Morita Cohomology Julian V. S. Holstein St John’s College Department of Pure Mathematics and Mathematical Statistics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specif- ically indicated in the text. No part of this dissertation has been submit- ted for any other qualification. iii Summary This work constructs and compares different kinds of categorified coho- mology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded cat- egory with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain com- plexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model struc- ture. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafify- ing constant presheaves. We call this categorified Cˇech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cˇech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples. iv Contents 1. Introduction 1 2. Morita cohomology 8 2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . 8 2.2. Further properties of dgCat . . . . . . . . . . . . . . . 22 2.3. Cohomology of presheaves of model categories . . . . 27 2.4. Sheafification of constant presheaves . . . . . . . . . . 38 2.5. Morita cohomology over general rings . . . . . . . . . 45 3. Homotopy locally constant sheaves 46 3.1. Strictification and computation of homotopy limits . . 46 3.2. Strictification for dg-categories . . . . . . . . . . . . . 53 3.3. Restriction to perfect complexes . . . . . . . . . . . . 58 3.4. Homotopy locally constant sheaves . . . . . . . . . . . 59 4. Infinity-local systems 65 4.1. Infinity-local systems . . . . . . . . . . . . . . . . . . 65 4.2. Loop space representations . . . . . . . . . . . . . . . 68 4.3. Cellular computations . . . . . . . . . . . . . . . . . . 72 4.4. Finiteness and Hochschild homology . . . . . . . . . . 79 v 5. Computation and Examples 84 5.1. Spheres . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2. Other topological spaces . . . . . . . . . . . . . . . . 87 A. Some technical results on dg-categories 91 A.1. A combinatorial model for dg-categories . . . . . . . . 91 A.2. Simplicial resolutions of dg-categories . . . . . . . . . 93 vi 1. Introduction The aim of this thesis is to construct and compare categorifications of the cohomology of topological spaces by considering coefficients in the category of differential graded categories. We begin with the calculation of RΓMorita(X, k), for k a field, the coho- mology of a locally contractible topological space X with coefficients in a constant sheaf. We categorify by considering the constant sheaf k not a as a sheaf of rings, but as a presheaf of dg-categories with one object, where we equip dg-categories with the Morita model structure. In this model structure k ' Chpe, which is fundamental to our construction. Hence we call this categorified Cˇech cohomology Morita cohomology. We writeH M(X) for RΓMorita(X, k). Let X be a locally contractible topological space. The following characterization follows once we establish a local model structure on presheaves of dg-categories. Theorem 2.21. Given a good hypercover {Ui}i∈I of X one can compute H M(X) ' holimi∈Iop Chpe. Remark 1.1. To prove existence of the local model structure we show that dgCatk is left proper if k is a field. This may be of independent interest. 1 Under some further assumptions on X we also identify Morita cohomology with a more intuitive category associated to X. Theorem 3.15. Let X have a bounded locally finite good hypercover. Then the dg-categoryH M(X) is quasi-equivalent to the dg-category of homotopy locally constant sheaves of perfect complexes. Remark 1.2. The homotopy category of the category of homotopy locally constant sheaves can be considered as the correct derived category of local systems on X in the sense that it contains the abelian category of local systems but its Ext-groups are given by cohomology of X with locally constant coefficients rather than group cohomology of the fundamental group. The proof uses strictification to write the homotopy limit as a category of homotopy cartesian sections of a constant Quillen presheaf. Homotopy cartesian sections are then identified with homotopy locally constant sheaves. Next we define a category of ∞-local systems on a simplicial set K by the action of simplicial sets on dg-categories, K 7→ ChdgK . This construction is well-known to give a Quillen adjunction from sSet to dgCat. For a topological space X one considers Y (X) B ChdgSing* X, which can be considered as a categorification of singular cohomology. We prove the following comparison theorem: Theorem 4.4. The category H M(X) is equivalent to the category of ∞-local systems (Chpe)Sing*(X). 2 Homotopy invariance and a Mayer–Vietoris theorem are easy to establish for∞-local systems and hence for Morita cohomology. The category of ∞-local systems is closely related to the loop space of X, as is shown by the next result: Theorem 4.8. If X is a pointed and connected topological space the category H M(X) is furthermore equivalent to the category of fibrant cofibrant representations in perfect complexes of chains on the based loop space ΩX. We then establish a method of computing ChpeC∗(ΩX) for a CW complex that allows us to calculate Morita cohomology for a number of examples. This characterization allows us to compute Hochschild homology of Morita cohomology in some cases. It follows for example from some results available in the literature that for a simply connected space HH∗(H M(X)) ' H∗(L X), where the right hand side is cohomology of the free loop space. The results of Chapters 3 to 5 still hold if k is an arbitrary commutative ring of characteristic 0. We indicate in Section 2.5 how to relate this to Chapter 2. 3 Relation to other work Here we collect some references to ideas and results in the literature which are related to our constructions. This is not meant to be an exhaustive list. The analogous statement to Theorem 3.15 for perfect complexes of coherent sheaves appears for example as Theorem 2.8 in [41] referring back to [24] and as an assertion in [53]. Carlos Simpson discusses non-abelian cohomology with coefficients in a stack as an internal hom-space in geometric stacks, for example in [40]. This construction also appears in work by Pantev, Toën, Vaquié, Vezzosi [35] who construct interesting additional structures on these mapping stacks. In particular in the topological situation they mention the derived stack Map(M,RPerf), where M is a manifold considered as a constant stack and RPerf is the moduli stack of perfect complexes. (One can of course use more general topological spaces, the manifold condition provides extra structure.) One can consider the construction of ∞-local systems in Chapter 4 as a non-geometric version of this, which is already somewhat interesting and more tractable then the mapping stack. The comparison of homotopy locally constant sheaves and ∞-local systems is a linear and stable version of results in [47] or [39], where the corresponding result for presheaves of simplicial sets is proved by going via the category of fibrations. 4 An A∞-category of ∞-local systems on a simplicial set is constructed in [7], where the authors go on to prove a Riemann-Hilbert theorem. Their explicit formulae can be seen to be equivalent to our construction, see Proposition A.3. Outline After briefly recalling some technical results and definitions in 2.1 we proceed in 2.2 to show that the two model structures on dg-categories are cellular and left proper. This allows us to define a local model structure on presheaves of dg-categories and define their cohomology in 2.3. We then explicitly sheafify the constant presheaf in 2.4 and use this to define Morita cohomology H M(X) and write down a formula as a homotopy limit of a constant diagram with fiber Chpe indexed by the distinct open sets of a hypercover. Chapter 2 closes with some comments on the situation if k is not a field in 2.5 We explain how strictification allows to compute a homotopy limit as a category of homotopy cartesian sections in 3.1 and prove a strictification result for dg-categories in 3.2. We restrict this correspondence to sections with perfect fibers to obtain a formula for Morita cohomology in 3.3. We then use this formula in 3.3 to identify H M(X) with the category of homotopy locally constant sheaves on X. Chapter 4 takes a different approach to Morita cohomology. We define a category of ∞-local systems from the cotensor action of simplicial sets on dg-categories, and show it es equivalent to H M(X) in 4.1. We 5 use this to identify H M(X) with representations of the loop space, which is another natural generalization of local systems, in 4.2. As a by-product we recover the well-known equivalence REndC∗(ΩX)(k, k) ' C∗(X, k). Section 4.3 is then concerned with providing an explicit method for computing the category of ∞-local systems. In 4.4 we collect some results about finiteness of H M(X) and show how to compute Hochschild (co)homology in some cases. Chapter 5 consists of some example computations. Finally, the appendix provides a combinatorial model category of dg-categories and a construction of explicit simplicial resolutions in dgCat from which we deduce an explicit formula for C K . Acknowledgements Firstly, I would like to thank my advisor Ian Grojnowski for many insightful comments and stimulating discussions as well as never- ending encouragement. Many thanks to Jon Pridham for some very helpful conversations. I am grateful to my colleagues, past and present, in the DPMMS who make it a wonderful place to do mathematics, and to St John’s College and Christ’s College which have both provided excellent working environments as well as great places to live. During this work I have been supported by a PhD studentship from the Engineering and Physical Sciences Research Council and by a Blyth fellowship at Christ’s College. 6 For invaluable non-mathematical support over the the course of this project I am greatly indebted to my family and friends, sine quibus non. I want to particularly thank my parents as well as Pit, Sarah and Shaul. 7 2. Morita cohomology 2.1. Preliminaries In this section we fix our notation and conventions and recall some definitions and preliminary results. Note that Lemma 2.2 does not seem to be explicitly available in the literature. 2.1.1. Notation and conventions We assume the reader is familiar with the theory of model categories, but will try to recall all the less well-known facts about them that we use. In any model category we write Q for functorial cofibrant replacement and R for functorial fibrant replacement. We write mapping spaces (with values in sSet) in a model categoryM as MapM (X,Y). All other enriched hom-spaces in a category D will be denoted as HomD (X,Y). In particular we use this notation for differential graded hom-spaces, internal hom-spaces and hom-spaces of diagrams enriched over the target category. It will always be clear from context which category we enrich in. 8 We will work over the (underived) commutative ground ring k. We assume characteristic 0 in order to freely use differential graded constructions. Remark 2.1. If k is not a field, for example if we we work over k = Z, then the category of dg-categories is not automatically k-flat, i.e. hom- spaces might not be cofibrant as chain complexes (see below). This means some technical results in Chapter 2 are unavailable, see Section 2.5. Ch = Chk will denote the model category of chain complexes over the ring k equipped with the projective model structure where fibrations are the surjections and weak equivalences are the quasi-isomorphisms. Note that we are using homological grading convention, i.e. the differential decreases degree. The degree is indicated by a subscript or the inverse of a superscript, i.e. Ci = C−i. We write Chpe for the subcategory of perfect complexes in Ch. Chdg denotes the dg-category whose object are fibrant and cofibrant objects of Ch. Note that there is a natural identification of the subcategory of compact objects in Chdg with Chpe. This follows since an object X in Chdg is compact if HomHo(Chdg)(A,−) commutes with arbitrary coproducts. Then compact objects are precisely perfect complexes, i.e. bounded complexes which are level-wise projective. But perfect complexes are automatically cofibrant in the projective model structure. There is a natural smart truncation functor τ≥0 from Ch to Ch≥0, the category of non-negatively graded chain complexes, which naturally has 9 the projective model structure. The functor τ≥0 is right Quillen with left adjoint the inclusion functor. 