Homotopy theory of bicomplexes

arXiv:1802.07610v1 [math.AT] 21 Feb 2018 HOMOTOPY THEORY OF BICOMPLEXES FERNANDO MURO AND CONSTANZE ROITZHEIM Abstract. We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure’s weak equivalences are detected by the E2-term of the spectral sequence associated to the filtration of the total complex by the horizontal degree. We then extend this result to twisted complexes. Contents Introduction 1 1. Complexes 3 2. Bicomplexes 4 3. The total model structure on bicomplexes 9 4. The Cartan–Eilenberg model structure on bicomplexes 12 5. Twisted complexes and their total model structure 16 References 25 Introduction The notion of chain complex is central to homological algebra, as they arise e.g. as resolutions of modules. Bicomplexes, in turn, arise as resolutions of chain complexes. These resolutions, first defined by Cartan and Eilenberg [CE56], are concentrated in a half-plane. They are used to compute derived functors between derived categories [GM03] in conjuction with the totalisation functor from bicomplexes to complexes. Homotopy theory of chain complexes is a well-known concept. Model categories provide a commonly used language to construct resolutions in the presence of a notion of equivalence weaker than isomorphisms. One such example is homology isomorphisms of chain complexes, also known as quasi-isomorphisms. A standard model structure on the category of chain complexes of modules over a ring kis the projective model structure, where weak equivalences are quasi-isomorphisms and cofibrant objects are also degreewise projective. The first goal of this paper is to provide useful model structures on the category of bicomplexes X∗,∗concentrated in the right half plane, i.e. Xp,q = 0 for p < 0. In the first one, the total model structure, the weak equivalences are exactly those morphisms whose totalisation induces an isomorphism in homology. In the second model structure, the Cartan–Eilenberg model structure, cofibrant resolutions The first author was partially supported by the Spanish Ministry of Economy under the grant MTM2016-76453-C2-1-P (AEI/FEDER, UE). 1 2 FERNANDO MURO AND CONSTANZE ROITZHEIM are Cartan–Eilenberg resolutions. Weak equivalences in this second model structure are the so-called “E2-equivalences”. Bicomplexes have horizontal and vertical differentials, dh:Xp,q −→ Xp−1,q, dv:Xp,q −→ Xp,q−1. These differentials anti-commute, i.e. dhdv+dvdh= 0. This condition implies that the horizontal differential dhinduces a differential on the vertical homology Hv ∗(X) of a bicomplex X, so one can consider Hh ∗(Hv ∗(X)). This is exactly the E2-term of a spectral sequence that converges strongly to the homology of the total complex Tot(X) of the bicomplex X. Therefore, the (Hh ∗◦ Hv ∗)-isomorphisms are called E2-equivalences, and they are special cases of (H∗◦ Tot)-isomorphisms. Both model structures will enjoy useful properties such as being combinatorial, proper, and monoidal. Furthermore, we will see that the first one is an abelian model structure, i.e. fibrations are surjections with fibrant kernel and cofibrations are injections with cofibrant cokernel. Cofibrant objects are also degreewise projective in both cases, so we can really see our model structures as generalisations of the projective model structure on chain complexes for different choices of weak equivalences. The total model structure, in addition, is Quillen equivalent to the model category of chain complexes. We will then generalise the total model structure on bicomplexes to the category of twisted complexes. While a bicomplex is a bigraded k-module with two differentials, a twisted complex is equipped with maps di:Xp,q −→ Xp−i,q+i−1, i ≥0, satisfying X i+j=n didj= 0, n ≥0. Naturally, making all the necessary definitions and calculations to obtain the total model structure on this category is a lot more involved than in the case of just two differentials. Another motivation for these model structures stems from the study of A∞-algebras, or “homotopy associative” algebras. Among other things, A∞-structures on the homology of a differential graded algebra allow us to see how many differential graded algebras realise this homology. However, this only works over a ground field [Kad80], or if all modules in question are projective. To circumnavigate this rather restrictive assumption, one can work in the context of derived A∞-algebras [Sag10]. These are bigraded objects, where the second degree allows to create a projective resolution compatible with any A∞-structure. Where A∞-algebras have an underlying chain complex, derived A∞-algebras have an underlying twisted chain complex concentrated in the right half plane. Furthermore, the homological perturbation lemma [Bro67] tells us that the vertical homology of every Cartan–Eilenberg resolution can be equipped with the structure of a twisted complex. Therefore, in order to understand the homotopy theory of derived A∞-algebras, specifically in an operadic context [LRW13, CESLW17], it is necessary to understand the homotopy theory of the underlying twisted complexes. HOMOTOPY THEORY OF BICOMPLEXES 3 This paper is organised as follows. In Section 1 we recall some basic definitions and results concerning chain complexes and the projective model structure, in particular on how the model structure is constructed using spheres and discs. In Section 2 we study the category bCh of bicomplexes concentrated in the right half-plane, define the bigraded analogue of spheres and discs and discuss the tensor product. Sections 3 and 4 give the total and the Cartan–Eilenberg model structures by showing that the generating cofibrations and trivial cofibrations defined using those spheres and discs together with the respective weak equivalences satisfy Smith’s recognition principle from [Hov99, Theorem 2.1.19]. Finally, we introduce the category of twisted complexes in Section 5, define spheres and discs, and obtain the desired model structure. 