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ENRICHMENTS of ADDITIVE MODEL CATEGORIES Contents 1 ENRICHMENTS OF ADDITIVE MODEL CATEGORIES DANIEL DUGGER AND BROOKE SHIPLEY Abstract. We prove that any stable, additive, combinatorial model category M has a canonical model enrichment over SpΣ(sAb) (symmetric spectra based on simplicial abelian groups). So to any object X ∈ M one can attach an endomorphism ring object in this category, denoted hEndad(X). Since the homotopy theory of ring objects in SpΣ(sAb) is equivalent to the homotopy theory of differential graded algebras, one also has a homotopy endomorphism dga hEnddga(X). We prove that the homotopy type of hEndad(X)—and thus also hEnddga(X)—is an invariant of Quillen equivalences between additive model categories. Contents 1. Introduction 1 2. Additive model categories 4 3. Universal additive model categories 8 4. Homotopy enrichments over SpΣ(sAb) 12 5. Chain enrichments 13 6. Homotopy theory of CI-categories 15 7. Adjoint modules 19 8. Applications of adjoint modules 24 9. A leftover proof 27 References 29 1. Introduction A model category is called additive if two conditions are satisfied. First, its hom-sets must have natural structures of abelian groups with respect to which composition is biadditive. Secondly, the abelian group structures on these hom- sets must interact well with the notion of ‘higher homotopies’. We give a precise definition in Section 2. Examples of additive model categories include chain com- plexes over a ring and differential graded modules over a differential graded algebra, as one should expect. Recall that a model category is called combinatorial if it is cofibrantly- generated and its underlying category is locally presentable. A model category is stable if it is pointed and the suspension functor is an auto-equivalence of the homotopy category. In [D4] it was shown that any stable, combinatorial model category could be naturally enriched over the category SpΣ of symmetric spectra. This enrichment is invariant under Quillen equivalences in a certain sense. Second author partially supported by NSF Grant No. 0134938 and a Sloan Research Fellowship. 1 2 DANIEL DUGGER AND BROOKE SHIPLEY In the present paper we extend the results of [D4] to show that any stable, com- binatorial, additive model category has a natural enrichment over SpΣ(sAb), the category of symmetric spectra based on simplicial abelian groups. This enrichment is not an invariant of Quillen equivalence, but it is preserved by Quillen equivalences which only involve additive model categories. Remark 1.1. The tools developed in this paper are applied in [DS2]. Two additive model categories M and N are called additively Quillen equivalent if there is a zig-zag of Quillen equivalences between M and N in which every intermediate step is additive. It is a strange fact, established in [DS2], that additive model categories can be Quillen equivalent but not additively Quillen equivalent. The demonstration of this fact uses the model enrichment developed in the present paper. Recall the notion of model enrichment and quasi-equivalence from [D4]. Let M be a model category and V be a symmetric monoidal model category. Briefly, a model enrichment is a bifunctor τ : Mop × M → V together with composition maps τ(Y, Z) ⊗ τ(X, Y ) → τ(X, Z) which are associative and unital. The bifunctor must interact well with the model category structure—see [D4] (or Section 4.1) for an explicit list of the necessary axioms. There is a notion of when two model enrichments of M by V are ‘quasi- equivalent’, which implies that they carry the same homotopical information. We let ME0(M, V) denotes the quasi-equivalence classes of model enrichments. If L: M N: R is a Quillen pair, there are induced functors denoted ∗ L∗ : ME0(M, V) → ME0(N, V) and L : ME0(N, V) → ME0(M, V). When (L, R) is a Quillen equivalence these are inverse bijections. Using the above language, we can state the basic results. These are proved in Sections 4 and 5. Theorem 1.2. If M is a stable, additive, combinatorial model category, then there Σ is a canonical element σM ∈ ME0(M, Sp (sAb)). If L: M → N is a Quillen ∗ equivalence then L∗(σM) = σN and L (σN) = σM. If X ∈ M, choose a cofibrant-fibrant object Xˆ which is weakly equivalent to ˆ ˆ Σ X. Then σM(X, X) gives a ring object in Sp (sAb). The resulting isomorphism class in Ho (Ring[SpΣ(sAb)]) only depends on the homotopy type of X. We write hEndad(X) for any ring object in this isomorphism class, and call it the additive homotopy endomorphism spectrum of X. Proposition 1.3. Let M and N be additive, stable, combinatorial model categories. Suppose M and N are Quillen equivalent through a zig-zag