), see Section A.11. Proof.We will only prove the results for L*; proofs for L* are entirely similar. Part (a) follows from Lemma 3.13, as the composite functor FN LQM preserves weak equivalences and has its~image in the cofibrant-fibrant~objects. For part (b), suppose Q1X -i X and Q2X -i X are two cofibrant-replacement functors for M. Write L*1and L*2for the resulting maps ME0(N, V) ! ME0(M, V). By Corollary 3.10 it suffices to show that L*1(o) = L*2(o) for any homotopy inv* *ariant enrichment o. Let Q3X = Q1X xX Q2X. There is a zig-zag of natural weak equivalences Q1 -~ Q3 -~!Q2. The result now follows by Lemma 3.14 applied to the composites F LQ1, F LQ3, and F LQ2. For part (c), it again suffices to prove L*(o) = (L0)*(o) in the case where o is homotopy invariant. The Quillen homotopy is a natural transformation L ! SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 11 L0 which is a weak equivalence on cofibrant objects. The result is then a dire* *ct application of Lemma 3.14. For (d) we will check that if o is a homotopy invariant enrichment of N then L*(L*o) = o in ME0(N, V). The enrichment L*(L*o) is the pullback of o along the composite functor F LQQRF :N ! N. There is a zig-zag of natural weak equivalences F LQQRF -~ LQQRF -~! F -~ Id (the second being the composite LQQRF ! LRF ! F , which is a weak equiva- lence because we have a Quillen equivalence). Each of the functors in the zig-z* *ag preserves weak equivalences, so the result follows from Lemma 3.14. Finally, we prove (e). By (d) it suffices just to prove L*oeM = oeN . The ass* *ump- tion gives us a natural isomorphism oeN (LA, X) ~=oeM (A, RX) (see Section A.11* *). One checks that the enrichments oeM (QRF X, QRF Y ) and oeN (F X, F Y ) are qua* *si- equivalent via the bimodule M(X, Y ) = oeM (QRF X, RF Y ) ~=oeN (LQRF X, F Y ). (The verification that this really is a bimodule requires some routine but tedi* *ous work, mainly using Remark A.13). But Proposition 3.9 says that oeN (F X, F Y ) * *is quasi-equivalent to oeN , so we are done. 4. Universal pointed model categories If C is a small category then there is a `universal model category' built fro* *m C. This was developed in [D1 ]. The present section deals with a pointed version * *of that theory. The category of functors from C to pointed simplicial sets plays t* *he role of a universal pointed model category built from C. 4.1. Basic definitions. Recall from [D1 ] that if C is a small category then UC denotes the model category of simplicial presheaves on C, with fibrations and w* *eak equivalences defined objectwise. One has the Yoneda embedding r :C ,! UC where rX is the presheaf Y 7! C(Y, X). Let U+ C be the category of functors from Cop into pointed simplicial sets, w* *ith the model structure where weak equivalences and fibrations are again objectwise. This can also be regarded as the undercategory (* # UC). There is a Quillen map UC ! U+ C where the left adjoint sends F to F+ (adding a disjoint basepoint) and the right adjoint forgets the basepoint. Write r+ for* * the composite C ,! UC ! U+ C. Finally, if S is a set of maps in UC then let S+ denote the image of S under UC ! U+ C. Note that if all the maps in S have cofibrant domain and codomain, then by [H , Prop. 3.3.18] one has an induced Quillen map UC=S ! U+ C=(S+ ). The following simple lemma unfortunately has a long proof: Lemma 4.2. Let S be a set of maps between cofibrant objects in UC, and suppose that the map ; ! * is a weak equivalence in UC=S. Then UC=S ! U+ C=(S+ ) is a Quillen equivalence. Proof.Write M = UC and M+ = U+ C = (* # M) (the lemma actually holds for any simplicial, left proper, cellular model category in place of UC). Write F :M AE M+ :U for the Quillen functors. We will start by showing that a map in M+ =(S+ ) is a weak equivalence if and only if it's a weak equivalence in M=* *S. Unfortunately the proof of this fact is somewhat lengthy. 12 DANIEL DUGGER An object X 2 M+ is (S+ )-fibrant if it is fibrant in M+ (equivalently, fibra* *nt in M) and if the induced map on simplicial mapping spaces M_M+ (B+ , X) ! M_M+(A+ , X) is a weak equivalence for every A ! B in S. By adjointness, how- ever, M_M+ (A+ , X) ~=M_M(A, X) (and similarly for B). It follows that X 2 M+ is (S+ )-fibrant if and only if X is S-fibrant in M. Suppose C is a cofibrant object in M. Using the fact that M=S is left proper and that ; ! * is a weak equivalence, it follows that C ! C q * is also a weak equivalence in M=S. As a consequence, if C ! D is a map between cofibrant objec* *ts which is a weak equivalence in M=S, then C+ ! D+ is also a weak equivalence in M=S. Now consider the construction of the localization functor LS+ for M+ =(S+ ). This is obtained via the small object argument, by iteratively forming pushouts along the maps [ n,k! n] + [A+ ! B+ ]. Here " + " denotes the simplicial tensor in the pointed category M+ , that is t* *o say K + A = (K+ A)=((* A) q (K+ *)) for K 2 sSet and A 2 M. The above maps are then readily identified with the maps h i ( n,k B) q n,k A( n A) ! ( n B)+ . i + As [( n,k B) q n,k A( n A) ! ( n B) is a map between cofibrant objects which is a weak equivalence in M=S, so is the displayed map above. It follows t* *hat for any X 2 M+ , the map X ! LS+X is a weak equivalence in M=S (in addition to being a weak equivalence in M+ =(S+ ), by construction). Let X ! Y be a map in M+ . Consider the square X ________//Y | | | | fflffl| fflffl| LS+X _____//LS+Y. The vertical maps are weak equivalences in both M=S and M+ =(S+ ). If X ! Y is a weak equivalence in M+ =(S+ ), then the bottom map is a weak equivalence in M+ . This is the same as being a weak equivalence in M, and therefore X ! Y is also a weak equivalence in M=S (going back around the square, using the 2-out-o* *f-3 property). Similarly, if X ! Y is a weak equivalence in M=S then so is the bott* *om map. But the objects LS+X and LS+Y are fibrant in M=S, so the bottom map is actually a weak equivalence in M (and also in M+ ). It follows that X ! Y is a weak equivalence in M+ =(S+ ). This completes the proof that a map in M+ =(S+ ) is a weak equivalence if and only if it is so in M=S. To show that M=S ! M+ =(S+ ) is a Quillen equivalence we must show two things. If A is a cofibrant object in M and A+ ! X is a fibrant replacement in M+ =(S+ ), we must show that A ! X is a weak equivalence in M=S. But from what we have already shown we know A ! A+ and A+ ! X are weak equivalences in M=S, so this is obvious. We must also show that if Z is a fibrant object in M+ =(S+ ) and B ! Z is a cofibrant replacement in