2.1.2. Differential graded categories Basic references for dg-categories are [29] and [49]. Many technical details are proven in [44]. Let dgCat denote the category of categories enriched in Ch. Given D ∈ dgCat we define the homotopy category H0(D) as the category with the same objects as D and HomH0(D)(A, B) = H0HomD (A, B). If D is a model category enriched in Ch we define LD as its subcategory of fibrant cofibrant objects. We say D is a dg-model category if the two structures are compatible, that is if they satisfy the pushout-product axiom, see for example the definitions in Section 3.1 of [49]. Then Ho(D) ' H0(LD), where we take the homotopy category in the sense of model categories on the left and in the sense of dg-categories on the right. We mostly consider dg-categories with unbounded hom-spaces, but there is a natural truncation functor τ≥0 : dgCat → dgCat≥0 that is just truncation on hom-spaces. Recall that there are two model structures on dgCat. Firstly there is the Dwyer–Kan model structure, denoted dgCatDK . Weak equivalences are quasi-equivalences, i.e. dg-functors that induce weak equivalences on hom-spaces and are essentially surjective on the homotopy category. Fibrations are those dg-functors F that are surjective on hom-spaces and have the property that every homotopy equivalence F(a) → b′ in the 10 codomain of F lifts to a homotopy equivalence a → b with F(b) = b′. A set of generating cofibrations is given by the following. • ∅ → k • S (n)→ D(n) for all n ∈ Z. Here S (n) is the linearization of the category a g→ b where g has degree n and is the only non-identity morphism. D(n) equals S (n) with additional morphisms k. f in degree n + 1 such that d f = g. Recall the functor D 7→ D-Mod sending a dg-category to its model category of modules, i.e. D-Mod is the category of functors D → Ch and strict natural transformations. This is naturally a cofibrantly generated model category enriched in Ch whose fibrations and weak equivalences are given levelwise. We usually consider its subcategory of fibrant and cofibrant objects, L(D-Mod). Remark 2.2. The construction of the model category D-Mod follows Chapter 11 of [23], but there are some changes since we are considering enriched diagrams. Let I and J denote the generating cofibrations and generating trivial cofibrations of Ch. The generating (trivial) cofibrations of D-Mod are then of the form hX ⊗ A → hX ⊗ B for A → B ∈ I (resp. J), where hX denotes the contravariant Yoneda embedding. As in Theorem 11.6.1 of [23] we transfer the model structure from Chdiscrete(D). This works since hX is compact in D-Mod and so are its tensor products with the domains of I, ensuring condition (1) of Theorem 11.3.2 holds. For the second condition we have to check that relative J ⊗ hX-cell complexes are weak equivalences. Pushouts are constructed levelwise. The generating trivial cofibrations of Ch are of 11 the form 0 → D(n). Since the pushout U ← 0 → D(n) is weakly equivalent to U we are done. Note that cofibrations in this model category need not be levelwise cofibrations, unless hom-spaces in D are cofibrant, in which case the map Hom(α, β)⊗A→ Hom(α, β)⊗B is a cofibration and the proof goes through just like in [23]. In fact, if hom-spaces are cofibrant this works for categories of functors enriched in any monoidal model category V . Remark 2.3. In order to satisfy the smallness assumption we will always assume that all our dg-categories are small relative to some larger universe. The homotopy category of the model category LDop-Mod is called the derived category of D and denoted D(D). Definition. We denote by dgCatMor the category of dg-categories with the Morita model structure, i.e. the Bousfield localization of dgCatDK along functors that induce equivalences of the derived categories, see Chapter 2 of [44]. Fibrant objects in dgCatMor are dg-categories A such that the homotopy category of A is equivalent (via Yoneda) to the subcategory of compact objects of D(A ) [29]. We can phrase this as: every compact object is quasi-representable. An object X ∈ D(A ) is called compact if HomD(A )(X,−) commutes with arbitrary coproducts. We denote by ()pe the subcategory of compact objects. Morita fibrant dg- categories are also called triangulated since their homotopy category is an (idempotent complete) triangulated category. 12 With these definitions D 7→ L(Dop-Mod)pe, often denoted the triangulated hull, is a fibrant replacement, for example k 7→ Chpe. The category dgCat is symmetric monoidal with tensor product D ⊗ E given as follows. The objects are ObD × ObE and HomD⊗E ((D, E), (D ′, E′)) B HomD (D,D ′) ⊗ HomE (E, E′). The unit is the one object category k, which is cofibrant in either model structure. While dgCat is not a monoidal model category there is a derived internal Hom space and the mapping spaces in dgCatMor can be computed as follows [48]: Let RHom(C ,D) be the dg-category of right- quasirepresentable C ⊗ Dop-modules, i.e. functors F : C ⊗Dop → Ch such that for any c ∈ C we have that F(c,−) is isomorphic in the homotopy category to a representable object in Dop-Mod and moreover cofibrant. Then RHom is right adjoint to the derived tensor product ⊗L. Moreover Map(C ,D) is weakly equivalent to the nerve of the subcategory of quasi-equivalences in RHom(C ,D). We will quote further properties of this construction as needed. We will need the following lemma relating the two model structures. The definition of homotopy limits will be recalled in Section 2.1.4 Lemma 2.1. Fibrant replacement as a functor from dgCatMor to dgCatDK preserves homotopy limits. Proof. We know that dgCatMor is a left Bousfield localization of dgCatDK , hence the identity is a right Quillen adjunction and its derived functor, given by fibrant replacement, preserves homotopy limits.  13 This means we can compute homotopy limits in dgCatMor by com- puting the homotopy limit of a levelwise Morita-fibrant replacement in dgCatDK . We will abuse notation and write R for the dg-algebra by the same name as well as for the 1-object dg-category with endomorphism space R concentrated in degree 0. Recall that there is a model structure on differential graded algebras over k with unbounded underlying chain complexes, which can be considered as the subcategory of one-object-categories in dgCatDK . Note that all objects of dgCatDK are fibrant and hence dgCatDK is right proper (i.e. pullbacks along fibrations preserve weak equivalences), while dgCatMor as a left Bousfield localization need not be and in fact is not, cf. Example 4.10 of [46]. We will consider the question of left properness (i.e. whether the pushout along a cofibration preserves weak equivalences) in 2.5. Recall also the category sModCatk of categories enriched over simplicial k-modules and the natural Dold–Kan or Dold–Puppe functor DK : dgCat≥0 → sModCat that is defined hom-wise. DK and its left adjoint, normalization, form a Quillen equivalence between non- negatively graded dg-categories and sModCat. For details see section 2.2 of [41] or [45]. Remark 2.4. While we are working with differential graded categories we are facing some technical difficulties for lack of good internal hom- spaces. It would be interesting to know if another model of stable linear (∞, 1)-categories could simplify our treatment. 14 2.1.3. Model V -categories and simplicial resolutions Model categories are naturally models for∞-categories and in fact have a notion of mapping spaces. Even if a model category is not enriched in sSet one can define mapping spaces in Ho(sSet). One way to do this is by defining simplicial resolutions, which we will make extended use of. Let ∆ be the simplex category and consider the constant diagram functor c : M → M ∆op . Then a simplicial resolution M• for M ∈ M is a fibrant replacement for cM in the Reedy model structure onM ∆ op . (For a definition of the Reedy model structure see for example Chapter 15 of [23].) For example, this construction allows us to compute mapping spaces: If cB→ B˜ is a simplicial framing inM ∆op and QA a cofibrant replacement in M then Map(A, B) ' Hom•(QA, B˜) ' R(Hom•(−, c−)), where the right-hand side uses the bifunctor Hom• : M op ×M ∆op → Set∆op that is defined levelwise. The dual notion is a cosimplicial resolution. Recall V is a symmetric monoidal model category if it is both symmetric monoidal and a model category and the structures are compatible, to be precise they satisfy the pushout-product axiom, see Definition 4.2.1 in [25]. This means in particular that tensor and internal Hom give rise to Quillen functors. We then call the adjunction of two variables satisfying the pushout-product axiom a Quillen adjunction of two variables. 15 Similarly a model V -category M is a model category M that is tensored, cotensored and enriched over V such that the pushout product axiom holds. We call a model Ch-category a dg-model category. For example a model sSet-category, better known as a simplicial model category, M consists of the data (M ,Map,⊗,map) where the enrichment Map : M op ×M → sSet, the cotensor (or power) map : sSetop ×M → M and the tensor ⊗ : sSet ×M → M satisfy the obvious adjointness properties (in other words, they form an adjunction of two variables). The pushout-product axiom says that the natural map fg : A ⊗ L qA⊗K B ⊗ K → B ⊗ L is a cofibration if f and g are and is acyclic if f or g is moreover acyclic. While not every model category is simplicial, every homotopy category of a model category is enriched, tensored and cotensored in Ho(sSet). In fact, M can be turned into a simplicial category in the sense that there is an enrichment Map and there are a tensor and cotensor which can be constructed from the simplicial and cosimplicial resolutions. Let a cosimplicial resolution A• ∈ M ∆ and a simplicial set K be given. Consider ∆K, the category of simplices of K, with the natural map u : ∆K → ∆ sending ∆[n] 7→ K to [n]. We define A• ⊗ K = colim∆K An to be the image of A• under colim ◦ u∗ : C ∆ → C ∆K → C . Similarly there is AK which is the image of the simplicial resolution A• ∈ M ∆op under lim ◦ v∗, where v : ∆Kop → ∆op. This can also be written as AK = limn( ∏ Kn An). IfM is a simplicial category one can use (RA)∆ n for An and (QA) ⊗ ∆n for An. 16 Remark 2.5. Note that AK can also be written as a homotopy limit, holim∆Kop An. (The definition of a homotopy limit is recalled below.) This follows for example from Theorem 19.9.1 of [23], the conditions are satisfied by Propositions 15.10.4 and 16.3.12. The functor (A,K) 7→ AK is adjoint to the mapping space construction A, B 7→ Hom(B, A•) ∈ sSet. Similarly (B,K) 7→ B ⊗ K is adjoint to the mapping space construction A, B 7→ Hom(B•, A) ∈ sSet, see Theorem 16.4.2 in [23]. Hence on the level of homotopy categories the two bifunctors together with Map give rise to an adjunction of two variables. This is of course not a Quillen adjunction, but it is sensitive enough to the model structure to allow for certain derived functors. We will quote further results about this construction as needed. 2.1.4. A very short introduction to homotopy limits Ordinary limits in a model category are not very well behaved, in particular they are not invariant under weak equivalence. A much better notion is provided by homotopy limits. Let I be a small category, M a model category and C : I → M a diagram. On the category of diagrams M I we can often define the injective model structure with levelwise weak equivalences and cofibrations. Limits are right adjoints of the constant diagram functor and with the injective model category structure on diagrams they become Quillen adjoints. Then homotopy limits are just their right derived functors. 17 in general, the injective model structure only exists if M is combina- torial (or if the index category is direct) and the dual projective model structure still needs M to be cofibrantly generated (or the index cate- gory to be inverse). But even if we only have M I as a category with weak equivalences, i.e. as a homotopical category, we can still define derived functors, see for example [15]. Simply put, the homotopy limit is the right adjoint to the constant functor Ho(M ) → Ho(M I) on the level of homotopy categories. Note that the derived functors of right Quillen functors preserve homotopy limits, and so do all Quillen equivalences. (The reason for the latter is that Quillen equivalences induce equivalences of the homotopy categories of diagram categories.) To compute homotopy limits explicitly there is a number of formulae available. Let us assume Ci is levelwise fibrant, and replace fibrantly if that is not the case. (Sometimes taking the homotopy limit of a levelwise fibrant replacement is called the corrected homotopy limit.) Then we will use the following, cf. e.g. Definition 19.1.4 in [23]. holim i Ci = eq ∏ i∈I CN(I↓i)i ⇒ ∏ h→ j CN(I↓h)j  Here we use a simplicial cotensor defined using a simplicial frame (Ci)• for Ci. Since Ci is assumed fibrant this is just a simplicial resolution such that Ci → (Ci)0 is an isomorphism, and since the construction is invariant under weak equivalences between fibrant objects we can take any simplicial resolution. 18 By contrast, for a much more computable example, let us consider a homotopy pullback. It is provided by replacing the target fibrantly and both maps by fibrations before taking the limit, i.e. holim(A→ B← C) ' lim(A˜→ B˜← C˜) where A˜ → B˜ and C˜ → B˜ are fibrations and A˜ ' A etc. If the model category is right proper, i.e. pushout along fibrations preserves weak equivalences, it suffices to replace one map by a fibration. Similarly we construct homotopy ends of bifunctors. Recall that an end is a particular kind of limit. Let α(I) denote the twisted arrow category of I: Objects are arrows, f : i → j, and morphisms are opposites of factorizations, i.e. ( f : i → j) ⇒ (g : i′ → j′) consists of maps i′ → i and j → j′ such that their obvious composition with f equals g. Then there are natural maps s and t (for source and target) from α(I) to Iop and I respectively. For a bifunctor F : Iop × I → C one defines the end∫ i F(i, i) to be limα(I)(s × t)∗F. Then the homotopy end is:∫ h i F(i, i) B holim α(I) (s × t)∗F Details on this view on homotopy ends can be found (dually) in [27]. The canonical example for an end is that natural transformations from F to G can be computed as ∫ A Hom(FA,GA). A similar example of the use of homotopy ends is provided by the computation of mapping spaces in the diagram category of a model category Map(A•, B•) '∫ h i Map(Ai, Bi). The case of simplicial sets is dealt with in [17]. 