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Derived A-infinity algebras and their homotopies

The notion of a derived A-infinity algebra, considered by Sagave, is a generalization of the classical notion of A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy, for non-negative integers r, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as multicomplexes) first since it is of independent interest and this simpler case clearly exemplifies the structure we study. We also give two new interpretations of derived A-infinity algebras as A-infinity algebras in twisted complexes and as A-infinity algebras in split filtered cochain complexes.


Introduction
The homotopy invariant version of an associative algebra, known as an A ∞ -algebra, has become an important idea in many areas of mathematics, including algebra, geometry and mathematical physics. An introduction to these structures and a discussion of various applications can be found in Keller's survey [Kel01]. Kadeishvili's work on minimal models used the existence of A ∞ -structures in order to classify differential graded algebras over a field up to quasi-isomorphism [Kad80].
The homotopy theory of these algebras has been studied by several authors, including Prouté, Grandis and Lefèvre-Hasegawa [Pro11], [Gra99], [LH03]. Grandis gave a notion of homotopy between morphisms of A ∞ -algebras via a functorial path construction. Working over a field, Lefèvre-Hasegawa establishes the structure of a "model category without limits" on the category of A ∞ -algebras.
In order to formulate a generalization of Kadeishvili's work over a general commutative ground ring, Sagave developed the notion of derived A ∞ -algebra [Sag10]. Sagave establishes the existence of minimal models for differential graded algebras (dgas) by showing that the structure of a derived A ∞ -algebra arises on some projective resolution of the homology of a differential graded algebra. Furthermore, the dga can be recovered up to quasi-isomorphism from this data.
In [LRW13] an operadic description of derived A ∞ -algebras was developed, working with nonsymmetric operads in the category vbC R of bicomplexes with zero horizontal differential. There is an operad dAs in this category encoding bidgas, which are simply monoids in bicomplexes. Derived A ∞ -algebras are precisely algebras over the operad dA ∞ = (dAs) ∞ = Ω((dAs) ¡ ).
Here (dAs) ¡ is the Koszul dual cooperad of the operad dAs, and Ω denotes the cobar construction. Further development of the operadic theory of these algebras was carried out in [ALR + 15]. The recent PhD thesis of Maes [Mae16] studies derived P ∞ -algebras, replacing the associative operad As with a suitable operad P .
A derived A ∞ -algebra has an underlying twisted complex, also known as a multicomplex or D ∞algebra. Twisted complexes arise as a natural generalization of the notion of double complex by considering a family of "differentials" indexed over the non-negative integers. These objects were first considered by Wall [Wal61] in his work on resolutions for extensions of groups and subsequently they have arisen in the work of many authors. They were studied by Gughenheim and May [GM74] in their approach to differential homological algebra. Meyer [Mey78] introduced homotopies between morphisms of twisted complexes and proved an acyclic models theorem for these objects. More recently, twisted complexes have proven to be an important tool in homological perturbation theory (see for example [Hue04]). Saneblidze [San07] introduced projective twisted complexes and showed that every (possibly unbounded) chain complex over an abelian category A is weakly equivalent to a projective multicomplex, provided that A has enough projectives and countable coproducts. This result provides a good description of the derived category of A. The work of Sagave on minimal models for differential graded algebras can be thought of as a multiplicative enhancement of Saneblidze's result [San07]. For twisted complexes over a field, Lapin [Lap03] gave a homotopy transfer theorem (see also [DSV15]).
In this paper, we initiate a study of the homotopy theory of derived A ∞ -algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy for r ≥ 0 and consider a related notion of E r -quasi-isomorphism. Denoting the set of r-homotopy equivalences by S r and the set of E r -quasi-isomorphisms by E r , we have the following inclusions.
We treat the special case of twisted complexes first, since it is of independent interest and the theory is simpler in this case. Every twisted complex has an associated spectral sequence, defined via the column filtration of its total complex. In fact, the totalization functor gives rise to an isomorphism of categories between the category of twisted complexes and the full subcategory of filtered complexes whose objects have split filtrations (Theorem 3.8). The class of E r -quasi-isomorphisms is given by those morphisms of twisted complexes inducing a quasi-isomorphism on the r-th stage of their associated spectral sequence. The notion of r-homotopy for twisted complexes that we consider corresponds to the notion of homotopy of order r introduced in [CE56] and further developed in [CG16] in the context of filtered complexes. We study the localized category of twisted complexes with respect to r-homotopies (Theorem 3.26). We present several equivalent formulations of r-homotopy: via a functorial path, via explicit formulas and an operadic approach (Theorem 3.37).
A substantial part of the paper is devoted to developing new interpretations of derived A ∞ -algebras. As well as being interesting in themselves, these new viewpoints are used in establishing properties of homotopy and they provide a key idea for the proof of the equivalence of the various formulations of r-homotopy in the derived A ∞ case. Firstly, we show that derived A ∞ -algebras can be interpreted as A ∞ -algebras in twisted complexes (Theorem 4.47). This formulation has the potential to be a very useful tool for the future development of different aspects of the theory of derived A ∞ -algebras. Secondly, under suitable boundedness conditions, we show that derived A ∞ -algebras can be viewed as split filtered A ∞ -algebras (Theorem 4.53). This result allows one to transfer known constructions in the category of A ∞ -algebras to the category of derived A ∞ -algebras, by checking compatibility with filtrations.
The context for these new interpretations is the theory of operadic algebras for monoidal categories over a base, as developed by Fresse [Fre09]. We endow the categories of twisted complexes and filtered complexes with a monoidal structure over the category of vertical bicomplexes and use this to enrich them over vertical bicomplexes. This allows us to formulate these algebra structures by means of an enriched endomorphism operad. The totalization functor extends to the enriched setting and gives an isomorphism between the vertical bicomplexes-enriched categories of twisted complexes and split filtered complexes (Theorem 4.37). We use this isomorphism to show that the different intepretations of derived A ∞ -algebras are equivalent.
We then turn to r-homotopy for derived A ∞ -algebras. Here, the class of E r -quasi-isomorphisms is defined by lifting E r -quasi-isomorphisms of the underlying twisted complexes. The notion of rhomotopy that we consider arises as a combination of the notion of r-homotopy for twisted complexes and the classical notion of homotopy between morphisms of A ∞ -algebras. We again present different approaches: via a functorial path construction, via explicit formulas and in operadic terms. For the operadic description we formulate the general notion of a (g, f )-coderivation for morphisms g, f of cofree coalgebras over a (non-symmetric) operad. In Theorem 5.31, we show that the different approaches are equivalent.
The results of this paper set up the foundations for a homotopy theory of derived A ∞ -algebras, contextualizing the ad-hoc notion of homotopy introduced by Sagave in a very particular case and generalizing the work of Grandis on functorial paths for A ∞ -algebras. We expect that both the new descriptions of derived A ∞ -algebras and the properties of homotopies developed here will allow us to endow the category of derived A ∞ -algebras with the structure of a model category without limits in the future, with weak equivalences being E r -quasi-isomorphisms.
The paper is organized as follows. Section 2 covers background material introducing some of the categories we work with. The notion of r-homotopy for twisted complexes is covered in Section 3. Our new interpretations of derived A ∞ -algebras are presented in Section 4. Finally, Section 5 studies r-homotopy for derived A ∞ -algebras.
Acknowledgements. The authors would like to thank the Banff International Research Station and the organizers of the Women in Topology workshop in April 2016 for bringing us together to work on this paper. We would also like to thank Gabriel Drummond-Cole, Fernando Muro, Emily Riehl, Agustí Roig and Sinan Yalin for helpful conversations.

Preliminaries
We first set up some notation which we will use throughout this paper.
Notation 2.1. Throughout this paper R will denote a commutative ring with unit. Unless stated otherwise, all tensor products will be taken over R. Let C be a category and A, B be arbitrary objects in C. We denote by Hom C (A, B) the set of morphisms from A to B in C. If (C, ⊗, 1) is symmetric monoidal closed, then we denote its internal hom-object by [ which is natural in A, B and C.
2.1. Filtered modules and filtered cochain complexes. We collect some algebraic preliminaries about filtered R-modules and filtered complexes of R-modules. Our complexes are cochain complexes and our filtrations are increasing.
Definition 2.2. A filtered R-module (A, F ) is given by a family of R-modules {F p A} p∈Z indexed by the integers such that F p−1 A ⊆ F p A for all p ∈ Z and A = ∪F p A. A morphism of filtered modules is a morphism f : A → B of R-modules which is compatible with filtrations: f (F p A) ⊂ F p B for all p ∈ Z.
Definition 2.3. The tensor product of two filtered R-modules (A, F ) and (B, F ) is a filtered Rmodule, with This makes the category of filtered R-modules into a symmetric monoidal category, where the unit is given by R with the trivial filtration 0 = F −1 R ⊂ F 0 R = R.
Denote by C R the category of cochain complexes of R-modules. The standard tensor product endows this with a symmetric monoidal structure, with unit R concentrated in degree zero.
We denote by fC R the category of filtered complexes of R-modules. Its morphisms are given by morphisms of complexes f : K → L compatible with filtrations: f (F p K) ⊂ F p L for all p ∈ Z. It is a symmetric monoidal category, with the filtration on the tensor product defined as above.
We next recall the notion of homotopy of order r due to Cartan and Eilenberg.
Definition 2.5. [CE56,p321] Let f, g : (K, F ) −→ (L, F ) be two morphisms of filtered complexes and let r ≥ 0 be an integer. A homotopy of order r from f to g is a map of graded R-modules H : K * → L * −1 such that dH + Hd = g − f and H(F p K n ) ⊂ F p+r L n−1 for all p, n ∈ Z.
