Homology, Homotopy and Applications, vol.5(1), 2003, pp.1–48

CATEGORY OF A_{∞}-CATEGORIES

VOLODYMYR LYUBASHENKO

(communicated by James Stasheff)

Abstract We define natural A_{∞}-transformations and construct A_{∞}-category of A_{∞}-functors. The notion of non-strict units in an A_{∞}-category is introduced. The 2-category of (unital) A_{∞}-categories, (unital) functors and transformations is described.

The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta63, II] Stasheff defines also A_{∞}-algebras and their homotopy-bar constructions. These algebras and their applications to topology were actively studied, for instance, by Smirnov [Smi80] and Kadeishvili [Kad80, Kad82]. We adopt some notations of Getzler and Jones [GJ90], which reduce the number of signs in formulas. The notion of an A_{∞}-category is a natural generalization of A_{∞}-algebras. It arose in connection with Floer homology in Fukaya’s work [Fuk93, Fuk] and was related by Kontsevich to mirror symmetry [Kon95]. See Keller [Kel01] for a survey on A_{∞}-algebras and categories.

In the present article we show that given two A_{∞}-categories 𝒜 and ℬ, one can construct a third A_{∞}-category A_{∞}(𝒜, ℬ) whose objects are A_{∞}-functors f : 𝒜 → ℬ, and morphisms are natural A_{∞}-transformations between such functors. This result was also obtained by Fukaya [Fuk] and by Kontsevich and Soibelman [KS], independently and, apparently, earlier. We describe compositions between such categories of A_{∞}-functors, which would allow us to construct a 2-category of unital A_{∞}-cat-egories. The latter notion is our generalization of strictly unital A_{∞}-categories (cf. Keller [Kel01]). We also discuss unit elements in unital A_{∞}-categories, unital natural A_{∞}-transformations, and unital A_{∞}-functors.

Plan of the article with comments and explanations. The first section describes some notation, sign conventions, composition convention, etc. used in the article. The ground commutative ring 𝕜 is not assumed to be a field. This is suggested by the development of homological algebra in [Dri02]. Working over a ring

The research was supported in part by grant 01.07/132 of State Fund for Fundamental Research of Ukraine Received June 5, 2002, revised December 24, 2002; published on January 27, 2003. 2000 Mathematics Subject Classification: 18D05, 18D20, 18G55, 57T30 Key words and phrases: A_{∞}-categories, A_{∞}-functors, A_{∞}-transformations, unit A_{∞}-transforma-tion, 2-category \begin{matrix}c \\ ◯ \end{matrix} 2003, Volodymyr Lyubashenko. Permission to copy for private use granted.

𝕜 instead of a field has strong consequences. For instance, one may not hope for Kadeishvili’s theorem on minimal models [Kad82] to hold for all A_{∞}-algebras over 𝕜.

In the second section we recall or give definitions of the main objects. A 𝕜-quiver is such a graph that the set of arrows (morphisms) between two vertices (objects) is a 𝕜-module (Definition 2.1). We view quivers as categories without multiplication and units. Cocategories are 𝕜-quivers and 𝕜-coalgebras with a matrix type decomposition into 𝕜-submodules, indexed by pairs of objects (Definition 2.2). A_{∞}-cate-gories are defined as a special kind of differential graded cocategories – the ones of the form of the tensor cocategory T 𝒜 of a 𝕜-quiver 𝒜 (Definition 2.3). A_{∞}-func-tors are homomorphisms of cocategories that commute with the differential (Definition 2.4). A_{∞}-transformations between A_{∞}-functors are defined as coderivations (Definition 2.6). They seem to make A_{∞}-category theory closer to ordinary category theory. Notice, however, that A_{∞}-transformations are analogs of transformations between ordinary functors, which do not satisfy the naturality condition. Natural A_{∞}-transformations are introduced in Definition 6.4. A_{∞}-functors and A_{∞}-trans-formations are determined by their components.

In the third section we study tensor products of cocategories and homomorphisms between them (Section 3.2). We concentrate on homomorphisms from the tensor product of tensor cocategories to another tensor cocategory. Given 𝕜-quivers 𝒜 and ℬ, we consider another 𝕜-quiver Coder(𝒜, ℬ), whose objects are the cocategory homomorphisms T 𝒜 → T ℬ and morphisms are coderivations (Section 3.2). We construct a cocategory homomorphism α : T 𝒜⊗T Coder(𝒜, ℬ) → T ℬ (Corollary 3.3 to Proposition 3.1), based on a map θ : T Coder(𝒜, ℬ) → Hom_{𝕜}(T 𝒜, T ℬ) (3.0.1). The homomorphism α is universal (Proposition 3.4), in other words, T Coder(𝒜, ℬ) is the inner hom-object Hom(T 𝒜, T ℬ) in the monoidal category generated by tensor cocategories.

This universality is exploited in the fourth section in order to show that the category of tensor cocategories is enriched in the monoidal category of graded cocategories. That is: there exists an associative unital multiplication M : T Coder(𝒜, ℬ)⊗ T Coder(ℬ, 𝒞) → T Coder(𝒜, 𝒞), which is a cocategory homomorphism (Proposition 4.1). Its explicit description uses the map θ.

The fifth section extends the results of the third section to differential graded tensor cocategories, that is, to A_{∞}-categories. With two A_{∞}-categories 𝒜, ℬ is associated a third A_{∞}-category A_{∞}(𝒜, ℬ) (Proposition 5.1). Its objects are A_{∞}-func-tors 𝒜 → ℬ, and its morphisms are coderivations. To reduce the number of signs in the theory we prefer to work with grading of a graded 𝕜-quiver or A_{∞}-cate-gory 𝒜 shifted by 1: s𝒜 = 𝒜[1]. In this notation the quiver A_{∞}(𝒜, ℬ) is a full subquiver of the quiver s^{−1} Coder(s𝒜, sℬ). The proof of Proposition 5.1 consists of constructing a differential B in the tensor cocategory of A_{∞}(𝒜, ℬ). The explicit formula (5.1.3) for B uses the map θ. A cocategory homomorphism from a tensor product of differential tensor cocategories to a single such cocategory is called an A_{∞}-functor in the generalized sense (Section 5.3). Restricting the cocategory homomorphism of Corollary 3.3 we get a homomorphism of differential graded cocategories T s𝒜 ⊗ T sA_{∞}(𝒜, ℬ) → T sℬ = T (sℬ) (Corollary 5.4). Its universality (Proposition 5.5) may be interpreted as T sA_{∞}(𝒜, ℬ) being the inner hom-object Hom(T s𝒜, T sℬ) in the monoidal category generated by differential graded tensor cocategories.

This universality is used in the sixth section to show that the category of A_{∞}-cat-egories is enriched in the monoidal category of differential graded cocategories. Namely, the multiplication M of Proposition 4.1 restricted to M : T sA_{∞}(𝒜, ℬ) ⊗ T sA_{∞}(ℬ, 𝒞) → T sA_{∞}(𝒜, 𝒞) is an A_{∞}-functor, that is, it commutes with the differential (equation (6.1.1)). By universality (Proposition 5.5) M corresponds to a unique A_{∞}-functor

A_{∞}(𝒜, ) : A_{∞}(ℬ, 𝒞) → A_{∞}(A_{∞}(𝒜, ℬ), A_{∞}(𝒜, 𝒞)).

We prove that it is strict and describe it in Proposition 6.2. Natural A_{∞}-transforma-tions are defined as cycles in the differential graded quiver of all A_{∞}-transformations (Definition 6.4).

Identifying cohomologous natural A_{∞}-transformations (that is, considering cohomology of the quiver of A_{∞}-transformations) in the seventh section, we get a non-2-unital 2-category A_{∞}, whose objects are A_{∞}-categories, 1-morphisms are A_{∞}-functors, and 2-morphisms are equivalence classes of natural A_{∞}-transforma-tions. Here non-2-unital means that unit 2-morphisms are missing in the 2-category A_{∞}. However, unit 1-morphisms are present – the identity A_{∞}-functors. Before constructing A_{∞} we construct a non-2-unital 2-category 𝒦A_{∞} enriched in 𝒦 – the homotopy category of differential graded complexes of 𝕜-modules (Proposition 7.1). Morphisms of 𝒦 are chain maps modulo homotopy. The notion of 2-category enriched in a symmetric monoidal category is discussed in Appendix A. The idea of the construction is that the binary operation becomes strictly associative if homotopic chain maps are identified. Similarly with other identities in a 2-category. The non-2-unital 2-category A_{∞} is obtained from 𝒦A_{∞} by taking the 0-th cohomology.

A_{∞}-categories are analogs of non-unital categories – categories without unit morphisms. We define a unital A_{∞}-category 𝒞 so that its cohomology H^{•}(𝒞) is a unital category, and for any representative 1_{X} ∈ 𝒞^{0}(X, X) of the unit class [1_{X}] ∈ H^{0}(𝒞(X, X)) the binary compositions with 1_{X} are homotopic to identity as chain maps 𝒞(X, Y) → 𝒞(X, Y) or 𝒞(Y, X) → 𝒞(Y, X) (Definition 7.3, Lemma 7.4). We prove that for a unital A_{∞}-category there exists a natural A_{∞}-transformation i^{𝒞} : id_{𝒞} → id_{𝒞} : 𝒞 → 𝒞 of the identity functor, whose square is equivalent to i^{𝒞} (Proposition 7.5). It is called a unit transformation of 𝒞 (Definition 7.6) and, indeed, it is a unit 2-morphism in the 2-category A_{∞}. Moreover, fi^{𝒞} : f → f is a unit 2-morphism of an A_{∞}-functor f : 𝒜 → 𝒞 (Corollary 7.9). The unit transformation i^{𝒞} is determined uniquely up to an equivalence (Corollary 7.10). If 𝒞 is unital, then A_{∞}(𝒜, 𝒞) is unital as well (Proposition 7.7).

The full 2-subcategory 𝒦^{u}A_{∞} (resp.^{u}A_{∞}) of 𝒦A_{∞} (resp. A_{∞}), whose objects are unital A_{∞}-categories, and 1-morphisms are all A_{∞}-functors, is 2-unital by Corollary 7.11 (resp. Corollary 7.12). Since ^{u}A_{∞} is an ordinary 2-category, the meaning of the statements ‘a natural A_{∞}-transformation is invertible’ and ‘an A_{∞}-functor is an equivalence’ is clear (Corollary 7.12). We show in Proposition 7.15 that a natural A_{∞}-transformation is invertible if and only if its 0-th component is invertible modulo boundary (in the sense of Section 7.13). The binary composition with a cycle invertible modulo boundary is homotopy invertible (Lemma 7.14).

In the eighth section we discuss unital A_{∞}-functors (between unital A_{∞}-cat-egories). Their first components map unit elements into unit elements modulo boundary (Definition 8.1). For a unital functor f : 𝒜 → ℬ we have an equivalence of natural A_{∞}-transformations i^{𝒜} f ≡ fi^{ℬ} (Proposition 8.2). An A_{∞}-functor isomorphic to a unital A_{∞}-functor is unital as well (8.2.4). Unital A_{∞}-categories, unital A_{∞}-functors and equivalence classes of natural A_{∞}-transformations form a 2-category A_{∞}^{u} ⊂ ^{u}A_{∞} (Definition 8.3), which is a close analog of the 2-category of usual categories. The construction of A_{∞}^{u} is the main point of the article. It is needed for developing a theory of free A_{∞}-categories, since it is expected that their universality properties are formulated in the language of 2-categories.