19 In general, we have the following lemma. Assume M I exists with the injective model structure and let Q and R denote cofibrant and fibrant replacement in this model category. Lemma 2.2. Consider a right Quillen functor H : M op ×M → V . Then there is a natural Quillen functor (F,G) → ∫ i H(Fi,Gi) from (M I)op ×M I to V whose derived functor is (F,G) 7→ ∫ i H(QFi,RGi) which is weakly equivalent to (F,G) 7→ ∫ h i RH(Fi,Gi) Proof. The V -structure exists by standard results in [30]. We have a model V -category by Lemma 2.3 below. Hence the derived functor is (F,G) 7→ ∫ i H(QFi,RGi). On the other hand ∫ i H(Fi,Gi) is the composition of levelwise hom- spaces with the limit, lim ◦ Hα(I) ◦ (s × t)∗ : (M I)op ×M I → (M op ×M )α(I) → V α(I) → V But then the derived functor is the composition of derived functors,∫ h i RH(Fi,Gi). This is a little subtle, since we do not want to fibrantly replace at the level of diagram categories. However, levelwise RH from (M op)I ×M I to V α(I) is a derived functor. This is the case since levelwise fibrant replacement gives a right deformation retract in the sense of 40.1 in [15] 20 since (s× t)∗ preserves all weak equivalences and levelwise H preserves weak equivalences between levelwise fibrant objects.  Remark 2.6. A slight modification of the lemma implies the formula for mapping spaces. We just have to replace H by Hom• : M op ×M ∆op → sSet and adjust the proof accordingly. Lemma 2.3. M I is a model V -category ifM is. Proof. Given cofibrations f : V → W in V and g : A → B in M I we have to show that whenever f and g are cofibrations then so is fg : (V ⊗ B) qV⊗A (W ⊗ A)→ W ⊗ B and fg is an acyclic cofibration if f or g is. Cofibrations and weak equivalences inM I are defined levelwise so it is enough to check ( fg)i. Colimits in diagram categories are also defined levelwise, so it is enough to check f(gi). But by assumption M is a model V -category.  These results generalise verbatim to categories of presections which will be defined in Section 3.1. Homotopy colimits etc. can be defined entirely dually. 21 2.2. Further properties of dgCat In this section we will show that dgCatMor is cellular and, if k is a field, left proper. This will be used to localize diagrams of dg-categories. It may also be of independent interest. Proposition 2.4. The categories dgCatDK and dgCatMor and their small diagram categories are cellular. Proof. Recall that a model category is cellular if it is cofibrantly gener- ated with generating cofibrations I and generating trivial cofibrations J such that the domains and codomains of the elements of I are compact, the domains of the elements of J are small relative to I and the cofibra- tions are effective monomorphisms. See Chapter 10 of [23] for precise definitions. Left Bousfield localization preserves being cellular, and so does taking the category of diagrams indexed by a small category I with the projective model structure, see Theorem 4.1.1 and Proposition 12.1.5 of [23]. So it is enough to show dgCatDK is cellular. The domains and codomains of elements of I are categories with at most two objects and perfect hom-spaces, so maps from these objects to relative I-complexes factor through small subcomplexes. So domains and codomains of I are compact. Similarly the domains of the elements of J have two objects and perfect hom-spaces. Hence taking maps from a domain of J commutes with filtered colimits. So domains of J are small relative to I. 22 We are left to check that relative I-cell complexes, i.e. transfinite compositions of pushouts of generating cofibrations, are effective monomorphisms, i.e. any relative I-cell complex f : X → Y is the equalizer of Y ⇒ Y qX Y . Note that we form the pushout along a generating cofibration by attaching maps freely. If we form C ′ and C ′′ from C by attaching maps freely then the equalizer will have the same objects and the hom-spaces are given by considering morphisms of the pushout that are in the image of both C ′ and C ′′. But these are precisely the hom-spaces of C .  Proposition 2.5. If k is a field the categories dgCatDK and dgCatMor and their small diagram categories are left proper.1 Proof. Left Bousfield localization preserves left properness, see Propo- sition 3.4.4 of [23], and so does taking the category of diagrams indexed by a small category I (with the injective or projective model structure) since pushout and pullback are constructed levelwise. So it is enough to show dgCatDK is left proper. The main work is showing that pushout along the generating cofibra- tions preserves quasi-equivalences. To see this suffices note firstly that transfinite compositions are just filtered colimits, and filtered colimits preserve quasi-equivalences as follows: A filtered colimit of categories can be computed set- theoretically on objects and morphisms. Now filtered colimits preserve weak equivalences of simplicial sets (since S n is compact) and hence 1Thanks to Jon Pridham for helpful discussions about this result. 23 of mapping spaces. They also preserve the homotopy category since a filtered colimit of equivalences of categories is an equivalence of categories and taking the homotopy category commutes with filtered colimits. Secondly, if pushout along some map preserves weak equivalences then so does pushout along a retract by functoriality of colimits. Since all cofibrations are retracts of transfinite compositions of generating cofibrations, it does indeed suffice to check generating cofibrations. It is clear that pushout along ∅ → k preserves quasi-equivalences. So consider the generating cofibration S (n) → D(n) with a map j : S (n) → C and a quasi-equivalence F : C → E . Let the objects of S (n) be denoted a, b and the non-identity morphism g. Then in forming the pushforward we adjoin a new map f with d f = j(g). We call the resulting category C ′. Then let E ′ be the pushout of S (n) → D(n) along F ◦ j. The pushout along j has the same objects as C . The morphism space is obtained by collecting maps from C to D, graded by how often they factor through j(a) → j(b). Write C (A, B) etc. for the enriched hom-spaces HomC (A, B) etc. Then the hom-spaces in C ′ are given as follows: C ′(C,D) = Tot⊕ ( C (C,D)⊕ (C ( j(b),D)⊗ k. f ⊗T ⊗C (C, j(a)))) (2.1) Here T = ∑ n(C ( j(b), j(a)) ⊗ k. f )⊗n and we introduce a horizontal degree n with C (C,D) in degree −1. The right hand side has a vertical differential given by the internal differential and a horizontal differential 24 given by f 7→ j(g) ∈ Hom( j(b), j(a)) composed with the necessary compositions. If the functor F is not the identity on objects from C to E we factor F = Q ◦ H : C → D → E where D has as objects the objects of C but HomD (A, B) = HomE (FA, FB). Then H is identity on objects and Q is an isomorphism on hom-spaces. We form the pushforward and obtain the factorization F′ = Q′ ◦ H′ through D ′. So it suffices to prove the following two lemmas.  Lemma 2.6. Q′ defined as above is a quasi-isomorphism if Q is. Proof. Q′ is quasi-essentially surjective if Q is since both D → D ′ and E → E ′ are essentially surjective as pushout along j does not change the set of objects. We filter the formula 2.1 for hom-spaces in E ′ by columns, i.e. by n. This filtration is bounded below and exhaustive for the direct sum total complex and hence the spectral sequence converges. Now we use the fact that Q induces isomorphisms on hom-spaces to obtain an isomorphism of spectral sequences, which in turn induces an isomorphism D ′(C,D)  E ′(QC,QD), so Q′ gives weak equivalences on mapping spaces. Note that since S (n) maps to E via D all the hom-spaces involved in computing E ′(QC,QD) are indeed images of hom-spaces in D .  Lemma 2.7. H′ defined as above is a quasi-isomorphism if H is. 25 Proof. H′ is quasi-essentially surjective if H is for the same reason that Q′ is. To consider the effect of H′ on mapping spaces proceed as in the previous lemma. Now H only induces weak equivalences on hom-spaces, but we know all hom-spaces are flat over k as k is a field. Hence the tensor product in 2.1 preserves weak equivalences. So the same spectral sequence argument applies.  Remark 2.7. Note that Dwyer and Kan prove left properness for simplicial categories on a fixed set of objects in [16]. Remark 2.8. If we have some non-flat hom-spaces then things go wrong even in the subcategory of dg-algebras. Consider Example 2.11 in [37]. In that paper the existence of a proper model for simplicial k-algebras is proven. A similar result for dg-categories may or may not be true, but is certainly beyond the scope of this work Hence the DK-model category of dg-categories is only left proper if all dg-categories are k-flat, i.e. if and only if k has flat dimension 0, which is the case if and only if k has no nilpotents and Krull dimension 0. In practice we may as well assume k is a field. See Section 2.5 for how to interpret some of the results in the remainder of this chapter for more general k. 26 2.3. Cohomology of presheaves of model categories Now we will define what we mean by cohomology of a sheaf with coefficients in a model category. Let us assumeM is a cellular and left proper model category. The case we are interested in isM = dgCatMor. We will consider the category M J of presheaves on a category Jop with values in a model category M . We will denote by M Jpro j the projective model structure on M J with levelwise weak equivalences and fibrations, and whose cofibrations are defined by the lifting property. IfM is cofibrantly generated this is well-known to be a model structure, which is cellular and left proper ifM J is. We are interested in enriching the model category M Jpro j. Let us start by recalling the case where the construction is straightforward. Let V be a monoidal model category and assume that it has a cofibrant unit. V = sSet,Ch are examples. Then if M is a model V -category, then so is M J. In particular if M is monoidal then M J is a model M -category. We can write Hom for the enriched hom-spaces, and the functor Hom : (M J)op × M J → M is right Quillen and there is a pleasant derived functor RHom obtained by fibrant and cofibrant replacement, see Lemma 2.2 If M is monoidal and a model category, but not a monoidal model category, then we can still construct an M -enrichment of M Jpro j as a plain category, which will of course not be a model category enrichment. We define HomMJ (A, B) = ∫ j Hom(A( j), B( j)), see [30]. 27 Note that this enrichment is not in general derivable, i.e. weak equivalences between cofibrant and fibrant pairs of objects do not necessarily go to weak equivalences. So defining a suitable substitute for RHom takes some care, see the proof of Lemma 2.9. We have to consider this case since our example of interest is M = dgCat, which a symmetric monoidal category and a model category, see [49], but not a symmetric monoidal model category. (The tensor product of two cofibrant objects need not be cofibrant.) Now fix a locally contractible topological space X, for example a CW complex, and consider presheaves on Op(X). We consider the Grothendieck topology induced by the usual topology on X and write the site as (SetOp(X) op , τ). In other words τ is just the collection of maps represented by open covers. (We will not use any more general Grothendieck topologies or sites.) We let J = Op(X)op. Our aim is to localize presheaves on Op(X) with respect to covers in τ. Recall that a left Bousfield localization of a model category N at a set of maps S is a left Quillen functor N → NS that is initial among left Quillen functors sending the elements of S to isomorphisms in the homotopy category. We need to know that left Bousfield localizations of M I exist. There are two general existence results: If M is combinatorial and left proper or if M is cellular and left proper. We have shown that diagrams in dgCat satisfy the latter condition in Section 2.2. Remark 2.9. In the appendix we prove that there is an equivalent subcategory of dgCatDK that is combinatorial, see Section A.1. Then 28 the existence of Bousfield localizations follows from Jeff Smith’s theorem, which is for example proven as Theorem 2.11 in [1]. Lemma 2.8. Assume N is a cellular and left proper model category and let S be a set of maps. Then NS exists. The cofibrations are equal to projective cofibrations, weak equivalences between are S -local weak equivalences and fibrant objects are S -local objects. Proof. This is Theorem 4.1.1 of [23].  Recall for future reference that an object P is S -local if it is fibrant inN and every f : A → B ∈ S induces a weak equivalence MapN (B, P) ' MapN (A, P). A map g : C → D is an S -local weak equivalence if it induces a weak equivalence MapN (D, P) ' MapN (C, P) for every S -local P. Given a set N we write N ·M B qN M ∈M for the tensor over Set and extend this notation to presheaves. Definition. LetM Jτ B (M Jpro j)Hτ denote the left Bousfield localization ofM Jpro j with respect to Hτ = {S · 1M → hW · 1M | S → hW ∈ τ} Here h− denotes the covariant Yoneda embedding X 7→ Hom(−, X). We have assumed M and hence M J is cellular and left proper. Since Hτ is a set the localizationM Jτ exists. 