Remark 2.6. Every filtered complex (K, d, F ) has an associated spectral sequence {E r (K), δ r } which is functorial for morphisms of filtered complexes. Given a homotopy of order r between morphisms of filtered complexes f, g : (K, F ) → (L, F ), the induced morphisms at the k > r stages of the spectral sequences coincide: f * k = g * k : E k (K) → E k (L) for all k > r (see [CE56,Proposition XV.3.1]). This result indicates how the notion of homotopy of order r is the suitable notion to study the r-derived category defined by inverting those morphisms of filtered complexes which induce an isomorphism at the E r+1 -stage of the associated spectral sequences (see [Par96], [CG16]).
2.2. Bigraded modules, vertical bicomplexes, sign conventions. We consider (Z, Z)-bigraded R-modules A = {A j i }, where elements of A j i are said to have bidegree (i, j). We sometimes refer to i as the horizontal degree and j the vertical degree. The total degree of an element a ∈ A j i is |a| = j − i. A morphism of bidegree (p, q) maps A j i to A j+q i+p . The tensor product of two bigraded R-modules A and B is the bigraded R-module A ⊗ B given by We introduce the following scalar product notation for bidegrees: for x, y of bidegree (x 1 , x 2 ), (y 1 , y 2 ) respectively, we let x, y = x 1 y 1 + x 2 y 2 . We follow the Koszul sign rule: if f : A → B and g : C → D are bigraded morphisms, then the morphism f ⊗ g : We denote by bgMod R the category of bigraded R-modules. It is symmetric monoidal with the above tensor product.
We will use a standard (vertical) shift S of bigraded modules, following Sagave's conventions, as in the first part of [Sag10,Section 4]. So S is the shift of bidegree (0, 1); it is an endofunctor on bigraded R-modules, where S(A) j i = A j+1 i and on morphisms Sf = (−1) v f , if f has bidegree (u, v). We write σ for the corresponding canonical isomorphism of bidegree (0, 1) from S to the identity; this means that Then Ψ k is the induced isomorphism from homomorphisms on a k-fold tensor power: Definition 2.7. A vertical bicomplex is a bigraded R-module A equipped with a vertical differential d A : A −→ A of bidegree (0, 1). A morphism of vertical bicomplexes is a morphism of bigraded modules commuting with the vertical differential.
We denote by vbC R the category of vertical bicomplexes. The tensor product of two vertical bicomplexes A and B is given by endowing the tensor product of underlying bigraded modules with vertical differential d A⊗B : u . This makes vbC R into a symmetric monoidal category.
The symmetric monoidal categories (C R , ⊗, R), (bgMod R , ⊗, R) and (vbC R , ⊗, R) are related by embeddings C R ֒→ vbC R and bgMod R ֒→ vbC R which are monoidal and full.
This gives an internal hom on vbC R , making it symmetric monoidal closed. It restricts to give the standard internal hom on the categories bgMod R and C R .
We denote by vbC R , C R , and bgMod R , the categories of vertical bicomplexes, complexes, and bigraded modules respectively, enriched over themselves via their symmetric monoidal closed structure.

Twisted complexes and r-homotopy
In this section, we recall some key properties of the category of twisted complexes. We define the totalization functor from twisted complexes to filtered cochain complexes and show that it induces an isomorphism of categories onto its image. We then introduce r-homotopies between morphisms of twisted complexes, consider their interplay with spectral sequences and study the localized category tC R [S −1 r ]. 3.1. The category of twisted complexes.
The composition of morphisms is given by (g • f ) m := i+j=m g i f j . A morphism f = {f m } m≥0 is said to be strict if f i = 0 for all i > 0. The identity morphism 1 A : A → A is the strict morphism given by 1 A 0 (x) = x. A morphism f = {f i } is an isomorphism if and only if f 0 is an isomorphism of bigraded R-modules. Indeed, an inverse of f is obtained from an inverse of f 0 by solving a triangular system.
Denote by tC R the category of twisted complexes of R-modules. Also, denote by bgMod ∞ R the full subcategory of tC R on twisted complexes with trivial structure i.e., d m = 0 for all m ≥ 0.
The following construction endows tC R with a symmetric monoidal structure.
Lemma 3.3. The category (tC R , ⊗, R) is symmetric monoidal, where the monoidal structure is given by the bifunctor and on morphisms is given by In particular, by the Koszul sign rule we have that . This functor describes a symmetric monoidal structure on bgMod ∞ R by restriction.
Similarly, one checks that f ⊗ g is a morphism of twisted complexes. It only remains to see that this construction is functorial. A direct computation shows that We extend the internal hom on bigraded modules to twisted complexes.
* with the structure of a twisted complex. Proof. It is a matter of calculation that into a twisted complex. This construction gives an internal hom and we denote by tC R the category of twisted complexes enriched over itself via this symmetric monoidal closed structure.
Definition 3.5. Let r ≥ 0 be a non-negative integer. An r-bigraded complex is a twisted complex (A, d m ) such that d m = 0 for all m = r.
Note that in this case, condition (A m1 ) implies that d r d r = 0. For r = 0, this coincides with the notion of vertical bicomplex. Denote by r-tC R the full subcategory of tC R of r-bigraded complexes. The prototypical example of such an object is given by the r-th term of the spectral sequence associated with a twisted complex, as we will see in Section 3.3.

3.2.
Total cochain complex of a twisted complex.
Definition 3.6. The total graded R-module Tot(A) of a bigraded R-module A = {A j i } is given by The column filtration of Tot(A) is the filtration given by F p Tot(A) n := i≤p A n+i i for all p, n ∈ Z.
We will show that, via the totalization functor, the category of twisted complexes is isomorphic to a full subcategory of that of filtered complexes. We use the following.
Definition 3.7. A filtered complex (K, d, F ) is said to be split if K = Tot(A) is the total graded module of a bigraded R-module A = {A j i } and F is the column filtration of Tot(A). We denote by sfC R the full subcategory of fC R whose objects are split filtered complexes.
Given a twisted complex (A, d m ), define a map d : Tot(A) → Tot(A) of degree 1 by letting where a i ∈ A n+i i denotes the i-th component of a, and d(a) j denotes the j-th component of d(a). Note that, for a given j ∈ Z there is a sufficiently large m ≥ 0 such that a j+m ′ = 0 for all m ′ ≥ m. Hence d(a) j is given by a finite sum. Also, for j sufficiently large, one has a j+m = 0 for all m ≥ 0, which implies d(a) j = 0.
Given a morphism f : Theorem 3.8. The assignments (A, d m ) → (Tot(A), d, F ), where F is the column filtration of Tot(A), and f → Tot(f ) define a functor Tot : tC R −→ fC R which is an isomorphism of categories when restricted to its image sfC R .
Proof. Let (A, d m ) be a twisted complex and let a = (a i ) i∈Z ∈ Tot(A) n . To see that (Tot(A), d) is a complex it suffices to note that: One easily verifies that F p−1 Tot(A) n ⊂ F p Tot(A) n and that d( Note that Tot(f ) is compatible with the filtration F and that Tot(f g) = Tot(f )Tot(g). This proves that Tot is a functor with values in the category of split filtered complexes. We next define a functor Tot −1 : sfC R → tC R inverse to the restriction of Tot onto its image. Let Since f is compatible with the filtration F , we have that f i = 0 for i < 0. Then the family {f m } m≥0 is a morphism of twisted complexes whose total morphism is f . It is straightforward to see that the above constructions define an inverse functor to the restriction of Tot.
Remark 3.9. Strict morphisms of twisted complexes correspond, via the above isomorphism of categories, to strict morphisms of split filtered complexes, that is, morphisms preserving the splittings.
We will also consider the following bounded versions of our categories, since the totalization functor has better properties when restricted to these. Definition 3.10. We let tC b R , vbC b R , bgMod b R be the full subcategories of (N, Z)-graded twisted complexes, vertical bicomplexes and bigraded modules respectively. We let fMod b R , sfMod b R , fC b R , sfC b R be the full subcategories of (split) non-negatively filtered modules, respectively complexes, i.e. the full subcategories of objects (K, F ) such that F p K n = 0 for all p < 0. We refer to all of these as the bounded subcategories of tC R , vbC R , bgMod R , fMod R , sfMod R , fC R and sfC R respectively.
In the following proposition, we show that the monoidal structures of twisted complexes and filtered complexes are compatible under the totalization functor.
Proposition 3.11. The functors Tot : bgMod R → fMod R and Tot : tC R → fC R are lax symmetric monoidal, with structure maps and given by ǫ = 1 R and for a = (a i ) i ∈ Tot(A) n 1 and b = (b j ) j ∈ Tot(B) n 2 , When restricted to the bounded case, the functors Tot : bgMod b R → fMod b R and Tot : tC b R → fC b R are strong symmetric monoidal functors.
Proof. Clearly ǫ is a map of filtered complexes and a direct computation shows that the same is true for µ A,B . We now show that µ A,B respects the symmetric structure i.e., that Diagram (1) commutes.
Let a ⊗ b ∈ Tot(A) n 1 ⊗ Tot(B) n 2 , with n 1 + n 2 = n. Then The fact that µ A,B defines a natural transformation, i.e. that Diagram (2) commutes, follows directly by restriction from the proof of Proposition 4.38. The coherence axioms are left to the reader. In the bounded case Tot is strong symmetric monoidal since ⊗ distributes over ⊕, therefore the natural transformation µ is a natural isomorphism.
We summarize the categories we study and their relations in the following commutative diagram.