There is a forgetful 2-functor k : A_{∞}^{u} → 𝒦-𝒞at, which takes a unital A_{∞}-category into the same differential quiver equipped with the binary composition, viewed as a 𝒦-category (Proposition 8.6). The 2-functor k reduces a unital A_{∞}-functor to its first component and a natural A_{∞}-transformation to its 0-th component. It turns out that many properties of an A_{∞}-functor are determined by its first component and many properties of a natural A_{∞}-transformation are determined by its 0-th component. For instance, if the first component of an A_{∞}-functor ϕ is homotopy invertible, then any natural A_{∞}-transformation y : fϕ → gϕ is equivalent to tϕ for a unique (up to equivalence) natural A_{∞}-transformation t : f → g (Cancellation Lemma 8.7). If an A_{∞}-functor ϕ : 𝒞 → ℬ to a unital A_{∞}-category ℬ has homotopy invertible first component, and each object of ℬ is “isomorphic modulo boundary” to an object from ϕ(Ob 𝒞), then ϕ is a unital equivalence, and 𝒞 is unital (Theorem 8.8). An equivalence between unital A_{∞}-categories is always unital (Corollary 8.9), which is not an immediate consequence of definitions.

As a first example of a unital A_{∞}-category we list strictly unital A_{∞}-categories (Section 8.11), which is a well-known notion. Other examples of unital A_{∞}-cate-gories are obtained via Theorem 8.8. For instance, if an A_{∞}-functor ϕ : 𝒞 → ℬ to a strictly unital A_{∞}-category ℬ is invertible, then 𝒞 is unital (Section 8.12). We stress again that taking the 0-th cohomology of a unital A_{∞}-category 𝒞 we get a 𝕜-linear category H^{0}(𝒞). This H^{0} can be viewed as a 2-functor (Section 8.13).

In Appendix A we define 2-categories enriched in a symmetric monoidal category. Non-2-unital 2-categories are described in Definition Appendix A.2. 2-unital (usual) 2-categories admit a concise Definition Appendix A.1 and an expanded Definition Appendix A.3+Appendix A.2.

In Appendix B we prove that the cone of a homotopical isomorphism is contractible.

1. Conventions

We fix a universe 𝒰 [GV73, Sections 0,1], [Bou73]. Many classes and sets in this paper will mean 𝒰 -small sets, even if not explicitly mentioned.

𝕜 denotes a (𝒰 -small) unital associative commutative ring. By abuse of notation it denotes also a chain complex, whose 0-th component is 𝕜, and the other components vanish. A 𝕜-module means a 𝒰 -small 𝕜-module. The tensor product ⊗ usually means ⊗_{𝕜} – the tensor product of graded 𝕜-modules. It turns the category of graded 𝕜-modules into a closed monoidal category. We will use its standard symmetry c : x ⊗ y ↦ (−)^{xy}y ⊗ x = (−)^{deg x·deg y}y ⊗ x. This paper contains many signs, and everywhere we abbreviate the usual (−1)^{(deg x)(deg y)} to (−)^{xy}. Similarly, (−)^{x} means (−1)^{deg x}, or, simply, (−1)^{x}, if x is an integer.

It is easy to understand the line

𝒜(X_{0}, X_{1}) ⊗_{𝕜} 𝒜(X_{1}, X_{2}) ⊗_{𝕜} ⋯ ⊗_{𝕜} 𝒜(X_{n−1}, X_{n}),

and it is much harder to understand the order in

𝒜(X_{n−1}, X_{n}) ⊗_{𝕜} ⋯ ⊗_{𝕜} 𝒜(X_{1}, X_{2}) ⊗_{𝕜} 𝒜(X_{0}, X_{1}).

That is why we use the right operators: the composition of two maps (or morphisms) f : X → Y and g : Y → Z is denoted by fg : X → Z. A map is written on elements as f : x ↦ xf = (x)f. However, these conventions are not used systematically, and f(x) might be used instead.

When f, g : X → Y are chain maps, f ∼ g means that they are homotopic. We denote by 𝒦 the category of differential graded 𝕜-modules, whose morphisms are chain maps modulo homotopy. A complex of 𝕜-modules X is called contractible if id_{X} ∼ 0, in other words, if X is isomorphic to 0 in 𝒦.

If C is a ℤ-graded 𝕜-module, then its suspension sC = C[1] is the same 𝕜-module with the shifted grading (sC)^{d} = C^{d+1}. The “identity” map C → sC of degree −1 is also denoted by s. We follow the Getzler–Jones sign conventions [GJ90], which include the idea to apply operations to complexes with shifted grading, and Koszul’s rule:

(x ⊗ y)(f ⊗ g) = (−)^{yf}xf ⊗ yg = (−1)^{deg y·deg f}xf ⊗ yg.

It takes its origin in Koszul’s note [Kos47]. The main notions of graded algebra were given their modern names in H. Cartan’s note [Car48]. See Boardman [Boa66] for operad-like approach to signs as opposed to closed symmetric monoidal category picture of Mac Lane [Mac63] (standard sign commutation rule). Combined together, these sign conventions make the number of signs in this paper tolerable.

If u : A → C, a ↦ au, is a chain map, its cone is the complex Cone(u) = C ⊕A[1], Cone^{k}(u) = C^{k} ⊕ A^{k+1}, with the differential (c, a)d = (cd^{C} + au, ad^{A[1]}) = (cd^{C} + au, −ad^{A}).

2. A_{∞}-categories, A_{∞}-functors and A_{∞}-transformations

2.1 Definition (Quiver). A graded 𝕜-quiver 𝒜 consists of the following data: a class of objects Ob 𝒜 (a 𝒰 -small set); a ℤ-graded 𝕜-module 𝒜(X, Y) = Hom_{𝒜} (X, Y) for each pair of objects X, Y. A morphism of 𝕜-quivers f : 𝒜 → ℬ is given by a map f : Ob 𝒜 → Ob ℬ, X ↦ Xf and by a 𝕜-linear map 𝒜(X, Y) → ℬ(Xf, Y f) for each pair of objects X, Y of 𝒜.

To a given graded 𝕜-quiver 𝒜 we associate another graded 𝕜-quiver – its tensor coalgebra T 𝒜, which has the same class of objects as 𝒜. For each sequence (X_{0}, X_{1}, X_{2}, …, X_{n}) of objects of 𝒜 there is the ℤ-graded 𝕜-module T^{n}𝒜 = 𝒜(X_{0}, X_{1}) ⊗_{𝕜} 𝒜(X_{1}, X_{2}) ⊗_{𝕜} ⋯ ⊗_{𝕜} 𝒜(X_{n−1}, X_{n}). For the sequence (X_{0}) with n = 0 it reduces to T^{0}𝒜 = 𝕜 in degree 0. The graded 𝕜-module T 𝒜(X, Y) = ⊕_{n⩾0}T^{n}𝒜 is the sum of the above modules over all sequences with X_{0} = X, X_{n} = Y. The quiver T 𝒜 is equipped with the cut comultiplication Δ : T 𝒜(X, Y) → ⊕_{Z∈Ob 𝒜} T 𝒜(X, Z) ⨂_{𝕜} T 𝒜(Z, Y), h_{1} ⊗h_{2} ⊗⋯⊗h_{n} ↦ ∑_{k=0}^{n} h_{1} ⊗⋯⊗h_{k} ⨂ h_{k+1} ⊗

⋯ ⊗ h_{n}, and the counit ε = (T 𝒜(X, Y) ^{pr_{0}}→ T^{0}𝒜(X, Y) → 𝕜), where the last map is id_{𝕜} if X = Y, or 0 if X ≠ Y (and T^{0}𝒜(X, Y) = 0). For this article it is the main example of the following notion:

2.2 Definition (Cocategory). A graded cocategory 𝒞 is a graded 𝕜-quiver 𝒞, equipped with a comultiplication – a 𝕜-linear map Δ_{X,Y}^{Z} : 𝒞(X, Y) → 𝒞(X, Z) ⊗_{𝕜} 𝒞(Z, Y) of degree 0 for all triples X, Y, Z of objects of 𝒞, and with a counit – a 𝕜-linear map ε_{X} : 𝒞(X, X) → 𝕜 of degree 0 for all objects X of 𝒞, such that the usual associativity equations and two counit equation hold.

Associated to each graded cocategory 𝒞 is a graded 𝕜-coalgebra C = ⊕_{X,Y ∈Ob 𝒞} 𝒞(X, Y). Vice versa, to a graded 𝕜-coalgebra, decomposed like that into 𝕜-submodules 𝒞(X, Y), X, Y ∈ Ob 𝒞, for some 𝒰 -small set Ob 𝒞, so that Δ(𝒞(X, Y)) ⊂ ⊕_{Z∈Ob 𝒞} 𝒞(X, Z) ⊗_{𝕜} 𝒞(Z, Y) for all pairs X, Y of objects of 𝒞, and ε(𝒞(X, Y)) = 0 for X ≠ Y, we associate a graded cocategory.

This interpretation allows one to define a cocategory homomorphism f : 𝒞 → 𝒟 as a particular case of a coalgebra homomorphism: a map f : Ob 𝒞 → Ob 𝒟, and 𝕜-linear maps 𝒞(X, Y) → 𝒟(Xf, Y f) for all pairs of objects X, Y of 𝒜, compatible with comultiplication and counit. Given cocategory homomorphisms f, g : 𝒞 → 𝒟 we say that a system of 𝕜-linear maps r : 𝒞(X, Y) → 𝒟(Xf, Y g), X, Y ∈ Ob 𝒞 is an (f, g)-coderivation, if the equation rΔ = Δ(f ⊗ r + r ⊗ g) holds.

In particular, these definitions apply to the tensor coalgebras T s𝒜 = T (s𝒜) of (suspended) 𝕜-quivers s𝒜. In this case cocategory homomorphisms and coderivations have a special form as we shall see below.

2.3 Definition (A_{∞}-category, Kontsevich [Kon95]). An A_{∞}-category 𝒜 consists of the following data: a graded 𝕜-quiver 𝒜; a differential b : T s𝒜 → T s𝒜 of degree 1, which is a (1,1)-coderivation, such that (T^{0}s𝒜)b = 0.

The definition of a (1,1)-coderivation bΔ = Δ(1 ⊗ b + b ⊗ 1) implies that a 𝕜-quiver morphism b is determined by a system of 𝕜-linear maps b pr_{1} : T s𝒜 → s𝒜 with components of degree 1

b_{n} : s𝒜(X_{0}, X_{1}) ⊗ s𝒜(X_{1}, X_{2}) ⊗ ⋯ ⊗ s𝒜(X_{n−1}, X_{n}) → s𝒜(X_{0}, X_{n}), n ⩾ 1,

via the formula

b_{kl} = (b|_{T^{k}s𝒜} ) pr_{l} : T^{k}s𝒜 → T^{l}s𝒜, b_{kl} = ∑_{\begin{matrix}r+n+t=k \\ r+1+t=l \end{matrix}} 1^{⊗r} ⊗ b_{n} ⊗ 1^{⊗t}.

Notice that the last condition of the definition implies b_{0} = 0. In particular, b_{k0} = 0, and k < l implies b_{kl} = 0. Since b^{2} is a (1,1)-coderivation of degree 2, the equation b^{2} = 0 is equivalent to its particular case b^{2} pr_{1} = 0, that is, for all k > 0

∑_{r+n+t=k}(1^{⊗r} ⊗ b_{n} ⊗ 1^{⊗t})b_{r+1+t} = 0 : T^{k}s𝒜 → s𝒜.  (2.3.1)

Using another, more traditional, form of components of b:

m_{n} = (𝒜^{⊗n} ^{s^{⊗n}} → (s𝒜)^{⊗n} ^{b_{n}}→ s𝒜 ^{s^{−1}} → 𝒜)

we rewrite (2.3.1) as follows:

∑_{r+n+t=k}(−)^{t+rn}(1^{⊗r} ⊗ m_{n} ⊗ 1^{⊗t})m_{r+1+t} = 0 : T^{k}𝒜 → 𝒜.  (2.3.2)

Notice that this equation differs in sign from [Kel01], because we are using right operators!