of additive (but not neces- sarily combinatorial) model categories. Let X ∈ M, and let Y ∈ Ho (N) correspond to X under the derived equivalence of homotopy categories. Then hEndad(X) and Σ hEndad(Y ) are weakly equivalent in Ring[Sp (sAb)]. Any ring object R in SpΣ(sAb) gives rise to a ring object in SpΣ by forgetting the abelian group structure—this is called the Eilenberg-MacLane spectrum asso- ciated to R. Recall that in [D4] it was shown how to attach to any X in a stable, combinatorial model category an isomorphism class in Ho (Ring[SpΣ]). This was called the homotopy endomorphism spectrum of X, and denoted hEnd(X). We have the following: Proposition 1.4. Given X ∈ M as above, the homotopy endomorphism spectrum hEnd(X) is the Eilenberg-MacLane spectrum associated to hEndad(X). ENRICHMENTS OF ADDITIVE MODEL CATEGORIES 3 Finally, we have two results explaining how to compute hEndad(X) when the model category M has some extra structure. Recall that if C is a symmetric monoidal model category then a C-model category is a model category equipped with compatible tensors, cotensors, and enriched hom-objects over C satisfying the analog of SM7. See [D4, Appendix A] for more detailed information. For X, Y ∈ M we denote the enriched hom-object by MC(X, Y ). Note that a SpΣ(sAb)-model category is automatically additive and stable. This follows from Corollary 2.9 below, and the appropriate analogue of [SS2, 3.5.2] or [GS, 3.2]. Proposition 1.5. Let M be a combinatorial SpΣ(sAb)-model category. Let X ∈ M be cofibrant-fibrant. Then hEndad(X) is weakly equivalent to the enriched hom- object MSpΣ(sAb)(X, X). In [S] it is shown that the model categories of rings in SpΣ(sAb) and in Ch are Quillen equivalent. This is recalled in Section 5. Note that the rings in Ch 0 are just dgas. The associated derived functors will be denoted H : DGA RingSpΣ(sAb):Θ0. (In [S] the functors H and Θ are functors between DGA and HZ-algebras with RingSpΣ(sAb) an intermediate category.) We then de- 0 fine the homotopy endomorphism dga of X to be Θ [hEndad(X)] and write hEnddga(X). Obviously, this carries exactly the same information as hEndad(X). 0 0 0 In fact, H [hEnddga(X)] is weakly equivalent to hEndad(X) since H and Θ are inverse equivalences on the homotopy category level. As above, we remark that a Ch-model category is automatically additive and stable, by Corollary 2.9 and the appropriate analogue of [SS2, 3.5.2]. Proposition 1.6. Let M be a combinatorial Ch-model category. Assume M has a generating set of compact objects, as defined in (5.3) below. Let X ∈ M be cofibrant-fibrant. Then MCh(X, X) is weakly equivalent to hEnddga(X). The assumption about the generating set in the above proposition is probably unnecessary, but we don’t know how to remove it. It is satisfied in most cases of interest. Remark 1.7. The study of dg-categories seems to be of current interest—see, for example, [Dr, T]. A dg-category is simply a category enriched over unbounded chain complexes Chk, where k is some commutative ground ring. We remark that the homotopy theory of dg-categories over Z is essentially the same as that of SpΣ(sAb)-categories (this follows from results of [S] and [SS3]). So the present paper may be regarded as associating to any stable, additive model category an underlying dg-category. 1.8. Organization of the paper. Sections 2 through 5 contain the main results on additive model categories and SpΣ(sAb)-enrichments. The rest of the paper develops the machinery needed to prove a technical lemma from Section 5. The main innovation here is the notion of ‘adjoint module’—developed in Sections 7 and 8—which allows one to compare enrichments over two different monoidal model categories. This notion has other applications, most notably in [GS]. 1.9. Notation and terminology. This paper is a sequel to [D4], and we will assume the reader is familiar with the machinery developed therein. In particular, we assume a familiarity with model enrichments and quasi-equivalences. 4 DANIEL DUGGER AND BROOKE SHIPLEY 2. Additive model categories In this section we define the notion of an additive model category, and prove some basic results for recognizing them. A category is preadditive if its hom-sets have natural structures of abelian groups for which the composition pairing is biadditive. A category is additive if it is preadditive and it has finite coproducts. This forces the existence of an initial object (the empty coproduct), which will necessarily be a zero object. See [ML, Section VIII.2]. A functor F : C → D between additive categories is an additive functor if F (f + g) = F (f) + F (g) for any two maps f, g : X → Y .
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