29 We have now localized with respect to Cˇech covers. We are interested in the local model structure which is obtained by localizing at all hypercovers. Remark 2.10. By way of motivation see [12] for the reasons that localizing at hypercovers gives the local model structure on simplicial presheaves, i.e. weak equivalences are precisely stalk-wise weak equivalences. Definition. A hypercover of an open set W ⊂ X is a simplicial presheaf U∗ on the topological space W such that: 1. For all n ≥ 0 the sheaf Un is isomorphic to a disjoint union of a small family of presheaves representable by open subsets of W. We can write Un = qi∈InhU(i)n for a set In where the U (i) n ⊂ W are open. 2. The map U0 → ∗ lives in τ, i.e. the U (i)0 form an open cover of W. 3. For every n ≥ 0 the map Un+1 → (cosknU∗)n+1 lives in τ. Here (cosknU)n+1 = MWn U is the n-th matching object computed in simplicial presheaves over W. Intuitively, the spaces occurring in U1 form a cover for the intersections of the U (i)0 , the spaces in U2 form a cover for the triple intersections of the U (i)1 etc. To every Cˇech cover one naturally associates a hypercover in which all Un+1 → (cosknU∗)n+1 are isomorphisms. Note that despite the notation Un is not an open set but a presheaf on open sets that is a coproduct of representables. 30 We denote by I = ∪In the category indexing the representables making up the hypercover. Associated to any hypercover of a topological space is the simplicial space n 7→ qi∈InU in which is also sometimes called a hypercover. Hypercovers are naturally simplicial presheaves. We work with presheaves with values in a more general model category. The obvious way to associate to a simplicial object in a model category a plain object is to take the homotopy colimit. Definition. Let the set of hypercovers inM J be defined as Hˇτ = {hocolim I (U∗ · 1M )→ hW · 1M | U∗ → hW a hypercover} where we take the levelwise tensor and the homotopy colimit in M J with the projective model structure. Since disjoint union commutes with cofibrant replacement we could equivalently take the limit of Un over ∆op, the opposite of the simplex category. Remark 2.11. Note that the homotopy colimit does not change if instead we use the localised model structureM Jτ . Left Bousfield localization is left Quillen and hence preserves homotopy colimits. Definition. Let the left Bousfield localization ofM Jτ at the hypercovers Hˇτ be denoted byM Jτˇ and call it the local model structure. The localization exists just as before. The fibrant objects are the Hˇτ- local objects ofM J. Note that Hom(hW ,F ) ' F (W) if the model structure on M J is enriched over M . So we sometimes write hypercovers as if they 31 are open sets. For example given a hypercover U∗ and a presheaf F ∈M J we writeF (Un) for Hom(Un,F ) etc. In particularF (Un) = F (qiU (i)n ) B ∏iF (U (i)n ). Definition. We call a presheaf F a sheaf (sometimes called hyper- sheaf ) if it satisfies F (W) ' holim Iop F (U∗) for every hypercover U∗ of every open W ⊂ X (2.2) The limit is over Iop = ∪In; we could write it holimn holimi∈In F (U (i)n ) which can be considered as holimn∈∆F (Un) using the convention above. This condition is also called descent with respect to hypercovers. Remark 2.12. If we give, say, the category of abelian groups the trivial model structure (only identities are weak equivalences and all maps are fibrations and cofibrations) we recover the usual notion of sheaf. Note that in general being a hypersheaf is a stronger condition than being a sheaf of underived objects. For the next Lemma we needM to have a certain homotopy enrichment over itself. For simplicity we specialise toM = dgCatMor. Lemma 2.9. Levelwise fibrant sheaves are fibrant in the above model structure. Proof. We need to show that for a levelwise fibrant presheaf F the sheaf condition on F implies that F is Hˇτ-local, i.e. that whenever  : hocolim(U∗ ·1)→ hW ·1 is in Hˇτ we have Map(hocolim(U∗ ·1),F ) ' 32 Map(hW · 1,F ). We will show that both sides are weakly equivalent to MapdgCatMor (1,F (W)). We need a suitable derived hom-space between sheaves of dg- categories with values in dg-categories. We define RHom′(A•, B•) B∫ h V RHom(AV , BV), where RHom is Toën’s internal derived Hom of dg- categories. First note that RHom′(hW · 1,F ) ' ∫ h V⊂W RHom(1,F (V)) ' holim V⊂W F (V) The first weak equivalence holds since hW(V) is just the indicator function for V ⊂ W and the second since the homotopy end over a bifunctor that is constant in the first variable degenerates to a homotopy limit, by comparing the diagrams. Then we observe holimV⊂W F (V) ' F (W) ifF satisfies the sheaf condition. We claim that this implies Map(hW · 1,F ) ' Map(1,F (W)). Note that in dgCat we have Map(A, B) ' Map(1,RHom(A, B)), see Corollary 6.4 of [48]. Moreover the mapping space in diagram categories is given by a homotopy end, see Lemma 2.2. Putting these together we see Map(A•, B•) = ∫ h V Map(1,RHom(AV , BV)). Then the claim follows since Map(1,−) commutes with homotopy lim- its and hence homotopy ends. Similarly we have Map(hocolim i (Ui · 1),F ) ' holim i Map(1,RHom′(Ui · 1,F )) ' Map(1, holim i holim V⊂Ui F (V)) 33 which is Map(1,F (W)) again by applying the sheaf condition twice.  Remark 2.13. The theory of enriched Bousfield localizations from [1] says that in the right setting M Jτˇ is an enriched model category and fibrant objects are precisely levelwise fibrant sheaves. However, this theory requires that we work with a category M that is tractable, left proper and a symmetric monoidal model category with cofibrant unit. The characterization of fibrant objects in particular depends on the enriched hom-space being a Quillen bifunctor. While dgCatMor is left proper by Proposition 2.5 and equivalent to a combinatorial and tractable subcategory by Section A.1, it is well-known dgCatMor is not symmetric monoidal. Tabuada’s equivalent category Lp of localizing pairs has a derivable internal Hom object, but is not a monoidal model category either. In fact, tensor product with a cofibrant object is not left Quillen. Recall the dg-categoryS (0) that is the linearization of a→ b. The exampleS (0) ⊗S (0) in dgCat gives rise to (∅ ⊂ S (0)) ⊗ (∅ ⊂ S (0)) ' (∅ ⊂ S (0) ⊗S (0)) which is again a tensor product of cofibrant objects that is not cofibrant. Then of course Hom(S (0),−) cannot be Quillen either. Lemma 2.10. Let M = dgCat. Assume that for two presheaves F andF ′ there is a hypercover V∗ on whichF andF ′ agree and which restricts to a hypercover of W for every open W. Then F and F ′ are weakly equivalent inM Jτˇ . 34 Proof. We need to show that there is a Hˇτ-local equivalence between F andF ′, i.e. MapM J (F ,G ) ' MapM J (F ′,G ) for any fibrant G. Specifically, we consider sets V in the hypercover of agreement con- tained in W. Then we know Map(F (V),G (V)) ' Map(F ′(V),G (V)). To compute Map(F ,G ) = ∫ h W Map(F (W),G (W)) note that the homo- topy end can be computed as follows:∫ h W Map(F (W),G (W)) ' ∫ Hom((Q•F )(W),RG (W)) Here we use fibrant replacement and a cosimplicial frame in M J. But now holimV G (V) = limV RG (V) by fibrancy of the diagram RG . So it suffices to consider ∫ W Hom(Q•F(W),RG (W)) where RG (W) = limV⊂W RG (V)). But an end is just given by the collection of all compatible maps, and every map from QiF (W) to RG (W) is determined by the maps from QiF (W) to RG (V), which factor through QiF (V). So the end over the V is the same as the end over all W and Map(F ,G ) ' ∫ V Hom(Q•F (V),RG (V)) ' ∫ V Hom(Q•F ′(V),RG (V)) ' Map(F ′,G ) This completes the proof.  Remark 2.14. If M is a symmetric monoidal model category then by Remark 2.13 fibrant objects are precisely levelwise fibrant sheaves and are again determined on a hypercover and Lemma 2.10 holds again. To compute cohomology we need to compute the derived functor of global sections. First we need to know that pushforward is right Quillen. 35 Lemma 2.11. Consider a map r : C → D of diagrams and a model category M. Then there is a Quillen adjunction r! : MCpro j  M D pro j : r ∗. Proof. We define r∗ by precomposition. Then r! exists as a Kan extension. Clearly r∗ preserves levelwise weak equivalences and fibrations.  Lemma 2.12. Given any map r : (C, τ)→ (D, σ) that preserves covers and hypercovers we get a Quillen adjunction r! : MCτˇ  M D σˇ : r ∗. The same adjunction works if we only localize with respect to Cˇech covers. Proof. To prove the result for the localization with respect to covers we use the universal property of localization applied to the map MC → MD → MDσ which is left Quillen and sends hypercovers to weak equivalences and hence must factor through MC → MCτ in the category of left Quillen functors, giving rise to r! ` r∗. To prove the result for the localization at hypercovers we repeat the same argument for MCτ → MCτˇ etc.  The arguments in the proofs of these two lemmas are Propositions 1.22 and 3.37 in [1]. Consider now locally contractible topological spaces X and Y with sites of open sets (Op(X), τ) and (Op(Y), σ). Given a map f : X → Y consider f −1 : (Op(X), τ) → (Op(Y), σ). Then f∗ B ( f −1)∗ and by the above it is a right Quillen functor. (This is the usual definition of pushforward: f∗F (U) B F ( f −1u).) As usual we write Γ or Γ(X,−) for (piX)∗ where piX : X → ∗. 36 Definition. Let C be a presheaf with values in a model categoryM and let C # be a fibrant replacement for C in the local model category M Jτˇ defined above. Then we define global sections as RΓ(X,C ) = C #(X) In Section 2.4 we will compute C # if C is constant. Since a sheaf satisfies F (X) = holimiF (Ui) for some cover {Ui} we can also think of global section as a suitable homotopy limit. A concise formulation of this is Theorem 2.21. Definition. Consider the presheaf k that is constant with value k ∈ dgCat and let k# be a fibrant replacement for k. Then we define Morita cohomology as RΓMorita(X, k) B RΓ(X,Chpe) = k#(X) in Ho(dgCatMor). We writeH M(X) B RΓMorita(X, k). We will also consider the version with unbounded fibers, RΓMorita(X,Ch). Remark 2.15. As usual RΓ(∅, k) ' 0, the terminal object of dgCat. Remark 2.16. The term cohomology is slightly misleading as our construction corresponds to the underlying complex and not the cohomology groups. The closest analogue to taking cohomology is probably semi-orthogonal decomposition, see for example [8]. 37 2.4. Sheafification of constant presheaves Our aim now is to compute a sheafification of the constant presheaf with values in a model category. We assume X is a locally contractible topological space. Fix a model category M that is cellular and left proper and that is moreover homotopy enriched over itself and has a cofibrant unit. We will also need that the derived internal hom-space commutes with homotopy colimits. The example we care about is M = dgCatMor. The fact that holim RHom(Ai, B) ' RHom(hocolim Ai, B) in dgCat follows from Corollary 6.5 of [48]. The one object dg-category k is a cofibrant unit. We write P for the constant presheaf with fiber P ∈M . First we will need two lemmas about comparing homotopy limits. Given a functor ι : I → J, recall the natural map e j : ( j ↓ ι) → J from the undercategory, sending (i, j→ ι(i)) to ι(i). Lemma 2.13. Let ι : I → J be functor between small categories such that for every j ∈ J the overcategory (ι ↓ j) is nonempty with a contractible nerve and let X : J → M be a diagram. Then the map holimJ X → holimI ι∗X is a weak equivalence. Lemma 2.14. Let ι : I → J be a functor between small categories and let X : J → M a diagram with values in a model category. Suppose that the composition X j → lim ( j↓ι) e∗j(X)→ holim( j↓ι) e ∗ j(X) 38 is a weak equivalence for every j. Then the natural map holimJ X → holimI ι∗X is a weak equivalence. Proofs . For topological spaces these are Theorems 6.12 and 6.14 of [11] and the proofs (in section 9.6) do not depend on the choice of model category.  We will also rely on the following results of [13]. The first statement is Theorem 1.3 and the second is a corollary of Proposition 4.6 as any basis is a complete open cover. Proposition 2.15. Consider a hypercover U∗ → X of a topological space as a simplicial space. Then the maps hocolim U∗ → |U∗| → X are weak equivalences in Top. The colimit here is over the category ∆op, but recall that hocolim∆op Un ' hocolimI U in. Proposition 2.16. Consider a basis U of a topological space X as a simplicial space. Then the map hocolimU∈U U → X is a weak equivalence in Top. Let X be locally contractible. Then we can define the (nonempty) set {Us}s∈S of all bases of contractible sets for X. Definition. Fix a basis of contractible sets Us for X. Let P be a constant presheaf with fiber P ∈M and define a presheafL sP by L sP (U) = holimV⊂U,V∈Us RP(V) 39 where P → RP is a fibrant replacement inM . Denote the natural map by λ : P → L sP . The restriction maps are induced by inclusion of diagrams. This construction proceeds via constructing rather large limits, so even the value ofL s on a contractible set is hard to make explicit. We will mainly be interested inL sk ' L sChpe . The following lemma is the first step towards showing that our construction does indeed give a sheaf. Lemma 2.17. Consider a constant presheaf P with fibrant fiber P ∈M on Op(X). Then on any contractible set U ⊂ Op(X) we have L sP (U) ' P. Proof. Consider U as a category. We need to show that holimUop P ' P. The crucial input is that the weak equivalences V → ∗ give rise to U ' hocolimV⊂U V ' hocolimU ∗ via Proposition 2.16. Now consider any N ∈ M and a cosimplicial resolution N•. Then we have the functor K 7→ N ⊗ K defined in the introduction which is left Quillen, as is shown in Corollary 5.4.4 of [25]. Hence it preserves homotopy colimits and we have: N = N ⊗ hocolim U ∗ ' hocolim U (N ⊗ ∗) = hocolim U N 40 Finally, we use the fact that M has internal hom-spaces. Replace N above be the cofibrant unit. Then we conclude: holim U∈Uop P(U) ' holim Uop RHom(1, P(U)) ' holim Uop RHom(1(U), P) ' RHom(hocolim U 1, P) ' RHom(1, P) ' P In the second line we use the fact that RHom(−, P) sends homotopy colimits to homotopy limits.  