2.4 Definition (A_{∞}-functor, e.g. Keller [Kel01]). An A_{∞}-functor f : 𝒜 → ℬ consists of the following data: A_{∞}-categories 𝒜 and ℬ, a cocategory homomorphism f : T s𝒜 → T sℬ of degree 0, which commutes with the differential b.

The definition of a cocategory homomorphism fΔ = Δ(f ⊗ f), fε = ε implies that f is determined by a map f : Ob 𝒜 → Ob ℬ, X ↦ Xf and a system of 𝕜-linear maps f pr_{1} : T s𝒜 → sℬ with components of degree 0

f_{n} : s𝒜(X_{0}, X_{1}) ⊗ s𝒜(X_{1}, X_{2}) ⊗ ⋯ ⊗ s𝒜(X_{n−1}, X_{n}) → sℬ(X_{0}f, X_{n}f),

n ⩾ 1, (note that f_{0} = 0) via the formula

f_{kl} = (f|_{T^{k}s𝒜} ) pr_{l} : T^{k}s𝒜 → T^{l}sℬ,  f_{kl} = ∑_{i_{1}+···+i_{l}=k} f_{i_{1}} ⊗f_{i_{2}} ⊗⋯⊗f_{i_{l}}. (2.4.1)

In particular, f_{00} = id : 𝕜 → 𝕜, and k < l implies f_{kl} = 0. Since fb and bf are both (f, f)-coderivations of degree 1, the equation fb = bf is equivalent to its particular case fb pr_{1} = bf pr_{1}, that is, for all k > 0

∑_{l>0;i_{1}+···+i_{l}=k}(f_{i_{1}} ⊗ f_{i_{2}} ⊗ ⋯ ⊗ f_{i_{l}})b_{l} = ∑_{r+n+t=k}(1^{⊗r} ⊗ b_{n} ⊗ 1^{⊗t})f_{r+1+t} : T^{k}s𝒜 → sℬ.

(2.4.2)

Using m_{n} we rewrite (2.4.2) as follows:

∑_{i_{1}+···+i_{l}=k}^{l>0}(−)^{σ}(f_{i_{1}} ⊗ f_{i_{2}} ⊗ ⋯ ⊗ f_{i_{l}})m_{l} =

∑_{r+n+t=k}(−)^{t+rn}(1^{⊗r} ⊗ m_{n} ⊗ 1^{⊗t})f_{r+1+t} : T^{k}𝒜 → ℬ,

σ = (i_{2} − 1) + 2(i_{3} − 1) + ⋯ + (l − 2)(i_{l−1} − 1) + (l − 1)(i_{l} − 1).

Notice that this equation differs in sign from [Kel01], because we are using right operators.

2.5 Example. An A_{∞}-category with one object is called an A_{∞}-algebra (Stasheff [Sta63]). An A_{∞}-functor between A_{∞}-algebras is called an A_{∞}-homomorphism (Kadeishvili [Kad82]). These notions are psychologically easier to deal with, than the general case. The following notion of A_{∞}-transformations also makes sense for A_{∞}-algebras, however, such a context seems too narrow for it, because an A_{∞}-trans-formation is an analog of a transformation between ordinary functors without the naturality condition. Needless to say, in ordinary category theory there is no reason to consider unnatural transformations. The reasons to do it for A_{∞}-version are given in Section 5.

2.6 Definition (A_{∞}-transformation). An A_{∞}-transformation r : f → g : 𝒜 → ℬ of degree d (pre natural transformation in terms of [Fuk]) consists of the following data: A_{∞}-categories 𝒜 and ℬ; A_{∞}-functors f, g : 𝒜 → ℬ; an (f, g)-coderivation r : T s𝒜 → T sℬ of degree d.

The definition of an (f, g)-coderivation rΔ = Δ(f ⊗ r + r ⊗ g) implies that r is determined by a system of 𝕜-linear maps r pr_{1} : T s𝒜 → sℬ with components of degree d

r_{n} : s𝒜(X_{0}, X_{1}) ⊗ s𝒜(X_{1}, X_{2}) ⊗ ⋯ ⊗ s𝒜(X_{n−1}, X_{n}) → sℬ(X_{0}f, X_{n}g),

n ⩾ 0, via the formula

r_{kl} = (r|_{T^{k}s𝒜} ) pr_{l} : T^{k}s𝒜 → T^{l}sℬ,
r_{kl} = ∑_{\begin{matrix}q+1+t=l \\ i_{1}+···+i_{q}+n+j_{1}+···+j_{t}=k \end{matrix}} f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{q}} ⊗ r_{n} ⊗ g_{j_{1}} ⊗ ⋯ ⊗ g_{j_{t}}.  (2.6.1)

Note that r_{0} is a system of 𝕜-linear maps _{X}r_{0} : 𝕜 → sℬ(Xf, Xg), X ∈ Ob 𝒜. In fact, the terms ‘A_{∞}-transformation’ and ‘coderivation’ are synonyms.

In particular, r_{0l} vanishes unless l = 1, and r_{01} = r_{0}. The component r_{kl} vanishes unless 1 ⩽ l ⩽ k + 1.

2.7 Examples. 1) The restriction of an A_{∞}-transformation r to T^{1} is

r|_{T^{1}s𝒜} = r_{1} ⊕ [(f_{1} ⊗ r_{0}) + (r_{0} ⊗ g_{1})],

where r_{1} : s𝒜(X, Y) → sℬ(Xf, Y g),

f_{1} ⊗ r_{0} : s𝒜(X, Y) = s𝒜(X, Y) ⊗ 𝕜 ^{f_{1}⊗r_{0}} → sℬ(Xf, Y f) ⊗ sℬ(Y f, Y g),
r_{0} ⊗ g_{1} : s𝒜(X, Y) = 𝕜 ⊗ s𝒜(X, Y) ^{r_{0}⊗g_{1}} → sℬ(Xf, Xg) ⊗ sℬ(Xg, Y g).

2) The restriction of an A_{∞}-transformation r to T^{2} is

r|_{T^{2}s𝒜} = r_{2} ⊕ [(f_{2} ⊗ r_{0}) + (f_{1} ⊗ r_{1}) + (r_{1} ⊗ g_{1}) + (r_{0} ⊗ g_{2})]⊕

⊕ [(f_{1} ⊗ f_{1} ⊗ r_{0}) + (f_{1} ⊗ r_{0} ⊗ g_{1}) + (r_{0} ⊗ g_{1} ⊗ g_{1})], where r_{2} : s𝒜(X, Y) ⊗ s𝒜(Y, Z) → sℬ(Xf, Zg),

f_{2} ⊗ r_{0} : s𝒜(X, Y) ⊗ s𝒜(Y, Z) ⊗ 𝕜 → sℬ(Xf, Zf) ⊗ sℬ(Zf, Zg),
f_{1} ⊗ r_{1} : s𝒜(X, Y) ⊗ s𝒜(Y, Z) → sℬ(Xf, Y f) ⊗ sℬ(Y f, Zg),
r_{1} ⊗ g_{1} : s𝒜(X, Y) ⊗ s𝒜(Y, Z) → sℬ(Xf, Y g) ⊗ sℬ(Y g, Zg),
r_{0} ⊗ g_{2} : 𝕜 ⊗ s𝒜(X, Y) ⊗ s𝒜(Y, Z) → sℬ(Xf, Xg) ⊗ sℬ(Xg, Zg),
f_{1} ⊗ f_{1} ⊗ r_{0} : s𝒜(X, Y) ⊗ s𝒜(Y, Z) ⊗ 𝕜 → sℬ(Xf, Y f) ⊗ sℬ(Y f, Zf) ⊗ sℬ(Zf, Zg),
f_{1} ⊗ r_{0} ⊗ g_{1} : s𝒜(X, Y) ⊗ 𝕜 ⊗ s𝒜(Y, Z) → sℬ(Xf, Y f) ⊗ sℬ(Y f, Y g) ⊗ sℬ(Y g, Zg),
r_{0} ⊗ g_{1} ⊗ g_{1} : 𝕜 ⊗ s𝒜(X, Y) ⊗ s𝒜(Y, Z) → sℬ(Xf, Xg) ⊗ sℬ(Xg, Y g) ⊗ sℬ(Y g, Zg).

The 𝕜-module of (f, g)-coderivations r is ∏_{n=0}^{∞} V_{n}, where

V_{n} = ∏_{X_{0},...,X_{n}∈Ob 𝒜} Hom_{𝕜}(s𝒜(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒜(X_{n−1}, X_{n}), sℬ(X_{0}f, X_{n}g))

(2.7.1)

is the graded 𝕜-module of n-th components r_{n}. It is equipped with the differential d : V_{n} → V_{n}, given by the following formula

r_{n}d = r_{n}b_{1} − (−)^{r_{n}} ∑_{α+1+β=n}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β})r_{n}.  (2.7.2)

3. Coderivations and cocategory homomorphisms

Let 𝒜, ℬ be graded 𝕜-quivers, and let f^{0}, f^{1}, …, f^{n} : T 𝒜 → T ℬ be cocategory homomorphisms. Consider n coderivations r_{1}, …, r_{n} as in

f^{0} ^{r^{1}}→ f^{1} ^{r^{2}}→ … f^{n−1} ^{r^{n}}→ f^{n} : T 𝒜 → T ℬ.

We construct the following system of 𝕜-linear maps from these data: θ = (r^{1} ⊗ ⋯ ⊗ r^{n})θ : T 𝒜(X, Y) → T ℬ(Xf^{0}, Y f^{n}) of degree deg r^{1} + ⋯ + deg r^{n}. Its components θ_{kl} = θ|_{T^{k}𝒜} pr_{l} : T^{k}𝒜 → T^{l}ℬ are given by the following formula

θ_{kl} = ∑ f_{i_{1}^{0}}^{0} ⊗⋯⊗f_{i_{m_{0}}^{0}}^{0} ⊗r_{j_{1}}^{1} ⊗f_{i_{1}^{1}}^{1} ⊗⋯⊗f_{i_{m_{1}}^{1}}^{1} ⊗⋯⊗r_{j_{n}}^{n} ⊗f_{i_{1}^{n}}^{n} ⊗⋯⊗f_{i_{mn}^{n}}^{n} , (3.0.1)

where summation is taken over all terms with

m_{0}+m_{1}+⋯+m_{n}+n = l, i_{1}^{0}+⋯+i_{m_{0}}^{0}+j_{1}+i_{1}^{1}+⋯+i_{m_{1}}^{1}+⋯+j_{n}+i_{1}^{n}+⋯+i_{m_{n}}^{n} = k.

The component θ_{kl} vanishes unless n ⩽ l ⩽ k + n. If n = 0, we set ()θ = f^{0}. If n = 1, the formula gives (r^{1})θ = r^{1}.