M=S, then B+ ! Z is a weak equivalence in M+ =(S+ ). This is the same as showing it's a weak equivalence in SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 13 M=S. But in the sequence B ! B+ ! Z, the first map and the composite are both equivalences in M=S; so the map B+ ! Z is an equivalence as well. 4.3. Basic properties. Proposition 4.4. Suppose that L: UC=S ! M is a Quillen map, where S is a set of maps between cofibrant objects. If M is pointed, there is a Quillen map L+ :U+ C=(S+ ) ! M such that the composite UC=S ! U+ C=(S+ ) ! M is L. If L is a Quillen equivalence, then so is L+ . Proof.For any A 2 C, write *A for the cosimplicial object [n] 7! L(rA n). Recall that the right adjoint to L sends an X 2 M to the simplicial presheaf A 7! M( *A, X). Since M is pointed, this simplicial presheaf is also pointed. L* *et Sing*: M ! U+ C be this functor. If F 2 U+ C define L+ (F ) to be the pushout of * L(*) ! L(F ). This is rea* *dily seen to be left adjoint to Sing*. It is also easy to check that L+ :U+ C ! M is* * a Quillen map and the composite UC ! U+ C ! M equals L. To obtain the map U+ C=(S+ ) ! M one only has to see that L+ maps elements of S+ to weak equivalences in M. But this is obvious: if A 2 UC then L+ (A q *)* * ~= L(A), and L takes elements of S to weak equivalences. Finally, assume that L is a Quillen equivalence. Since M is pointed, it follo* *ws that ; ! * is a weak equivalence in UC=S (using that L(;) = * and R(*) = *). So by the above lemma, UC=S ! U+ C=(S+ ) is a Quillen equivalence; therefore L+ is one as well. The next two propositions of this section accentuate the roll of U+ C as the * *uni- versal pointed model category built from C. These results are direct generaliza* *tions of [D1 , Props. 2.3, 5.10]. Proposition 4.5. Let C be a small category, and let fl :C ! M be a functor from C into a pointed model category M. Then fl "factors" through U+ C, in the sense that there is a Quillen pair L: U+ C AE M: R and a natural weak equivalen* *ce L O r+ -~! fl. Moreover, the category of all such factorizations_as defined in * *[D1 , p. 147]_is contractible. Proof.The result follows from [D1 , Prop. 2.3] and Proposition 4.4 above. Proposition 4.6. Suppose L: U+ C=S ! N is a Quillen map, and P :M -~! N is a Quillen equivalence between pointed model categories. Then there is a Quil* *len map L0:U+ C=S ! M such that P O L0is Quillen homotopic to L. Moreover, if M is simplicial then L0can be chosen to be simplicial. Proof.The first statement follows directly from [D1 , Prop. 5.10] and Proposi- tion 4.4 above. The second statement was never made explicit in [D1 ], but foll* *ows at once from analyzing the proof of [D1 , Prop. 2.3]. To define F one first get* *s a map f :C ! M with values in the cofibrant objects, and then F can be taken to be the unique colimit-preserving functor characterized by F (rA K) = f(A) K, where A 2 C and K 2 sSet. This is clearly a simplicial functor. Proposition 4.7. Let M be a pointed, combinatorial model category. (a)There is a Quillen equivalence U+ C=S ! M for some C and S. 14 DANIEL DUGGER (b)Let N be a pointed model category, and let M -~ M1 -~! . .~.-Mn -~! N be a zig-zag of Quillen equivalences (where the intermediate model categories are not necessarily pointed or combinatorial). Then there is a simple zig-za* *g of Quillen equivalences M -~ U+ C=S -~! N for some C and S. (c)In the context of (b), the simple zig-zag can be chosen so that the derived * *equiv- alence Ho (M) ' Ho (N) is isomorphic to the derived equivalence specified by the original zig-zag. In part (b), note that we have replaced a zig-zag of Quillen equivalences_in which the intermediate steps are not necessarily pointed_by one in which the in- termediate steps are pointed. For (c), recall that two pairs of adjoint functo* *rs L: C AE D: R and L0:C AE D: R0are said to be isomorphic if there is a natural i* *so- morphism LX ~=L0X for all X 2 C (equivalently, if there is a natural isomorphism RY ~=R0Y for all Y 2 D). Proof.Let M be a pointed, combinatorial model category. By [D1 , Th. 6.3] there is a Quillen equivalence UC=S ! M for some C and S. Proposition 4.4 shows there is an induced Quillen equivalence U+ C=(S+ ) ! M. This proves (a). Parts (b) and (c) follow in the same way from [D1 , Cor. 6.5], or directly by applying Proposition 4.6. 4.8. Application to stabilization. Suppose M is a stable model category, and we happen to have a Quillen equivalence U+ C=S ! M. It follows in particular that U+ C=S is also stable. Now, U+ C=S is a simplicial, left proper, cellular * *model category. So using [Ho1 , Secs. 7,8] we can form the corresponding category of symmetric spectra Sp (U+ C=S) (with its stable model structure). This comes with a Quillen map U+ C=S ! Sp (U+ C=S), and since U+ C is stable this map is a Quillen equivalence [Ho1 , Th. 9.1]. Finally, the category U+ C=S satisfies* * the hypotheses of [Ho1 , Th. 8.11], and so Sp (U+ C=S) is a spectral model category* * (in the sense of Section A.8). We have just proven part (a) of the following: Proposition 4.9. Let M be a stable model category, and suppose U+ C=S ! M is a Quillen equivalence. (a)There is a zig-zag of Quillen equivalences M -~ U+ C=S -~! Sp (U+ C=S). (b)If U+ D=T ! M is another Quillen equivalence, there is a diagram of Quillen equivalences M oo___cU+cC=S_____//SpF(U+FC=S) FFFF | | FF | | F fflffl| fflffl| U+ D=T ____//_Sp (U+ D=T ) where the left vertical map is a simplicial adjunction, the right vertical m* *ap is a spectral adjunction, the square commutes on-the-nose, and the triangle commutes up to a Quillen homotopy. Proof.We have left only to prove (b). Given Quillen equivalences L1: U+ C=S ! M and L2: U+ D=T ! M, it follows from Proposition 4.6 that there is a Quillen map F :U+ C=S ! U+ D=T making the triangle commute up to Quillen homotopy. Since SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 15 U+ D=T is a simplicial model category, we can choose F to be simplicial. But th* *is ensures that Sp (U+ C=S) ! Sp (U+ D=T ) is spectral. 5.The main results In this section we attach to any stable, combinatorial model category M a mod* *el enrichment oM over symmetric spectra. This involves choices, but these choices only affect the end result up to quasi-equivalence. We also show that a zig-zag* * of Quillen equivalences between model categories M and N must carry oM to oN . So the canonical enrichments o give rise to invariants of model categories up to Q* *uillen equivalence. Finally, we specialize all these results to establish basic proper* *ties of homotopy endomorphism spectra. The present results are all direct consequences of work from previous section* *s. Our only job is to tie everything together. 