All hooked arrows are embeddings; arrows with a * are full embeddings. We embed bigraded modules in vertical bicomplexes and twisted chain complexes by assigning them trivial differentials. The forgetful functors U forget the differential structure. All functors are strong symmetric monoidal, except for Tot. This is strong symmetric monoidal when restricted to the full subcategories of bounded objects; it is only lax symmetric monoidal otherwise.
3.3. Spectral sequence associated to a twisted complex. Every twisted complex (A, d m ) has an associated spectral sequence , which is functorial for morphisms of twisted complexes. Denote by δ r the differential of the r-th term. We choose the bigrading in such a way that for all r ≥ 0, the pair (E r (A, d m ), δ r ) is an r-bigraded complex, so we have a functor E r : , δ r−1 ) and the map δ r depends on the maps d m for m ≤ r.
The map δ r is induced by d r only on those classes that have a representative a ∈ A j i for which d k (a) = 0 for all k < r. In particular, for r = 1 we have The morphism of spectral sequences is given by E 0 (f ) = f 0 and E r (f ) = H(E r−1 (f ), δ r−1 ) for r ≥ 1. In particular, for r = 1 we have E 1 (f ) = H d 0 (f 0 ). We refer to [Boa99] and [Hur10] for further properties of the spectral sequence associated to a twisted complex.
For the rest of this section, let r ≥ 0 be an integer. We shall consider the following notion of weak equivalence in the category of twisted complexes.
at the r-stage of the associated spectral sequence is a quasi-isomorphism of r-bigraded complexes (that is, E * , * r+1 (f ) is an isomorphism). Denote by E r the class of E r -quasi-isomorphisms of tC R . This class is closed under composition and contains all isomorphisms of tC R . Denote by Ho r (tC R ) := tC R [E −1 r ] the r-homotopy category of twisted complexes defined by inverting E r -quasi-isomorphisms. Since E r ⊂ E r+1 for all r ≥ 0, we have a chain of functors Ho 0 (tC R ) −→ Ho 1 (tC R ) −→ · · · −→ Ho r (tC R ) −→ · · · Remark 3.13. The class E 1 of E 1 -quasi-isomorphisms corresponds to the class of weak multiequivalences defined by Huebschmann [Hue04] and the class of E 2 -equivalences considered by Sagave [Sag10].
3.4. r-homotopies and r-homotopy equivalences. We next define a collection of functorial paths indexed by an integer r ≥ 0 on the category of twisted complexes, giving rise to the corresponding notions of r-homotopy.
Definition 3.14. The r-path of a twisted complex (A, d m ) is the twisted complex given by We have strict morphisms of twisted complexes given by ∂ − A (x, y, z) = x, ∂ + A (x, y, z) = z and ι A (x) = (x, 0, x). We will denote by ∂ 0 A : P r (A) → A the map of bidegree (r, r − 1) given by (x, y, z) → y. We will often omit the subscripts of these maps when there is no danger of confusion.
Definition 3.15. The r-path of a morphism f : The above definitions give rise to a functorial path P r : tC R → tC R in the category of twisted complexes. This gives a natural notion of homotopy.
Definition 3.16. Let f, g : A → B be two morphisms of twisted complexes. An r-homotopy from f to g is given by a morphism of twisted complexes h : A → P r (B) such that ∂ − B • h = f and ∂ + B • h = g. We use the notation h : f ≃ r g. Remark 3.17. Let Λ r be the r-bigraded complex generated by e − , e + in bidegree (0, 0) and u in bidegree (−r, 1 − r), with the differential δ r (e − ) = −u, δ r (e + ) = u and δ i (e ± ) = 0 for all i = r. Then the assignment (x, y, z) → e − ⊗ x + u ⊗ y + e + ⊗ z defines a strict isomorphism of twisted complexes from the r-path (P r (A), D m ) of a twisted complex (A, d m ) to the twisted complex ( be two morphisms of twisted complexes. Giving an r-homotopy h : f ≃ r g is equivalent to giving a collection of morphisms h m : It is a matter of verification to see that the family { h m } m≥0 satisfies (H m1 ) for all m ≥ 0. Conversely, one may check that given a family { h m } m≥0 satisfying (H m1 ), then the family h m (x) : Remark 3.19. For r = 1 we recover the notion of homotopy between morphisms of twisted complexes first introduced by Meyer [Mey78], also considered by Saneblidze [San07] and Huebschmann [Hue04].
be morphisms of twisted complexes. Giving an r-homotopy h : f ≃ r g is equivalent to giving a homotopy of order r H : Proof. Given an r-homotopy h : A → P r (B) from f to g we obtain a morphism of filtered complexes Proposition 3.21. The notion of r-homotopy defines an equivalence relation on the set of morphisms between two given twisted complexes, which is compatible with the composition.
Proof. The homotopy relation defined by a functorial path is reflexive and compatible with the composition (see for example [ Using the equivalent notion of r-homotopy of Proposition 3.18 we get an r-homotopy h ′′ by letting Denote by S r the class of r-homotopy equivalences of tC R . This class is closed under composition and contains all isomorphisms.
Proposition 3.23. For all r ≥ 0, we have S r ⊂ S r+1 .
Proof. Using the equivalent notion of homotopy of Proposition 3.18, it is straightforward to see that given an r-homotopy h from f to g, we obtain an (r + 1)-homotopy h ′ from f to g by letting h ′ 0 = 0 and h ′ m = h m−1 for m > 0.
Proposition 3.24. For all r ≥ 0, we have S r ⊂ E r .
Proof. By Lemma 3.20, an r-homotopy from f to g in tC R gives a chain homotopy H from Tot(f ) to Consider the morphisms h m : P r (A) → P r (A) of bidegree (−m + r, −m + r − 1) defined by h 0 (x, y, z) = (0, 0, y) and h i = 0 for all i > 0. It only remains to verify condition (H r1 ) of Proposition 3.18. We have Theorem 3.26. The localized category tC R [S −1 r ] is canonically isomorphic to the quotient category π r (tC R ) := tC R / ≃ r .
Proof. Denote by γ r : tC R → tC R [S −1 r ] the localization functor. It suffices to show that if h : f ≃ r g then γ r (f ) = γ r (g) (see [GNPR10, Proposition 1.3.3]). Consider the following diagram of morphisms of twisted complexes. B By Lemma 3.25 the morphism ι B is an r-homotopy equivalence. Hence the above diagram is a hammock between the S r -zigzags f and g in the sense of [DHKS04]. This gives f = g in tC R [S −1 r ]. 3.5. The r-translation and the r-cone. The r-path construction is related to a translation functor depending on r (see [CG16] and [CG14] for similar constructions in the categories of filtered complexes and filtered commutative dgas respectively). Furthermore, the cone obtained via this translation allows one to detect E r -quasi-isomorphisms, as we shall see next.
Definition 3.28. The r-cone of a morphism f : where we adopt the convention that f <0 = 0.
and (a, b) → a respectively. These fit into a short exact sequence The following is a matter of verification.
Lemma 3.29. Let w : A → B be a morphism of twisted complexes and X a twisted complex. Giving a morphism τ : C r (w) → X of twisted complexes is equivalent to giving a pair (f, h) where f : B → X is a morphism of twisted complexes and h : 0 ≃ r f w is an r-homotopy from 0 to f w. Proof. Let τ : C r (w) → X be a morphism of twisted complexes. Define a morphism of twisted complexes f : B → X by letting f m (b) := τ m (0, b). Let h m : A → X be defined by h m (a) := (−1) m τ m (a, 0). By Proposition 3.18 this gives an r-homotopy h from 0 to f w. Conversely, given Proposition 3.30. Let r ≥ 0 and let f : be a morphism of twisted complexes. We have a long exact sequence In particular, the morphism f is an E r -quasi-isomorphism if and only if the r-cone of f is E r -acyclic, that is, E * , * r+1 (C r (f )) = 0. Proof. For every p we have a short exact sequence of complexes → 0, which induces a long exact sequence in cohomology. This proves the result for r = 0. Assume that , so the contribution of f to the differential vanishes. This gives a direct sum decomposition E p,q r (C r (f )) ∼ = E p−r,q−r+1 r (A) ⊕ E p,q r (B) inducing a long exact sequence in cohomology.
3.6. Operadic approach. In this section we recall how to view twisted complexes as algebras over the operad D ∞ . We then study r-homotopy from this point of view.
Let D be the operad of dual numbers in vertical bicomplexes. Here D = R[ǫ]/(ǫ 2 ), where the bidegree of ǫ is (−1, 0). This has trivial vertical differential and contains only arity one operations, so it can be thought of as simply a bigraded R-algebra.
The category of twisted complexes tC R is isomorphic to the category of D ∞ -algebras in vertical bicomplexes (see [LRW13,Section 3.1] or [LV12,10.3.17] for the singly-graded analogue). Using the so-called Rosetta Stone [LV12], this means that twisted complexes can be studied via structure on cofree coalgebras over the Koszul dual cooperad D ¡ .
We first recall some details from [ALR + 15, 3.4] about D ¡ -coalgebras. We then make explicit how twisted complexes and their morphisms may be encoded via cofree coalgebras, before putting rhomotopies into this context.
The Koszul dual D ¡ of D is again concentrated in arity one and can be thought of as just an Rcoalgebra. We have A D ¡ -coalgebra is a (left)-comodule C over this coalgebra and this turns out to just be a pair (C, f ), where C is an R-module and f is a linear map f : C → C of bidegree (1, 1). (Given a coaction ρ : In particular, if d has bidegree (0, 1) then it anti-commutes with f . Remark 3.31. As an example, the cofree conilpotent D ¡ -coalgebra generated by a bigraded module A is given by It will be useful to introduce the following basic object.