3.1 Proposition. For each n ⩾ 0 the map θ satisfies the equation

(r^{1} ⊗ r^{2} ⊗ ⋯ ⊗ r^{n})θΔ = Δ ∑_{k=0}^{n}(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θ. (3.1.1)

Proof. Let us write down the required equation using ⨂ to separate the two copies of T ℬ in T ℬ ⨂ T ℬ, and ⊗ to denote the multiplication in T ℬ. We have to prove that for each n, x, y, z ⩾ 0 (T^{x}𝒜 ^{(r^{1}⊗···⊗r^{n})θ_{x,y+z}} → T^{y+z}ℬ ^{Δ_{yz}} → T^{y}ℬ ⨂ T^{z}ℬ)

= ∑_{k=0}^{n} ∑_{u+v=x}(T^{x}𝒜 ^{Δ_{uv}} → T^{u}𝒜 ⨂ T^{v}𝒜 ^{(r^{1}⊗···⊗r^{k})θ_{uy} ⨂(r^{k+1}⊗···⊗r^{n})θ_{vz}} → T^{y}ℬ ⨂ T^{z}ℬ).

Substituting (3.0.1) for θ in the above equation we come to an identity, which is proved by inspection. Indeed, skipping all the intermediate steps, we get the following equation:

∑_{k=0}^{n} ∑_{\begin{matrix}m_{0}+m_{1}+···+m_{n}+n=y+z \\ m_{0}+m_{1}+···+m_{k−1}+k−1<y⩽m_{0}+m_{1}+···+m_{k}+k \\ i_{1}^{0}+···+i_{m0}^{0} +j_{1}+i_{1}^{1}+···+i_{m1}^{1} +···+j_{n}+i_{1}^{n}+···+i_{mn}^{n} =x \end{matrix}}

f_{i_{1}^{0}}^{0}⊗⋯⊗f_{i_{m_{0}}^{0}}^{0} ⊗⋯⊗r_{j_{k}}^{k}⊗f_{i_{1}^{k}}^{k}⊗⋯⊗f_{i_{w}^{k}}^{k} ⨂ f_{i_{w+1}^{k}}^{k}⊗⋯⊗f_{i_{m_{k}}^{k}}^{k} ⊗r_{j_{k+1}}^{k+1}⊗⋯⊗f_{i_{1}^{n}}^{n}⊗⋯⊗f_{i_{mn}^{n}}^{n}

= ∑_{k=0}^{n} ∑_{u+v=x} ∑_{\begin{matrix}m_{0}+m_{1}+···+m_{k−1}+k+w=y \\ i_{1}^{0}+···+i_{m_{0}}^{0} +···+j_{k}+i_{1}^{k}+···+i_{w}^{k}=u \end{matrix}} ∑_{\begin{matrix}t+m_{k+1}+···+m_{n}+n−k=z \\ l_{1}+···+l_{t}+j_{k+1}+···+i_{1}^{n}+···+i_{mn}^{n} =v \end{matrix}}

f_{i_{1}^{0}}^{0}⊗⋯⊗f_{i_{m_{0}}^{0}}^{0} ⊗⋯⊗r_{j_{k}}^{k}⊗f_{i_{1}^{k}}^{k}⊗⋯⊗f_{i_{w}^{k}}^{k} ⨂ f_{l_{1}}^{k}⊗⋯⊗f_{l_{t}}^{k}⊗r_{j_{k+1}}^{k+1}⊗⋯⊗f_{i_{1}^{n}}^{n}⊗⋯⊗f_{i_{mn}^{n}}^{n} .

In the left hand side w denotes the expression y − (m_{0} + m_{1} + ⋯ + m_{k−1} + k) and lies in the interval 0 ⩽ w ⩽ m_{k}. Identifying m_{k} in the left hand side with w + t in the right hand side, we deduce that the both sides are equal.

3.2. Cocategory homomorphisms. Graded 𝕜-coalgebras form a symmetric monoidal category. The tensor product C⊗_{𝕜}D of 𝕜-coalgebras C, D is equipped with the comultiplication C⊗D ^{Δ⊗Δ} → C⊗C⊗D⊗D ^{1⊗c⊗1} → C⊗D⊗C⊗D, using the standard symmetry c of graded 𝕜-modules. Since graded cocategories are, in fact, graded coalgebras with a special decomposition, they also form a symmetric monoidal category gCoCat. If 𝒞 and 𝒟 are cocategories, then the class of objects of their tensor product 𝒞 ⊗ 𝒟 is Ob 𝒞 × Ob 𝒟, and 𝒞 ⊗ 𝒟(X × U, Y × W) = 𝒞(X, Y) ⊗_{𝕜} 𝒟(U, W).

Let ϕ : T 𝒜 ⊗ T 𝒞 → T ℬ be a cocategory homomorphism of degree 0. It is determined uniquely by its composition with pr_{1}, that is, by a family ϕ pr_{1} = (ϕ_{nm})_{n,m⩾0}, ϕ_{nm} : T^{n}𝒜 ⊗ T^{m}𝒞 → ℬ, ϕ_{00} = 0, with the same underlying map of objects Ob 𝒜 × Ob 𝒞 → Ob ℬ. Indeed, for given families of composable arrows f^{0} ^{p^{1}}→ f^{1} ^{p^{2}}→ … f^{n−1} ^{p^{n}}→ f^{n} of 𝒜 and g^{0} ^{t^{1}}→ g^{1} ^{t^{2}}→ … g^{m−1} ^{t^{m}}→ g^{m} of 𝒞 we have (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ϕ

= ∑_{\begin{matrix}i_{1}+···+i_{k}=n \\ j_{1}+···+j_{k}=m \end{matrix}}(−)^{σ}(p^{1} ⊗ ⋯ ⊗ p^{i_{1}} ⊗ t^{1} ⊗ ⋯ ⊗ t^{j_{1}})ϕ_{i_{1}j_{1}}

⊗ (p^{i_{1}+1} ⊗ ⋯ ⊗ p^{i_{1}+i_{2}} ⊗ t^{j_{1}+1} ⊗ ⋯ ⊗ t^{j_{1}+j_{2}})ϕ_{i_{2}j_{2}}

⊗ ⋯ ⊗ (p^{i_{1}+···+i_{k−1}+1} ⊗ ⋯ ⊗ p^{i_{1}+···+i_{k}} ⊗ t^{j_{1}+···+j_{k−1}+1} ⊗ ⋯ ⊗ t^{j_{1}+···+j_{k}})ϕ_{i_{k}j_{k}}. (3.2.1)

The sign depends on the parity of an integer

σ = (t^{1} + ⋯ + t^{j_{1}})(p^{i_{1}+1} + ⋯ + p^{i_{1}+···+i_{k}})+

(t^{j_{1}+1} + ⋯ + t^{j_{1}+j_{2}})(p^{i_{1}+i_{2}+1} + ⋯ + p^{i_{1}+···+i_{k}})

+ ⋯ + (t^{j_{1}+···+j_{k−2}+1} + ⋯ + t^{j_{1}+···+j_{k−1}})(p^{i_{1}+···+i_{k−1}+1} + ⋯ + p^{i_{1}+···+i_{k}}), (3.2.2)

where each coderivation has to be replaced with its degree. Recall that in our notation (Section 1) we abbreviate (−1)^{(deg t)(deg p)} to (−)^{tp}. By definition the homomorphism ϕ satisfies the equation

\begin{matrix}T 𝒜 ⊗ T 𝒞 & \overline{^{ϕ} →} & T ℬ & \overline{^{Δ} →} & T ℬ ⊗ T ℬ \\ T 𝒜 ⊗ T 𝒜 ⊗ T 𝒞 ⊗ T 𝒞 & ↓^{Δ⊗Δ} & \overline{^{1⊗c⊗1} →} & T 𝒜 ⊗ T 𝒞 ⊗ T 𝒜 ⊗ T 𝒞 & ^{↑^{ϕ⊗ϕ}} \end{matrix} (3.2.3)

Introduce 𝕜-linear maps (t^{1} ⊗ ⋯ ⊗ t^{m})χ : T 𝒜 → T ℬ by the formula a[(t^{1} ⊗ ⋯ ⊗ t^{m})χ] = (a ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ϕ, a ∈ T 𝒜. Then the above equation is equivalent to

(t^{1} ⊗ t^{2} ⊗ ⋯ ⊗ t^{m})χΔ = Δ ∑_{k=0}^{m}(t^{1} ⊗ ⋯ ⊗ t^{k})χ ⊗ (t^{k+1} ⊗ ⋯ ⊗ t^{m})χ. (3.2.4)

for all m ⩾ 0.

When 𝒜, ℬ are graded 𝕜-quivers, we define a new 𝕜-quiver Coder(𝒜, ℬ), whose objects are cocategory homomorphisms f : T 𝒜 → T ℬ. These homomorphisms are determined by a system f pr_{1} = (f_{n})_{n⩾1} of morphisms of 𝕜-quivers f_{n} : T^{n}𝒜 → ℬ of degree 0 with the same underlying map Ob 𝒜 → Ob ℬ, see (2.4.1). The 𝕜-module of morphisms between f, g : T 𝒜 → T ℬ consists of (f, g)-coderivations:

[Coder(𝒜, ℬ)(f, g)]^{d} = {r : T 𝒜 → T ℬ | rΔ = Δ(f ⊗r+r⊗g), deg r = d}, d ∈ ℤ.

Such a coderivation r is determined by a system of 𝕜-linear maps r pr_{1} = (r_{n})_{n⩾0}, r_{n} : T^{n}𝒜(X, Y) → ℬ(Xf, Y g) of degree d as in (2.6.1).

3.3 Corollary (to Proposition 3.1). A map α : T 𝒜 ⊗ T Coder(𝒜, ℬ) → T ℬ, a ⊗ r^{1} ⊗ ⋯ ⊗ r^{n} ↦ a[(r^{1} ⊗ ⋯ ⊗ r^{n})θ], is a cocategory homomorphism of degree 0.

Proof. Equation (3.1.1) means that equation (3.2.4) holds for χ = θ, which is equivalent to (3.2.3) for ϕ = α, 𝒞 = Coder(𝒜, ℬ).

3.4 Proposition. For any cocategory homomorphism ϕ : T 𝒜 ⊗ T 𝒞^{1} ⊗ T 𝒞^{2} ⊗ ⋯ ⊗ T 𝒞^{q} → T ℬ of degree 0 there is a unique cocategory homomorphism ψ : T 𝒞^{1} ⊗ T 𝒞^{2} ⊗ ⋯ ⊗ T 𝒞^{q} → T Coder(𝒜, ℬ) of degree 0, such that

ϕ = (T 𝒜 ⊗ T 𝒞^{1} ⊗ T 𝒞^{2} ⊗ ⋯ ⊗ T 𝒞^{q} ^{1⊗ψ} → T 𝒜 ⊗ T Coder(𝒜, ℬ) ^{α}→ T ℬ).