5.1. Construction of spectral enrichments. Let M be a stable, combinatorial model category. By Proposition 4.9(a) there is a zig-zag of Quillen equivalences M -L U+ C=S -F! Sp (U+ C=S). The right-most model category comes equipped with a spectral enrichment oe. We define oM 2 ME0(M, Sp ) to be L*(F *oe). Proposition 5.2. The element oM 2 ME0(M, Sp ) doesn't depend on the choice of C, S, or the Quillen equivalence U+ C=S -~! M. Proof.Applying ME0(-, Sp ) to the diagram from Proposition 4.9(b) gives a com- mutative diagram of bijections, by Proposition 3.15. The result follows immedia* *tely from chasing around this diagram and using Proposition 3.15(e). Choose a homotopy invariant enrichment quasi-equivalent to oM . By Corol- lary 3.11 this induces an enrichment of Ho (M) by Ho (Sp ), and different choic* *es lead to equivalent enrichments. This proves Corollary 1.4. We now turn our attention to functoriality: Proposition 5.3. Suppose L: M ! N is a Quillen equivalence between stable, combinatorial model categories. Then L*(oN ) = oM and L*(oM ) = oN . Proof.Choose a Quillen equivalence U+ C=S ! M, by Proposition 4.5. We then have a diagram of Quillen equivalences Sp (U+ C=S)oo___U+ C=S ____//_M____//N. Applying ME0(-, Sp ) to the diagram yields a diagram of bijections by Proposi- tion 3.15. The result follows from chasing around this diagram. Remark 5.4. The above result is more useful in light of Proposition 4.7(b). Sup- pose M and N are stable, combinatorial model categories which are Quillen equiv- alent. This includes the possibility that the Quillen equivalence occurs throug* *h a zig-zag, where the intermediate steps may not be combinatorial or pointed. So t* *he above result doesn't apply directly. However, Proposition 4.7(b) shows that any such zig-zag may be replaced by a simple zig-zag where the intermediate step is both combinatorial and pointed (hence also stable). One example of this techniq* *ue is given in the proof of Theorem 1.6 below. 16 DANIEL DUGGER Proposition 5.5. Assume that M is stable, combinatorial, and a spectral model category. Then oM is quasi-equivalent to the enrichment oe provided by the spe* *ctral structure. Proof.As M is spectral, it is in particular simplicial (cf. A.8). So one may ch* *oose a Quillen equivalence L: U+ C=S ! M consisting of simplicial functors (see discus* *sion in the proof of Proposition 4.9). We have the Quillen maps U+ C=S________//M | | fflffl| Sp (U+ C=S). We claim there is a spectral Quillen equivalence Sp (U+ C=S) ! M making the triangle commute. This immediately implies the result we want: applying ME0(-, Sp ) to the triangle gives a commutative diagram of bijections by Propo- sition 3.15(d), and the diagonal map sends the canonical spectral enrichment of Sp (U+ C=S) to the given spectral enrichment of M by Proposition 3.15(e). We are reduced to constructing the spectral Quillen map Sp (U+ C=S) ! M. Note that objects in Sp (U+ C) may be regarded as presheaves of symmetric spec- tra on C. That is, we are looking at the functor category Func(Cop, Sp ). By Proposition A.14, the composite C ! U+ C ! M induces a spectral Quillen map Re :Func(Cop, Sp ) AE M: Sing, where the functor category is given the `objectwise' model structure. Note that the composite of right adjoints M ! Sp (U+ C=S) ! U+ C=S is indeed the right adjoint of L. We need to check that (Re , Sing) give a Quillen map Sp (U+ C=S) ! M. By Proposition B.1, the domain model category is identical to (Sp U+ C)=Sstab(no- tation as in Appendix B). But to show a Quillen map Sp (U+ C) ! M descends to (Sp U+ C)=Sstab, it is sufficient to check that the left adjoint sends eleme* *nts of Sstabto weak equivalences in M. A typical element of Sstabis a map Fi(A) ! Fi(B) where A ! B is in S (Fi(-) is defined in Appendix B). Certainly Re sends F0A ! F0B to a weak equivalence, since ReOF0 is the map L: U+ C ! M and this map sends elements of S to weak equivalences by construction. For i 1, note that the ith suspension of FiA ! FiB is F0A ! F0B. Since M is a stable model category, the fact that Re sends F0A ! F0B to a weak equivalence therefore immediately implies that it does the same for FiA ! FiB. 5.6. Homotopy endomorphism spectra. Let M be a stable, combinatorial model category, and let X 2 M be a cofibrant-fibrant object. Consider the ring spectrum oM (X, X). By Corollary 3.6, the isomorphism class of this ring spectr* *um in Ho (RingSpectra) only depends on oM up to quasi-equivalence. Now let W be an arbitrary object in M, and let X1 and X2 be two cofibrant- fibrant objects weakly equivalent to W . Then there exists a weak equivalence f :X1 ! X2. Let I be the category with one object and an identity map, and consider the two functors I ! M whose images are X1 and X2, respectively. Ap- plying Corollary 3.7 to this situation, we find that oM (X1, X1) and oM (X2, X2) are weakly equivalent ring spectra. So the corresponding isomorphism class in Ho(RingSpectra) is a well-defined invariant of W . We will write hEnd(W ) for a* *ny ring spectrum in this isomorphism class. SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 17 The two main results about homotopy endomorphism ring spectra were stated as Theorem 1.6 and Proposition 1.7. We now give the proofs: Proof of Theorem 1.6.If two stable, combinatorial model categories M and N are Quillen equivalent through a zig-zag, then by Proposition 4.7(b,c) there is a s* *imple zig-zag M -~ U+ C=S -~! N inducing an isomorphic derived equivalence of the homotopy categories. Now we apply Proposition 5.3 (twice) to connect oM to oN . Finally, the required equivalence of homotopy endomorphism ring spectra follows from Corollary 3.6. Proof of Proposition 1.7.This is a special case of Proposition 5.5. 6.A leftover proof In this section we complete the proof of Proposition 3.5. Essentially this am* *ounts to just explaining why the proof has already been given in [SS2, Lemma A.2.3]. * *The differences between our situation and that of [SS2] are (1) our indexing catego* *ries are not necessarily discrete (i.e., they have maps other than identities), and * *(2) we are dealing with a general symmetric monoidal model category rather than symmetric spectra. It turns out that neither difference is significant. 