Proposition 2.18. For two choices Ut and Us there is a chain of quasi- isomorphisms between L tP and L s P . Hence there is a presheaf LP well defined in the homotopy category. Proof. By considering the union of of Us and Ut it suffices to show the result if Ut is a subcover of Us. By Lemma 2.14 it then suffices to fix Ui ∈ Ut and check that holimi/ι P ' P where ι is the natural inclusion map. But the arrow category stands for the opposite of the category of all the elements of Us contained in Ui. These form a basis and hence the homotopy limit is given be Lemma 2.17.  Proposition 2.19. For any choice of Us the presheaf L sP is fibrant, i.e. it is Hˇ-local. Proof. By Lemma 2.9, it is enough to show L sP is levelwise fibrant (immediate from definition) and satisfies the sheaf condition. Given a hypercover {Wi}i∈I of U we may assume that any element of Us is a subset of one of the Wi. Then we consider for every i the basis 41 of contractibles Us(i) for Wi of elements of Us that are contained in Wi. Then we obtain the following: holim Iop L sP (Wi) ' holimi∈Iop holimU∈Us(i)op P(U)← holimU∈Usop P(U) And our aim is to show the arrow on the right is a weak equivalence. By considering RHom(1⊗hocolim ∗, P) as in the proof of Lemma 2.17 it suffices to show hocolimi∈I hocolimV∈Us(i) V → hocolimU∈Us U is a weak equivalence. But if we apply Proposition 2.16 this is weakly equivalent to hocolimi∈I Wi → X, which is a weak equivalence by Proposition 2.15.  If X is locally contractible then it has a basis of contractible open sets. Moreover one can associate a hypercover to any basis. For details on the construction see Section 4 of [13] and note that a basis is a complete cover. Proposition 2.20. If P is constant then the natural map P → LP is a weak equivalence of presheaves. Proof. To show that L resolves P it is enough to observe that LP(U) 'P for contractible U by Lemma 2.17. Now the contractible opens give rise to a hypercover on which P and LP agree and that restricts to a hypercover on every open set. By Lemma 2.10 that suffices to prove the proposition.  Remark 2.17. An objectF inM Jτˇ is called a homotopy locally constant presheaf if there is a hypercover U∗ such that all restrictions F |U(i)n are weakly equivalent to constant presheaves. 42 IfP is only homotopy locally constant we still have the construction of LP and Proposition 2.20 holds as well as Lemma 2.17 for small enough contractible open sets. We expect that P → LP will still be a fibrant replacement. However, the proof of Proposition 2.19 relies explicitly on the fact that we are considering constant presheaves, so it does not readily adapt to the more general case. With Proposition 2.20 we can compute RΓ(X, P) as LP(X). Note that since we have not used functorial factorization this is not a functor on the level of model categories but only on the level of homotopy categories. Definition. We will call a cover in (SetOp(X) op , τ) a good cover if all its elements and all their finite intersections are contractible. Correspondingly a good hypercover is a hypercover such that all its open sets U (i)n are contractible. We will now consider a good hypercover {Ui}i∈I . For computations it is easier not to consider the full simplicial presheaf given by open sets in the cover but only the semi-simplicial diagram of nondegenerate open sets, i.e. leaving out identity inclusions. This becomes particularly relevant when we consider locally finite covers in the Chapter 3. Theorem 2.21. Let U∗ → hX be a good hypercover of a topological space X. Let P be a constant presheaf on X. Then RΓ(X, P) = holimIop0 P where I0 indexes the distinct contractible sets of U∗. 43 Proof. We consider a fibrant replacement LP as in Definition 2.4. Let I index the connected open sets of U∗. Then we have: RΓ(X, P) ' LP(X) ' holimLP(U∗) ' holim Iop LP(U (i)n ) ' holimIop P Here we use Lemma 2.17 to identify LP(U (i) n ) and P. Now consider ι : Iop0 ⊂ Iop and note that all the overcategories ι ↓ i are trivial (any i ∈ I is isomorphic to some j ∈ I0) so by Lemma 2.13 we have RΓ(X,U) ' holim Iop0 P  Remark 2.18. Note that we can of course take the hypercover associated to a Cˇech cover in this theorem. In fact, since we are concerned with locally constant presheaves it makes very little difference whether we use the Cˇech or local model structure for computations. Considering hypercovers simplifies the theory and Cˇech covers make for simpler examples. We conclude this section with some results on functoriality. Lemma 2.22. Let f : X → Y be a continuous map and let PX or Y denote the constant presheaf with fiber P on X or Y. Then RΓ(X, P) ' RΓ(Y,R f∗(P)). Proof. The fact that RΓ◦R f∗ ' RΓ follows immediately from piY,∗ ◦ f∗ = piX,∗ and the fact that all these maps preserve fibrations.  Lemma 2.23. In the setting of the previous lemma there is a functor RΓ(Y, PY)→ RΓ(X, PX). 44 Proof. Γ is a covariant functor. From Lemma 2.22 we have a natural weak equivalence RΓ(Y,R f∗(PX))→ RΓ(X, PX). Let P• →P• be a fibrant replacement. It is then enough to construct a map f • : PY → R f∗(PX) of sheaves on Y . On any open set U this is given by PY(U) = P→ f∗PX(U)→ f∗PX(U).  Remark 2.19. With P = Chpe this gives functoriality for Morita cohomology if we use functorial factorizations. Note that our computation using good covers is not functorial unless we pick compatible covers. However, if X and Y have bases of contractible sets which are suitably compatible we can just write down the comparison map between homotopy limits. 2.5. Morita cohomology over general rings Everything we have done since Section 2.2 was built on the assumption that dgCat is left proper, which is only the case if k is of flat dimension zero, see Remark 2.8. Nevertheless, we can consider the question of what Morita cohomology should be over other ground rings. The obvious way out is to use Γ(X,LChpe) as our definition of Morita cohomology if k has positive flat dimension. All pertinent results in the remainder of the chapter then still apply, in particular Theorem 2.21, and we can prove equivalence with the category of homotopy locally constant sheaves in Chapter 3 and with∞-local systems in Section 4.1. 45 3. Homotopy locally constant sheaves 3.1. Strictification and computation of homotopy limits In this chapter we will show that Morita cohomology of X is equivalent to the category of homotopy locally constant sheaves of perfect complexes on X. We begin in this section with generalities on strictification and the computation of homotopy limits. But first we set up the situation with which we will be concerned. Fix throughout this chapter a topological space X and assume that X has a good hypercover U = {Ui}i∈I satisfying certain finiteness conditions. Specifically we assume that U satisfies the following two conditions, which we sum up by saying U is bounded locally finite. • U is locally finite. (Every point has a neighbourhood meeting only finitely many elements of U.) • There is some positive integer n such that no chain of distinct open sets in U has length greater than n. 46 Remark 3.1. If X is a finite-dimensional CW complex it has a bounded locally finite cover. One can show this by induction on the n-skeleta using collaring, see Lemma 1.1.7 in [20], to extend a bounded locally finite hypercover on Xn to one on a neighbourhood in Xn in Xn+1. Then one extends over the n + 1-cells. Note that by Theorem 2.21 we can compute cohomology as the homotopy limit of a diagram indexed by I0 ⊂ I, the category of non- degenerate objects. In the next section we will use strictification to compute this small homotopy limit explicitly as a category of homotopy cartesian sections. Let us consider the fiber Ch instead of Chpe at first, which has the advantage of being a model category. Model categories are often a convenient model to do computations with ∞-categories. However, as the category of model categories is not itself a model category there exist no homotopy limits of model categories. Instead one can compute categories of homotopy cartesian sections and strictification results compare them to homotopy limits of the∞-categories associated with the model categories in question. Generally speaking, using strictification to compute a homotopy limit proceeds as follows. As ingredients we need some localization functor L : MC → ∞Cat from model categories to some model of (∞, 1)-categories and a method, call it hsect, of computing homotopy cartesian sections of a Quillen presheaf, i.e. of a suitable diagram of model categories. Then given a diagram (Mi) of model categories indexed by I one proves holimi∈Iop LMi ' L hsect(I,Mi). 47 We will proceed by adapting the strictification result for inverse diagrams of simplicial categories from Spitzweck [42] to dg-categories. J is an inverse category if one can associate to every element a non- negative integer, called the degree, and every non-identity morphism lowers degree. This is certainly the case for I0 if U is bounded locally finite. We then have to restrict to compact objects in the fibers to compute RΓ(X,Chpe) rather than RΓ(X,Ch). Remark 3.2. There is a wide range of strictification results in the literature: For simplicial sets [18, 52], simplicial categories [42], Segal categories (Theorem 18.6 of [24]) and complete Segal spaces [4, 5]. Most of the above results make fewer assumptions on the index cate- gory, for example Theorem 18.6 of [24] proves strictification of Segal categories with general Reedy index categories, and a generalization to arbitrary small simplicial index categories is mentioned in Theorem 4.2.1 of [51]. But since it is unclear how to adapt this proof to the dg-setting and since the existence of a bounded locally finite good hy- percover for X does not seem a very restrictive assumption we stay with it. We will deal with model categories that are already enriched in some symmetric monoidal model category V and our ∞-categories will be V -categories. (Think V = sSet or Ch.) Definition. Denote by L the localization functor L : V MC → V Cat that sends M to Mc f , the subcategory of fibrant cofibrant objects of M. 48 The fibrant cofibrant replacement is necessary to ensure that the V -hom spaces are invariant under weak equivalences. In the case V = sSet compare the homotopy equivalence between LM and the Dwyer–Kan localization of M. Let us set up the machinery: Definition. A left Quillen presheaf on a small category I is a contravariant functor M• : I → Cat, written as i 7→ Mi such that for every i ∈ Ob(I) the category Mi is a model category and for every map f : i→ j in I the map f ∗ : M j → Mi is left Quillen. (One can similarly define right Quillen presheaves.) Definition. The constant left Quillen presheaf with fiber M, denoted as M is the Quillen presheaf with Mi = M for all i and f ∗ = 1M for all f . Remark 3.3. One can define Quillen presheaves in terms of pseudofunc- tors instead of functors, see [42]. The complicated definition is in [25]. One then rectifies the pseudofunctor to turn it into a suitable functor, i.e. into a left Quillen presheaf as defined above. Definition. Let M• be a left Quillen presheaf of model categories. We define a left section to be a tuple consisting of (Xi, φ f ) for i ∈ Ob(I) and f ∈ Mor(I) where Xi ∈ Mi and φ f : f ∗X j → Xi, satisfies φg ◦ (g∗φ f ) = φ f◦g : ( f ◦ g)∗Xk → Xi for composable pairs g : Xi → X j, f : X j → Xk. A morphism of sections consists of mi : Xi → Yi in Mi making the obvious diagrams commute. We write the category of sections of 49 M• as psect(I,M•). The levelwise weak equivalences make it into a homotopical category. Definition. A homotopy cartesian section is a section for which all the comparison maps φ f : R f ∗X j → Xi are isomorphisms in Ho(Mi). We write the category of homotopy cartesian sections of M as hsect(I,M•). If I is an inverse category or M is combinatorial then the category of left sections psect(I,M) has an injective model structure, just like a diagram category, in which the weak equivalences and cofibrations are defined levelwise, cf. Theorem 1.32 of [1]. We write L hsect(I,M•) for the subcategory of homotopy coherent sections whose objects are moreover fibrant and cofibrant. Note that hsect(I,M•) is not itself a model category since it is not in general closed under limits. Remark 3.4. One would like homotopy cartesian sections to be the fibrant cofibrant objects in a suitable model structure. If we are working with the projective model structure of right sections then (under reasonable conditions) there exists a Bousfield localization, the so-called homotopy limit structure (cf. Theorem 2.44 of [1]). The objects of L hsectR(I,M) (which are projective fibrant) are precisely the fibrant cofibrant objects of (psectR)holim(I,M). The homotopy limit structure on left sections is subtler. It is the subject matter of [3]. Assuming the category of left sections is a right proper model category Bergner constructs a right Bousfield localization where the cofibrant objects are the homotopy cartesian ones in Theorem 3.2 50 of [5]. Without the hard properness assumption the right Bousfield localization only exists as a right semimodel category, cf. [2]. Note that we will still use model category theory, all we are losing is a conceptually elegant characterization of the subcategory we are interested in. In the remainder of this section we recall the construction of enrichments of presections and presheaves that will be used in the proof of strictification. Assume that V is a symmetric monoidal model category and that the we are given a left Quillen presheaf such that all the Mi are model V - categories. Note that V will be the category Ch in our application. Lemma 3.1. If M• is as above and the comparison functors are V - functors then psect(I,M•) is a model V -category. Proof. Tensor and cotensor can be defined levelwise. We define Hompsect(X•,Y•) as the end ∫ i Hom(Xi,Yi). The same reasoning as in diagram categories applies, see the discussion before Lemma 2.2. Since cofibrations and weak equivalences in psect(I,M•) are defined levelwise the pushout product axiom holds, cf. Lemma 2.3, and we have a model V -structure.  