1. Complexes We briefly recall a couple of facts about the model categories Ch of unbounded (chain) complexes and Ch≥0of complexes concentrated in non-negative degrees. We use the convention that differentials shift the degree by −1. Throughout this paper, kdenotes a commutative ground ring. Further conditions on kwill be imposed when necessary. Tensor product will always be taken over k. As a category, Ch is locally finitely presentable. Limits and colimits are computed pointwise. It is also a closed symmetric monoidal category with respect to the tensor product. The symmetry constraint uses the Koszul sign convention, and the inner Hom is the graded module HomCh(X, Y )n=Y m∈Z Homk(Xm, Ym+n) endowed with the following differential d(f) = df −(−1)|f|fd. Here Homkdenotes the inner Hom in the category of modules. The category Ch is also a combinatorial proper model category. Weak equivalences are quasi-isomorphisms and fibrations are (pointwise) surjections. Let us recall the generating (trivial) cofibrations. For a k-module A, we define the chain complex Dn(A) to be · · · → 0→A1 →A→0→ · · · concentrated in degrees nand n−1. Similarily we define Sn(A) to just consist of Aconcentrated in degree n. This in fact gives us adjoint functor pairs evn:k-mod −−→ ←− Ch : Dn and Zn:k-mod −−→ ←− Ch : Sn where evndenotes evaluation at degree nand Zn(X) = ker[d:Xn→Xn−1] denotes the cycles in degree n. (Note that when we write adjunctions, the top arrow is always the left adjoint.) We now define the n-sphere Snto be Sn(k) and the n-disk Dnto be Dn(k) for short. This is used to construct the projective model structure on Ch, which is the model structure we will consider throughout this paper. Define sets ICh and JCh as ICh ={Sn−1֒→Dn}n∈Z, JCh ={0֒→Dn}n∈Z. Here Sn−1֒→Dnis the identity in degree n−1. Furthermore, let Wdenote the class of H∗-isomorphisms, i.e. quasi-isomorphisms. 4 FERNANDO MURO AND CONSTANZE ROITZHEIM Recall that for a class of maps I, the class I-inj is given by all the maps that have the right lifting property with respect to I. Furthermore, I-cof is the class of all maps that have the left lifting properties with respect to all maps in I-inj. The class I-cell is given by all transfinite compositions of pushouts of elements of I. This class satisfies I-cell ⊆I-cof. Then in our case ICh,JCh, and Wsatisfy the following properties: • W has the two-out-of-three property and is closed under retracts, •JCh-cell ⊆ W ∩ ICh-cof, •ICh-inj ⊆ W ∩ JCh-inj, •either W ∩ ICh-cof ⊆JCh-cof or W ∩ JCh-inj ⊆ICh-inj (a posteriori both), plus some set-theoretic conditions. By [Hov99, Theorem 2.1.19], this means that there is a cofibrantly generated model structure with weak equivalences W, generating cofibrations ICh, and generating trivial cofibrations JCh. (Trivial) fibrations are the maps in JCh-inj (resp. ICh-inj), and cofibrations are retracts of maps in JCh-cell. It can furthermore be shown using the adjunctions defined earlier, that a map is a fibration if and only if it is a degreewise surjection, and that cofibrations are the degreewise monomorphisms with cofibrant cokernel. The last property means that this model structure is abelian in the sense of [Hov07], i.e. a cofibration is a monomorphism with cofibrant cokernel and a fibration is an epimorphism with fibrant kernel. Cofibrant objects do not have an easy characterization, but they are known to be pointwise projective. Furthermore, it is compatible with the monoidal structure in the sense of [Hov99, Definition 4.2.6]. The tensor unit S0is actually cofibrant since it is the cokernel of the generating cofibration S−1֒→D0. The monoid axiom of Schwede and Shipley [SS00, Definition 3.3] is also satisfied. The full subcategory Ch≥0⊂Ch of complexes concentrated in non-negative degrees inherits a monoidal model structure with the same tensor product and weak equivalences. The inner HomCh≥0(X, Y ) is the non-negative truncation of HomCh(X, Y ). The former is a subcomplex of the latter, both complexes coincide in (strictly) positive degrees, and HomCh≥0(X, Y )0= Ch≥0(X, Y ) = Z0(HomCh(X, Y )). The fibrations in Ch≥0are the maps which are surjective on positive degrees but not necessarily in degree 0. Cofibrations are precisely the maps with pointwise projective cokernel. Sets of generating (trivial) cofibrations are ICh≥0={0֒→S0} ∪ {Sn−1֒→Dn}n≥1, JCh≥0={0֒→Dn}n≥1. The model structure on Ch≥0is proper, monoidal, with cofibrant monoidal tensor unit S0, and it satisfies the monoid axiom. It is not abelian, though, since fibrations need not be surjective. The inclusion Ch≥0⊂Ch is a left Quillen functor. Its right adjoint is the non-negative truncation. We will use the outline of these well-known results as a blueprint for the model structures on bicomplexes, respectively twisted chain complexes, that we are going to construct in the subsequent chapters. 