Proof. Let us start with a simple case q = 1, 𝒞 = 𝒞^{1}. Each object g of 𝒞 induces a cocategory morphism gψ : a ↦ (a ⊗ g)ϕ. We set ψ_{0} = 0. Each element p ∈ 𝒞(g, h) induces a coderivation (p)ψ_{1} = pψ : a ↦ (a ⊗ p)ϕ. Suppose that ψ_{i} are already found for 0 ⩽ i < n. Then we find ψ_{n} from the sought identity χ = ψθ. Namely, for g^{0} ^{p^{1}}→ g^{1} ^{p^{2}}→ … g^{n−1} ^{p^{n}}→ g^{n} we have to satisfy the identity

(p^{1}⊗⋯⊗p^{n})χ = (p^{1}⊗⋯⊗p^{n})ψ_{n}+∑_{l=2}^{n} ∑_{i_{1}+···+i_{l}=n}[(p^{1}⊗⋯⊗p^{n}).(ψ_{i_{1}}⊗ψ_{i_{2}}⊗⋯⊗ψ_{i_{l}})]θ,

which expresses ψ via its components ψ_{k}. Notice that the unknown ψ_{n} occurs only in the singled out summand, corresponding to l = 1. The factors ψ_{i} in the sum are already known, since i < n. So we define (p^{1} ⊗ ⋯ ⊗ p^{n})ψ_{n} : T 𝒜 → T ℬ as the difference of (p^{1} ⊗ ⋯ ⊗ p^{n})χ and the sum in the right hand side. Assume that ψ is a cocategory homomorphism up to the level n, that is,

(p^{1} ⊗ ⋯ ⊗ p^{m})ψ = ∑_{l=1}^{m} ∑_{i_{1}+···+i_{l}=m}(p^{1} ⊗ ⋯ ⊗ p^{m}).(ψ_{i_{1}} ⊗ ψ_{i_{2}} ⊗ ⋯ ⊗ ψ_{i_{l}}) (3.4.1)

for all 0 ⩽ m ⩽ n. Taking into account equations (3.1.1), we see that (3.2.4) is equivalent to an equation of the form

(p^{1} ⊗ ⋯ ⊗ p^{n})ψ_{n}Δ = Δ[g^{0}ψ ⊗ (p^{1} ⊗ ⋯ ⊗ p^{n})ψ_{n} + (p^{1} ⊗ ⋯ ⊗ p^{n})ψ_{n} ⊗ g^{n}ψ + μ].

Moreover, if (p^{1} ⊗⋯⊗p^{n})ψ_{n} were a (g^{0}ψ, g^{n}ψ)-coderivation, it would imply (3.2.4) by Section 3.2. We deduce that, indeed, the above μ = 0, and (p^{1} ⊗ ⋯ ⊗ p^{n})ψ_{n} is a (g^{0}ψ, g^{n}ψ)-coderivation. Thus, we have found a unique (p^{1} ⊗ ⋯ ⊗ p^{n})ψ_{n} ∈ Coder(𝒜, ℬ) and (3.4.1) for m = n defines uniquely an element (p^{1} ⊗ ⋯ ⊗ p^{n})ψ ∈ T Coder(𝒜, ℬ).

The case q > 1 is similar to the case q = 1, however, the reasoning is slightly obstructed by a big amount of indices. So we explain in detail the case q = 2 only, and in the general case no new phenomena occur. Further we shall use the obtained formulas in the case q = 2.

Let 𝒞 = 𝒞^{1}, 𝒟 = 𝒞^{2}. We consider a cocategory homomorphism ϕ : T 𝒜 ⊗ T 𝒞 ⊗ T 𝒟 → T ℬ, a ↦ a[(c ⊗ d)χ]. We have to obtain from it a unique cocategory homomorphism ψ : T 𝒞 ⊗ T 𝒟 → T Coder(𝒜, ℬ).

A pair of objects f ∈ Ob 𝒞, g ∈ Ob 𝒟 induces a cocategory morphism (f, g)ψ : a ↦ (a⊗1_{f} ⊗1_{g})ϕ. We set ψ_{00} = 0. An object f ∈ Ob 𝒞 and an element t ∈ 𝒟(g^{0}, g^{1}) induce an ((f, g^{0})ψ, (f, g^{1})ψ)-coderivation (f ⊗ t)ψ_{01} = (f ⊗ t)ψ : a ↦ (a ⊗ 1_{f} ⊗ t)ϕ. An element p ∈ 𝒞(f^{0}, f^{1}) and an object g ∈ Ob 𝒟 induce an ((f^{0}, g)ψ, (f^{1}, g)ψ)- coderivation (p ⊗ g)ψ_{10} = (p ⊗ g)ψ : a ↦ (a ⊗ p ⊗ 1_{g})ϕ. Suppose that ψ_{ij} are already found for 0 ⩽ i ⩽ n, 0 ⩽ j ⩽ m, (i, j) ≠ (n, m). Then we find ψ_{nm} from the

sought identity χ = ψθ. Namely, for f^{0} ^{p^{1}}→ f^{1} ^{p^{2}}→ … f^{n−1} ^{p^{n}}→ f^{n} in 𝒞 and g^{0} ^{t^{1}}→ g^{1} ^{t^{2}}→ … g^{m−1} ^{t^{m}}→ g^{m} in 𝒟 we have to satisfy the identity

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})χ = (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ψ_{nm}

+ ∑_{\begin{matrix}i_{1}+···+i_{k}=n \\ j_{1}+···+j_{k}=m \end{matrix}}^{k>1}(−)^{σ}[(p^{1} ⊗ ⋯ ⊗ p^{i_{1}} ⊗ t^{1} ⊗ ⋯ ⊗ t^{j_{1}})ψ_{i_{1}j_{1}}

⊗ (p^{i_{1}+1} ⊗ ⋯ ⊗ p^{i_{1}+i_{2}} ⊗ t^{j_{1}+1} ⊗ ⋯ ⊗ t^{j_{1}+j_{2}})ψ_{i_{2}j_{2}}

⊗ ⋯ ⊗ (p^{i_{1}+···+i_{k−1}+1} ⊗ ⋯ ⊗ p^{i_{1}+···+i_{k}} ⊗ t^{j_{1}+···+j_{k−1}+1} ⊗ ⋯ ⊗ t^{j_{1}+···+j_{k}})ψ_{i_{k}j_{k}}]θ, (3.4.2)

which is nothing else but (3.2.1). The sign is determined by (3.2.2). All terms of the sum are already known. So we define a map (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ψ_{nm} : T 𝒜 → T ℬ as the difference of the left hand side and the sum in the right hand side. The fact that ϕ is a homomorphism is equivalent to the identity for the map χ for all n, m ⩾ 0:

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})χΔ = Δ ∑_{k=0}^{n} ∑_{l=0}^{m}(−)^{(p^{k+1}+···+p^{n})(t^{1}+···+t^{l})}

(p^{1} ⊗ ⋯ ⊗ p^{k} ⊗ t^{1} ⊗ ⋯ ⊗ t^{l})χ ⊗ (p^{k+1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{l+1} ⊗ ⋯ ⊗ t^{m})χ, (3.4.3)

similarly to Section 3.2. From it we get an equation for (p^{1}⊗⋯⊗p^{n}⊗t^{1}⊗⋯⊗t^{m})ψ_{nm}

(p^{1}⊗⋯⊗p^{n}⊗t^{1}⊗⋯⊗t^{m})ψ_{nm}Δ = Δ[(f^{0}, g^{0})ψ⊗(p^{1}⊗⋯⊗p^{n}⊗t^{1}⊗⋯⊗t^{m})ψ_{nm}

+ (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ψ_{nm} ⊗ (f^{n}, g^{m})ψ + μ].

Notice that if ψ is indeed a homomorphism and ψ_{nm} is its component, then ϕ is a homomorphism, hence, (3.4.3) holds. Thus, the above equation with μ = 0 implies (3.4.3) (and it happens only for one value of μ). Since we know that (3.4.3) holds, it implies μ = 0. Therefore, (p^{1}⊗⋯⊗p^{n}⊗t^{1}⊗⋯⊗t^{m})ψ_{nm} is a ((f^{0}, g^{0})ψ, (f^{n}, g^{m})ψ)- coderivation. Thus, we have found a unique (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ψ_{nm} ∈ Coder(𝒜, ℬ), and

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ψ = (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})ψ_{nm}

+ ∑_{\begin{matrix}i_{1}+···+i_{k}=n \\ j_{1}+···+j_{k}=m \end{matrix}}^{k>1}(−)^{σ}(p^{1} ⊗ ⋯ ⊗ p^{i_{1}} ⊗ t^{1} ⊗ ⋯ ⊗ t^{j_{1}})ψ_{i_{1}j_{1}}

⊗ (p^{i_{1}+1} ⊗ ⋯ ⊗ p^{i_{1}+i_{2}} ⊗ t^{j_{1}+1} ⊗ ⋯ ⊗ t^{j_{1}+j_{2}})ψ_{i_{2}j_{2}}

⊗ ⋯ ⊗ (p^{i_{1}+···+i_{k−1}+1} ⊗ ⋯ ⊗ p^{i_{1}+···+i_{k}} ⊗ t^{j_{1}+···+j_{k−1}+1} ⊗ ⋯ ⊗ t^{j_{1}+···+j_{k}})ψ_{i_{k}j_{k}}

defines uniquely an element of T Coder(𝒜, ℬ). The above formula implies that ψ is a homomorphism.

A generalization to q > 2 is straightforward.

We interpret the above proposition as the existence of inner hom-objects Hom(T 𝒜, T ℬ) = T Coder(𝒜, ℬ) in the monoidal category of cocategories of the form T 𝒞^{1} ⊗ T 𝒞^{2} ⊗ ⋯ ⊗ T 𝒞^{r}.

4. A category enriched in cocategories

Let us show that the category of tensor coalgebras of graded 𝕜-quivers is enriched in gCoCat.

Let 𝒜, ℬ, 𝒞 be graded 𝕜-quivers. Consider the cocategory homomorphism given by the upper right path in the diagram

T 𝒜 ⊗ T Coder(𝒜, ℬ) ⊗ T Coder(ℬ, 𝒞) \overline{→^{α⊗1}} T ℬ ⊗ T Coder(ℬ, 𝒞)

=

T 𝒜 ⊗ T Coder(𝒜, 𝒞) ↓^{1⊗M}  \overline{^{α}  →} T 𝒞 ↓^{α}

(4.0.1)

By Proposition 3.4 there is a graded cocategory morphism of degree 0

M : T Coder(𝒜, ℬ) ⊗ T Coder(ℬ, 𝒞) → T Coder(𝒜, 𝒞).

Denote by 𝟙 a graded 1-object-0-morphisms 𝕜-quiver, that is, Ob 𝟙 = {∗}, 𝟙(∗, ∗) = 0. Then T 𝟙 = 𝕜 is a unit object of the monoidal category of graded cocategories. Denote by r : T 𝒜 ⊗ T 𝟙 → T 𝒜 and l : T 𝟙 ⊗ T 𝒜 → T 𝒜 the corresponding natural cocategory isomorphisms. By Proposition 3.4 there exists a unique cocategory morphism η_{𝒜} : T 𝟙 → T Coder(𝒜, 𝒜), such that

r = (T 𝒜 ⊗ T 𝟙 ^{1⊗η_{𝒜}}→ T 𝒜 ⊗ T Coder(𝒜, 𝒜) ^{α}→ T 𝒜).

Namely, the object ∗ ∈ Ob 𝟙 goes to the identity homomorphism id_{𝒜} : 𝒜 → 𝒜, which acts as the identity map on objects, and has only one non-vanishing component (id_{𝒜} )_{1} = id : 𝒜(X, Y) → 𝒜(X, Y).

4.1 Proposition (See also Kontsevich and Soibelman [KS]). The multiplication M is associative and η is its two-sided unit:

T Coder(𝒜, ℬ) ⊗ T Coder(ℬ, 𝒞) ⊗ T Coder(𝒞, 𝒟)  \overline{→^{M⊗1}} T Coder(𝒜, 𝒞) ⊗ T Coder(𝒞, 𝒟)

T Coder(𝒜, ℬ) ⊗ T Coder(ℬ, 𝒟) ↓^{1⊗M}  \overline{^{M}  →} T Coder(𝒜, 𝒟) ↓^{M}

Proof. The cocategory homomorphism

T 𝒜 ⊗ T Coder(𝒜, ℬ) ⊗ T Coder(ℬ, 𝒞) ⊗ T Coder(𝒞, 𝒟)

α⊗1⊗1 → T ℬ ⊗ T Coder(ℬ, 𝒞) ⊗ T Coder(𝒞, 𝒟) ^{α⊗1} → T 𝒞 ⊗ T Coder(𝒞, 𝒟) ^{α}→ T 𝒟

can be written down as

(α ⊗ 1 ⊗ 1)(1 ⊗ M)α = (1 ⊗ 1 ⊗ M)(α ⊗ 1)α = (1 ⊗ 1 ⊗ M)(1 ⊗ M)α,

or as

(1 ⊗ M ⊗ 1)(α ⊗ 1)α = (1 ⊗ M ⊗ 1)(1 ⊗ M)α.