6.1. Modules. Let V be a symmetric monoidal category. Let C be a category, and let oe be an enrichment of C by V. A left oe-module is a collection of obje* *cts M(c) 2 V (for each c 2 C) together with maps oe(a, b) M(a) ! M(b) such that the following diagrams commute: oe(b, c) oe(a, b) _M(a)//_oe(b, c) M(b)S M(a)__//Ooe(a, a) M(a) | | OOOOO | | | OOO | fflffl| fflffl| OO''|fflffl oe(a, c) M(a)_________//_M(c) M(a) As for the case of bimodules (see Section 2.4), M inherits a natural structure * *of a functor C ! V. (An SC-module is precisely a functor M :C ! V, and so the map SC ! oe gives every left oe-module a structure of functor by restriction). Remark 6.2. A more concise way to phrase the above definition is to say that a left oe-module is a V-functor from the V-enriched category C to the V-enriched category V. We now record several basic facts about modules and functors. To begin with, one can check that colimits and limits in the category of oe-modules are the sa* *me as those in the category of functors Func(C, V). For each c 2 C, note that the functor oe(c, -): C ! V has an obvious structure of left oe-module. It is the `free' module determined by c. For A 2 V we write oe(c, -) A for the module a 7! oe(c, a) A. The canonical map SC ! oe induces a forgetful functor from oe-modules to SC- modules, which is readily checked to have a left adjoint: we'll call this adjo* *int oe (-). Let T :(SC - mod) ! (SC - mod) be the resulting cotriple. It's useful* * to note that if M :C ! V is a functor then oe M is the coequalizer of a a oe(b, -) M(a) ' oe(a, -) M(a) a!b a 18 DANIEL DUGGER (the coequalizer can be interpreted either in the category of oe-modules or the* * cat- egory of functors, as they coincide). GivenQtwo functors M,QN :C ! V, one can define F (M, N) 2 V as the equalizer of aV(M(a), N(a)) ' a!bV(M(a), N(b)). Together with the `objectwise' def- initions of the tensor and cotensor, this makes Func(C, V) into a closed V-modu* *le category (see Appendix A for terminology). If M :C ! V is a functor and X 2 V, one notes that there is a canonical isomorphism T (M X) ~=(T M) X; this follows from the explicit description of oe (-) given above. The map of functors M F (M, N) ! N therefore gives rise to a map T M F (M, N) ! T N, or a map jM,N :F (M, N) ! F (T M, T N) by adjointness. If M and N are oe-modules then they come equipped with maps of functors T M ! M and T N ! N. One defines Foe(M, N) 2 V as the equalizer of the two obvious maps F (M, N) ' F (T M, N) (to define one of the maps one uses jM,N ). With this definition_as well as the objectwise definitions for the ten* *sor and cotensor_the category of oe-modules becomes a closed V-module category. The adjunction (SC - mod) AE (oe - mod) is a V-adjunction. Using this together with the observation that oe(a, -) = oe SC(a, -), one sees that there are nat* *ural isomorphisms Foe(oe(a, -), M) ~=M(a). Proposition 6.3. Assume C is small and V is a combinatorial, symmetric monoidal model category satisfying the monoid axiom. Let oe be an enrichment of M by V. Then there is a cofibrantly-generated model structure on the category of left oe-modules in which a map M ! M0 is a weak equivalence or fibration pre- cisely when M(a) ! M0(a) is a weak equivalence or fibration for every a 2 C. Th* *is makes the category of left oe-modules into a V-model category. If the unit S 2 * *V is cofibrant, then the free modules oe(a, -) are cofibrant. Proof.Take the generating cofibrations (resp. trivial cofibrations) to be maps oe(a, -) A ! oe(a, -) B where A ! B is a generating cofibration (resp. triv- ial cofibration) of V and a 2 C is any object. Checking that this gives rise t* *o a cofibrantly-generated model structure is a routine application of [H , Th. 11.* *3.1]. The other statements are routine verifications as well. See also [SS2, Th. A.1.* *1]. Remark 6.4. Of course everything above also works for right oe-modules. 6.5. Bimodules. Suppose oe is an enrichment of C by V, and o is an enrich- ment of D by V. Define oe o to be the enrichment on C x D given by (oe o)((c1, d1), (c2, d2)) = oe(c1, c2) o(d1, d2). Define oeop to be the en* *richment of Cop given by oeop(a, b) = oe(b, a). Finally, define a oe - o bimodule to be* * a left oop oe-module. Remark 6.6. Upon unraveling the above definition, the reader will find that it * *is equivalent with the more naive (and concrete) version given in Section 3 for the case C = D. The notational conventions of that naive definition dictated the us* *e of oop oe rather than oe oop in the above definition. It follows from Proposition 6.3 that the category of oe - o bimodules has a m* *odel structure in which weak equivalences and fibrations are determined objectwise. Note that if M is a oe - o bimodule, then for any a 2 C the functor M(a, -) i* *s a left oe-module and the functor M(-, a) is a right o-module. SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 19 6.7. The main proof. Exactly following [SS2, Lem. A.2.3], we can now conclude the Proof of Proposition 3.5.We will sketch the proof for the reader's convenience. Suppose oe and o are model enrichments of M by V, defined over some small cate- gory I consisting of cofibrant-fibrant objects. Assume there is a quasi-equival* *ence between them given by the pointed bimodule M. If the composites oe(a, b) S ! oe(a, b) M(a, a) ! M(a, b) are all trivial fibrations (or if the correspondin* *g maps o(a, b) ! M(a, b) are all trivial fibrations) then the proof is exactly as in [* *loc. cit]. For the general case, we first replace M with a fibrant model in the category of oe - o bimodules over I; this makes M objectwise fibrant. For each a 2 I, the distinguished map S ! M(a, a) gives a map of right o-modules Fa = o(-, a) ! M(-, a). We apply our functorial factorization in the model category of right o- ~ modules to obtain Fa ae Na -i M(-, a). As the factorization is functorial, for every map a ! b in I there is an induced map of right o-modules Na ! Nb. Note that each Na is both cofibrant and fibrant as a o-module: the fibrancy is immediate, but the cofibrancy uses that Fa is cofibrant (which in turn depends * *on the unit S 2 V being cofibrant). Let E be the model enrichment of I given by E(a, b) = Fo(Na, Nb). Define U to be the oe - E bimodule U(a, b) = Fo(Na, M(-, b)) and W to be the E-o bimodule given by W (a, b) = Fo(Fa, Nb). The fact that W is a right o-module uses the existence of maps o(i, j) ! Fo(Fi, Fj), which is easily established. * *One sees that U and W are naturally pointed, and give quasi-equivalences between oe and E, and between E and o, respectively. Moreover, we are now in the case hand* *led by the first paragraph of this proof, because for U and W the appropriate maps are trivial fibrations. So we