2. Bicomplexes This section consists of elementary definitions and examples which are relevant for later computations. We consider (N×Z)-graded bicomplexes made of anticommutative squares. HOMOTOPY THEORY OF BICOMPLEXES 5 Definition 2.1. Abicomplex Xis a bigraded module X={Xp,q}p,q∈Zwith Xp,q = 0 for p < 0, equipped with horizontal and vertical differentials dh:Xp,q −→ Xp−1,q, dv:Xp,q −→ Xp,q−1, respectively, satisfying dvdh+dhdv= 0. A morphism of bicomplexes f:X→Yis a family of maps fp,q :Xp,q →Yp,q compatible with the horizontal and vertical differentials dhf=fdh, dvf=fdv. We denote the category of bicomplexes by bCh. For any bigraded module X, given x∈Xp,q, we say that |x|h=pis the horizontal degree of x, and |x|v=qis its vertical degree. The bidegree of xis (|x|h,|x|v) = (p, q) and the total degree is |x|=|x|h+|x|v=p+q. The category bCh is clearly abelian and locally finitely presentable. Limits and colimits are computed pointwise. Remark 2.2.The equation dvdh+dhdvsays that the following squares are anticommutative in a bicomplex Xp−1,q dv  Xp,q dv  dh oo Xp−1,q−1Xp,q−1 dh oo Some readers will probably prefer that these squares commute. If we denote by X′ the underlying bigraded module of Xendowed with the same horizontal differential d′ h=dhand the new vertical differential d′ vdefined by d′ v(x) = (−1)|x|hx, we obtain a bicomplex X′with commuting differentials d′ hd′ v=d′ vd′ h. If bCh′ denotes the category of bicomplexes with commuting differentials, we obtain an isomorphims of categories bCh ∼ =bCh′ X7→ X′. Definition 2.3. Let Xbe a bicomplex. The vertical cycles Zv(X) are the elements in the kernel of the vertical differential of X, and the vertical boundaries Bv(X) are the elements in the image of dh. The vertical homology is Hv(X) = Zv(X) Bv(X). We can regard Zv(X), Bv(X), and Hv(X) as bicomplexes with trivial vertical differential. Their vertical differentials are induced by that of X. We similarly define the horizontal cycles Zh(X), boundaries Bh(X), and homology Hh(X). Let Abe a k-module. Then we define Dp,q(A), p > 0, q∈Z, to be the bicomplex whose underlying bigraded module is Dp,q p,q (A) = Dp,q p−1,q(A) = Dp,q p,q−1(A) = Dp,q p−1,q−1(A) = A and zero elsewhere. Its four nontrivial differentials are given by the identity except for dv=−1 : Dp,q(A)p,q −→ Dp,q(A)p,q−1. 6 FERNANDO MURO AND CONSTANZE ROITZHEIM Furthermore, we define ∂hDp,q(A) and ∂vDp,q(A) to be the horizontal resp. vertical boundaries of Dp,q(A). We also define Sp,q(A) to be the bicomplex with Ain bidegree (p, q) and zero in all other degrees. The following can be easily verified. Lemma 2.4. The above definitions give rise to adjunctions evp,q :k-mod −−→ ←− bCh: Dp,q Zh p−1,q :k-mod −−→ ←− bCh: ∂hDp,q Zv p−1,q :k-mod −−→ ←− bCh: ∂vDp,q Zh p◦Zv q:k-mod −−→ ←− bCh: Sp,q Here evp,q denotes evaluation at bidegree (p, q). We define the (p, q)-disc Dp,q as Dp,q(k). We can view it as the bicomplex freely generated by a single element xp,q ∈Dp,q. The free k-module generators are xp,q, dh(xp,q), −dv(xp,q), and dvdh(xp,q) = −dhdv(xp,q), respectively. The horizontal and vertical boundaries of the (p, q)-disk will again be denoted by ∂hDp,q and ∂vDp,q, respectively. The (p, q)-sphere is kconcentrated in bidegree (p, q). These bicomplexes look as follows, q q−1 k k k k −1 Dp,q ∂vDp,q ∂hDp,q Sp−1,q−1 p−1p Here, unlabelled arrows are identities. Remark 2.5.The bicomplex ∂hDp,q is freely generated by the horizontal cycle dh(xp,q) in bidegree (p−1, q). Similarly, ∂vDp,q is freely generated by the vertical cycle −dv(xp,q) in bidegree (p, q −1). Since our bicomplexes are concentrated in non-negative horizontal degree, ∂hD1,q is freely generated by the element dh(x1,q) in bidegree (0, q). Indeed, morally, we can define D0,q =∂hD1,q and ∂vD0,q =S0,q−1. The reader can check that most of the properties of Dp,q and ∂vDp,q for p > 0 extend to the case p= 0 with these definitions, but we have preferred two avoid two different notations for the same object. As a consequence of Lemma 2.4, we have the following useful natural isomorphisms, which we list for convenience. HOMOTOPY THEORY OF BICOMPLEXES 7 Corollary 2.6. For any bicomplex X,p > 0, and q∈Z, there are natural isomorphisms bCh(S0,q−1֒→∂hD1,q, X)∼ =Ch(Sq−1֒→Dq, X0,∗), bCh(∂vDp,q ֒→Dp,q, X)∼ =Ch(Sq−1֒→Dq, Xp,∗), bCh(0 ֒→∂hD1,q, X)∼ =Ch(0 ֒→Dq, X0,∗) ∼ =Ch(0 ֒→S0, X∗,q), bCh(0 ֒→S0,q−1, X)∼ =Ch≥0(0 ֒→S0, Zv ∗,q−1(X)), bCh(Sp−1,q−1֒→∂vDp,q, X)∼ =Ch≥0(Sp−1֒→Dp, Zv ∗,q−1(X)), bCh(∂hDp,q ֒→Dp,q, X)∼ =Ch≥0(Sp−1֒→Dp, X∗,q), bCh(0 ֒→∂vDp,q, X)∼ =Ch≥0(0 ֒→Dp, Zv ∗,q−1(X)).  We now consider the monoidal structure on bicomplexes. Definition 2.7. The tensor product X⊗Yof two bicomplexes Xand Yis the bicomplex defined as (X⊗Y)p,q =M m+s=p n+t=q Xm,n ⊗Ys,t with horizontal and vertical differentials defined as dh(x⊗y) = dh(x)⊗y+ (−1)|x|x⊗dh(y), dv(x⊗y) = dv(x)⊗y+ (−1)|x|x⊗dv(y). Note that both formulas use the total degree in their sign conventions. This tensor product endows bCh with a closed symmetric monoidal structure with obvious associativity and unit constraints. The tensor unit is kconcentrated in bidegree (0,0). The symmetry constraint uses the Koszul sign rule with respect to the total degree, X⊗Y∼ =Y⊗X, x⊗y7→ (−1)|x||y|y⊗x. The mapping objects HombCh(X, Y ) in bCh, adjoints to the tensor product, are defined by the k-modules HombCh(X, Y )p,q =Y s≥0 t∈Z Homk(Xs,t, Ys+p,t+q), p > 0, q ∈Z, and the submodules HombCh(X, Y )0,q ⊂Y s≥0 t∈Z Homk(Xs,t, Ys,t+q), q ∈Z, formed by the elements fsuch that dhf= (−1)|f|fdh. The horizontal and vertical differentials are defined by dh(f) = dhf−(−1)|f|fdh, dv(f) = dvf−(−1)|f|fdv. Remark 2.8.The previous definition would not work for bicomplexes with commuting differentials (see Remark 2.2). The readers which prefer commuting