It follows that the derived internal hom-spaces can be computed by cofi- brantly and fibrantly replacing source and target, RHompsect(X•,Y•) =∫ i Hom((QX)i, (RY)i), cf. Lemma 2.2. 51 In particular if all Mi are dg-model categories then psect(I,M) is a dg- model category. Definition. If M is enriched in V let V Psh(M) be the category of V - functors from M to V , i.e. functors such that the induced map on hom- spaces is a morphism in V . V Psh(M) is a model category if V = Ch or if V has cofibrant hom- spaces, see Remark 2.2. Lemma 3.2. V Psh(M) is enriched, tensored and cotensored over V . Proof. We can tensor and cotensor levelwise. For the enrichment we have to define an object in V of V -natural transformations between two V -functors F,G : A → B. Recall that a V -natural transformation is, for every object A ∈ A a morphism 1V → B(FA,GA). And morphism spaces in V live themselves in V so Nat(F,G) is a limit (to be specific, an end) in V . This is of course entirely standard, see Chapter 1 of [30].  Lemma 3.3. There is an enriched Yoneda embedding M → V Psh(M). If V has a cofibrant unit and fibrant hom-spaces then the Yoneda embedding factors through the subcategory of fibrant cofibrant objects. Proof. For the existence of the embedding see 2.35 in [30]. It is clear from the projective model structure that the objects in the image are fibrant. To see the image consists of cofibrations, we just note that the maps 0→ hX ⊗ 1 are generating cofibrations.  The conditions of the lemma are satisfied in Ch. 52 3.2. Strictification for dg-categories Our goal now is to prove the following theorem: Theorem 3.4. Let I be a direct category. Let Mi be a presheaf of model categories enriched in Ch. Then L hsect(I,M•)  holimi∈Iop LMi in Ho(dgCatDK). This theorem allows us to characterize Morita cohomology of X. From Theorem 2.21 we immediately obtain the following: Corollary 3.5. Let {Vi}i∈I be a locally finite good hypercover of X. Then RΓ(X,Chdg) ' L hsect(I0,Ch). We are mainly interested in restricting attention to Chpe. In this case we have to be careful about defining the right hand side. We will consider this situation in Section 3.3. To show Theorem 3.4 we closely follow the method of proof in [42], replacing enrichments in simplicial sets by enrichment in chain complexes wherever appropriate. For easier reference we write in terms of V -categories, where V = Ch for our purposes and V = sSet in [42]. One simplification is that we are assuming the model categories we start with are already enriched in Ch, so that we can use restriction to fibrant cofibrant objects instead of Dwyer–Kan localization as the localization functor. There are two times two steps to the proof: First we define homotopy embeddings ρ1 and ρ2 of the two sides into L psect(I,V PS h(RLM•)). 53 We then show that their images are given by homotopy cartesian section whose objects are in the image of Mi. The first pair of steps are quite formal. The second pair is given by explicit constructions using induction along the degree of the index category. The proof of the strictification result depends on setting up a comparison between the limit construction and presections. Since the fibrant replacement of LM• is not a Quillen presheaf we have to embed everything into a presheaf of enriched model categories. This is achieved by using the Yoneda embedding. We write RLM• for i 7→ (RLM)i, where R stands for fibrant replacement in the injective model structure on diagrams of V -categories and L is taking fibrant cofibrant objects of every Mi. Lemma 3.6. Let Di be an Iop-diagram of V -categories. We have a canonical full V -embedding: ρ2 : holim D• = lim RD• ↪→ L psect(I,V PS h(RD•)) Proof. The map to psect(I,V Psh(D f•)) is obtained by composing the Yoneda embedding with the map of V -categories limi Ci → psect(I,C•) that sends a to {pii(a)} if pii : lim j C j → Ci are the universal maps. (C• is not a model category, but we can still take psect with the obvious meaning, the comparison maps are identities by definition.) Recall that Ob(lim Ci) consists of collections {ci ∈ ObCi} such that C( f )(c j) = ci for f : i → j and Mor(lim Ci) consist of collections {gi ∈ Mor(Ci)} such that C( f ) ◦ gi = g j ◦ C( f ), i.e. the hom-space 54 between {ci} and {di} is given by ∫ i Hom(ci, di). If the Ci are enriched then Homlim Ci(a, b) = ∫ Hom(pii(a), pii(b)) in V . The universal property is clear. Hence the objects are a subset of the objects of presections, namely the ones with identities for comparison maps. By definition of the morphisms in psect(I,D•) we have the following isomorphisms of hom-spaces: Hompsect(c•, d•)  ∫ i HomV Psh(RD•)i(ci, di)  ∫ i HomRDi(ci, di)  Homlim RD•({ci}), {di}) Here we use that the enriched Yoneda embedding is indeed an enriched embedding. We abusively write ci for the image of ci in V Psh(RDi). This proves there is an embedding of the homotopy limit into psect(I, •). To show this embedding factors through fibrant cofibrant objects note first that cofibrations are defined levelwise. For fibrations one uses the fibrancy of RLM•, this is Lemma 6.3 of [42].  It follows from this embedding that homotopy equivalences in the homotopy limit are determined levelwise since in L psect homotopy equivalences are weak equivalences and weak equivalences are defined levelwise. This is Corollary 6.5 in [42]. From now on we will write ρ2 for the case Di = LMi. We also have the following: 55 Lemma 3.7. There is a natural homotopy V -embedding ρ1 : L hsect M• ↪→ L psect(I,V PS h(RLM•)) Proof. We have an embedding hsect ↪→ psect and homotopy embeddings Mi ↪→ V Psh(RLMi) which give a homotopy embedding when we apply L psect(I,−) since the hom-spaces of presections between fibrant cofibrant objects are given by homotopy ends, which are invariant under levelwise weak equivalence.  Remark 3.5. Note that the situation is a little more intricate in [42] where simplicial localization and restriction to fibrant cofibrant objects are a priori distinct and need to be compared through another embedding. Next we have to identify the images of ρ1 and ρ2. The explicit computation is done in Lemma 6.6 of [42]. The only use of special properties of the category sCat made in this lemma (and the results needed for it) is the characterization of fibrations in terms of lifting homotopy equivalences. But this characterization is also valid for fibrations in dgCatDK . We provide the argument here for convenience and future reference. Lemma 3.8. The image of ρ1 consists of homotopy cartesian sections X• ∈ L psect(I,V Psh(RLM•)) such that all Xi are in the image of Mi. Proof. We proceed by induction on the degree of the indexing category. Let X• be given as in the statement and assume by induction we have a 56 levelwise equivalence Y 1. In other words S (n − 1) provides a model for singular chains on ΩS n equipped with the Pontryagin product. This is of course well-known: H∗(ΩX) is a polynomial algebra on a generator in degree n − 1. A computation via the James construction is in 3.C and 4.J of [22]. So there is a natural map S (n− 1)  H∗(ΩS n)→ NΩS n which is a quasi-isomorphism. (We will see in Example 5.2 how to show directly that S (n − 1) ' NΩS n.) We also need to know that there is a map D(n) → NΩBn compatible with S (n − 1) → D(n). This follows by the lifting property of the cofibration S (n − 1) → D(n) with respect to the trivial fibration NΩBn → ∗. These are the building blocks needed to associate to any connected CW-complex X without 1-cells a dg-algebraB(X) quasi-isomorphic to C∗(ΩX) that approximates the way X is glued from cells. Lemma 4.10. The inclusion of dg-algebras as dg-categories with one object preserves pushouts. Proof. Constructing the pushout of dg-categories with one object we obtain a dg-category with a single object. Furthermore maps from a one-object dg-category to other dg-categories are just maps from the dg-algebra of endomorphism to the dg-algebra of endomorphisms of the image.  74 In the proof of the next theorem we need to compute some homotopy pushouts. We assume k is a field so that the model category dgAlgk is proper and we can compute homotopy pushouts as pushouts whenever one of the constituent maps is a cofibration. Theorem 4.11. Assume k is a field. Associated to every connected CW complex X with cells in dimension ≥ 2 there is a cofibrant dg-algebra B(X) with one generator in degree n − 1 for every n-cell, that is quasi- equivalent to N(ΩX). In particular Y (X) ' ChB(X). Proof. We proceed by induction on the cells, let α be the index. We begin with the 2-skeleton of X which by assumption is a wedge of s spheres. Since the derived versions of N, k⊗−, G and Sing* all preserve homotopy colimits, so does NΩ on cofibrant spaces in CGHauss. This gives NΩ( ∨ s S 2) ' ⊗sk[x1], i.e. a free dg-algebra on s generators in degree 1. We defineB(X2) B ⊗sk[x1]. From here on we can compute N(ΩX) inductively by forming, for every pushout X≤α = colim(en+1α ← S n → X<α), the diagram of dg-categories N(ΩBn+1) ← N(ΩS n) → N(ΩX<α). Our aim is to show the pushout of the second diagram is weakly equivalent to N(ΩX≤α). Since CGHauss and dgCatDK are proper the pushouts of both diagrams are homotopy pushouts and hence matched up by NΩ. Hence N(ΩX≤α) is determined by the map N(ΩS n) → N(ΩX<α). Since the left-hand side is weakly equivalent to a free dg-algebra on a single generator in degree n − 1 it suffices to specify its image, which is a homology class on the right-hand side, i.e. an element of Hn−1(ΩX<α). 75 Inductively assume there is a weak equivalence B(X<α) → N(ΩX<α). In particular there is an isomorphism on Hn−1 and hence there is a map S (n − 1) → B(X<α) corresponding to the attachment map S n → X<α. This gives a weak equivalence between pushout diagrams. We define B(X≤α) to be colim(D(n) ← S (n − 1) → B(X<α)). By the previous lemma the pushout computed in dgAlg is the same as the one in dgCat. Since the pushouts are homotopy pushouts of diagrams which are levelwise weakly equivalent they agree up to homotopy and we have B(X≤α) ' N(ΩX≤α). To extend to infinite CW-complexes we have to check the same argument goes through for filtered colimits. Since the maps X<α → X≤α are cofibrations the filtered colimit is a homotopy colimit and commutes with NΩ. So NΩX≤λ ' hocolimα<λ NΩXα and we can defineB(X≤λ) as colimα<λB(X≤α).  Proposition 4.12. Let k be a commutative ring of characteristic 0. Assume the CW complex X as above is such that all attachment maps are cofibrations. ThenB(X) constructed as above is weakly equivalent to NΩ(X). Proof. The only place we used that k is a field was in asserting that pushouts are homotopy pushouts if one of the maps is a cofibration. With our new assumption both N(ΩBn) → N(ΩS n) and N(ΩS n) → N(ΩXα) are cofibrations, so the pushout is a homotopy pushout without assuming properness.  76 Remark 4.6. To use this computation in practice we need to identify the degree n−1 element y ofB(X<α) that corresponds to the image of S n−1. Then we adjoin a new generator x with dx = y. This can of course be quite non-trivial. There are some examples in the next section. Next we deal with the case of 1-cells. Remark 4.7. The main difficulty in considering ChS 1 arises as follows. It is clear that C∗(ΩS 1) ' k[Z]. However, k[Z] is not a cofibrant dg- algebra, so cannot be used for computing homotopy pushouts. Theorem 4.13. Associated to every connected CW complex X there is a dg-algebra B(X) with one generator in degree n − 1 for every n-cell with n ≥ 2, and with two inverse generators in degree 0 for every 1-cell, such that Y (X) ' ChB(X). Proof. Let us define S ∗(0) = k[a, a−1] and D∗(1) = k[a, a−1, b 7→ a − 1] and consider the cofibration S ∗(0) ↪→ D∗(0). Of course D∗(0) ' k. Then we have compatible quasi-isomorphisms NΩS 1 → S ∗(0) and NΩB2 → D∗(1). The first is induced by projection to connected components G Sing* S 1 → Z, the second map exists since D∗(1)→ 0 is a trivial fibration and NΩS 1 → NΩB2 is a cofibration. Let X1 be the 1-skeleton of X and defineB(X1) = B( ∨ s S 1) := ⊗sS ∗(0) which is weakly equivalent to C∗(Ω( ∨ s S 1)) There is an obvious map from S ∗(0) to B(X1) for any attachment map S 1 → X1. Assume first that X is obtained from X1 by attaching a 2-cell. Then we define B(X) = colim (D∗(1)← S ∗(0)→ B(X1)) 77 Now Y (X) is the homotopy pullback of Y (B2) ← Y (S 1) → Y (X1). But this diagram is weakly equivalent to ChNΩB 2 → ChNΩS 1 ← ChNΩX1 which is in turn weakly equivalent to ChD ∗(1) → ChS ∗(0) ← ChB(X1). These are all pullback diagrams of fibrant objects with one map a fibration, hence they are homotopy pullbacks as dgCatDK is right proper. Since the diagrams are levelwise quasi-equivalent their pullbacks are quasi-equivalent, and thus also isomorphic in Ho(dgCatMor). But since D 7→ ChD sends colimits to limits it also follows that Y (X) ' holim ( Y (B2)→ Y (S 1)← Y (X1) ) ' holim ( ChD ∗(1) → ChS ∗(0) ← ChB(X1) ) ' lim ( ChD ∗(1) → ChS ∗(0) ← ChB(X1) ) ' Chcolim(D∗(1)←S ∗(0)→B(X1)) The colimit in the exponent is how we have definedB(X). Now consider the general case. First to obtain B(X2) note that any attachment map from S 1 factors through X1, so we can repeat the previous step as often as required. Attachment of higher-dimensional cells works in exactly the same manner, we just have to replace S ∗(0) by S (n − 1) and D∗(1) by D(n). The argument extends to filtered colimits just like in the proof of Theorem 4.11.  