Definition 3.32. Let A, B, C be bigraded modules. We denote by bgMod R (A, B) the bigraded module given by is the inner hom-object of bigraded modules. More precisely, g ∈ bgMod R (A, B) v u is given by g := (g 0 , g 1 , g 2 , . . . ), where g j : A → B is a map of bigraded modules of bidegree (u − j, v − j). Moreover, we define a composition morphism In the next section, we will develop this much further, in particular defining the enriched category bgMod R .
We explain how structure in the world of D ¡ -coalgebras corresponds to the explicit twisted complex notions. We write U for the forgetful functor from D ¡ -coalgebras to bigraded modules, left adjoint to the cofree coalgebra functor D ¡ .
Proposition 3.33. Let A, B and C be bigraded modules.
(1) We have bijections of underlying sets Here a map of bidegree (u, v) of bigraded modules F : Furthermore if B = A then F is also a coderivation of D ¡ -coalgebras. We associate to such a map F the collection of maps f n : A → B given by f n (a) = F (x n ⊗ a).
, corresponds to composition of morphisms of twisted complexes. Proof.
(1) As above, let ∆ : Then one checks easily that ∆ F = D ¡ ( F )∆, for F to be a morphism or a coderivation.
One obtains the map f m : A → B by setting f m (a) = π B F (x m ⊗ a) and similarly for any (4) One checks the statement about composition similarly.
Remark 3.34. The fact that the condition is the same to be a morphism of coalgebras as to be a coderivation arises because the cooperad D ¡ has only unary operations.
In order to formulate r-homotopy in this context we use a certain kind of shift operation on morphisms.
We write S r for the r-th iterate of this operation.
) v u under the bijection of Proposition 3.33, then Sf corresponds to F d A x . Proof. Let G be the map corresponding to Sf . Then, for all n ≥ 0 and all a ∈ A, which is equivalent to the r-homotopy condition, by Proposition 3.18.

New interpretations of derived A ∞ -algebras
In this section we reinterpret derived A ∞ -algebras both as A ∞ -algebras in twisted chain complexes and as split filtered A ∞ -algebras. First we recall the basic notions regarding derived A ∞ -algebras. Next we endow the categories of twisted complexes and filtered complexes with a monoidal structure over a base, in the sense of Fresse [Fre09], and explicitly describe the enrichments that these structures induce. Then we show that the totalization functor and its properties extend to this enriched setting and use this to prove our main results.
4.1. The category of derived A ∞ -algebras. We begin by recalling the basic definitions for derived A ∞ -algebras, also known as dA ∞ -algebras. Then the composite morphism f g : A → C of dA ∞ -algebras has components Proof. This is a direct consequence of Equations (4) and (5) in [LRW13, Theorem 2.8].
Proof. If f g = 1, then f 01 g 01 = 1, so f 01 is an isomorphism and g 01 = f −1 01 . Then the equation giving the (uk) component of the composite has a "top term" f 01 g uk , with all other summands involving components g ij with i < u or j < k. Thus we can successively solve for each g uk if and only if f 01 is an isomorphism. This gives a right inverse g for f if and only if f 01 is an isomorphism. The same argument shows that g also has a right inverse, and this must be f , so f and g are two-sided inverses.
Denote by dA ∞ (R) the category of dA ∞ -algebras over R.
Example 4.6 (Underlying twisted complex). There is a forgetful functor U : dA ∞ (R) −→ tC R defined by sending a dA ∞ -algebra (A, m ij ) to the twisted complex (A, m i1 ) and a morphism f = {f ij } of dA ∞ -algebras to the morphism of twisted complexes given by We define E r -quasi-isomorphism for dA ∞ -algebras via their underlying twisted complexes.
Denote by E r the class of E r -quasi-isomorphisms of dA ∞ (R) and by Ho The forgetful functor induces a functor U : Ho r (dA ∞ (R)) −→ Ho r (tC R ).

4.2.
Monoidal categories over a base. In the following sections, all our notions of enriched category theory follow [Bor94]. Our new descriptions of dA ∞ -algebras will use certain enriched categories coming from monoidal categories over a base as defined in [Fre09]. We recall this notion first.
Definition 4.8. Let (V , ⊗, 1) be a symmetric monoidal category and (C, ⊗, 1) be a monoidal category. We say that C is a monoidal category over V if we have an external tensor product * : V × C → C such that we have natural isomorphisms: Remark 4.9. If we have, in addition, a bifunctor C (−, −) : (for example, if * preserves colimits on the left and certain smallness conditions hold) we get a Venriched category C with the same objects as C and with hom-objects given by C (−, −). The unit morphism u A : 1 → C (A, A) corresponds to the identity map in C under the adjunction and the composition morphism is given by the adjoint of the composite where ev AB is the adjoint of the identity C (A, B) → C (A, B). Note that by construction, the underlying category of C is C. Furthermore, C is a monoidal V -enriched category, namely we have an enriched functor where C × C is the enriched category with objects Ob(C ) × Ob(C ) and hom-objects In particular we get maps in V given by the adjoint of the composite The monoidal structures of vertical bicomplexes and twisted complexes are compatible. More precisely, we can use the tensor product in twisted complexes to define monoidal structures over a base by restriction.
Lemma 4.10. The category tC R is a monoidal category over vbC R and the category bgMod ∞ R is a monoidal category over bgMod R . In both cases, the external tensor product is given by restricting the tensor product in twisted complexes. We use the notation ⊗ instead of * since the external tensor product coincides with the internal tensor product in tC R .
Proof. The axioms of Definition 4.8 hold because ⊗ is a symmetric monoidal structure on tC R and the vertical arrows in the following diagram are monoidal embeddings.
We can also endow filtered complexes with a monoidal structure over vbC R . We use the following.  Lemma 4.13. The category fC R is monoidal over vbC R with external tensor product given by * : On morphisms it is given by the assignment (f, g) → Tot c (f ) ⊗ g. This induces by restriction a monoidal structure on fMod R over bgMod R .
Proof. The assignments define a bifunctor since Tot c : vbC R → fMod R is a functor. Furthermore, the axioms of Definition 4.8 hold since ⊗ is a symmetric monoidal product in fC R and Tot c is strong symmetric monoidal since ⊗ distributes over ⊕. Finally, since fMod R ֒→ fC R and bgMod R ֒→ vbC R are full embeddings, this construction induces by restriction a bifunctor * : bgMod R ×fMod R → fMod R which gives a monoidal structure on fMod R over bgMod R .
4.3. Enrichments from monoidal structures over a base. We now explain how to give enrichments to the categories of bigraded modules, filtered modules, twisted complexes and filtered complexes, using their monoidal structure over a base. This will give the vbC R -enriched categories tC R and fC R and bgMod R -enriched categories bgMod R and fMod R . We emphasize that bgMod R is different from its standard enrichment coming from its symmetric monoidal closed structure. Recall from Definition 3.32 that for bigraded modules A, B, C, we have already defined a bigraded module bgMod R (A, B) and a composition Lemma 4.14. The composition morphism respects the identity and is associative. c(g, h)).
where < f, d A > is the scalar product for the bidegrees and c is the composition morphism described in Definition 3.32. More precisely, Lemma 4.16. The following equations hold where the bidegree of f is (u, v). Proof. One has The last equation follows from the associativity of c.
Since δ is of bidegree (0, 1), Lemma 4.16 allows us to make the following definition. and Moreover, the functor − ⊗ B : vbC R → tC R is left adjoint to the functor tC R (B, −) : tC R → vbC R , i.e., for all A ∈ vbC R , B, C ∈ tC R we have natural isomorphisms given by f →f where for a ∈ A v u ,f (a) m is given by b → (−1) m|a| f m (a ⊗ b).
Proof. If f is a map in tC R then it is of bidegree (0, 0) and δ(f ) = 0 and thus by (2) δ(c(f, g)) = c(δ(f ), g) + (−1) v c(f, δ(g)) = c(f, δ(g)), showing that tC R (B, f ) is a map of vertical bicomplexes. A similar argument shows that tC R (f, B) is a map of vertical bicomplexes. Finally, the fact that tC R (−, −) is a bifunctor follows directly from Lemmas 4.14 and 4.16. Now, to see the adjointness property we describe a map Hom tC R (A ⊗ B, C) → Hom vbC R (A, tC R (B, C)), which sends a map of twisted complexes f = (f m ) : It is clear thatf is a bidegree (0, 0) map of bimodules. To show that it is a map of vertical bicomplexes we will show that δf =f The inverse map is constructed in a similar fashion.  C)). Proof. This follows directly from the facts that if B, C ∈ bgMod ∞ R then tC R (B, C) has trivial differential and that the functors bgMod R ֒→ vbC R and bgMod ∞ R ֒→ tC R are full embeddings. This construction gives us our different enrichments of twisted complexes and bigraded modules, which we describe now.
Definition 4.21. The vbC R -enriched category of twisted complexes tC R is the enriched category given by the following data.
(1) The objects of tC R are twisted complexes.
(2) For A, B twisted complexes the hom-object is the vertical bicomplex tC R (A, B) (3) The composition morphism c : tC R (B, C) ⊗ tC R (A, B) → tC R (A, C) is given by Definition 3.32.
(4) The unit morphism R → tC R (A, A) is given by the morphism of vertical bicomplexes sending 1 ∈ R to 1 A : A → A, the strict morphism of twisted complexes given by the identity of A.
Definition 4.22. We denote by bgMod R the bgMod R -enriched category of bigraded modules given by the following data.
(1) The objects of bgMod R are bigraded modules.
(2) For A, B bigraded modules the hom-object is the bigraded module bgMod R (A, B).