differentials will probably find more natural to consider the horizontal and vertical degrees in the 8 FERNANDO MURO AND CONSTANZE ROITZHEIM definition of the horizontal and vertical differentials of the tensor product. Indeed, this yields a bicomplex X′⊗′Y′with commuting differentials d′ h(x′⊗y′) = d′ h(x′)⊗y′+ (−1)|x′|hx′⊗d′ h(y′), d′ v(x′⊗y′) = d′ v(x′)⊗y′+ (−1)|x′|vx′⊗d′ v(y′). The underlying bigraded module of X′⊗′Y′is obviously defined as for X⊗Y. In this case it ismore sensible to use the Koszul sign rule with respect to the horizontal and vertical degrees separately in the definition of the symmetry constraint, X′⊗′Y′∼ =Y′⊗′X′, x′⊗y′7→ (−1)|x′|h|y′|h+|x′|v|y′|vy′⊗x′. This endows the category bCh′of bicomplexes with commuting differentials with a closed symmetric monoidal structure. The isomorphism of categories bCh ∼ =bCh′in Remark 2.2 together with the natural isomorphism (X⊗Y)′∼ =X′⊗′Y′, x⊗y7→ (−1)|x|v|y|hx⊗y, defines a symmetric monoidal isomorphism. We will work with bCh since certain computations are simpler here. Definition 2.9. Given a bigraded module X, the graded module Tot(X) is defined as Totn(X) = M p+q=n Xp,q. If Xis a bicomplex, Tot(X) equipped with the differential dTot =dh+dv is called the total complex of X. This construction defines the totalisation exact functor Tot: bCh −→ Ch . Remark 2.10.The totalisation functor is strong symmetric monoidal in the obvious naive way. In addition, Tot preserves (co)limits, since they are computed pointwise both in bCh and Ch. If we had used bicomplexes with commuting differentials, we would have had to include signs in the natural isomorphism comparing the tensor products in the source and in the target of Tot. Remark 2.11.For any bigraded module X, Tot(X) has a natural increasing nonnegative exhaustive filtration defined by FmTotn(X) = M p+q=n p≤m Xp,q. If Xis a bicomplex, this filtration of the total complex Tot(X) is compatible with the differential. Since our bicomplexes are concentrated in the right half-plane, the associated spectral sequence converges strongly to the homology of Tot(X). The E2-term is Hh(Hv(X)) [McC01, Theorem 2.15], E2 p,q =Hh p,q(Hv(X)) =⇒Hp+q(Tot(X)). This will play a central role in the model structures to be defined later. HOMOTOPY THEORY OF BICOMPLEXES 9 3. The total model structure on bicomplexes Let Wdenote the class of (H∗◦Tot)-isomorphisms in bCh, i.e. maps which induce a quasi-isomorphism in totalisation, and let ITot ={S0,q−1֒→∂hD1,q}q∈Z∪ {∂vDp,q ֒→Dp,q}p>0 q∈Z , JTot ={0֒→∂hD1,q}q∈Z∪ {∂vDp,q ֒→Dp,q}p>0 q∈Z . In the same way that one constructs the projective model structure on chain complexes outlined in Section 1, we are going to use Smith’s recognition principle to show that this choice defines a cofibrantly generated model structure on bCh. We will then further characterise its cofibrations and fibrations. Theorem 3.1. The category of bicomplexes bCh can be endowed with a proper combinatorial abelian model category structure called the total model structure with the following properties: •a morphism f:X→Yis a weak equivalence if Tot(f)is a quasi-isomorphism in Ch, i.e. the class of weak equivalences is W, •a morphism f:X→Yis a (trivial) fibration if it is pointwise surjective and Hv p,∗(f)is an isomorphism for all p > 0(resp. p≥0), •the cofibrations are the injective maps with cofibrant cokernel. Cofibrant implies pointwise projective. Furthermore, its generating cofibrations and trivial cofibrations are given by ITot and JTot, respectively. Proof. The proposed weak equivalences in bChTot are clearly closed under retracts and satisfy the 2-out-of-3 property. We will now verify the various lifting properties we require for our proof, making use of the identities in Lemma 2.4 and Corollary 2.6. We use them in combination with the model structures on Ch and Ch≥0reviewed in the previous sections, whose generating (trivial) cofibrations we know. A map f:X→Yin bCh has the right lifting property with respect to 0 →∂hD1,q if and only if f0,∗:X0,∗→Y0,∗has the right lifting property with respect to 0 →Dq. This happens for all q∈Zwhenever f0,∗is a fibration in Ch, i.e. degreewise surjective. For p > 0, having the right lifting property with respect to ∂vDp,q →Dp,q is equivalent to fp,∗:Xp,∗→Yp,∗having the right lifting property with respect to all Sq−1→Dq. This happens for all q∈Zprecisely when fp,∗is a trivial fibration in Ch, i.e. degreewise surjective as well as a homology isomorphism. Similarily, having the right lifting property with respect to S0,q−1→∂hD1,q is equivalent to f0,∗:X0,∗→Y0,∗having the right lifting property with respect to Sq−1→Dqin Ch. This happens for all q∈Zwhenever f0,∗is a trivial fibration in Ch. We can summarise our findings as follows. ITot-inj = {degreewise surjective fsuch that Hv p,∗(f) is an isomorphism for all p≥0} and JTot-inj = {degreewise surjective fsuch that Hv p,∗(f) is an isomorphism for all p > 0.} This is consistent with our claims about the (trivial) fibrations in bChTot. 