The uniqueness part of Proposition 3.4 implies that (1 ⊗ M)M = (M ⊗ 1)M. Similarly one proves that η is a unit for M.

By (2.4.1) we find

gCoCat(T 𝟙, T Coder(𝒜, ℬ)) = Maps({∗}, Ob Coder(𝒜, ℬ)) = gCoCat(T 𝒜, T ℬ).

Thus we can interpret Proposition 4.1 as saying that the category of tensor coalgebras of graded 𝕜-quivers admits an enrichment in gCoCat.

Let us find explicit formulas for M. It is defined on objects as composition: if f : 𝒜 → ℬ and g : ℬ → 𝒞 are cocategory morphisms, then (f, g)M = fg : 𝒜 → 𝒞. On coderivations M is specified by its composition with pr_{1} : T Coder(𝒜, 𝒞) → Coder(𝒜, 𝒞). Let us write in this section (h, k) as a shorthand for Coder(𝒜, 𝒞)(h, k), the 𝕜-module of (h, k)-coderivations. The components of M are

M_{nm} = M|_{T^{n}⊗T^{m}} pr_{1} : T^{n} Coder(𝒜, ℬ) ⊗ T^{m} Coder(ℬ, 𝒞) → Coder(𝒜, 𝒞),

M_{nm} : (f^{0}, f^{1}) ⊗ ⋯ ⊗ (f^{n−1}, f^{n}) ⊗ (g^{0}, g^{1}) ⊗ ⋯ ⊗ (g^{m−1}, g^{m}) → (f^{0}g^{0}, f^{n}g^{m}),

where f^{0}, …, f^{n} : 𝒜 → ℬ and g^{0}, …, g^{m} : ℬ → 𝒞 are cocategory morphisms. We have M_{00} = 0.

According to proof of Proposition 3.4 the component M_{nm} is determined recursively from equation (3.4.2):

(p^{1} ⊗ ⋯ ⊗ p^{n})θ(t^{1} ⊗ ⋯ ⊗ t^{m})θ = (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})M_{nm}

+ ∑_{\begin{matrix}i_{1}+···+i_{k}=n \\ j_{1}+···+j_{k}=m \end{matrix}}^{k>1} (−)^{σ}[(p^{1} ⊗ ⋯ ⊗ p^{i_{1}} ⊗ t^{1} ⊗ ⋯ ⊗ t^{j_{1}})M_{i_{1}j_{1}}

⊗ (p^{i_{1}+1} ⊗ ⋯ ⊗ p^{i_{1}+i_{2}} ⊗ t^{j_{1}+1} ⊗ ⋯ ⊗ t^{j_{1}+j_{2}})M_{i_{2}j_{2}}

⊗ ⋯ ⊗ (p^{i_{1}+···+i_{k−1}+1} ⊗ ⋯ ⊗ p^{i_{1}+···+i_{k}} ⊗ t^{j_{1}+···+j_{k−1}+1} ⊗ ⋯ ⊗ t^{j_{1}+···+j_{k}})M_{i_{k}j_{k}}]θ. (4.1.1)

Since (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})M_{nm} is a coderivation, it is determined by its composition with projection pr_{1}. Composing (4.1.1) with pr_{1} we get

(p^{1} ⊗⋯⊗p^{n})θ(t^{1} ⊗⋯⊗t^{m})θ pr_{1} = (p^{1} ⊗⋯⊗p^{n} ⊗t^{1} ⊗⋯⊗t^{m})M_{nm} pr_{1} . (4.1.2)

Therefore, if m = 0 and n is positive, M_{n0} is given by the formula:

M_{n0} : (f^{0}, f^{1}) ⊗ ⋯ ⊗ (f^{n−1}, f^{n}) ⊗ 𝕜_{g^{0}} → (f^{0}g^{0}, f^{n}g^{0}),
r^{1} ⊗ ⋯ ⊗ r^{n} ⊗ 1 ↦ (r^{1} ⊗ ⋯ ⊗ r^{n} | g^{0})M_{n0},
[(r^{1} ⊗ ⋯ ⊗ r^{n} | g^{0})M_{n0}] pr_{1} = (r^{1} ⊗ ⋯ ⊗ r^{n})θg^{0} pr_{1},

where | separates the arguments in place of ⊗. If m = 1, then M_{n1} is given by the formula:

M_{n1} : (f^{0}, f^{1}) ⊗ ⋯ ⊗ (f^{n−1}, f^{n}) ⊗ (g^{0}, g^{1}) → (f^{0}g^{0}, f^{n}g^{1}),
r^{1} ⊗ ⋯ ⊗ r^{n} ⊗ t^{1} ↦ (r^{1} ⊗ ⋯ ⊗ r^{n} ⊗ t^{1})M_{n1},
[(r^{1} ⊗ ⋯ ⊗ r^{n} ⊗ t^{1})M_{n1}] pr_{1} = (r^{1} ⊗ ⋯ ⊗ r^{n})θt^{1} pr_{1} .

Explicitly we write

[(r^{1} ⊗ ⋯ ⊗ r^{n} | g^{0})M_{n0}]_{k} = ∑_{l}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}g_{l}^{0},  (4.1.3)
[(r^{1} ⊗ ⋯ ⊗ r^{n} ⊗ t^{1})M_{n1}]_{k} = ∑_{l}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}t_{l}^{1}.  (4.1.4)

Finally, M_{nm} = 0 for m > 1, since the left hand side of (4.1.2) vanishes.

4.2 Examples. 1) The component M_{01} is the composition: (f^{0} | t^{1})M_{01} = f^{0}t^{1}. 2) The component M_{10} is the composition: (r^{1} | g^{0})M_{10} = r^{1}g^{0}.

3) If r : f → g : 𝒜 → ℬ and p : h → k : ℬ → 𝒞 are A_{∞}-transformations, then (r ⊗ p)M_{11} : fh → gk : 𝒜 → 𝒞 has the following components:

[(r ⊗ p)M_{11}]_{0} = r_{0}p_{1},
[(r ⊗ p)M_{11}]_{1} = r_{1}p_{1} + (f_{1} ⊗ r_{0})p_{2} + (r_{0} ⊗ g_{1})p_{2}, etc.

4) If f ^{r}→ g ^{p}→ h : 𝒜 → ℬ are A_{∞}-transformations, and k : ℬ → 𝒞 is an A_{∞}-functor, then (r ⊗ p | k)M_{20} : fk → hk : 𝒜 → 𝒞 has the following components:

[(r ⊗ p | k)M_{20}]_{0} = (r_{0} ⊗ p_{0})k_{2},
[(r ⊗ p | k)M_{20}]_{1} = (r_{1} ⊗ p_{0})k_{2} + (r_{0} ⊗ p_{1})k_{2}

+ (r_{0} ⊗ p_{0} ⊗ h_{1})k_{3} + (r_{0} ⊗ g_{1} ⊗ p_{0})k_{3} + (f_{1} ⊗ r_{0} ⊗ p_{0})k_{3}, etc.

5) If f ^{r}→ g ^{p}→ h : 𝒜 → ℬ and t : k → l : ℬ → 𝒞 are A_{∞}-transformations, then (r ⊗ p ⊗ t)M_{21} : fk → hl : 𝒜 → 𝒞 has the following components:

[(r ⊗ p ⊗ t)M_{21}]_{0} = (r_{0} ⊗ p_{0})t_{2},
[(r ⊗ p ⊗ t)M_{21}]_{1} = (r_{1} ⊗ p_{0})t_{2} + (r_{0} ⊗ p_{1})t_{2}

+ (r_{0} ⊗ p_{0} ⊗ h_{1})t_{3} + (r_{0} ⊗ g_{1} ⊗ p_{0})t_{3} + (f_{1} ⊗ r_{0} ⊗ p_{0})t_{3}, etc.

5. A_{∞}-category of A_{∞}-functors

Let us construct a new A_{∞}-category A_{∞}(𝒜, ℬ) out of given two 𝒜 and ℬ. Its underlying graded 𝕜-quiver is a full subquiver of s^{−1} Coder(s𝒜, sℬ). The objects of A_{∞}(𝒜, ℬ) are A_{∞}-functors f : 𝒜 → ℬ. Given two such functors f, g : 𝒜 → ℬ we define the graded 𝕜-module A_{∞}(𝒜, ℬ)(f, g) as the space of all A_{∞}-transformations r : f → g, namely,

[A_{∞}(𝒜, ℬ)(f, g)]^{d+1} =
{r : f → g | A_{∞}-transformation r : T s𝒜 → T sℬ has degree d}.

In this section we use the notation (f, g) = sA_{∞}(𝒜, ℬ)(f, g) = Coder(s𝒜, sℬ)(f, g) for the sake of brevity. The degree of r as an element of (f, g) will be exactly d:

(f, g)^{d} = {r : f → g | A_{∞}-transformation r : T s𝒜 → T sℬ has degree d}.

We will use only this (natural) degree of r in order to permute it with other things by Koszul’s rule.

Notice that even if 𝒜, ℬ have one object (and are A_{∞}-algebras), the quiver A_{∞}(𝒜, ℬ) has several objects. Thus theory of A_{∞}-algebras leads to the theory of A_{∞}-categories.

5.1 Proposition (See also Fukaya [Fuk], Kontsevich and Soibelman [KS02, KS] and Lefèvre-Hasegawa [LH02]). Let 𝒜, ℬ be A_{∞}-categories. Then there exists a unique (1,1)-coderivation B : T sA_{∞}(𝒜, ℬ) → T sA_{∞}(𝒜, ℬ) of degree 1, such that B_{0} = 0 and

(r^{1} ⊗ ⋯ ⊗ r^{n})θb = [(r^{1} ⊗ ⋯ ⊗ r^{n})B]θ + (−)^{r^{1}+···+r^{n}}b(r^{1} ⊗ ⋯ ⊗ r^{n})θ (5.1.1)

for all n ⩾ 0, r^{1} ⊗ ⋯ ⊗ r^{n} ∈ (f^{0}, f^{1}) ⊗ ⋯ ⊗ (f^{n−1}, f^{n}). It satisfies B^{2} = 0, thus, it gives an A_{∞}-structure of A_{∞}(𝒜, ℬ).