(3) The composition morphism c : bgMod R (B, C) ⊗ bgMod R (A, B) → bgMod R (A, C) is given by Definition 3.32. (4) The unit morphism R → bgMod R (A, A) is given by the morphism of bigraded modules that sends 1 ∈ R to 1 A : A → A, the strict morphism given by the identity of A.
Lemma 4.23. tC R and bgMod R are well-defined and their enrichments are the ones induced by the external tensor products ⊗ : vbC R × tC R → tC R and ⊗ : bgMod R × bgMod ∞ R → bgMod ∞ R . Therefore, these are also monoidal enriched categories and their underlying categories are tC R and bgMod ∞ R respectively.
Proof. This follows directly from Propositions 4.18 and 4.20. Notice in particular that in the case of tC R , the fact that the composition morphism c is a map of vertical bicomplexes is equivalent to equation (2) in Lemma 4.16.
Remark 4.24. There is an interpretation of bgMod ∞ R and bgMod R via a standard categorical construction, the co-Kleisli category for a comonad. Recall that D ¡ denotes the cofree D ¡ -coalgebra functor from bgMod R to the category of D ¡ -coalgebras, with left adjoint the forgetful functor U . Then bgMod ∞ R is the co-Kleisli category of bgMod R for the comonad U D ¡ . And this construction enriches. Namely, recall that bgMod R denotes the category of bigraded modules enriched over itself via its symmetric monoidal closed structure. Then bgMod R is the enriched co-Kleisli category of bgMod R for the enriched comonad U D ¡ . To see this, note that the objects are the same, Proposition 3.33 gives the isomorphism on morphisms or hom-objects and this isomorphism respects the composition.
We can describe the monoidal structures of tC R and bgMod R explicitly.
Lemma 4.25. The monoidal structure of tC R is given by the following map of vertical bicomplexes.
The monoidal structure of bgMod R is given by the restriction of this map.
Proof. The same argument as in the proof of Proposition 4.18 shows that this is indeed a map of vertical bicomplexes. To see that this is indeed the enriched monoidal structure induced by the monoidal structure of tC R over vbC R , one can follow closely Remark 4.9.
Next we introduce enriched structures on filtered modules and filtered complexes. We enrich filtered modules over bigraded modules in the following way.
Definition 4.26. The bgMod R -enriched category of filtered modules fMod R is the enriched category given by the following data.
(1) The objects of fMod R are filtered modules.
(2) For filtered modules (K, F ) and (L, F ), the bigraded module fMod R (K, L) is given by The composition morphism is given by c(f, g) = (−1) u|g| f g, where f has bidegree (u, v).
(4) The unit morphism is given by the map R → fMod R (K, K) given by 1 → 1 K .
We denote by sfMod R the full subcategory of fMod R on objects split filtered modules.
showing that the composition morphism is a map in bgMod R and 1 K ∈ fMod R (K, K) 0 0 . It is a short computation to show that the associativity and unit axiom hold.
Remark 4.28. Notice that morphisms f ∈ fMod R (K, L) v 0 correspond precisely to degree v morphisms of filtered modules which respect the filtration.
We will enrich filtered complexes over vertical bicomplexes analogously.
Definition 4.29. Let (K, d K , F ) and (L, d L , F ) be filtered complexes. We define fC R (K, L) to be the vertical bicomplex whose underlying bigraded module is fMod R (K, L) with vertical differential Note that d K ∈ fMod R (K, K) 1 0 and d L ∈ fMod R (L, L) 1 0 . So δ(f ) has bidegree (u, v + 1). Also δ 2 = 0 and thus fC R (K, L) is indeed a vertical bicomplex. (1) The objects of fC R are filtered complexes.
(2) For K, L filtered complexes the hom-object is the vertical bicomplex fC R (K, L).
(3) The composition morphism is given as in fMod R in Definition 4.26.
(4) The unit morphism is given by the map R → fC R (K, K) given by 1 → 1 K .
We denote by sfC R the full subcategory of fC R on objects split filtered complexes.
Lemma 4.32. The above definition gives a well-defined vbC R -enriched category, fC R .
The associativity and unit axiom hold, because they hold in fMod R .
Lemma 4.33. The enrichment of filtered complexes and filtered modules is the one induced by the external tensor products * : vbC R × fC R → fC R and * : bgMod R × fMod R → fMod R . Therefore, the enriched categories fC R and fMod R are also monoidal enriched categories and their underlying categories are fC R and fMod R respectively.
Proof. Since fC R is a well-defined vbC R -enriched category we have a bifunctor It is left to show that we have natural isomorphisms Hom fC R (A * K, L) ∼ =Hom vbC R (A, fC R (K, L)). In one direction we have a map (a ⊗ k)).

In the inverse direction we have a map
Hom vbC R (A, fC R (K, L)) −→Hom fC R (Tot c (A) ⊗ K, L) g : a → g a → g : (a i ) ⊗ k → g a i (k).
These constructions are inverse to each other and natural.
We can define the monoidal structure of fC R explicitly.
Lemma 4.34. The monoidal structure of fMod R is given by the following map of vertical bicomplexes.
Proof. A direct computation using the Koszul rule gives which shows that the construction is functorial. To see that this is indeed the enriched monoidal structure induced by the monoidal structure of fC R over vbC R , one can follow closely Remark 4.9. This justifies our sign conventions. We now describe this structure explicitly and show that the properties of the totalization functor also hold in the enriched sense. In order to do this, we first extend the definition of Tot to morphisms of any bidegree.
where the last map is a projection and multiplication with the indicated sign. Proof. One has to prove that, for all m, q ∈ Z, Tot(f )(F q (Tot(A) m )) ⊂ F q+u (Tot(B) m+v−u ). Let a ∈ F q (Tot(A) m ), that is to say a k = 0 for k > q. Hence for j > q one has Tot(f )(a) j+u = 0. Hence Tot(f )(a) ∈ F q+u (Tot(B) m+v−u ). Recall that in bgMod R and in fMod R , the compositions c(f, g) and c(Tot(f ), Tot(g)), with f and g of bidegree (u, v) and (u ′ , v ′ ) respectively, are defined by c(Tot(f ), Tot(g)) = (−1) u|g| Tot(f )Tot(g).
Let a ∈ Tot(A) n . f → Tot(f ) define a bgMod R -enriched functor Tot : bgMod R → fMod R which restricts to an isomorphism onto its image sfMod R . Furthermore, this functor extends to a vbC R -enriched functor Tot : tC R → fC R which restricts to an isomorphism onto its image sfC R .
Proof. By construction, Tot A,B is a morphism in bgMod R . To show that Tot defines an enriched functor it is only left to check that this assignment is associative and unital. This holds by Lemma 4.36. To see it restricts to an isomorphism onto its image we construct a bgMod R -enriched functor which is inverse to the restriction of Tot onto its image. Let K, L be split filtered modules, then we define Tot −1 to be given by the assignments Tot −1 (K) := Tot −1 (K) and We have that, Tot(Tot −1 (K)) = K and Tot −1 (Tot(A)) = A. Now we check that Tot −1 (Tot(f )) = f . Write π l for the projection l≤j+u B m+v−u+l l ։ B m+v−u+l l (without a sign). For f of bidegree (u, v) and a j ∈ A m+j j one has that showing that Tot −1 (Tot(f )) = f . To see that Tot(Tot −1 (g)) = g, let g : (Tot(A)) n → (Tot(B)) n+v−u be a map of bidegree (u, v). For (a) ∈ (Tot(A)) n we write g(a) = (g k (a)) k∈Z where g k (a) ∈ B n+v−u+k k .
Recall that ∀q, we have that g(F q ((Tot(A)) n ) ⊂ F q+u (Tot(B)) n+v−u . Fix q ∈ Z. If (a) ∈ (Tot(A)) n then write a = α q−1 + m≥0 a q+m where α q−1 ∈ r≤q−1 A n+r r = F q−1 (Tot(A)) n and m≥0 a q+m is finite. As a consequence g q+u (a) = m≥0 g q+u (a q+m ), Tot −1 (g) m (a j+m ) = (−1) (m+u)n g j+u (a j+m ). Hence showing that Tot(Tot −1 (g)) = g. Finally, it follows that Tot −1 is associative and unital since it is the inverse to Tot and thus it defines a bgMod R -enriched functor Tot −1 : sfMod R −→ bgMod R which is inverse to the restriction of Tot onto its image.
Now to see that both Tot and Tot −1 extend to vbC R -enriched functors notice first that for any Tot(δ(f )). Thus, δTot = Totδ. Therefore, the maps defining Tot are maps in vbC R . The same argument shows that Tot −1 is a vbC R -enriched functor. Proof. We show that Tot : tC R → fC R is lax symmetric monoidal in the enriched sense. The same argument will show by restriction that Tot : bgMod R → fMod R is also enriched lax symmetric monoidal. Let (A, B) be an object in tC R × tC R . The vbC R -enriched natural transformations µ and ǫ are given by the following assignments: where µ A,B is given as in Proposition 3.11. That is, for a = (a i ) i ∈ Tot(A) n 1 and b = (b j ) j ∈ Tot(B) n 2 , Note first that since µ A,B is a map of complexes of degree 0 that respects the filtration, µ (A,B) is indeed a map in vbC R , and similarly for ǫ. To show that µ indeed defines an enriched natural transformation amounts to showing that for Let a ⊗ b ∈ Tot(A) n 1 ⊗ Tot(B) n 2 , with n 1 + n 2 = n. Calculating the left-hand side we get where ǫ r =k 1 n 2 + m 1 n 1 + m 2 n 2 + un + (k 1 + n 1 )|g| + u ′ n 2 .