16 FERNANDO MURO AND CONSTANZE ROITZHEIM Moreover, if fis a cofibration in Ch≥0and Uis a trivially cofibrant object in Ch regarded as a bicomplex concentrated in horizontal degree 0, then U⊗zq(f) is a trivial cofibration in bChCE. Indeed, the complex U, being trivially cofibrant, is a retract of a direct sum of copies of Dt,t∈Z, i.e. Uregarded as a bicomplex is a retract of a direct sum of copies of ∂hD1,t,t∈Z. Moreover, ∂hD1,t ⊗zq(f)∼ =ct+q(f), hence U⊗zq(f) is retract of a direct sum factors of the form ct(f), t∈Z, which are trivial cofibrations. Recall also that trivially cofibrant objects in Ch are closed under tensor products since the model category Ch is monoidal. If we combine the previous observations with the fact that the map ∂hDp,q ֒→Dp,q is the same as ∂hD1,1⊗(Sp−1,q−1֒→∂vDp,q), we conclude that the push-out product of two maps in ICE is a cofibration, and that the push-out product of a map in ICE and a map in JCE is a trivial cofibration. Hence bChCE is monoidal. Let us now consider the monoid axiom [SS00, Definition 3.3]. Since Cartan– Eilenberg equivalences are closed under transfinite compositions, it suffices to prove that, given two bicomplexes Xand Y, the push-out of Xalong a map fin Y⊗JCE is a weak equivalence. All maps in JCE are pointwise split monomorphisms, hence maps in Y⊗JCE are injective and therefore the push-out along a map fin Y⊗JCE is a weak equivalence if and only if the cokernel of fis acyclic. The cokernels of maps in Y⊗JCE are of the form Y⊗∂vDp,q and Y⊗∂hDp,q,p > 0, q∈Z. They are acyclic in bChCE because Hh(Hv(Y⊗∂vDp,q)) = Hh(Hv(Y)⊗∂vDp,q) = 0, Hv(Y⊗∂hDp,q) = 0. This concludes the verification of the monoid axiom. The tensor unit S0,0is cofibrant since 0 ֒→S0,0is one of the generating cofibrations.  5. Twisted complexes and their total model structure In this section, we generalise the total model structure on bCh to the category of twisted complexes. The notion of twisted complex goes back to Wall [Wal61]. They have proven useful in many contexts in the construction of small resolutions. Definition 5.1. Atwisted complex X, also known as multicomplex, is a bigraded module X={Xp,q}p,q∈Zwith Xp,q = 0 for p < 0 equipped with maps di:Xp,q −→ Xp−i,q+i−1, i ≥0, satisfying X i+j=n didj= 0, n ≥0. Abusing terminology, we also call the maps didifferentials, despite they do not square to zero in general. HOMOTOPY THEORY OF BICOMPLEXES 17 d0 d1 d2 d3 d4 A morphism of bicomplexes f:X→Yis a family of maps fp,q :Xp,q →Yp,q compatible with the differentials. We denote the category of twisted complexes by tCh. Definition 5.2. We define the total complex of the twisted complex Xas the chain complex which in degree nis Tot(X)n=M p+q=n Xp,q. The differential in Tot(X) is then d=X i≥0 di. Totalisation defines an functor Tot: tCh −→ Ch . Note that the sum in the differential is finite on each bidegree (p, q) since the target of diis trivial for i > p. We see that the differential on Tot(X) is compatible with the filtration by the horizontal degree considered in Remark 2.11. Moreover, any such differential in Tot(X) comes from a unique twisted complex structure on X. The category tCh is clearly abelian and locally finitely presentable. Limits and colimits are computed pointwise. The totalisation functor is exact and preserves (co)limits. Remark 5.3.A bicomplex is the same as a twisted complex with di= 0 for i≥2. The horizontal and vertical differentials are dh=d1and dv=d0. This observation defines a fully faithful functor bCh →tCh which has a left adjoint tCh →bCh sending Xto X.X i≥2 di(X). The first equations defining a general twisted complex Xare d2 0= 0, d0d1+d1d0= 0, d0d2+d2 1+d2d0= 0. Hence d0, which points vertically downwards, is always a differential, so we can define the vertical cycles Zv(X) and vertical homology Hv(X). Moreover, d1in- duces a differential on Hv(X) ponting horizontally to the left, hence we can define its horizontal homology Hh(Hv(X)). As in Remark 2.11, this is the second term of the spectral sequence of the filtered complex Tot(X), converging strongly to its homology, E2 p,q =Hh p,q(Hv(X)) =⇒Hp+q(Tot(X)). 18 FERNANDO MURO AND CONSTANZE ROITZHEIM Definition 5.4. We define the twisted (p, q)-disc ˜ Dp,q,p≥0, q∈Z, as the twisted complex freely generated by a single element xp,q ∈˜ Dp,q p,q . More precisely, as a k-module ˜ Dp,q p,q =kgenerated by xp,q, and for 0 ≤s≤pand n≥1, ˜ Dp,q p−s,q+s−nis the quotient of the free module generated by {di1···din(xp,q)}i1+···+in=s by the relations X ij+ij+1 =m di1···dijdij+1 ···din(xp,q)1≤j<n i1+···+ij−1+m+ij+2+···+in=s Elsewhere, ˜ Dp,q is trivial. These relations provide a way of taking any d0in di1···din(xp,q) to the right. In particular, di1···din(xp,q) is trivial as long as there are two 0 subscripts, since d2 0= 0. Therefore ˜ Dp,q s,t = 0 if t < q −1. Moreover, we can take any word di1···din(xp,q) with exactly one d0in it and uniquely rewrite it as the sum of words without any d0and a word with exactly one d0at the right. More precisely, if the only trivial subscript is ij= 0 then di1···dij−1d0dij+1 ···din(xp,q) = (−1)n−jdi1···dij−1dij+1 ···dind0(xp,q) + n−j X u=1 X s+t=ij+u s,t>0 (−1)udi1···dij−1dij+1 ···dij+u−1 |{z } u−1 factors dsdtdij+2 dij+u+1 ···din(xp,q) |{z } n−j−ufactors . One can straightforwardly check that the rewriting process consisting of replacing the word on the left with the sum on the right, removing words di1···din(xp,q) with two 0 subscripts, and not changing words with no 0 subscript, sends the defining relations of ˜ Dp,q to 0. Consequently, for 0 ≤s≤pand 1 ≤n≤s+ 1, ˜ Dp,q p−s,q+s−n is freely generated by {di1···din(xp,q)}i1,...,in>0 i1+···+in=s ∪ {di1···din−1d0(xp,q)}i1,...,in−1>0 i1+···+in−1=s . Hence, the rank of this ˜ Dp,q p−s,q+s−nis s−1 n−1+s−1 n−2if 1 < n ≤s, and 1 if n= 1 or n=s+ 1. The following picture gives a rough idea of how ˜ D4,0looks like 1 1 1 1 1 1 1 2 3 4 1 3 6 1 4 1 The nodes indicate the rank of the non-trivial parts of the underlying bigraded module. The non-trivial arrows are also depicted. HOMOTOPY THEORY OF BICOMPLEXES 19 Remark 5.5.One can define an analogous