Proof. For n = 0 (5.1.1) reads as f^{0}b = (f^{0})B +bf^{0}, hence, (f^{0})B = f^{0}b−bf^{0} = 0. In particular, we may set B_{0} = 0. Assume that the coderivation components B_{j} for j < n are already found, so that (5.1.1) is satisfied up to n − 1 arguments. Let us determine a 𝕜-linear map (r^{1} ⊗ ⋯ ⊗ r^{n})B_{n} : T s𝒜 → T sℬ from equation (5.1.1), rewritten as follows:

(r^{1} ⊗ ⋯ ⊗ r^{n})B_{n} = (r^{1} ⊗ ⋯ ⊗ r^{n})θb − (−)^{r^{1}+···+r^{n}}b(r^{1} ⊗ ⋯ ⊗ r^{n})θ

− ∑_{q+j+t=n}^{j<n}[(r^{1} ⊗ ⋯ ⊗ r^{n})(1^{⊗q} ⊗ B_{j} ⊗ 1^{⊗t})]θ. (5.1.2)

Let us show that (r^{1} ⊗ ⋯ ⊗ r^{n})B_{n} is a (f^{0}, f^{n})-coderivation. Indeed,

(r^{1} ⊗ ⋯ ⊗ r^{n})B_{n}Δ_{ℬ} = (r^{1} ⊗ ⋯ ⊗ r^{n})θbΔ_{ℬ} − (−)^{r^{1}+···+r^{n}}b(r^{1} ⊗ ⋯ ⊗ r^{n})θΔ_{ℬ}

− ∑_{q+j+t=n}^{j<n}[(r^{1} ⊗ ⋯ ⊗ r^{n})(1^{⊗q} ⊗ B_{j} ⊗ 1^{⊗t})]θΔ_{ℬ}

= (r^{1} ⊗ ⋯ ⊗ r^{n})θΔ_{ℬ} (1 ⊗ b + b ⊗ 1)

− (−)^{r^{1}+···+r^{n}}bΔ_{𝒜} ∑_{k=0}^{n}(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θ

− ∑_{q+j+t=n}^{j<n} Δ_{𝒜} {[(r^{1} ⊗ ⋯ ⊗ r^{n})(1^{⊗q} ⊗ B_{j} ⊗ 1^{⊗t})]Δ_{A_{∞}(𝒜,ℬ)} (θ ⊗ θ)}

= Δ_{𝒜} {∑_{k=0}^{n}(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θ(1 ⊗ b + b ⊗ 1)

− (−)^{r^{1}+···+r^{n}}(1 ⊗ b + b ⊗ 1) ∑_{k=0}^{n}(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θ

− ∑_{k+v+j+t=n}^{j<n}[(r^{1} ⊗ ⋯ ⊗ r^{n})(1^{⊗k} ⨂ 1^{⊗v} ⊗ B_{j} ⊗ 1^{⊗t})](θ ⨂ θ)

− ∑_{q+j+w+u=n}^{j<n}[(r^{1} ⊗ ⋯ ⊗ r^{n})(1^{⊗q} ⊗ B_{j} ⊗ 1^{⊗w} ⨂ 1^{⊗u})](θ ⨂ θ)}

= Δ_{𝒜} {∑_{k=0}^{n}(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θb

− ∑_{k=0}^{n}(−)^{r^{k+1}+···+r^{n}}(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ b(r^{k+1} ⊗ ⋯ ⊗ r^{n})θ

− ∑_{k+v+j+t=n}^{j<n}(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})(1^{⊗v} ⊗ B_{j} ⊗ 1^{⊗t})θ

+ ∑_{k=0}^{n}(−)^{r^{k+1}+···+r^{n}}(r^{1} ⊗ ⋯ ⊗ r^{k})θb ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θ

− ∑_{k=0}^{n}(−)^{r^{1}+···+r^{n}}b(r^{1} ⊗ ⋯ ⊗ r^{k})θ ⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θ

− ∑_{k=0}^{n}(−)^{r^{k+1}+···+r^{n}} ∑_{q+j+w=k}^{j<n}(r^{1} ⊗ ⋯ ⊗ r^{k})(1^{⊗q} ⊗ B_{j} ⊗ 1^{⊗w})θ

⊗ (r^{k+1} ⊗ ⋯ ⊗ r^{n})θ}

= Δ_{𝒜} [f^{0} ⊗ (r^{1} ⊗ ⋯ ⊗ r^{n})B_{n} + (r^{1} ⊗ ⋯ ⊗ r^{n})B_{n} ⊗ f^{n}].

The last three sums cancel out for all k < n due to (5.1.2), and for k = n they give (r^{1} ⊗ ⋯ ⊗ r^{n})B_{n} ⊗ f^{n} due to the same equation. Similarly for the previous three sums. Therefore, (5.1.2) is, indeed, a recursive definition of components B_{n} of a coderivation B. The uniqueness of B is obvious.

Clearly, B^{2} : T sA_{∞}(𝒜, ℬ) → T sA_{∞}(𝒜, ℬ) is a (1,1)-coderivation of degree 2. From (5.1.1) we find

[(r^{1} ⊗ ⋯ ⊗ r^{n})B^{2}]θ = [(r^{1} ⊗ ⋯ ⊗ r^{n})B]θb − (−)^{r^{1}+···+r^{n}+1}b[(r^{1} ⊗ ⋯ ⊗ r^{n})B]θ

= (r^{1} ⊗ ⋯ ⊗ r^{n})θb^{2} − (−)^{r^{1}+···+r^{n}}b[(r^{1} ⊗ ⋯ ⊗ r^{n})θ]b

− (−)^{r^{1}+···+r^{n}+1}b[(r^{1} ⊗ ⋯ ⊗ r^{n})θ]b − b^{2}(r^{1} ⊗ ⋯ ⊗ r^{n})θ = 0.

Composing this equation with pr_{1} : T sℬ → sℬ we get

0 = [(r^{1} ⊗ ⋯ ⊗ r^{n})B^{2}]θ pr_{1} = (r^{1} ⊗ ⋯ ⊗ r^{n})[B^{2}]_{n} pr_{1} .

Therefore, all components of the (f^{0}, f^{n})-coderivation (r^{1} ⊗ ⋯ ⊗ r^{n})[B^{2}]_{n} vanish. We deduce that the coderivations (r^{1} ⊗ ⋯ ⊗ r^{n})[B^{2}]_{n} vanish, hence, all [B^{2}]_{n} = 0. Finally, B^{2} = 0.

Let us find explicitly the components of B, composing (5.1.1) with pr_{1} : T sℬ → sℬ:

B_{1} : (f, g) → (f, g), r ↦ (r)B_{1} = [r, b] = rb − (−)^{r}br,

B_{n} : (f^{0}, f^{1}) ⊗ ⋯ ⊗ (f^{n−1}, f^{n}) → (f^{0}, f^{n}), r^{1} ⊗ ⋯ ⊗ r^{n} ↦ (r^{1} ⊗ ⋯ ⊗ r^{n})B_{n}, for n > 1,

where the last transformation is defined by its composition with pr_{1}:

[(r^{1} ⊗ ⋯ ⊗ r^{n})B_{n}] pr_{1} = [(r^{1} ⊗ ⋯ ⊗ r^{n})θ]b pr_{1} .

In the other terms, for n > 1

[(r^{1} ⊗ ⋯ ⊗ r^{n})B_{n}]_{k} = ∑_{l}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}b_{l}.  (5.1.3)

Since B^{2} = 0, we have, in particular,

∑_{r+n+t=k}(1^{⊗r} ⊗ B_{n} ⊗ 1^{⊗t})B_{r+1+t} = 0 : T^{k}sA_{∞}(𝒜, ℬ) → sA_{∞}(𝒜, ℬ).

5.2 Examples. 1) When n = 1, r : f → g : 𝒜 → ℬ, we find the components of the A_{∞}-transformation (r)B_{1} : f → g : 𝒜 → ℬ as follows (see Examples 2.7):

[(r)B_{1}]_{0} = r_{0}b_{1},
[(r)B_{1}]_{1} = r_{1}b_{1} + (f_{1} ⊗ r_{0})b_{2} + (r_{0} ⊗ g_{1})b_{2} − (−)^{r}b_{1}r_{1},
[(r)B_{1}]_{2} = r_{2}b_{1} + (f_{2} ⊗ r_{0})b_{2} + (f_{1} ⊗ r_{1})b_{2} + (r_{1} ⊗ g_{1})b_{2} + (r_{0} ⊗ g_{2})b_{2}
+ (f_{1} ⊗ f_{1} ⊗ r_{0})b_{3} + (f_{1} ⊗ r_{0} ⊗ g_{1})b_{3} + (r_{0} ⊗ g_{1} ⊗ g_{1})b_{3} − (−)^{r}b_{2}r_{1}
− (−)^{r}(1 ⊗ b_{1})r_{2} − (−)^{r}(b_{1} ⊗ 1)r_{2}.

2) When n = 2, f ^{r}→ g ^{p}→ h : 𝒜 → ℬ, we find the components of the A_{∞}-transformation (r ⊗ p)B_{2} : f → h : 𝒜 → ℬ as follows:

[(r ⊗ p)B_{2}]_{0} = (r_{0} ⊗ p_{0})b_{2},
[(r ⊗ p)B_{2}]_{1} = (r_{1} ⊗ p_{0})b_{2} + (r_{0} ⊗ p_{1})b_{2}
+ (r_{0} ⊗ p_{0} ⊗ h_{1})b_{3} + (r_{0} ⊗ g_{1} ⊗ p_{0})b_{3} + (f_{1} ⊗ r_{0} ⊗ p_{0})b_{3},
[(r ⊗ p)B_{2}]_{2} = (r_{2} ⊗ p_{0})b_{2} + (r_{1} ⊗ p_{1})b_{2} + (r_{0} ⊗ p_{2})b_{2} + (r_{1} ⊗ p_{0} ⊗ h_{1})b_{3}
+ (r_{0} ⊗ p_{1} ⊗ h_{1})b_{3} + (r_{1} ⊗ g_{1} ⊗ p_{0})b_{3} + (r_{0} ⊗ g_{1} ⊗ p_{1})b_{3}
+ (f_{1} ⊗ r_{1} ⊗ p_{0})b_{3} + (f_{1} ⊗ r_{0} ⊗ p_{1})b_{3} + (r_{0} ⊗ p_{0} ⊗ h_{2})b_{3}
+ (r_{0} ⊗ g_{2} ⊗ p_{0})b_{3} + (f_{2} ⊗ r_{0} ⊗ p_{0})b_{3} + (r_{0} ⊗ p_{0} ⊗ h_{1} ⊗ h_{1})b_{4}
+ (r_{0} ⊗ g_{1} ⊗ g_{1} ⊗ p_{0})b_{4} + (r_{0} ⊗ g_{1} ⊗ p_{0} ⊗ h_{1})b_{4}
+ (f_{1} ⊗ r_{0} ⊗ p_{0} ⊗ h_{1})b_{4} + (f_{1} ⊗ r_{0} ⊗ g_{1} ⊗ p_{0})b_{4}
+ (f_{1} ⊗ f_{1} ⊗ r_{0} ⊗ p_{0})b_{4}.

3) When n = 3, f ^{r}→ g ^{p}→ h ^{t}→ k : 𝒜 → ℬ, we find the components of the A_{∞}-transformation (r ⊗ p ⊗ t)B_{3} : f → k : 𝒜 → ℬ as follows:

[(r ⊗ p ⊗ t)B_{3}]_{0} = (r_{0} ⊗ p_{0} ⊗ t_{0})b_{3},
[(r ⊗ p ⊗ t)B_{3}]_{1} = (r_{1} ⊗ p_{0} ⊗ t_{0})b_{3} + (r_{0} ⊗ p_{1} ⊗ t_{0})b_{3} + (r_{0} ⊗ p_{0} ⊗ t_{1})b_{3}
+ (r_{0} ⊗ p_{0} ⊗ t_{0} ⊗ k_{1})b_{4} + (r_{0} ⊗ p_{0} ⊗ h_{1} ⊗ t_{0})b_{4}
+ (r_{0} ⊗ g_{1} ⊗ p_{0} ⊗ t_{0})b_{4} + (f_{1} ⊗ r_{0} ⊗ p_{0} ⊗ t_{0})b_{4}.