Remark 4.8. By constructionB(X) is Morita-equivalent to NΩX, but it does not follow from the construction whether the two dg-algebras are isomorphic in Ho(dgAlg). 78 4.4. Finiteness and Hochschild homology In this section we consider conditions for Morita cohomology to satisfy various finiteness properties, and determine Hochschild homology in some cases by quoting relevant results from the literature. Let us first make some definitions. Here R denotes fibrant replacement in dgCatMor. Specifically, RB = L(Bop-Mod)pe. We say a dg-category D is locally proper if the hom-space between any two objects is a perfect complex. D is proper if moreover the triangulated category H0(RD) has a compact generator, i.e. a compact object which detects all objects. Recall an object X in a model category is homotopically finitely presented if Map(X,−) commutes with filtered colimits. D is smooth if it is homotopically finitely presented as a Dop ⊗ D-module. D is saturated if it is smooth, proper and Morita fibrant. D is of finite type if there is a homotopically finitely presented dg- algebra B such that RD ' R(Bop). These definitions are Morita-invariant (except for the condition of being Morita fibrant). Toën shows in Lemma 2.6 of [50] that a dg-category has a compact generator if and only if RD ' RBop for some dg-algebra B and is moreover proper if and only if the underlying complex of B is perfect. Moreover any dg-category of finite type is smooth (Proposition 2.14 of [50]). 79 Remark 4.9. One reason to be interested in these finiteness conditions is that if a dg-category is saturated there is a nice moduli stack of objects, this is the main result of [50]. Proposition 4.14. Y u(X) is triangulated and has a compact generator. If X is a finite CW-complex without 1-cells then Y u is smooth. If moreover H∗(ΩX) is of finite type then Y u(X) is saturated. Proof. Note first that as a homotopy limit Y u(X) is fibrant and the compact generator is given by C∗(ΩX). Theorem 4.13 implies that in the absence of 1-cells the dg-algebraB(X) is homotopically finitely presented. So the category Y u(X) is of finite type and hence smooth. If H∗(ΩX) is of finite type, then B(X) is a perfect complex over k, and Y u is moreover proper and we find that Y u(X) is saturated.  By contrast if X is an infinite CW-complex then B(X) is usually not homotopically finitely presented. For example consider B(CP∞) ' k[x1]/(x21). Any cofibrant replacement as infinitely many generators and hence the identity does not factor through any subobject of finite type. Next we consider properness for Y (X). The categoryH M(X) is locally proper if all cohomology groups of X with coefficients in local systems are finite dimensional and concentrated in finitely many degree. This is for example the case if X has a finite good cover. Then the hom-spaces are finite limits of perfect chain complexes. 80 This is in contrast to Ext-groups of local systems which can be large even if X is very well behaved, for example if X is a smooth projective variety [10]. The example X = S 1 shows that we cannot expect Y (X) to be proper in general. ChS 1 is the category of complexes of Z-representations, with infinitely many connected components, see Example 5.1. Proposition 4.15. If pi1(X) has only finitely many irreducible repre- sentations which are all finite dimensional then there exists a com- pact generator A and Y (X) ' L(End(A)op-Mod)pe. Y (X) is proper if C∗(X,End(A)) is a perfect complex. Proof. We define A to be the sum of all the irreducibles. Then A maps to the lowest nontrivial homology group of any object in Y (X) and hence generates the dg-category since objects with trivial homology are quasi-isomorphic to 0. By Lemma 2.6 of [50] L(Y (X)op-Mod) ' L(EndY (X)(A)op-Mod). Since Y (X) ' L(Y (X)op-Mod)pe we deduce that Y (X) is the subcategory of compact objects in End(A)-Mod. The second statement is clear.  The proposition applies for example if the fundamental group is finite. Then we can take A to be the group ring. Example 4.2. Let X be simply connected. Then we can take A = k and find End(A) ' RHom ΩX(k, k) ' C∗(X, k) by earlier results. In particular Y (X) ' C∗(X, k) in dgCatMor. Then Y (X) is proper if and only 81 if C∗(X, k) is a perfect complex. If C∗(X, k) is homotopically finitely presented then Y (X) is moreover smooth and saturated. If Y (X) has a compact generator it becomes much easier to compute secondary invariants. In particular we can compute Hochschild homology and cohomology. For definitions and a summary of results see [29]. Since Hochschild homology and cohomology are Morita- invariant we can compute them on a generator of a dg-category if there is one. Example 4.2 implies the following proposition. Here HH stands for either HH∗ or HH∗. Proposition 4.16. Let X be simply connected then HH(Y (X))  HH(C∗(X)). So we can compute Hochschild (co)homology of Morita cohomology from minimal models (in the sense of Sullivan). Proposition 4.17. HH(Y u(X))  HH(C∗(ΩX))  HH(B(X)). Proof. The second isomorphism follows since Hochschild (co)homology is Morita-invariant.  The following applications follows from results readily available in the literature. Proposition 4.18. Let X be simply connected then HH∗(Y (X))  H∗(L X). If M is a simply connected closed oriented manifold of 82 dimension d then HH∗(Y (M))  H∗+d(LM) as graded algebras with the Chas-Sullivan product on the right hand side. Proof. If X is simply connected it is well known (see [32]) that HH∗(C∗(X, k))  H∗(L X) whereL X is the free loop space. The second part follows since the Hochschild cohomology ring of singular cochains on M (with the cup product) is isomorphic to its loop homology with the Chas-Sullivan product, cf. [9].  Proposition 4.19. HH∗(Y u(X))  H∗(L X). If X is simply connected HH∗(Y u(K))  H∗(L X) as graded algebras. Proof. We find HH∗(Y u(X))  HH∗Ω(X)  H∗(L X) from 7.3.14 in [31]. For a the result that HH∗ Sing* ΩX  H ∗(L X) (as graded algebras) for a simply connected CW-complex X, see [34].  Note that we do not expect Hochschild homology of Y (X) to be particularly tractable if X is not simply connected. For example Y (S 1) has |k∗| simple objects with no morphisms between them, cf. Example 5.1. Hence it follows from the explicit definition in [29] that Hochschild homology consists of |k∗| copies of HH∗(k[y]) where y lives in degree 1 and has square 0. 83 5. Computation and Examples In this chapter we compute some examples of Morita cohomology. In the following whenever an element has a subscript, this will denote its degree. For simplicity we will often not mention the restriction to fibrant cofibrant objects in representations of C∗(ΩX) orB(X). 5.1. Spheres Example 5.1. We begin with the case X = S 1. Clearly H M(S 1) is equivalent to the category of representations of Z ' ΩS 1. We can also view this as the category of bounded chain complexes of local systems on S 1 since S 1 is K(Z, 1) and the cohomological dimension of Z is 1, so that all non-trivial extensions in chain complexes of Z-modules are extensions in the abelian category of Z-modules. We can also characterizeH M(S 1) as the explicit limit (Chpe)I ×Chpe×Chpe Chpe Here ChpeI is the path object in dg-categories, which is (Chpe)1 in the simplicial resolution constructed in the appendix, see Example A.2. 84 The limit then comes out as the category of pairs (M, φ ∈ Aut(M)) with morphisms ( f , g, h) : (M, φ) → (N, ψ) in Hom(M,N)⊕2 ⊕ Hom(M,N)[−1] with differential ( f , g, h) 7→ (d f , dg, dh − (−1)|g|gφ + ψ f ) In particular Hom∗(k, k)  k⊕ k[1], which is exactly cohomology of S 1, as predicted. We can also compute H M(S 1) with a Cˇech cover. Using a good cover by three open sets and their intersections and an explicit model for Ch1, see above, we find that an object is given by three chain complexes with weak equivalences between them. We can use two weak equivalences to identify the complexes and are left with a single homotopy invertible map. As we have stated before, the categoryH M(S 1) is highly disconnected, in fact isomorphism classes of simple objects are naturally in bijection with k∗. Of course k∗ has a geometric structure, and one way of interpreting large sets of isomorphism classes of objects is to consider a moduli stack of objects of H M(X). We will not follow this direction here. Example 5.2. If n > 1 then H M(S n) ' ChpeS (n), i.e. the category of perfect chain complexes with an endomorphism in degree n− 1 that are fibrant and cofibrant as such modules. Proof 1. This is immediate from the quasi-isomorphism S (n) → N(Ω Sing* S n) which was mentioned in the last section.  85 Proof 2. We can also compute B(S 2) using the method of Theorem 4.13 by gluing two copies of B2 along S 1. The resulting dg-algebra has one invertible generator with two trivialising homotopies, which is quasi-isomorphic to k[x1] = S (1). Once we know the case n = 2 we can inductively compute S n = Dn qS n−1 Dn and note that S (n) ' D(n) ⊗LS (n) D(n). Note that we can use this construction ofB(S n) in the proof of Theorem 4.13. There is no circularity as we only need a model for spheres in smaller dimensions to computeB(S n).  Example 5.3. Next consider some more detail for n = 2. Since k is a generator and REndC∗(ΩS 2) ' C∗(S 2) ' k[x2, x3 d→ x22] =: A we can characterize Y (S 2) as compact objects in A-Mod. An example of an object of RΓMorita(S 2, k) is the chain complex associated to the Hopf fibration p : S 3 → S 2. As a homotopy locally constant sheaf we can consider this as Rp∗ Sing*(S 3). As a representation of ΩS 2 this can be written as k⊕k[−1] with the canonical map of degree 1. Since pi1(S 2) is trivial, we can also view RΓMor(S 2, k) as generated by the trivial local system and the information H∗(S 2,−) provides about (iterated) extensions. This provides a slightly different viewpoint on Morita cohomology. Specifically, consider the forgetful map D → Ch. The objects in the fibre over M  ⊕ Mi[−i] are all the homotopy locally constant sheaves with homology M. They can be determined 86 iteratively. For example, over M0 ⊕ M1[−1] the fiber is parametrized by C∗(X,Hom1(M1[−1],M0). 5.2. Other topological spaces Example 5.4. H M(BG) is just the dg-category of perfect complexes with an action of G. Example 5.5. RΓ(RP2,Chpe) is given by representations of B(RP2) on perfect complexes, and B(RP2) has generators a0, a−10 , b1 such that db1 = a0 ◦ a0 − 1. This follows from Theorem 4.13. The identification db1 = a0 ◦ a0 − 1 is induced by the attaching map from the boundary of the 2-cell to RP1. If we are working over the field Q Morita cohomology has certain similarities to rational homotopy theory, cf. the duality between C∗(ΩX) and C∗(X) in the simply connected case. On the other hand we see that RP2 has trivial minimal model, but its Morita cohomology is a dg- category with two simple objects corresponding to the irreducible reps of Z/2. We can obtainB(RP3) fromB(RP2) by adding c2 with dc2 = 0. Example 5.6. Next we compute the map p∗ : H M(S 2) → H M(S 3) induced by the Hopf fibration. On the level of loop spaces we see that the map is induced by Ωp∗ : H∗(ΩS 3) → H∗(ΩS 2) which is given by x2 7→ y21 on the generators. 87 With this in mind we can work outH M(CP2) explicitly by considering the following diagram: H M(B4) i∗−→H M(S 3) p ∗ ←−H M(CP1) On the level of dg-algebras we have D(3) i∗←− S (3) p∗−→ B(S 2)  S (2) The attaching map p∗, is induced by the Hopf fibration. As we have just seen it corresponds to the map H∗(ΩS 3) → H∗(ΩS 2) given by sending x2 7→ y21. Hence we find: B(CP2) ' k[α1, α3 | dα3 = α21] Example 5.7. We can generalise this to CPn, every extension over a 2i-cell corresponding to another map α2i−1 in degree 2i − 1. We find d : α3 7→ α21;α5 7→ α3α1 + α1α3;α7 7→ α5α1 + α23 + α1α5 etc. Let us compare this with a computation of H∗(ΩCPn), which is done e.g. in [38]. The fibration ΩS 2n+1 → ΩCPn → S 1 is a direct product. Hence H∗(ΩCPn)  Λ(y1) ⊗ k[y2n] as a Hopf algebra, in particular the Pontryagin products agree. C∗(ΩCPn) is moreover formal since C∗(ΩS 2n+1) and C∗(ΩS 1) are. To relate this to the above description identify y2n = α2n−1α1 + · · · + α1α2n−1. The dg-algebra B(X) is larger since it is quasi-free (i.e. the underlying graded associative algebra is free), while H∗(ΩCPn) is only quasi-free as a commutative dg-algebra. 