algorithm to move the d0in a word to the left. Therefore, {di1···din(xp,q)}i1,...,in>0 i1+···+in=s ∪ {d0di1···din−1(xp,q)}i1,...,in−1>0 i1+···+in−1=s is an alternative basis of ˜ Dp,q p−s,q+s−nfor 0 ≤n−1≤s≤p. For any k-module A, we can easily define ˜ Dp,q(A) by degreewise tensoring with A, which gives an analogous construction with copies of Ainstead of copies of kat each node. By definition, this gives us an adjoint functor pair ˜ Dp,q :k-mod −−→ ←− tCh : evp,q, where evp,q(X) = Xp,q. In particular, for any twisted complex X, we have a natural isomorphism tCh( ˜ Dp,q, X) = Xp,q. Lemma 5.6. Twisted disks have trivial total homology, H∗(Tot( ˜ Dp,q)) = 0. Proof. Using the bases in Remark 5.5, the vertical homology d0applied to a basis element is either zero or of the form d0di1···din−1(xp,q). Thus, their vertical homology vanishes, Hv(˜ Dp,q) = 0, since ˜ Dp,q is vertically contractible. Using the spectral sequence in Remark 5.3, this implies that the homology of the total complex is also trivial.  Definition 5.7. We define the vertical boundary ∂v˜ Dp,q of the twisted (p, q)-disc, p≥0, q∈Z, as the twisted complex freely generated by a single element yp,q−1∈ ∂v˜ Dp,q p,q−1satisfying dv(yp,q−1) = 0 (i.e. a vertical cycle). Arguing as in Definition 5.4, ∂v˜ Dp,q p,q−1=kgenerated by yp,q−1, and for 0 ≤s≤p and 1 ≤n≤s,∂v˜ Dp,q p−s,q−1+s−nis freely generated by {di1···din(yp,q−1)}i1,...,in>0 i1+···+in=s . Hence its rank is s−1 n−1. Elsewhere, ∂v˜ Dp,q is trivial. We depict ∂v˜ D4,0with the same conventions as in Definition 5.4, 11 1 1 1 1 2 3 1 3 1 Again, we can define ∂v˜ Dp,q(A) for a k-module Ain the analogous way and obtain the following. 20 FERNANDO MURO AND CONSTANZE ROITZHEIM Lemma 5.8. The pair of functors ∂v˜ Dp,q :k-mod −−→ ←− tCh : Zv p,q−1 is an adjoint pair. The inclusion of vertical boundaries ∂v˜ Dp,q ֒→˜ Dp,q is the morphism representing the natural map d0:Xp,q →Zv p,q−1(X). Remark 5.9.On bidegree (p, q −1), the inclusion of vertical boundaries is defined by yp,q−17→ d0(xp,q). Hence it is clearly injective since it maps bijectively the bases in Definition 5.7 to the second factors of the unions defining the bases in Definition 5.4. Moreover, this observation also proves that the cokernel of ∂v˜ Dp,q ֒→˜ Dp,q is ∂v˜ Dp,q+1. Corollary 5.10. We have the following natural isomorphisms for any twisted complex X,p≥0, and q∈Z: tCh(∂v˜ Dp,q ֒→˜ Dp,q, X) = Ch(Sq−1֒→Dq, Xp,∗). We need to know that, analogously to our previous model categories, the vertical boundary of the disc is actually acyclic, which requires more work than proving it for the disc itself. Lemma 5.11. For p > 0,H∗(Tot(∂v˜ Dp,q)) = 0. Proof. The vertical differential d0on ∂v˜ Dp,q is given by d0di1···din(yp,q−1) = n X j=1 (−1)jX ij,1+ij,2=ij di1···dij−1dij,1dij,2dij+1 ···din(yp,q−1), see Definition 5.4 and recall that d0(yp,q−1) = 0. We will see that, up to degree shift and change of sign in the differential, the chain complex ∂v˜ Dp,q p−s,∗,s≥2, is isomorphic to C∗(∆s−2, k), the coaugmented simplicial cochain complex of the indicated simplex with coefficients in our ground ring, which is contractible. Recall that the augmented simplicial chain complex C∗(∆n, k), n≥0, is freely generated in each degree −1≤t≤nby the strictly increasing sequences [v0,...,vt] of length n+ 1 formed by integers 0 ≤vi≤n. The rank of Ct(∆n, k) is therefore n+1 t+1 for −1≤t≤n. The complex is zero elsewhere. Its differential is defined by d([v0,...,vt]) = t X i=0 (−1)i[v0,...,bvi,...,vt], where bvimeans viremoved. The cochain complex C∗(∆n, k) is the k-linear dual of C∗(∆n, k), hence it is freely generated by the dual basis elements [v0,...,vt]∗, whose differentials are d([v0, . . . , vt]∗) = t+1 X i=0 X vi−1<u<vi (−1)i[...,vi−1, u, vi,...]∗. Here, in the index of the second summation we understand that v−1=−1 and vt+1 =n+ 1. As we have seen in Definition 5.7, the rank of ∂v˜ Dp,q p−s,q−1+s−nis s−1 n−1, the same as the rank of Cn−2(∆s−2, k). Moreover, we have an isomorphism ∂v˜ Dp,q p−s,q−1+s−n∼ =Cn−2(∆s−2, k) given by the basis bijection di1···din(yp,q−1)7→ i2−1, i2+i3−1,··· , n X j=2 ij−1∗ . HOMOTOPY THEORY OF BICOMPLEXES 21 This isomorphism is clearly compatible with differentials up to a −1 sign. Therefore, Hv p−s,∗(∂v˜ Dp,q) = 0 if s≥2. The remaining non-trivial part of Hv(∂v˜ Dp,q) reduces to Hv p,q−1(∂v˜ Dp,q) = Hv p−1,q−1(∂v˜ Dp,q) = k. The differential d1on Hv(∂v˜ Dp,q) is an isomorphism between these two modules, i.e. Hv(∂v˜ Dp,q) = ∂vDp,q, hence Hh(Hv(∂v˜ Dp,q)) = 0. As this is the E2-term of the spectral sequence in Remark 5.3, this implies H∗(Tot(∂v˜ Dp,q)) = 0 as claimed.  We note that Lemma 5.6 for p > 0 could also have been derived from Lemma 5.11 and Remark 5.9. Furthermore, the isomorphism ∂v˜ Dp,q p−s,∗∼ =C∗(∆s−2, k), s ≥2, in the previous proof is of course a great convenience, but we are not aware of a conceptual reason as to why simplicial cochains appear in this proof. We now consider the monoidal structure on twisted complexes, attributed by Meyer [Mey78] to Liulevicius [Liu67]. Definition 5.12. The tensor product X⊗Yof two twisted complexes Xand Y is the twisted complex with the same underlying bigraded module as in Definition 2.7, such that the differential on Tot(X⊗Y) = Tot(X)⊗Tot(Y) is the standard tensor product differential. This is equivalent to saying that the maps dion X⊗Y are given by di(x⊗y) = di(x)⊗y+ (−1)|x|x⊗di(y), i ≥0. This tensor product equips tCh with a closed symmetric monoidal structure with obvious associativity and unit constraints. The unit and symmetry constraints are the same as in Definition 2.7. The mapping objects are defined by the submodules HomtCh(X, Y )p,q ⊂Y s≥0 t∈Z Homk(Xs,t, Ys+p,t+q), p ≥0, q ∈Z, formed by the elements fsuch that, for i > p, dif= (−1)|f|fdi. The maps diare defined by di(f) = dif−(−1)|f|fdi. The totalisation functor on twisted complexes is strong symmetric monoidal in the obvious naive way. We will now construct the total model structure on tCh by checking that the given weak equivalences, generating cofibrations, and generating trivial cofibrations satisfy [Hov99, Theorem 2.1.29]. 