On the other hand, evaluating the right-hand side we get where ǫ l = k 1 n 2 + m 1 n 1 + m 2 n 2 + un + (k 1 + n 1 )|g| + u ′ n 2 , showing that the equality holds. That this construction respects the symmetric structure was already shown in the proof of Proposition 3.11. The coherence axioms are left to the reader.

4.5.
Derived A ∞ -algebras as A ∞ -algebras in twisted complexes. We reinterpret the category of derived A ∞ -algebras as the category of A ∞ -algebras in twisted complexes. Generally, given an operad P on a symmetric monoidal category one studies P-algebra structures on objects of the same category. We can extend this to the case of monoidal categories over a base via the following definition due to [Fre09].
Definition 4.39. Let C be a monoidal category over V and let P be an operad in V . A P-algebra in C consists of an object A ∈ C, together with maps for which the unit and associativity axioms hold.
We can give an equivalent definition by means of an enriched endomorphism operad.
Definition 4.40. Let C be a monoidal V -enriched category and A an object of C . We define End A to be the collection in V given by  and composition where n = n 1 + · · · + n r . The first morphism is given by the monoidal structure of C and the second is the composition of the symmetry morphism of V with the composition morphism of C .
Proof. The appropriate diagrams commute by associativity of composition in an enriched category. We also refer the reader to [Fre09, Definition 3.4.1].
Example 4.42. For any twisted complex A and any filtered complex K we have operads in vertical bicomplexes End A and End K .
The following result gives an equivalent interpretation of P-algebras in monoidal categories over a base.
Proposition 4.43. [Fre09,Proposition 3.4.3] Let C be a monoidal category over V , let P be an operad in V and A an object in C. Then there is a one-to-one correspondence between P-algebra structures on A and morphisms of operads P → End A .
We reinterpret dA ∞ -algebras as A ∞ -algebras in twisted complexes using the structure of tC R as a monoidal category over vbC R .  Let (A, d A ) be a twisted complex, A its underlying bigraded module and consider A ∞ as an operad in vbC R sitting in horizontal degree zero. There is a one-to-one correspondence between A ∞ -algebra structures on (A, d A ) and dA ∞ -algebra structures on A which respect the twisted complex structure of A. More precisely, let End A be the operad in vbC R corresponding to the bigraded module A. We have a natural isomorphism of sets where vbOp denotes the category of operads in vertical bicomplexes and Hom vbOp,d A denotes the subset of morphisms which send We first compute the left-hand side of (4). Since δ( Then by setting the notation gives us the relations u=i+p v=v+q+t j=1+r+t By setting m ij = (−1) ij m ij one obtains the relation u=i+p v=v+q+t j=1+r+t which by [LRW13] is equivalent to giving a map dA ∞ → End A of operads in vbC R .
Remark 4.45. Note that as in the case of A ∞ -algebras in C R we have two equivalent descriptions of A ∞ -algebras in tC R .
(1) A twisted complex (A, d A ) together with a morphism A ∞ → End A of operads in vbC R , which is determined by a family of elements M A i ∈ tC R (A ⊗i , A) 2−i 0 for i ≥ 2 for which the (A ′ 0i ) relations hold, where the composition is the one prescribed by the composition morphisms of tC R .
(2) A bigraded module A together with a family of elements M i ∈ bgMod R (A ⊗i , A) 2−i 0 for i ≥ 1 for which the (A 0i ) relations hold, where the composition is the one prescribed by the composition morphisms of bgMod R .
Here (A ′ 0i ) and (A 0i ) are the A ∞ relations for i ≥ 2 or i ≥ 1 respectively. Since the composition morphism in bgMod R is induced from the one in tC R by forgetting the differential, these two presentations are equivalent.
We now consider infinity morphisms and composition.
q s ) and c is described in Definition 3.32.
The category of A ∞ -algebras in twisted complexes, denoted A tC ∞ (R), is the category with objects A ∞ -algebras in twisted complexes and whose morphisms are A ∞ -morphisms in twisted complexes.
Theorem 4.47. The construction above extends to a functor Ψ : A tC ∞ (R) → dA ∞ (R) which is an isomorphism of categories.
Proof. On objects Ψ(A, M A i ) = (A, m A ij ) takes an A ∞ -algebra in twisted complexes and associates to it its corresponding dA ∞ -algebra as described in Proposition 4.44.
On morphisms, consider F : Multiplying the equation by (−1) (i+p)(j+q) = (−1) u(v+1) and setting f ij = (−1) ij f ij and m pq = (−1) pq m pq one gets the equation Let us compute the signs modulo 2 σ l = rq + t + (i + p)(j + q) + ij + pq + iq = rq + t + pj, This gives exactly the relation defining morphisms of dA ∞ -algebras: p s + q s ). Moreover, any morphism between dA ∞ -algebras can be constructed in this way. Therefore this construction is a bijection on morphisms. Finally, this construction is functorial. The relations (C 0v ) translate to p s + q s ), which is the composition of morphisms of dA ∞ -algebras.
4.6. Derived A ∞ -algebras as filtered A ∞ -algebras. From the fact that tC b R and sfMod b R are isomorphic vbC R -enriched monoidal categories, we now reinterpret dA ∞ -algebras, in the bounded case, in terms of split filtered A ∞ -algebras. First we define filtered A ∞ -algebras and their morphisms.
Definition 4.48. A filtered A ∞ -algebra is an A ∞ -algebra (A, m i ) together with a filtration {F p A i } p∈Z on each R-module A i such that for all i ≥ 1 and all p 1 , . . . , p i ∈ Z and n 1 , . . . , n i ≥ 0, Such a filtered A ∞ -algebra is said to be split if A = Tot(B) is the total graded module of a bigraded R-module B = {B j i } and F is the column filtration of Tot(B). Remark 4.49. Consider A ∞ as an operad in filtered complexes with the trivial filtration and let K be a filtered complex. There is a one-to-one correspondence between filtered A ∞ -structures on K and morphisms of operads in filtered complexes A ∞ → End K . To see this, notice that if one forgets the filtrations such a map of operads gives an A ∞ structure on K. The fact that this is a map of operads in filtered complexes implies that all the m i s respect the filtrations.
Denote by fA ∞ (R) the category of filtered A ∞ -algebras. Composition is given as in the unfiltered case (this respects the filtration). We consider the following full subcategories of fA ∞ (R).
• fA b ∞ (R): the subcategory on non-negatively filtered A ∞ -algebras. • sfA b ∞ (R): the subcategory on split non-negatively filtered A ∞ -algebras. That is, we have full embeddings Lemma 4.51. For any twisted complex A there is a morphism of operads which is an isomorphism of operads if A is bounded.
Proof. In each arity n this map is given by the composition of the following morphisms in vertical bicomplexes where µ n is the map of vertical bicomplexes constructed from µ (A,A) recursively as the following composite These assemble to a morphism of operads since Tot is a vbC R -enriched functor and µ is a vbC Renriched natural transformation. Furthermore, if A is bounded all maps are isomorphisms and thus the map gives an isomorphism of operads.
Proposition 4.52. Let (A, d A ) ∈ tC b R be an (N, Z)-graded twisted complex and A its underlying bigraded module. There is a one-to-one correspondence between filtered A ∞ -algebra structures on Tot(A) and dA ∞ -algebra structures on A which respect the twisted complex structure of A. This bijection is induced by a one-to-one correspondence between filtered A ∞ -algebra structures on Tot(A) and A ∞ -algebra structures on (A, d A ). More precisely we have natural isomorphisms of sets where vbOp and fCOp denote the categories of operads in vbC R and fC R respectively, and Hom vbOp,d A denotes the subset of morphisms which send µ i1 to d A i . We view A ∞ as an operad in vbC R sitting in horizontal degree zero or as an operad in filtered complexes with trivial filtration.
Proof. The first isomorphism holds by Proposition 4.44. The second isomorphism follows directly from Lemma 4.51. Finally, to see the third isomorphism let f : A ∞ → End Tot(A) be a map of operads in vbC b R . Again, since A ∞ is quasi-free, this is equivalent to maps in vbC b which are determined by elements M n := f (µ n ) ∈ End Tot(A) (n) of bidegree (0, 2 − n) such that Since A is (N, Z)-graded, Tot is symmetric monoidal, and thus we have that So these maps give the complex Tot(A) the structure of an A ∞ -algebra. Moreover, all of the M n s respect the filtration since they have horizontal degree zero. Therefore, the map f gives Tot(A) the structure of a split filtered A ∞ -algebra and it is clear that any filtered A ∞ -algebra structure on Tot(A) can be described by such f .
This construction extends to infinity morphisms.
Theorem 4.53. The totalization functor extends to a functor Φ : A tC ∞ (R) → fA ∞ (R) which in the bounded case restricts to an isomorphism between the categories of bounded A ∞ -algebras in twisted complexes and split non-negatively filtered A ∞ -algebras.
Proof. The functor on objects is given as in Proposition 4.51. Here we describe this explicitly on elements. Note first that for any i the map m i has horizontal degree 0 thus it respects the filtrations. Now, we compute v=v+q+t j=1+r+t Here the first equality holds by definition, the second by naturality of µ and the third because the M i s satisfy the relations (A 0v ). Thus (Tot(A), m i ) is a filtered A ∞ -algebra. The same computation gives the result on morphisms and this is stable under the composition of morphisms giving the functor Φ. Furthermore, when A ∈ tC b R this functor restricts to an isomorphism by Proposition 4.52.
Corollary 4.54. Let Tot denote the composite . The functor Tot restricts to an isomorphism between the category of (N, Z)-graded dA ∞ -algebras and the category of split non-negatively filtered A ∞ -algebras. Furthermore, this functor fits into a commutative diagram of categories where the horizontal arrows are forgetful functors and the vertical arrows are full embeddings.