get a zig-zag of four direct equivalences between* * oe and o. 7. The additive case: Homotopy enrichments over Sp (sAb) We'll say that a model category is additive if its underlying category has a * *zero object and is enriched over abelian groups. If M is an additive, stable, combin* *atorial model category, we will produce a model enrichment of M by Sp (sAb). This allows us in particular to attach to every object X 2 M an isomorphism class in Ho (Ring[Sp (sAb)]). Write hEndad(X) for any object in this isomorphism class. By [S], the homotopy category of Ring[Sp (sAb)] is the same as the homotopy category of dgas over Z. So hEndad(X) can be regarded as a `homotopy endomor- phism dga'. Unlike the homotopy endomorphism spectra of Section 5.6, however, this dga is not an invariant of Quillen equivalence. It does act as an invarian* *t if one restricts to strings of Quillen equivalences involving only additive model cate* *gories, though. Here are the basic results (see (7.4) for additional terminology): Proposition 7.1. Given X 2 M as above, the homotopy endomorphism spectrum hEnd(X) is the Eilenberg-MacLane spectrum associated to hEndad(X). Proposition 7.2. Let M and N be additive, stable, combinatorial model categorie* *s. Suppose M and N are Quillen equivalent through a zig-zag of additive (but not n* *eces- sarily combinatorial) model categories. Let X 2 M, and let Y 2 Ho (N) correspond 20 DANIEL DUGGER to X under the derived equivalence of homotopy categories. Then hEndad(X) and hEndad(Y ) are weakly equivalent in Ring[Sp (sAb)]. Proposition 7.3. Let M be additive, stable, combinatorial, and an Sp (sAb)- model category. Let X 2 M be cofibrant-fibrant. Then hEndad(X) is weakly equiv- alent to the cotensor object F (X, X). The proofs of the above two results are for the most part similar to the corr* *e- sponding results for homotopy endomorphism spectra. One difference is that they depend on developing a theory of universal additive model categories. Another, more important, difference is the following. Recall from Proposition 4.7(b) th* *at any zig-zag of Quillen equivalences between two pointed model categories (with the intermediate steps not necessarily pointed) could be replaced by a simple z* *ig- zag where the third model category is also pointed. In contrast to this, it is * *not generally true that a zig-zag of Quillen equivalences between two additive model categories (with intermediate steps not necessarily additive) can be replaced b* *y a simple zig-zag where the middle step is also additive. This is only true if we * *assume that all the intermediate steps are additive in the first place. 7.4. Background. If M is a monoidal model category which is combinatorial and satisfies the monoid axiom, then by [SS1, Th. 4.1(3)] the category of monoids in M has an induced model structure where the weak equivalences and fibrations are the same as those in M. We'll write Ring[M] for this model category. If N is another such monoidal model category and L: M AE N: R is a Quillen pair which is weak monoidal in the sense of [SS3, Def. 3.6], then there is an induced Quil* *len map Ring[M] ! Ring[N]. This is a Quillen equivalence if M ! N was a Quillen equivalence and the units in M and N are cofibrant [SS3, Th. 3.12]. The adjunction Set* AE Ab is strong monoidal, and therefore induces strong monoidal Quillen functors Sp (sSet*) AE Sp (sAb). Therefore one gets a Quillen pair F :Ring[Sp ] AE Ring[Sp (sAb)]: U. By the Eilenberg-MacLane ring spectrum associated to an R 2 Ring[Sp (sAb)] we simply mean the ring spectrum UR. 7.5. Universal additive model categories. Let C be a small, semi-additive cat- egory. This means the Hom-sets of C have a natural structure of abelian groups, and C has a zero-object [M , VIII.2]_the `semi' is to indicate that C need not * *have direct sums. One says that a functor F :Cop! Ab is additive if F (0) ~=0 and for any two maps f, g :X ! Y in C one has F (f + g) = F (f) + F (g). Note that for every X 2 C, the representable functor rX defined by U 7! C(U, X) is additive. Let Func(Cop, Ab) denote the category of all functors. The Yoneda Lemma does not hold in this category: that is, if F 2 Func(Cop, Ab) one need not have Hom (rX, F ) ~=F (X) for all X 2 C. But it is easy to check that this does hold when F is an additive functor. Let Funcad(Cop, Ab) denote the full subcategory of additive functors. The fol- lowing lemma records several basic facts about this category, most of which fol* *low from the Yoneda Lemma. Lemma 7.6. Let C be a small, semi-additive category. (a)Colimits and limits in Funcad(Cop, Ab) are the same as those in Func(Cop, Ab* *). SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 21 (b)Every additive functor F 2 Func(Cop, Ab) is isomorphic to its canonical coli* *mit with respect to the embedding r :C ,! Func(Cop, Ab). That is, the natural map colimrX!FrX ! F is an isomorphism. (c)The additive functors in Func(Cop, Ab) are precisely those functors which are colimits of representables. (d)The inclusion i: Funcad(Cop, Ab) ,! Func(Cop, Ab) has a left adjoint Ad (for `additivization'), and the composite AdOi is naturally isomorphic to the ide* *ntity. (e)Suppose given a co-complete, additive category A and an additive functor fl :C ! A. Define Sing:A ! Funcad(Cop, Ab) by letting Sing(a) be the functor a 7! A(flc, a). Then Singhas a left adjoint Re, and there are natural isomor- phisms Re(rX) ~=fl(X). Proof.Left to the reader. By [H , Th. 11.6.1] the category Func(Cop, sAb) has a cofibrantly-generated m* *odel structure in which the weak equivalences and fibrations are defined objectwise.* * We will need the analagous result for the category of additive functors: Lemma 7.7. The category Funcad(Cop, sAb) has a cofibrantly-generated model structure in which the weak equivalences and fibrations are defined objectwise.