22 FERNANDO MURO AND CONSTANZE ROITZHEIM These are W={f∈tCh |Tot(f) is a quasi-isomorphism in Ch}, I={∂v˜ Dp,q ֒→˜ Dp,q}p≥0 q∈Z , J={0֒→˜ D0,q}q∈Z∪ {∂v˜ Dp,q ֒→˜ Dp,q}p>0 q∈Z . Theorem 5.13. The category of twisted complexes tCh can be equipped with a proper combinatorial abelian model category structure such that: •f:X→Yis a weak equivalence if Tot(f)is a quasi-isomorphism in Ch, •fis a (trivial) fibration if it is pointwise surjective and Hv p,q(f)is an isomorphism for all p > 0(resp. p≥0) and q∈Z, •f:X֌Yis a cofibration in tCh if and only if it is injective with cofibrant cokernel. Cofibrant implies degreewise projective. Proof. This proof is essentially the same as that of Theorem 3.1. The characterisation of (trivial) fibrations in the statement follows from Lemma 5.10 instead of Lemma 2.4. (Trivial) fibrations and surjective weak equivalences can be detected by their kernels for exactly the same reason. A pushout along a generating cofibration adds copies of kin certain degrees, see Remark 5.9. Therefore, cofibrations are monomorphisms with pointwise projective cokernel. Properties (4), (5), and (6) in [Hov99, Theorem 2.1.19] follow by the same arguments, using here the spectral sequence in Remark 5.3, which is the twisted analog of that in Remark 2.11. The argument for properness is literally the same. The fact that Tot takes Ito cofibrations in Ch follows easily from Remark 5.9.  Remark 5.14.As in the total model structure on bicomplexes, a twisted complex Xis fibrant in tCh whenever its vertical homology is concentrated in horizontal degree 0, and it is trivially fibrant if the vertical homology vanishes completely. Proposition 5.15. The total model structure on tCh is monoidal, has a cofibrant tensor unit, and satisfies the monoid axiom. Proof. As tCh is an abelian model category, we can use [Hov07, Theorem 4.2] to prove monoidality. Hypothesis (1) follows from the fact that cofibrant twisted complexes are pointwise projective. The tensor unit is cofibrant since it is the cokernel of the generating cofibration ∂v˜ D0,0֒→˜ D0,0, which gives us (4). With (2) and (3) we do not proceed in the same way as in the proof of Theorem 3.1, since it would be even more complicated than what follows. By adjunction, the claims are equivalent to prove that, for any cofibrant twisted complex Xand any (trivial) fibration f, HomtCh(X, f) is a (trivial) fibration of the mapping objects, and for any trivially cofibrant twisted complex Yand any fibration f, HomtCh(Y, f) is a trivial fibration, compare [Hov99, Lemma 4.2.2]. As remarked in the proof of Theorem 3.1, it suffices to take Xto be the cokernel of a generating cofibration and to take Yto be the cokernel of a generating trivial cofibration. This strategy has the advantage that we do not have to have an explicit characterisation of the cofibrations in tCh. By Remark 5.9, those cokernels are X=∂v˜ D0,q, Y =∂v˜ Dp,q, Y =˜ D0,q, p > 0, q ∈Z. HOMOTOPY THEORY OF BICOMPLEXES 23 We start with the two easy cases. The twisted complex ∂v˜ D0,q is kconcentrated in bidegree (0, q −1), hence for any A∈tCh we have natural isomorphisms HomtCh(∂v˜ D0,q, A)s,t =As,t+q−1, Hv s,t(HomtCh(∂v˜ D0,q, A)) = Hv s,t+q−1(A). The claim for X=∂v˜ D0,q is an obvious consequence of these formulas as a map is a (trivial) fibration if and only if it is surjective and an isomorphism on Hv p,q for p > 0 (resp. p≥0). Next, since ˜ D0,q is Dqconcentrated in horizontal degree 0, we have natural isomorphisms of chain complexes, s≥0, HomtCh(˜ D0,q, A)s,∗= HomCh(Dq, As,∗), the first having differential d0. The object Dqis trivially cofibrant in Ch and hence mapping out of it preserves (trivial) fibrations in Ch. If f:A→Bis a fibration in tCh then fs,∗:As,∗−→ Bs,∗ is a fibration of chain complexes, which implies that HomCh(Dq, fs,∗) is a trivial fibration of complexes for all s≥0. Hence HomtCh(˜ D0,q, A) is a trivial fibration in tCh. Let p > 0. We now consider the most difficult case, Y=∂v˜ Dp,q, which requires more calculations. Recall the basis of the underlying bigraded module of ∂v˜ Dp,q in Definition 5.7. For each s≥0, we consider the bigraded submodule ∂v˜ Dp,q,s ⊂∂v˜ Dp,q generated by the elements yp,q−1and di1···din(yp,q−1) with i1≤s. The inclusion ∂v˜ Dp,q,s ⊂∂v˜ Dp,q is compatible with the vertical differential d0, see the formula at the beginning of the proof of Lemma 5.11. Using this basis and the definition of the mapping object tCh we see that, for all Z∈tCh, s≥0, and t∈Z, the composite HomtCh(∂v˜ Dp,q, Z)s,t ⊂Y u≥0 w∈Z Homk(∂v˜ Dp,q u,w, Zu+s,w+t)։Y u≥0 w∈Z Homk(∂v˜ Dp,q,s u,w , Zu+s,w+t) is an isomorphism. In particular, HomtCh(∂v˜ Dp,q,−) preserves surjections since ∂v˜ Dp,q,s is pointwise free. Our model structure is abelian, which means that (trivial) fibrations are exactly the surjections with (trivially) fibrant kernel. Hence, it suffices to prove that HomtCh(∂v˜ Dp,q, Z) is trivially fibrant for any fibrant twisted complex Z. A twisted complex is trivially fibrant if it has trivial vertical homology. For any s≥0, the previous isomorphism yields an identification of chain complexes HomtCh(∂v˜ Dp,q, Z)s,∗=Y u≥0 HomCh(∂v˜ Dp,q,s u,∗, Zu+s,∗), the first having differential d0again. The twisted complex ∂v˜ Dp,q is concentrated in horizontal degrees ≤p. Hence the previous product is actually indexed by 0≤u≤p. We must prove that each factor of the product is acyclic. We distinguish the possible cases. 