Proof. This follows directly from Theorems 4.47 and 4.53.

Derived A ∞ -algebras and r-homotopy
The main goal of this section is to study different but equivalent interpretations of the notion of r-homotopy for derived A ∞ -algebras. We first define r-homotopy by constructing a functorial r-path object. Then, we study some of the properties of r-homotopy. Most notably, we show that 0-homotopy defines an equivalence relation and we study the localized category dA ∞ (R)[S −1 r ]. Finally, we give an operadic interpretation of r-homotopy and show that both notions are equivalent. 5.1. Twisted dgas and tensor product. In general, the tensor product of two dA ∞ -algebras does not inherit a natural dA ∞ -algebra structure giving rise to a monoidal structure on dA ∞ (R). The construction works if one of the components is a twisted differential graded algebra, as we next show.
Definition 5.1. A twisted dga is a dA ∞ -algebra (A, µ ij ) whose only non-zero structure morphisms are µ i1 for i ≥ 0 and µ 02 .
Lemma 5.2. For (A, µ ij ) a twisted dga, the following hold.
Proposition 5.3. Let (Λ, µ ij ) be a twisted dga and let (A, m ij ) be a dA ∞ -algebra. The bigraded module Λ ⊗ A is endowed with a dA ∞ -algebra structure given by Here τ j : (Λ ⊗ A) ⊗j → Λ ⊗j ⊗ A ⊗j denotes the standard isomorphism given by the symmetric monoidal structure and µ j is defined in Lemma 5.2.
Using this, for all u ≥ 0 and all v ≥ 2, we have u=i+p, v=j+q−1 j=1+r+t, Proposition 5.4. Let (Λ, µ ij ) be a twisted dga. The above construction gives rise to a functor Λ ⊗ − : Furthermore, for a dA ∞ -algebra (A, m ij ), the construction above gives rise to a functor − ⊗ A sending a strict morphism f : Proof. The proof follows exactly the same lines of computation as the preceding proof. 5.2. r-homotopies and r-homotopy equivalences. We next define a collection of functorial paths indexed by an integer r ≥ 0 on the category of dA ∞ -algebras, giving rise to the corresponding notions of r-homotopy.
We will use specific twisted dgas defined in the following proposition whose proof is left to the reader.
Note that for a dA ∞ -algebra A, one has P r ( . Hence we may identify P r (A) with the bigraded R-module given by P r (A) j i = A j i ⊕ A j+r−1 i+r ⊕ A j i as in Definition 3.14. More precisely, the triple (x, y, z) is identified with e − ⊗ x + u ⊗ y + e + ⊗ z.
Note that under the identification above, the map t n is obtained as the composite As a consequence, combining Propositons 5.3 and 5.4 one obtains the following.
Proposition 5.7. The r-path (P r (A), M ij ) of a dA ∞ -algebra (A, m ij ) is given by the bigraded module P r (A) together with the morphisms M ij : P r (A) ⊗j → P r (A) of bidegree (−i, 2 − i − j) given by and the morphisms The r-path of a morphism f : . The structure morphisms of the r-path . It follows directly from the above proposition that the r-path is compatible with the forgetful functor U : dA ∞ (R) −→ tC R . Also, if (A, m 0j ) is a dA ∞ -algebra concentrated in horizontal degree 0, then its 0-path P 0 (A) coincides with its path object as an A ∞ -algebra as defined by Grandis in [Gra99]. Hence the 0-path is compatible with the inclusion A ∞ (R) ֒→ dA ∞ (R).
Definition 5.8. Let f, g : A → B be two morphisms of dA ∞ -algebras. An r-homotopy from f to g is given by a morphism of dA ∞ -algebras h : A → P r (B) such that ∂ − B • h = f and ∂ + B • h = g. We use the notation h : f ≃ r g. We postpone until later giving an explicit version of r-homotopy, in terms of a collection of morphisms h ij : A ⊗j → B; see Proposition 5.32.
In the category of A ∞ -algebras, the notion of homotopy defines an equivalence relation on the sets of A ∞ -morphisms (see [Pro11], see also [Gra99]). We next prove an analogous result in the context of dA ∞ -algebras, for 0-homotopies. Our proof is an adaptation of the proof given by Grandis for A ∞ -algebras.
Proposition 5.9. The notion of r-homotopy is reflexive and compatible with the composition. Furthermore, for r = 0 it defines an equivalence relation on the set of morphisms of dA ∞ -algebras from A to B, provided A and B are (N, Z)-graded.
Proof. Since the notion of r-homotopy is defined via a functorial path, it is reflexive and compatible with the composition. To show that 0-homotopy is symmetric we will define a natural reversion morphism of the r-path ζ : P 0 (A) → P 0 (A) of a dA ∞ -algebra (A, m ij ) such that ∂ ± ζ = ∂ ∓ . Then, given a 0-homotopy h : f ≃ 0 g we will have a 0-homotopy ζh : g ≃ 0 f . Consider the filtered A ∞ -algebra defined by applying Tot to P 0 (A). This is given by: with structure morphisms for j ≥ 2. We will next define a morphism of filtered A ∞ -algebras ζ : Tot(P 0 (A)) → Tot(P 0 (A)). Note that such a map is determined by its composition with the three maps ∂ − , ∂ 0 and ∂ + defined by projection to each of the direct summands of Tot(P 0 (A)). We let ∂ − ζ 1 = ∂ + , ∂ 0 ζ 1 = −∂ 0 and ∂ − ζ 1 = ∂ + . For j > 1, we let ∂ ± ζ j = 0 and define ∂ 0 ζ j inductively by where all indices in the sum are integers ≥ 1 and S nqxp is a sign coefficient (see [Gra99,p56]). By [Gra99,Theorem 7.1], the family {ζ j } j≥1 is a morphism of A ∞ -algebras. Since for all p ∈ Z, ∂ ǫ (F p Tot(P 0 (A)) ⊂ F p P 0 (A) for ǫ ∈ {−, 0, +}, the morphism ζ is compatible with filtrations. Therefore by Corollary 4.54 it gives the desired reversion of dA ∞ -algebras.
Remark 5.10. The proof of symmetry and transitivity of 0-homotopies given above does not extend to r-homotopies, due to the fact that for r > 0, the projection ∂ 0 : Tot(P r (A)) → Tot(A) is not necessarily compatible with filtrations. Note that we have ∂ 0 (F p Tot(P r (A))) ⊂ F p+r (Tot(A)).
Denote by ∼ r the congruence of dA ∞ (R) generated by r-homotopies: f ∼ r g if and only if there is a chain of r-homotopies f ≃ r · · · ≃ r g from f to g or a chain g ≃ r · · · ≃ r f from g to f . Definition 5.11. A morphism of dA ∞ -algebras f : A → B is called an r-homotopy equivalence if there exists a morphism g : B → A satisfying f • g ∼ r 1 B and g • f ∼ r 1 A . Denote by S r the class of r-homotopy equivalences of dA ∞ (R). This class is closed under composition and contains all isomorphisms. Since the r-path commutes with the forgetful functor U : dA ∞ (R) −→ tC R , we have S r = U −1 (S tC R r ). Note as well, that S r ⊂ S r+1 and S r ⊂ E r for all r ≥ 0. Proof. Note first that the category of twisted dgas together with strict morphisms is a monoidal category, hence Λ r ⊗ Λ r is a twisted dga. Let ∆ : Λ r → Λ r ⊗ Λ r be the map given by ∆(e − ) = e − ⊗ (e − + e + ) + e + ⊗ e − , ∆(e + ) = e + ⊗ e + , and ∆(u) = u ⊗ e + + e + ⊗ u.
That ∆ is a strict morphism of dgas is a matter of computation. Furthermore one has Consequently ∆ ⊗ 1 A : P r (A) → P r (P r (A)) is an r-homotopy from ι A • ∂ − A to the identity. Theorem 5.13. The localized category dA ∞ (R)[S −1 r ] is canonically isomorphic to the quotient category π r (dA ∞ (R)) := dA ∞ (R)/ ∼ r . Proof. The proof is analogous to that of Proposition 3.26, using Lemma 5.12. 5.3. Operadic approach. Since dA ∞ -algebras are algebras for the operad (dAs) ∞ , one expects to be able to describe homotopies in terms of structure on cofree (dAs) ¡ -coalgebras, where (dAs) ¡ is the Koszul dual cooperad. We carry this out in this section, giving several equivalent formulations and showing that they agree with the definition via a path object presented above. 5.3.1. (dAs) ¡ -coalgebras and coderivations. In this setting, a homotopy between morphisms g and f should be a coderivation homotopy, that is, it satisfies two conditions, a usual homotopy relation together with a condition of compatibility with the comultiplication, called a (g, f )-coderivation condition. (See [ALR + 15] for the coderivation notion in an operadic context and [LH03] for (g, f )coderivations in the setting of A ∞ -algebras.) First we recall the (g, f )-coderivation condition for coassociative coalgebras.
Definition 5.14. Let (A, ∆ A ), (B, ∆ B ) be coassociative R-coalgebras and let f, g : A → B be coalgebra morphisms. A (g, f )-coderivation is an R-linear map h : A → B such that Next we define (g, f )-coderivations in a suitable operadic setting.