* * This model structure is simplicial, left proper, and cellular. Proof.The proof uses the adjoint pair (Ad, i) to create the model structure, as in [H , Th. 11.3.2]. Recall that the model category Func(Cop, sAb) has generati* *ng trivial cofibrations J = {rX Z[ n,k] ! rX Z[ n] | X 2 C}. Our notation is that if K 2 sSet then Z[K] 2 sAb is the levelwise free abelian group on K; and if A 2 sAb then rX A denotes the presheaf U 7! C(U, X) A (with the tensor performed levelwise). To apply [H , 11.3.2] we must verify that the functor i takes relative Ad(J)-* *cell complexes to weak equivalences. However, note that the domains and codomains of maps in J are all additive functors (since representables are additive), and* * so Ad(J) = J. The fact that forming pushouts in Funcad(Cop, sAb) and Func(Cop, sAb) give the same answers (by Lemma 7.6(a)) therefore shows that the Ad(J)-cell com- plexes are indeed weak equivalences in Func(Cop, sAb). Finally, it is routine to check that the resulting model structure is simplic* *ial, left proper, and cellular. From now on we will write UadC for the category Funcad(Cop, sAb) with the model structure provided by the above lemma. The reason for the notation is provided by the next result. Theorem 7.8. Let M be an additive model category. (a)Suppose C is a small, semi-additive category and fl :C ! M is an additive functor. Then there is a Quillen pair Re :UadC AE M: Sing together with a natural weak equivalence Re Or -~!fl. (b)If M is combinatorial then there is a Quillen equivalence UadC=S -~! M for some small, semi-additive category C and some set of maps S in UadC. (c)Suppose M -~ M1 -~!. .-.~ Mn -~! N is a zig-zag of Quillen equivalences in which all the model categories are additive. If M is combinatorial, there* * is a simple zig-zag of equivalences M -~ UadC=S -~! N 22 DANIEL DUGGER such that the derived equivalence Ho (M) ' Ho (N) is isomorphic to the deriv* *ed equivalence given by the original zig-zag. Proof.The proofs for (a) and (c) are simple, and exactly follow the case for UC (see [D1 , Prop. 2.3, Cor. 6.5]). The proof of (b) is slightly more complicated* *, and will be postponed until the end of this section. Remark 7.9. The result in (c) is false if one does not assume that all the Mi's are additive. For an example, let R be the dga Z[e; de = 2]=(e4) and let T be the dga Z=2[x; dx = 0]=(x2), where both e and x have degree 1. Let M and N be the categories of R- and T -modules, respectively. These turn out to be Quil* *len equivalent, but they cannot be linked by a zig-zag of Quillen equivalences betw* *een additive model categories. A verification of these claims can be found in [DS , Example 6.10]. 7.10. Endomorphism objects. Let M be an additive, stable, combinatorial model category. By Theorem 7.8 there is a Quillen equivalence UadC=S ! M for some small, additive category C and some set of maps S in UadC. The category UadC=S is simplicial, left proper, and cellular, so we may form Sp (UadC=S). Si* *nce UadC=S is stable (since M was), we obtain a zig-zag of Quillen equivalences M -~ UadC=S -~! Sp (UadC=S). The category UadC is an sAb-model category, and therefore Sp (UadC=S) is an Sp (sAb)-model category. We can transport this enrichment onto M via the Quillen equivalences, and therefore get an element oeM 2 ME0(M, Sp (sAb)). Just as in Section 5, one shows that this quasi-equivalence class does not depend on* * the choice of C, S, or the Quillen equivalence UadC=S -~! M. Let X 2 M, and let "Xbe a cofibrant-fibrant object weakly equivalent to X. We write hEndad (X) for any object in Ring[Sp (sAb)] having the homotopy type of oeM (X", "X), and we'll call this the additive homotopy endomorphism object of X. By Corollaries 3.6 and 3.7, this homotopy type depends only on the homotopy type of X and the quasi-equivalence class of oeM _and so it is a well-defined i* *nvariant of X and M. Proof of Proposition 7.2.This is entirely similar to the proof of Theorem 1.6. Proof of Proposition 7.3.Same as the proof of Proposition 5.5. Proof of Proposition 7.1.We know that there exists a zig-zag of Quillen equiv- alences M -~ UadC=S -~! Sp (UadC=S). Therefore, using Theorem 1.6 and Proposition 7.2 we may as well assume M = Sp (UadC=S). This is an Sp (sAb)- model category, and so for any object X we have a ring object F (X, X) in Sp (sAb). The adjoint functors Set*AE Ab induce a strong monoidal adjunction F :Sp (sSet*) AE Sp (sAb): U. The Sp (sAb)-structure on M therefore yields an induced Sp -structure as well (see Section A.6). In this structure, the end* *o- morphism ring spectrum of X is precisely U[F (X, X)]. Proposition 5.5 tells us this has the homotopy type of the ring spectrum hEnd (X), at least when X is cofibrant-fibrant. And Proposition 7.3 says that F (X, X) has the homotopy type of hEndad(X). This is all we needed to check. SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 23 7.11. Additive presentations. We turn to the proof of Theorem 7.8(b). This will be deduced from the work of [D2 ] plus some purely formal considerations. Let M be a combinatorial model category. By [D2 , Prop. 3.3], there is a small category C and a functor C ! M such that the induced map L: UC ! M is homotopically surjective (see [D2 , Def. 3.1] for the definition). Then [D2 ,* * Prop. 3.2] shows that there is a set of maps S in UC which the derived functor of L t* *akes to weak equivalences, and such that the resulting map UC=S ! M is a Quillen equivalence. Now suppose that M was also an additive model category. By examining the proof of [D2 , Prop. 3.3] one sees that C may be chosen to be a semi-additive category and the functor fl :C ! M an additive functor (the category C is a cer* *tain full subcategory of the cosimplicial objects over M). By Theorem 7.8(a) there is an induced map F :UadC ! M. Again using [D2 , Prop. 3.2], it will be enough to prove that this map is homotopically surjective. Consider now the following sequence of adjoint pairs: __Z_//_ __Ad//_ _F__//_ Func(Cop, sSet)oo_Func(Cop, sAb)oo__Funcad(Cop, sAb)oo__M U i The composite of the right adjoints is clearly the right adjoint of L, so the c* *omposite of the left adjoints is L. We have constructed things so that this composite is homotopically surjective, and we are trying to show that F is also homotopically surjective. Lemma 7.12. If X 2 C then Ad(Z(rX)) ~= rX (or to be more precise, Ui(Ad(Z(rX))) ~=rX). Proof.This is clear, since the functors F 7! Funcad(Ad(Z(rX)), F ) and F 7! Funcad(rX, F ) are both naturally isomorphic to F (X). Let G 2 Funcad(Cop, sAb). Let QG be the simplicial presheaf whose nth level is a (rXn) rXn!rXn-1!...!rX0!Gn where the coproduct is in Func(Cop, sSet). The simplicial presheaf QG is treate* *d in detail in [D1 , Sec. 2.6], as it is a cofibrant-replacement functor for UC. Lik* *ewise, let QadG be the simplicial presheaf whose nth level is M (rXn) rXn!rXn-1!...!rX0!Gn where the coproduct is now in Func(Cop, sAb). The proof of [D2 , Prop. 2.8] sho* *wing that Q is a cofibrant-replacement functor for UC adapts verbatim to show that Q* *ad is a cofibrant-replacement functor for UadC. Note that by Lemma 7.12 we have QadG = Ad(Z(QG)), since Ad and Z(-) are left adjoints and therefore preserve coproducts. Finally we are in a position to conclude the Proof of Theorem 7.8(b).We have reduced to showing that F :UadC ! M is ho- motopically surjective. Let Singbe the right adjoint of F . Then we must show t* *hat for every fibrant object X 2 M the induced map F Qad(SingX) ! X is a weak equivalence. 