24 FERNANDO MURO AND CONSTANZE ROITZHEIM s= 0 If s= 0, ∂v˜ Dp,q,0 p,∗is kconcentrated in vertical degree q−1 and ∂v˜ Dp,q,0 u,∗= 0 for u6=p. In the latter case, HomCh(∂v˜ Dp,q,0 u,∗, Zu,∗) = 0, and in the former, Ht(HomCh(∂v˜ Dp,q,0 p,∗, Zp,∗)) = Ht+q−1(Zp,∗) = Hv p,t+q−1(Z) = 0. This is indeed zero since p > 0 and Zis a fibrant twisted complex. s > 0, u =p, p −1 Both ∂v˜ Dp,q,s p,∗and ∂v˜ Dp,q,s p−1,∗are kconcentrated in vertical degree q−1, son we can check as right above that HomCh(∂v˜ Dp,q,s u,∗, Zu+s,∗) is acyclic. s > 0, p −s≤u≤p−2 The restriction i1≤sis empty in horizontal degrees ≥p−sin ∂v˜ Dp,q because in bidegree (p−j, q −1 + j−n) it is generated by di1···din(yp,q−1) with i1,···in>0 and i1+···+in=j. Hence, ∂v˜ Dp,q,s u,∗=∂v˜ Dp,q u,∗for u≥p−s. In the proof of Lemma 5.11, for u≤p−2 we have established an identification between ∂v˜ Dp,q u,∗and the coaugmented cochain complex of a simplex C∗(∆p−u−2, k), which is trivially cofibrant in Ch. Therefore HomCh(∂v˜ Dp,q,s u,∗, Zu+s,∗) is also acyclic for p−s≤u≤p−2. s > 0, u =p−s−1 This is the first case where the condition i1≤sis meaningful, but it only discards ds+1(yp,q−1), the topmost non-trivial free generator of the trivially cofibrant complex ∂v˜ Dp,q p−s−1,∗. Therefore, ∂v˜ Dp,q,s p−s−1,∗contains kconcentrated in vertical degree q+s−2 as a strong deformation retract. Hence, Ht(HomCh(∂v˜ Dp,q,s p−s−1,∗, Zp−1,∗)) = Ht+q+s−2(Zp−1,∗) = Hv p−1,t+q+s−2(Z) = 0, since Zis fibrant and u≥0, so p≥s+ 1 >1, i.e. p−1>0. s > 0,0≤u < p −s−1 Using the explicit identification of ∂v˜ Dp,q u,∗with C∗(∆p−u−2, k) in the proof of Lemma 5.11, we can identify the vertical subcomplexes ∂v˜ Dp,q,s u,∗∼ =C∗(∆p−u−2,∆p−u−s−2, k). This follows from the fact that, by the constraint Pn j=1 ij=p−ufor the elements di1···din(yp,q−1) of the basis of ∂v˜ Dp,q,s u,∗, the condition i1≤sis equivalent to n X j=2 i2−1≥p−u−s−1. Hence the basis elements of ∂v˜ Dp,q,s u,∗correspond to the duals of the simplices in ∆p−u−2which are not in ∆p−u−s−2. The latter simplex makes sense by the upper bound of u. This shows that ∂v˜ Dp,q,s u,∗is trivially cofibrant under the current hypotheses, so HomCh(∂v˜ Dp,q,s u,∗, Zu+s,∗) is also acyclic. Therefore, we have finally proved that tCh is monoidal. HOMOTOPY THEORY OF BICOMPLEXES 25 Lastly, the functor Tot takes generating trivial cofibrations in tCh to cofibrations in Ch, compare Remark 5.9, which are trivial by Lemmas 5.6 and 5.11. Hence the monoid axiom for tCh follows from the validity of the monoid axiom in Ch.  Proposition 5.16. The inclusion of chain complexes as twisted complexes concentrated in horizontal degree 0is the left adjoint of a strong symmetric monoidal Quillen equivalence, Ch ⇄tCh . Proof. As in the proof of Proposition 3.4, the left adjoint obviously preserves the tensor product and the tensor unit, and the right adjoint is tCh →Ch: X7→ X0,∗. Clearly, this right adjoint preserves (trivial) fibrations. This shows that the adjoint pair is a Quillen pair. The same argument as in Proposition 3.4 shows that it is a Quillen equivalence. Here we should use the spectral sequence in Remark 5.3 instead of Remark 2.11.  Corollary 5.17. The adjoint pair in Remark 5.3 defines a strong symmetric monoidal Quillen equivalence tCh ⇄bChTot . Proof. The right adjoint, which is the full inclusion bCh ⊂tCh, obviously preserves (trivial) fibrations, so the adjoint pair in the statement is a Quillen pair. This Quillen pair and the Quillen equivalence in Proposition 5.16 Ch ⇄tCh ⇄bChTot compose to the Quillen equivalence in Proposition 3.4. Hence the corollary follows from the 2-out-of-3 property for Quillen equivalences [Hov99, Corollary 1.3.15].  References [Bro67] R. Brown, The twisted Eilenberg-Zilber theorem, p. 33?37, 1967. [CE56] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480 [CESLW17] J. Cirici, D. Egas Santander, M. Livernet, and S. Whitehouse, Derived A∞-algebras and their homotopies, arXiv:1609.08077 (2017), 1–47. [GM03] S. I. Gelfand and Y. I. Manin, Methods of homological algebra, second ed., Springer Monographs in Mathematics, Springer-Verlag, 2003. [Hir03] P.S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR MR1944041 (2003j:18018) [Hov99] M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. [Hov02] ,Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553–592. MR 1938704 [Hov07] ,Cotorsion pairs and model categories, Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 277–296. MR 2355778 [Kad80] T. V. Kadeishvili, On the theory of homology of fiber spaces, Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 35 (1980), no. 3(213), 183?188, International Topology Conference (Moscow State Univ., Moscow, 1979). [Liu67] A. Liulevicius, Multicomplexes and a general change of rings theorem, Mimeographed notes, Univ. of Chicago., 1967. [LRW13] M. Livernet, C. Roitzheim, and S. Whitehouse, Derived A∞-algebras in an operadic context, Algebr. Geom. Topol. 13 (2013), no. 1, 409–440. MR 3031646 [McC01] J. McCleary, A user’s guide to spectral sequences, second ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR MR1793722 (2002c:55027)