Definition 5.15. Let C be a non-symmetric cooperad in vertical bicomplexes. For X, Y, Z vertical bicomplexes, the vertical bicomplex C(X; Y ; Z) is given by If f : X → X ′ , h : Y → Y ′ and g : Z → Z ′ are maps of vertical bicomplexes, the map is defined as the direct sum of the maps 1 ⊗ f ⊗a ⊗ h ⊗ g ⊗b . In order to understand very explicitly what such a thing looks like in the (dAs) ¡ case, we first recall Proposition 3.2 of [ALR + 15] and then extend it to (g, f )-coderivations. As there, it is slightly simpler to use cooperadic suspension and work with the suspended cooperad Λ(dAs) ¡ .
Proposition 5.17. [ALR + 15, Proposition 3.2] Cooperadic suspension gives rise to an isomorphism of categories between the category of conilpotent coalgebras over the cooperad (dAs) ¡ and the category of triples (C, ∆, f ) as above.
Example 5.18. [ALR + 15, Example 3.3] As an example we give the structure corresponding to the cofree Λ(dAs) ¡ -coalgebra cogenerated by C. We have Λ(dAs) ¡ (C) ∼ = R[x] ⊗ T C, where T C denotes the reduced tensor coalgebra on C.
Let π 0 denote the projection of R[x] ⊗ T C onto kx 0 ⊗ T C ∼ = T C. Then where the first ∆ is the usual deconcatenation product defined on T C.
The linear map f will be denoted d x in this cofree case and Proposition 5.19. Let g and f be maps of (dAs) ¡ -coalgebras (dAs) ¡ (A) → (dAs) ¡ (B) and let h : (dAs) ¡ (A) → (dAs) ¡ (B) be an r-(g, f )-coderivation. If f , g correspond under the above isomorphism of categories to coalgebra morphismsf ,g : Proof. After applying cooperadic suspension, the (g, f )-coderivation condition gives the commutative diagram Let C = SA and D = SB. As in [ALR + 15, Example 3.3], the structure map ρ C of the cofree coalgebra on C is completely determined by ∆ = ρ 0,2 and d x = ρ 1,1 , where ρ i,j : C → C ⊗j is the following composite Post-composing the horizontal maps with π 0,2 in the commutative diagram above, we find that we obtain a commutative diagram and the outer commuting square gives the r-(g,f )-coderivation condition forh. Similarly, one may post-compose the horizontal maps with π 1,1 and check that the resulting condition is Next we want to pass from morphisms on R[x] ⊗ T C, commuting with d x , to families of morphisms on T C. As in Example 3.3 of [ALR + 15], given a morphism of bigraded modules f : where f n,i : T (C) → T (D) and a ∈ C ⊗j . Then commuting with the map d x means that f is completely determined by the family of maps f n,0 .
Define f n : T C → T C by f n (a) = (−1) nj f n,0 (a) = (−1) nj π 0 f (x n ⊗ a), where a ∈ C ⊗j . The correspondence gives a bijection between the subset of morphisms in bgMod R (R[x]⊗T C, R[x]⊗ T D) which commute with d x and bgMod ∞ R (T C, T D). Recall from [ALR + 15] that under this assignment, a square-zero coderivation δ on R[x] ⊗ T C corresponds to a twisted complex structure on T C.
Proof. We extract the i-th map in the families corresponding to the two sides of the given equation. On the right-hand side, by Propostion 5.22, we obtain g i−r − f i−r for i ≥ r and 0 for i < r.
Recall from [ALR + 15], that one can write δ i,j (a) = l δ i,j,l (a), with δ i,j,l (a) ∈ D ⊗l , and similarly for h i,j (a). Anti-commuting with the map d x implies that δ i,j,l (a) = (−1) j(n+l+1) δ i−j,0,l (a) for a ∈ C ⊗n , and similarly for h.
On the left-hand side we calculate to check the signs. For a ∈ C ⊗n , where δ i,j (a) = l δ i,j,l (a), with δ i,j,l (a) ∈ D ⊗l , and similarly for h i,j (a). Continuing the above calculation, we obtain as required.
Thus an r-shifted version of the operadic notion of coderivation homotopy corresponds to a sequence of maps of bigraded R-modules h n : T C → T D, where h n has bidegree (r − n, r − n − 1), satisfying both the condition in Proposition 5.20 and the condition in Proposition 5.23. We are going to show that this is equivalent to an r-homotopy of dA ∞ -algebras, as defined via the path construction.
As an intermediate step, we reformulate the conditions using the composition in bgMod R .

Proof.
(1) We have that c(δ, δ) = 0 if and only if the δ i satisfy the twisted complex relations. And the coderivation condition on δ translates into the coderivation condition on the individual δ i . In more detail, Comparison with path definition. We will show that the path definition of r-homotopy (Definition 5.8) corresponds to imposing conditions (3) and (4) in Proposition 5.27. In section 5.3.1 we have checked that these match up with the corresponding (dAs) ¡ -coalgebra notions. Let f, g : A → B be two morphisms of dA ∞ -algebras and let h : A → P r (B) be an r-homotopy from f to g. Recall that this is a morphism of dA ∞ -algebras satisfying ∂ − B • h = f and ∂ + B • h = g. Let F, G : T SA → T SB be the coalgebra morphisms in bgMod R corresponding to f, g and let H : T SA → T SP r (B) be the coalgebra morphism in bgMod R corresponding to h.
We have three projection maps from P r (B) to B, on to the left, middle and right copies of B, denoted ∂ − B , ∂ 0 B and ∂ + B respectively. We denote the corresponding maps SP r (B) to SB by π L , π M and π R respectively. Then let π ∈ bgMod R (T SP r (B), T S B) be the strict map s,t π ⊗s L ⊗ π M ⊗ π ⊗t R . We claim this is a commutative diagram in the category bgMod R . The top square is such a commutative diagram since H is a coalgebra morphism in bgMod R . In the bottom square all the morphisms are strict, so composition and tensor agree with the usual ones and we just need to see that this commutes in the usual sense, which can be easily checked. Finally, one can check that c(T S (∂ + B ), H) = G and c(T S (∂ − B ), H) = F , so the commuting of the outer square reads c(G ⊗c(π, H) + c(π, H) ⊗F ), ∆) = c(∆, c(π, H)), which is the (G, F )-coderivation condition for c(π, H).
When we expand out the composition, there are three types of terms appearing, according to whether the input to m ij is of the form π ⊗j L , π ⊗a L ⊗ π M ⊗ π ⊗b R or π ⊗j R . And where M ij Ψ j (M ij ) : SP r (B) ⊗j → SP r (B). Again there are three sorts of terms, according to whether they involve π L M ij , π M M ij or π R M ij . From the definition of the M ij for P r (B), we have, for all (i, j), , for (i, j) = (r, 1), Sign calculations with the suspension show that these convert to π L M ij = m ij π ⊗j L , π R M ij = m ij π ⊗j R , for all (i, j) and (−1) r+i+1 π M M ij = a+b+1=j m ij (π ⊗a L ⊗ π M ⊗ π ⊗b R ), π M M r1 = − m r1 π M + π R − π L .
Using the above, one may now check that in c(π, d Pr B ) + (−1) r c(d B , π) i almost all terms cancel pairwise, with the exception, when i = r, of the extra terms in c(π, d Pr B ) coming from the special form of M r1 . These contribute s,t 1 ⊗s ⊗ (π R − π L ) ⊗ 1 ⊗t = j π ⊗j R − π ⊗j L = (T (π R ) − T (π L )).  We now collect everything together.
Theorem 5.31. Let f, g : A → B be two morphisms of dA ∞ -algebras. Then h : A → P r (B) being an r-homotopy from f to g in the sense of Definition 5.8 is equivalent to conditions (3) and (4) of Proposition 5.27 on c(π, H) : T SA → T SB, where H : T SA → T S P r (B) is the coalgebra morphism in bgMod R corresponding to h.
Proof. We have already seen that c(π, H) does satisfy conditions (3) and (4) of Proposition 5.27. It is also straightforward to see that we can recover h from c(π, H). Indeed, from c(π, H) we can extract G, F and c(π M , H 1 ), where H 1 : T SA → SP r (B) is the composite of H with the strict projection T SP r (B) → SP r (B) and c(π, H) is uniquely determined by this data. Now H 1 is also uniquely determined by G, F and c(π M , H 1 ) and we haveh ij = H 1 ij .
Thus the notion of r-homotopy defined via the path construction coincides with the operadic one.
Proof. We have seen that the path definition of r-homotopy for dA ∞ -algebras agrees with the rcoderivation-family and r-homotopy of twisted complex conditions on the corresponding families of maps T SA → T SB.
Suppose that the dA ∞ -algebra structures of A and B are encoded in families of coderivations d A i and d B i making T SA and T SB respectively into twisted chain complexes. Suppose also that f corresponds to the coalgebra-family of mapsf i : T SA → T SB, giving a morphism of twisted complexes; similarly g corresponds to a coalgebra-family of mapsg i : T SA → T SB.
The r-homotopy condition on (h i ) between maps of twisted complexes is equivalent to, for all m ≥ 0, This is an equality of maps T SA → T SB. Here the bidegree ofh i is (r − i, r − i − 1), that of d A i and of d B i is (−i, −i + 1) and that off i and ofg i is (−i, −i). For each m ≥ 0 and for k ≥ 1, we consider the "component of equation (H m1 ) from (SA) ⊗k → SB". That is, we pre-compose with the inclusion (SA) ⊗k → T SA and post-compose with the projection to SB. We will show that this gives, after shifting, the required statement.
Then it is a matter of direct calculation that the extraction of the relevant component maps from equation (H m1 ), together with removing the shifts using the isomorphism Ψ k , gives the required result.
Remark 5.33. Sagave [Sag10, Definition 4.9] defined homotopy of morphisms from A to B of dA ∞algebras only in the special case where A is minimal and B is a bidga. One may check that his definition is equivalent to our 0-homotopy in that case.