24 DANIEL DUGGER However, the fact that L: UC ! M is homotopically surjective says that LQ(Ui SingX) ! X is a weak equivalence in M. And we have seen above that F QadUi SingX = F Ad(Z(Q SingX)) = LQ SingX, so we are done. Appendix A. D-model categories In the body of the paper we need to deal with spectral model categories. These are model categories which are enriched, tensored, and cotensored over the model category of symmetric spectra, and where the analog of SM7 holds. In this appen* *dix we briefly review some very general material relevant to this situation. We ass* *ume the reader already has some experience in this area (for instance in the settin* *g of simplicial model categories), and for that reason only give a broad outline. A.1. Basic definitions. Let D be a closed symmetric monoidal category. The `symmetric monoidal' part says we are given a bifunctor , a unit object 1D , together with associativity, commutativity, and unital isomorphisms making cert* *ain diagrams commute (see [Ho2 , Defs. 4.1.1, 4.1.4] for a nice summary). The `clos* *ed' part says that there is also a bifunctor (d, e) 7! D_(d, e) 2 D together with a* * natural isomorphism D(a, D_(d, e)) ~=D(a d, e). Note that, in particular, this gives us isomorphisms D(1D , D_(d, e)) ~=D(1D d* *, e) ~= D(d, e). We define a closed D-module category to be a category M equipped with natural constructions which assign to every X, Z 2 M and d 2 D objects X d 2 M, F (d, Z) 2 M, and M_D (X, Z) 2 D. One requires, first, that there are natural isomorphisms (X d) e ~=X (d * * e) and X 1D ~=X making certain diagrams commute (see [Ho2 , Def. 4.1.6]). One also requires natural isomorphisms (A.1) M(X d, Z) ~=M(X, F (d, Z)) ~=D(d, M_D(X, Z)) (see [Ho2 , 4.1.12]). Remark A.2. Taking d = 1D , note that we obtain isomorphisms M(X, Z) ~= M(X 1, Z) ~=D(1, M_D(X, Z)). Proposition A.3. Suppose D is a symmetric monoidal category, and M is a closed D-module category. Then one has canonical isomorphisms M_D(X d, Z) ~=M_D(X, F (d, Z)) ~=D_(d, M_D(X, Z)) of objects in D. Applying D(1D , -) to these isomorphisms yields the isomorphis* *ms in (A.1). Proof.The Yoneda Lemma says that two objects a, b 2 D are isomorphic if and only if there is a natural isomorphism D(e, a) ~=D(e, b), for e 2 D. The proof * *of the proposition is straightforward using this idea. SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 25 Proposition A.4. Suppose D is a symmetric monoidal category, and M is a closed D-module category. Then there are `composition' maps M_D(Y, Z) M_D(X, Y ) ! M_D(X, Z), natural in X, Y , and Z. These maps satisfy associativity and unital condition* *s. The induced map D_(1, M_D(Y, Z)) D_(1, M_D(X, Y )) ! D_(1, M_D(X, Z)) coincides with the composition in M under the isomorphisms from Remark A.2. Proof.We will only construct the maps, leaving the other verifications to the r* *eader. The adjointness isomorphisms from (A.1) give rise to natural maps X M_(X, Y ) ! Y (adjoint to the identity M_(X, Y ) ! M_(X, Y )). There is a corresponding map Y M_(Y, Z) ! Z. Now consider the composite X [M_(X, Y ) M_(Y, Z)] ~=[X M_(X, Y )] M_(Y, Z) ! Y M_(Y, Z) ! Z. Adjointness now gives M_(X, Y ) M_(Y, Z) ! M_(X, Z), and finally one uses that D is symmetric monoidal. Remark A.5. The basic definition of a D-module category doesn't really need D to be symmetric monoidal. In fact, in [Ho2 ] this is not assumed. However, the above propositions definitely need the symmetric hypothesis. A symmetric monoidal model category consists of a closed symmetric monoidal category M, together with a model structure on M, satisfying two condi- tions: (1)The analog of SM7, as given in either [Ho2 , 4.2.1] or [Ho2 , 4.2.2(2)]. (2)A unit condition given in [Ho2 , 4.2.6(2)]. Finally, let D be a symmetric monoidal model category. A D-model category is a model category M which is also a closed D-module category and where the two conditions from [Ho2 , 4.2.18] hold: these are again the analog of SM7 and a un* *it condition. A.6. Lifting module structures. Suppose that C and D are symmetric monoidal model categories, and that L: C AE D: R is a Quillen pair. One says this adjunc- tion is strong symmetric monoidal if there are isomorphisms L(1C) ~=1D and L(X Y ) ~=LX LY compatible with the associativity, commutativity, and unital isomorphisms in C and D. Lemma A.7. Assume that L: C AE D: R is a strong symmetric monoidal Quillen adjunction. Let M be a D-model category. Then M also becomes a C-model category by setting X c = X L(c), FC(c, Y ) = F (Lc, Y ), and M_C(X, Y ) = R M_D(X, Y ) . Proof.Routine. 26 DANIEL DUGGER A.8. Spectral model categories. Let Sp = Sp (sSet+) be the usual category of symmetric spectra [HSS ]. This is a symmetric monoidal model category. We wi* *ll call an Sp -model category simply a spectral model category. Note that there are adjoint functors sSet+ AE Sp where the left adjoint is K 7! 1 (K) and the right adjoint is Ev 0, the functor sending a spectrum to the space in its 0th level. The functor 1 is called F0 in [HSS ]. These funct* *ors are strong symmetric monoidal (see [HSS , 2.2.6]). Therefore any spectral model category becomes an sSet+-model category in a natural way, via Lemma A.7. The adjoint functors sSet AE sSet+ (which are also strong monoidal) in turn show that any sSet+-model structure gives rise to an underlying simplicial model structure. A.9. Diagram categories. Let I be a small category. If D is cofibrantly- generated, then DI has a model structure in which the weak equivalences and fibrations are defined objectwise. If X 2 DI and d 2 D, define the two objects X d, F (d, X) 2 DI as follows: X d: i 7! X(i) d, F (d, X): i 7! F (d, X(i)). Also, if X, Z 2 DI define DI_D(X, Z) 2 D to be the equalizer of Y Y D_(X(i), Z(i)) ' D_(X(j), Z(k)). i j!k Lemma A.10. Assume D is a cofibrantly-generated, symmetric monoidal model category. With the above definitions, DI is a D-model category. Proof.Straightforward. A.11. Adjunctions. Lemma A.12. Let M and N be closed D-module categories, and let L: M AE N: R be adjoint functors. The following are equivalent: (a)There are natural isomorphisms N_D(LX, Y ) ~=M_D(X, RY ) which after apply- ing D(1D , -) reduce to the adjunction N(LX, Y ) ~=M(X, RY ). (b)There are natural isomorphisms L(X d) ~= L(X) d which reduce to the canonical isomorphism for d = 1D . (c)There are natural isomorphisms R(F (d, Z)) ~=F (d, RZ) which reduce to the canonical isomorphism when d = 1D . Proof.Left to the reader. In the situation of the above lemma, we'll say that the adjoint pair is a D- adjunction between M and N. When M and N are D-model categories we'll say that M ! N is a D-Quillen map (resp. D-Quillen equivalence) if it is both a Quillen map (resp. Quillen equivalence) and a D-adjunction. In this paper we mostly need simplicial and spectral Quillen functors, i.e. the cases where D = * *sSet or D = Sp . Remark A.13. Note that in the situation of a D-adjunction one may form the following composite, for any A, B 2 N: ~= N_(A, B) ! N_(LRA, B) -! M_(RA, RB). SPECTRAL ENRICHMENTS OF MODEL CATEGORIES 27 Similarly, one has a natural map M_(X, Y ) ! N_(LX, LY ) for X, Y 2 M. It is a * *rou- ~= tine exercise to check that the adjunction isomorphism N_(LA, X) -! M_(A, RX) is equal to the composite N_(LA, X) ! N_(RLA, RX) ! N_(A, RX), just as for ordinary adjunctions. Let D be a cofibrantly-generated, symmetric monoidal model category, and let M be a D-model category. Suppose I is a small category and fl :I ! M is a funct* *or. Define Sing:M ! Func(Iop, D) by sending X 2 M to the functor i 7! M_D(fl(i), X). This has a left adjoint Re :Func(Iop, D) ! M which sends a functor A to the coequalizer a a fl(j) A(k) ' fl(i) A(i). j!k i Proposition A.14. The adjoint pair Re : Func(Iop, D) AE M: Sing is a D- adjunction. Proof.One readily checks condition (c) in Lemma A.12. Appendix B. Stabilization and localization Let M be an sSet+-model category which is pointed, left proper, and cellular. Under these conditions one may form the stabilized model category Sp M [Ho1 ], and this is again a left proper and cellular model category. Recall that there * *are Quillen pairs Fi:M AE Sp M: Evi, for every i 0 (F0X is also written 1 X, and FiX is morally the ith desuspension of F0X). If S is a set of maps between cofibrant objects in M, let Sstab= {Fi(A) ! Fi(B) | A ! B 2 S and i 0}. Our goal is the following basic result about commuting stabilization and locali* *za- tion: Proposition B.1. In the above situation, the model categories Sp (M=S) and (Sp M)=Sstabare identical. Proof.The stable model structure on Sp M is formed in two steps. One starts with the projective model structure SpprojM where fibrations and weak equivalences a* *re levelwise (and cofibrations are forced). Then one localizes this projective str* *ucture at a specific set of maps given in [Ho1 , Def. 8.7]. Call this set TM . It is i* *mportant that TM depends only on the generating cofibrations of M. So Sp (M=S) is the localization of Spproj(M=S) at the set TM=S . Likewise, (Sp M)=Sstabis the localization of (SpprojM)=Sstabat the set of maps TM . But as the generating cofibrations of M and M=S are the same, we have TM = TM=S . In this way we have reduced the proposition to the statement that the model struct* *ures Spproj(M=S) and (SpprojM)=Sstabare identical. The trivial fibrations in a model category and its Bousfield localization are* * always the same. This shows that the trivial fibrations in the following categories ar* *e the same: Spproj(M=S), SpprojM, (SpprojM)=Sstab. An immediate corollary is that the cofibrations are also the same in these three model categories. Note also that these are all simplicial model categories, wi* *th 28 DANIEL DUGGER simplicial structure induced by that on M_and in particular that the simplicial structures are identical. Since the trivial fibrations in Spproj(M=S) and (SpprojM)=Sstabare the same, it will suffice to show that trivial cofibrations are also the same. But a cofi* *bration A ae B is trivial precisely when the induced map on simplicial mapping spaces Map (B, X) ! Map (A, X) is a weak equivalence for every fibrant object X. Since the model categories have the same simplicial structures, we have reduced to sh* *ow- ing that they have the same class of fibrant objects. A fibrant object in Spproj(M=S) is a spectrum E such that each Eiis fibrant in M=S; this means Ei is fibrant in M, and for every A ! B in S the induced map Map (B, Ei) ! Map (A, Ei) is a weak equivalence (recall that S consists of maps between cofibrant objects). A fibrant object in (SpprojM)=Sstabis a fibrant spectrum E 2 SpprojM (mean- ing only that each Ei is fibrant in M) which is Sstab-local. The latter condit* *ion means that for every A ! B in S and for every i, the map Map (Fi(B), E) ! Map (Fi(A), E) is a weak equivalence. But the adjoint pair (Fi, Evi) is a simpl* *icial adjunction_one readily checks condition (b) or (c) of Lemma A.12. So we have Map (Fi(B), E) ~=Map (B, Evi(E)), and the same for A. This verifies that the two classes of fibrant objects are the same, and completes the proof. References [AR] J. Adamek and J. Rosicky, Locally presentable and accessible categories, * *London Math. Society Lecture Note Series 189, Cambridge University Press, 1994. [B] F. Borceux Handbook of categorical algebra 2: Categories and structures, * *Cambridge University Press, 1994. [D1] D. Dugger, Universal homotopy theories, Adv. Math. 164, no. 1 (2001), 144* *-176. [D2] D. Dugger, Combinatorial model categories have presentations, Adv. Math. * *164, no. 1 (2001), 177-201. [DS] D. Dugger and B. Shipley, Topological equivalences for differential grade* *d algebras, preprint, 2004. [DK] W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topol* *ogy 19 (1980), 427-440. [H] P. Hirschhorn, Model categories and their localizations, Mathematical Sur* *veys and Mono- graphs, vol. 99, Amer. Math. Soc., 2003. [Ho1] M. Hovey, Spectra and symmetric spectra in general model categories, J. P* *ure Appl. Algebra 165 (2001), no. 1, 63-127. [Ho2] M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63,* * Amer. Math. Soc., 1999. [HSS] M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, Jour. Amer. Math. * *Soc. 13 (1999), no. 1, 149-208. [M] S. MacLane, Categories for the working mathematician, Second edition, Spr* *inger-Verlag New York, 1998. [SS1] S. Schwede and B. Shipley, Algebras and modules in monoidal model categor* *ies, Proc. London Math. Soc. 80 (2000), 491-511. [SS2] S. Schwede and B. Shipley, Stable model categories are categories of modu* *les, Topology 42 (2003), 103-153. [SS3] S. Schwede and B. Shipley, Equivalences of monoidal model categories, Alg* *ebr. Geom. Topol. 3 (2003), 287-334. [S] B. Shipley, HZ-algebra spectra are differential graded algebras, preprint* *, 2004. Department of Mathematics, University of Oregon, Eugene, OR 97403 E-mail address: ddugger@math.uoregon.edu