88 Example 5.8. Taking the limit we findB(CP∞). Of course the homology algebra of ΩCP∞ is just that of S 1. Indeed k[α1, α3, . . . ] is a quasi-free model for k[z1]. We conclude with a few examples that give some insight into what Morita cohomology (doesn’t) tell us about a space. From [47] we know that a CW-complex X is determined up to weak equivalence by the ∞-category of homotopy locally constant sheaves of spaces with its fiber functor. We can think of this category as RΓ(X, sSet) and it is natural to compare with RΓ(X,Chpe). The main differences are linearization, stabilization and restriction to compact fibers. Example 5.9. An example of a space with trivial Morita cohomology is provided by the classifying space of Higman’s 4-group H. This is known to be a finite CW complex and H is an acyclic group without non-trivial finite dimensional representations. For references and other examples see e.g. [6]. To show that the Morita cohomology of BH is trivial we have to show it is Morita equivalent to Ch. Now given an object M of H M(BH) we can take homology. As H has no non-trivial finite-dimensional representations this is a direct sum of shifted trivial representations. We can now show by induction that M must be a trivial extension, as there is no cohomology H>0(H, k). Since quasi-isomorphisms are detected on the underlying complex an object with trivial homology is quasi- isomorphic to 0. It follows thatH M(BH)  Ch in Ho(dgCatMor). 89 Example 5.10. Whenever a group G and its algebraic completion Galg have the same cohomology with coefficients in their finite-dimensional representations then they have the same Morita cohomology. By definition this is true for algebraically good groups, see [28]. Example 5.11. Next we consider an example of information in the category of simplicial fibrations with fiber of finite type that is not detected by Morita cohomology. Consider the outer automorphism group of the free group on four generators, Out(F4). This group is know to be finitely presented (since Aut(F4) is), and nonlinear [19]. It is also isomorphic to pi0Map(F, F) if F = ∨ 4 S 1. Hence there is a homotopy locally constant sheaf of simplicial sets of finite type (with fiber F) on BOut(F4) that does not descend to any BG for G a quotient of Out(F4). On the other hand the associated complex of local systems must descend to the classifying space of the image of Out(F4) in its linearization. Remark 5.1. Contrast this with Y u(X) which loses little information about the weak homotopy type of X. Since ΩX is an H-space it is nilpotent and there is a Whitehead theorem for integral homology for nilpotent spaces. So if NΩX and NΩY are quasi-equivalent (rather than just Morita equivalent) then X ' Y . 90 A. Some technical results on dg-categories A.1. A combinatorial model for dg-categories For some technical questions it is convenient to work with combinato- rial model categories. (For example it is easier to prove the existence of localizations and injective model structures.) One does not expect dgCatMor to be combinatorial as it is too large, but we show in this section that it is equivalent to a subcategory that is combinatorial. Definition. A model category is combinatorial if the underlying category is locally presentable. Being locally presentable is a finiteness condition. Definition. Let λ be a regular cardinal. An object A in a category D is λ-presentable if it is small with respect to λ-filtered colimits, i.e. if for every λ-filtered colimit colim Bi the map colim Hom(A, Bi) → Hom(A, colim Bi) is an isomorphism. We say A is presentable if it is λ-presentable for some λ. A cocomplete category is locally presentable if for some regular cardinal λ it has a set S of λ-presentable objects such that every object is a λ-directed colimit of objects in S . 91 Proposition A.1. The categories dgCatDK and dgCatMor are Quillen equivalent to combinatorial subcategories. Proof. This follows immediately from the proof of the main theorem of [36]. Let D denote either of the two model structures. Let S be the collection of objects that are domains or codomains of the generating cofibrations and generating trivial cofibrations. Clearly S is a set. Let S denote the full subcategory of D with objects S . Define ηS (X) to be the colimit of the forgetful diagram (s → A) 7→ s indexed by the overcategory S ↓ A. Then an object A ∈ D is S -generated if it is isomorphic to ηS (X). Now by the proof of Theorem 1.1 in [36] the subcategory of S - generated objects of D is a model category DS which is Quillen equivalent to the original one. Moreover, by Proposition 3.1 of [36],DS is locally presentable if every object in S is presentable. But this is clear since they have finitely many objects and generating morphisms.  Remark A.1. Note that Vopenka’s principle is not needed here since the objects of S are presentable. We get another finiteness condition for free. A model category is called tractable if the generating cofibrations and generating trivial cofibrations can be chosen to have cofibrant domains. Corollary A.2. The categories DS are tractable. 92 Proof. By Corollary 1.12 in [1] it is enough to check all the generating cofibrations can be chosen to have cofibrant domains. This is immediate in our example.  A.2. Simplicial resolutions of dg-categories In this section we will construct explicit simplicial resolutions C 7→ C• in dgCatDK to improve our understanding of the homotopy theory of dgCatDK . Such a resolution can be used for explicit, if unwieldy, computations. The resolutions we construct will be simplicial frames for a fibrant replacement. Our construction is directly motivated by Simpson’s construction of global sections of a presheaf of dg-categories as a dg-category of Maurer–Cartan elements, cf. section 5.4 of [41]. Remark A.2. In fact, the construction of Cn below corresponds to considering the constant presheaf of dg-categories on a covering of |∆n| by n + 1 open sets (corresponding to leaving out one of the faces). Define Cn as follows. We think of this as C ∆ n , but recall that the precise definition of C ∆ n differs by a a limit over degenerate simplices. Definition. AssumeC is fibrant, replace fibrantly otherwise. ThenCn is a dg-category with objects given by pairs (E, η) where E is a collection E0, . . . , En ∈ ObC and η is a collection of ηI = η(I) ∈ Homk−1(Ei0 , Eik) for all multi-indices I = (i0, . . . , ik) with 1 ≤ k ≤ n. The case k = 0 is subsumed by the differential on E. (We interpret η(i) = 0 where it 93 comes up in computation.) These pairs must satisfy the Maurer–Cartan condition: δη + η2 = 0, explained below. We also demand that all ηi ∈ Hom(Ei, Ei) are weak equivalences in C . Let us spell out the Maurer–Cartan condition. Intuitively, η provides all the comparison maps as well as homotopies between the different compositions. We define the differential (δη)(i0, . . . , ik) B d(η(i0, . . . , ik)) + (−1)|η| k−1∑ j=1 (−1) jη(i0, . . . , î j, . . . , ik) which lives in Homk(Ei0 , Eik). We write δ = d + ∆. Here we define |η| = 1. The product is: (φ ◦ η)(i0, . . . , ik) B k∑ j=0 (−1)|φ| jφ(i j, . . . , ik) ◦ η(i0, . . . , i j) Both definitions follow section 5.2 of [41], with some corrections to the signs. We leave out the terms in δη corresponding to leaving out i0 and ik as they do not live in the correct hom-spaces. One can now check that ∆d = −d∆ (and hence δ2 = 0) and we have the following Leibniz rule: δ(φ ◦ η) = (−1)|η|(δφ) ◦ η + φ ◦ (δη) The same equation holds for the summands d and ∆. (The unusual sign appears because of the backward notation for compositions.) Example A.1. For n = 1 we have (δη+ η2)01 = d(η01) + 0, the expected cycle condition. For n = 2 we have for example (δη + η2)012 = d(η012) + η02 − η12 ◦ η01 ∈ Hom1(E0, E2) 94 So an element of D2 is of the form (E, η) where E = (E0, E1, E2) and η = (η01, η02, η12; η013) satisfies dη+η2 = 0, which comes out to dηi j = 0 and dη012 = −η02 + η12 ◦ η01. This agrees with our intuition that η012 is a homotopy from η12 ◦ η01 to η02. Morphisms from (E, η) to (F, φ) are as follows. Hom−mCn ((E, η), (F, φ)) = {a(i0, . . . , ik)} where a(i0, . . . , ik) ∈ Homm−k(Ei0 , Fik). (Here C−m = Cm.) We write m = |a| for the degree of a morphism. We have a differential dη,φ defined by (dη,φ(a))(i0, . . . , ik) = δ(a) + φ ◦ a − (−1)|a|a ◦ η where composition and differential are defined as above. The Maurer– Cartan condition on η and φ together with the Leibniz rule ensures (dη,φ)2 = 0. Example A.2. For example C1 agrees with the path object in dgCat as constructed in Section 3 of [46]. Indeed, objects are homotopy invertible morphisms η : A → B and morphisms from η to φ are given by triples (a0, a1, a01) with differential δ : (a0, a1, a01) 7→ (da0, da1, da01 + φ ◦ a0 − (−1)|a0 |a0 ◦ η) Notation. Given an object or morphism α and a positive integer k we write α[k] for the collection of all αi0...ik . Before we embark on the somewhat technical proof that C• is a simplicial resolution, we note the following application. We can extend 95 the definitions of the differentials and composition to functions defined on general simplices. (That is, we replace “leaving out the i-th term” by the map induced by ∂i etc.) Proposition A.3. Given a simplicial set K we can construct C K as the dg-category with objects (E, η) where E ∈ (ObC )K0 and η assigns to every k-simplex in K≥1 a map in Homk−1(E(∂ k 0σ), E(∂ k maxσ)) satisfying the Maurer–Cartan equations. Hom-spaces are defined similarly to hom-spaces in C•. Proof. This follows from the construction of C K = lim∆Kop C•.  Remark A.3. This shows that the construction of ∞-local systems as C 7→ C K corresponds to the ∞-local systems of [7], or the A∞-functor of [26] if K is the nerve of a category, or to the Cˇech globalization in [41] if K is the nerve of an open covering. Proposition A.4. The inclusion from the constant simplicial dg- category cC to C• is a levelwise weak equivalence. Proof. We have to check the inclusion map ι : cC → Cn is a quasi- equivalence. Let us first show that ι induces weak equivalences on hom-complexes. We have to show that HomC ∆n ((E, η), (F, φ)) ' HomC (E, F) when both η and φ are of the form (1, 0), i.e. the constituent morphisms in degree 0 are the identity and all others are 0. 96 Write (H, dH) B Hom(E, F) and note that from the definitions we can write Hom((E, 0), (F, 0)) ' (H[1] ⊗ ∧ 〈e0, . . . , en〉,D) Here the ei all have degree 1 and we identify H.ei0 ∧ · · · ∧ eik with the a(i0, . . . , ik). The differential D is dH + ι∑ ei where the second term denotes contraction. This complex is a resolution of (H, dH). Next we show ι is quasi-essentially surjective, i.e. show that any object (E, η) is equivalent to an object (F0, (1, 0)) where F0 is of the form (F0, . . . , F0). We can deduce this if we can show that every (E, η) is equivalent to some (F, φ) such that all compositions which agree up to homotopy by δφ+φ2 = 0 agree strictly, i.e. φ = (φ[0], 0), and that any such (F, (φ[0], 0)) is equivalent to (F0, (1, 0)). The second part of this is immediate: We define a map from (F0, (1, 0)) to (F, (φ[0], 0)) by sending F0 to Fi via φ(0, i) = φ(i − 1, i) · · · φ(0, 1). Since all φ( j, j + 1) are homotopy invertible there is a homotopy inverse. We will now show that any (E, η) is equivalent to (F, φ) where φ has no higher homotopies. Let F = E and let φ(i, j) = η( j − 1, j) · · · η(i, i + 1). We may assume by induction on n that all η(i0, . . . ik) with ik < n are 0. We define the homotopy equivalence H : (E, η)→ (E, φ) as follows: H(i) = 1 H(i0, . . . , ik) = (−1)k−1η(i0, . . . , ik−1, n − 1, n) if ik = n and in−1 , n − 1 H(i0, . . . , ik) = 0 otherwise 97 And define H− to be equal to H in degree 0 and −H in degree > 0, i.e. the sign of H(i0, . . . , ik) is always (−1)k. Then it is clear that H and H− are inverses. Since H(i0, . . . , in) is zero unless in = n there are no nontrivial compositions and the composition 1 ◦ H(. . . ) and H−(. . . ) ◦ 1 cancel in degrees greater than 0. So it remains to show that dH = dH− = 0 to show we have a genuine homotopy equivalence. We consider H first. Putting together our definitions we find the following. Let us first assume ik−1 , n − 1 and ik = n. To obtain the correct signs recall that |H| = 0 and |η| = |φ| = 1. (dH)(i0, . . . , ik) = d(H(i0, . . . , ik)) + k−1∑ j=1 (−1) jH(i0, . . . , iˆ j, . . . , in) + k∑ j=0 (−1) jφ(i j . . . in) ◦ H(i0, . . . , i j) − k∑ j=0 H(i j, . . . , ik) ◦ η(i0, . . . , i j) = (−1)k−1dη(i0, . . . , n − 1, n) + (−1)k−2 ∑ j (−1) jη(i0, . . . , iˆ j, . . . , n − 1, n) + 0 − (−1)k−2η(i1, . . . , n − 1, n) ◦ η(i0, i1) − 1 ◦ η(i0, . . . , ik) = 0 The last equality holds since the penultimate term is of the form (−1)k−1(δη + η2)(i0, . . . , ik−1, n − 1, n) 98 This becomes clear if we write η(i0, . . . , ik) = η(i0, . . . , n̂ − 1, n) and observe that all the other terms we expect in δη + η2 are 0. The other cases are easier. If ik , n all terms in the differential are 0 and if ik−1 = n − 1 and ik = n there are only two nonzero terms, which cancel. When we consider dH− the sign of the term η(i0, . . . , ik) changes, as it now comes from η ◦H and not H ◦ η. This cancels the effect of the sign of H(i) also changing by a factor of −1. There are no other occurrences of the sign of H(i) unless k = 1 when all but the last two terms are zero and the last two terms cancel.  Proposition A.5. C• is Reedy fibrant. Proof. Write η