5.3. Differentials. Let 𝒜, 𝒞^{1}, 𝒞^{2}, …, 𝒞^{q}, ℬ be A_{∞}-categories. Let ϕ : T s𝒜 ⊗ T s𝒞^{1} ⊗T s𝒞^{2} ⊗⋯⊗T s𝒞^{q} → T sℬ, (a⊗c^{1} ⊗c^{2} ⊗⋯⊗c^{q}) ↦ a.(c^{1} ⊗c^{2} ⊗⋯⊗c^{q})χ be a cocategory homomorphism of degree 0. If the homomorphism ϕ commutes with the differential:

ϕb = ( ∑_{r+t=q} 1^{⊗r} ⊗ b ⊗ 1^{⊗t})ϕ,

then ϕ is called an A_{∞}-functor (in a generalized sense, extending Definition 2.4). This condition is fulfilled if and only if χ commutes with the differential:

(c^{1} ⊗ c^{2} ⊗ ⋯ ⊗ c^{q})χb

= ∑_{k=1}^{q}(−)^{c^{k+1}+···+c^{q}}(c^{1} ⊗ ⋯ ⊗ c^{k}b ⊗ ⋯ ⊗ c^{q})χ + (−)^{c^{1}+···+c^{q}}b(c^{1} ⊗ c^{2} ⊗ ⋯ ⊗ c^{q})χ. (5.3.1)

In particular, for q = 1 we get the equation

(c)χb = (cb)χ + (−)^{c}b(c)χ.

5.4 Corollary (to Proposition 5.1). There is a unique A_{∞}-category structure for A_{∞}(𝒜, ℬ), such that the action homomorphism α : T s𝒜 ⊗ T sA_{∞}(𝒜, ℬ) → T sℬ is an A_{∞}-functor.

Proof. The homomorphism α : a⊗r^{1} ⊗⋯⊗r^{n} ↦ a[(r^{1} ⊗⋯⊗r^{n})θ] of Corollary 3.3 uses χ = θ. Hence, α is an A_{∞}-functor if and only if (r)θb = (rB)θ + (−)^{r}b(r)θ for r = r^{1} ⊗ ⋯ ⊗ r^{n}, n ⩾ 0, that is, if and only if equations (5.1.1) hold.

5.5 Proposition. For any A_{∞}-functor ϕ : T s𝒜⊗T s𝒞^{1} ⊗T s𝒞^{2} ⊗⋯⊗T s𝒞^{q} → T sℬ there is a unique A_{∞}-functor ψ : T s𝒞^{1} ⊗ T s𝒞^{2} ⊗ ⋯ ⊗ T s𝒞^{q} → T sA_{∞}(𝒜, ℬ), such that

ϕ = (T s𝒜⊗T s𝒞^{1}⊗T s𝒞^{2}⊗⋯⊗T s𝒞^{q} ^{1⊗ψ} → T s𝒜⊗T sA_{∞}(𝒜, ℬ) ^{α}→ T sℬ). (5.5.1)

Proof. If f^{i} ∈ Ob 𝒞^{i}, then (5.3.1) implies that the cocategory homomorphism (f^{1}, f^{2}, …, f^{q})ψ = (f^{1}, f^{2}, …, f^{q})χ of degree 0 commutes with the differential b. Hence, it is an A_{∞}-functor, that is, an object of A_{∞}(𝒜, ℬ). By Proposition 3.4 there exists a unique cocategory homomorphism ψ : T s𝒞^{1} ⊗ T s𝒞^{2} ⊗ ⋯ ⊗ T s𝒞^{q} → T sA_{∞}(𝒜, ℬ), of degree 0, such that (5.5.1) holds. We have to prove that ψ commutes with the differential.

Using (5.1.1) and (5.3.1) we find for χ = ψθ the equation

(c^{1} ⊗c^{2} ⊗⋯⊗c^{q})ψBθ = (c^{1} ⊗c^{2} ⊗⋯⊗c^{q})ψθb−(−)^{c^{1}+···+c^{q}}b(c^{1} ⊗c^{2} ⊗⋯⊗c^{q})ψθ

= (c^{1} ⊗ c^{2} ⊗ ⋯ ⊗ c^{q}) ∑_{k=1}^{q}(1^{⊗k−1} ⊗ b ⊗ 1^{⊗q−k})ψθ. (5.5.2)

Notice that ψB and κ =^{def} ∑_{k=1}^{q}(1^{⊗k−1} ⊗ b ⊗ 1^{⊗q−k})ψ are both (ψ, ψ)-coderivations. Let c^{i} be an element of T^{n^{i}}s𝒞^{i} = s𝒞^{i}(f^{i}, −) ⊗ ⋯ ⊗ s𝒞^{i}(−, g^{i}) for 1 ⩽ i ⩽ q. Composing (5.5.2) with pr_{1} : T sℬ → sℬ we get

(c^{1} ⊗ c^{2} ⊗ ⋯ ⊗ c^{q})[ψB]_{n^{1}...n^{q}} pr_{1} = (c^{1} ⊗ c^{2} ⊗ ⋯ ⊗ c^{q})ψBθ pr_{1}

= (c^{1} ⊗ c^{2} ⊗ ⋯ ⊗ c^{q})κθ pr_{1} = (c^{1} ⊗ c^{2} ⊗ ⋯ ⊗ c^{q})κ_{n^{1}...n^{q}} pr_{1} .

Since all components of ((f^{1}, f^{2}, …, f^{q})ψ, (g^{1}, g^{2}, …, g^{q})ψ)-coderivations (c^{1} ⊗c^{2} ⊗ ⋯⊗c^{q})[ψB]_{n^{1}...n^{q}} and (c^{1}⊗c^{2}⊗⋯⊗c^{q})κ_{n^{1}...n^{q}} coincide, these coderivations coincide as well. Therefore, all the components of (ψ, ψ)-coderivations ψB and κ coincide. We conclude that these coderivations coincide as well: ψB = κ = ∑_{k=1}^{q}(1^{⊗k−1} ⊗ b ⊗ 1^{⊗q−k})ψ.

6. Enriched category of A_{∞}-categories

6.1 Definition (Differential graded cocategory). A differential graded cocategory 𝒞 is a graded cocategory equipped with a (1,1)-coderivation b = (b : 𝒞(X, Y) → 𝒞(X, Y))_{X,Y ∈Ob 𝒞} of degree 1, such that b^{2} = 0.

As in Section 2.2 a differential graded cocategory can be identified with a differential graded 𝕜-coalgebra, decomposed in a special way. An example of a differential graded cocategory is given by T s𝒜, where 𝒜 is an A_{∞}-category.

Differential graded cocategories form a symmetric monoidal category dgCoCat. If 𝒞 and 𝒟 are differential graded cocategories then their tensor product is the graded cocategory 𝒞 ⊗ 𝒟, equipped with the differential 1 ⊗ b + b ⊗ 1. We want to show that the category of A_{∞}-categories is enriched in dgCoCat.

Let 𝒜, ℬ, 𝒞 be A_{∞}-categories. There is a graded cocategory morphism of degree 0

M : T sA_{∞}(𝒜, ℬ) ⊗ T sA_{∞}(ℬ, 𝒞) → T sA_{∞}(𝒜, 𝒞),

defined in Section 4 via diagram (4.0.1). Since all cocategory morphisms α, α ⊗ 1 in this diagram commute with the differential by Corollary 5.4, the cocategory morphism M also commutes with the differential:

(1 ⊗ B + B ⊗ 1)M = MB  (6.1.1)

by Proposition 5.5. Therefore, M is an A_{∞}-functor. The unit η_{𝒜} : T 𝟙 → T Coder(s𝒜, s𝒜), 1 ↦ id_{𝒜} also is an A_{∞}-functor for trivial reasons. The set

dgCoCat(T 𝟙, T sA_{∞}(𝒜, ℬ)) = Maps({∗}, Ob A_{∞}(𝒜, ℬ)) = dgCoCat(T s𝒜, T sℬ) is the set of A_{∞}-functors 𝒜 → ℬ. We summarize the above statements as follows: the category of A_{∞}-categories is enriched in dgCoCat. Moreover, it is enriched in the monoidal subcategory of dgCoCat generated by T s𝒞, where 𝒞 are A_{∞}-categories.

Let us apply Proposition 5.5 to the A_{∞}-functor M. From that result we deduce the existence of a unique A_{∞}-functor

A_{∞}(𝒜, ) : A_{∞}(ℬ, 𝒞) → A_{∞}(A_{∞}(𝒜, ℬ), A_{∞}(𝒜, 𝒞)),

such that

M = [T sA_{∞}(𝒜, ℬ) ⊗ T sA_{∞}(ℬ, 𝒞) ^{1⊗A_{∞}(𝒜, )} →

T sA_{∞}(𝒜, ℬ) ⊗ T sA_{∞}(A_{∞}(𝒜, ℬ), A_{∞}(𝒜, 𝒞)) ^{α}→ T sA_{∞}(𝒜, 𝒞)]. (6.1.2)

Let us find the components of A_{∞}(𝒜, ).

6.2 Proposition. The A_{∞}-functor A_{∞}(𝒜, ) is strict. It maps an object of A_{∞}(ℬ, 𝒞), an A_{∞}-functor g : ℬ → 𝒞, to the object of the target A_{∞}-category (1 ⊗ g)M : A_{∞}(𝒜, ℬ) → A_{∞}(𝒜, 𝒞) (which is also an A_{∞}-functor). The first component A_{∞}(𝒜, )_{1} maps an element t of sA_{∞}(ℬ, 𝒞)(g, h), a (g, h)-coderivation t : T sℬ → T s𝒞, to the ((1⊗g)M, (1⊗h)M)-coderivation (1⊗t)M : T sA_{∞}(𝒜, ℬ) → T sA_{∞}(𝒜, 𝒞), an element of sA_{∞}(A_{∞}(𝒜, ℬ), A_{∞}(𝒜, 𝒞))((1 ⊗ g)M, (1 ⊗ h)M).

Proof. Clearly, A_{∞}(𝒜, ) gives the mapping of objects g ↦ (1 ⊗ g)M as described. To prove that A_{∞}(𝒜, )_{1} : t ↦ (1 ⊗ t)M and A_{∞}(𝒜, )_{k} = 0 for k > 1 it suffices to substitute a cocategory homomorphism with those components into (6.1.2) and to check this equation (see Proposition 3.4). Let

p^{1} ⊗ ⋯ ⊗ p^{n} ∈ sA_{∞}(𝒜, ℬ)(f^{0}, f^{1}) ⊗ ⋯ ⊗ sA_{∞}(𝒜, ℬ)(f^{n−1}, f^{n}),
t^{1} ⊗ ⋯ ⊗ t^{m} ∈ sA_{∞}(ℬ, 𝒞)(g^{0}, g^{1}) ⊗ ⋯ ⊗ sA_{∞}(ℬ, 𝒞)(g^{m−1}, g^{m}).  (6.2.1)

The equation to check is

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})M = (p^{1} ⊗ ⋯ ⊗ p^{n}).[(t^{1} ⊗ ⋯ ⊗ t^{m})A_{∞}(𝒜, )]θ,

that is,

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})M = (p^{1} ⊗ ⋯ ⊗ p^{n}).[(1 ⊗ t^{1})M ⊗ ⋯ ⊗ (1 ⊗ t^{m})M]θ.

The left hand side is a cocategory homomorphism. Let us prove that the right hand side

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})L =^{def} (p^{1} ⊗ ⋯ ⊗ p^{n}).[(1 ⊗ t^{1})M ⊗ ⋯ ⊗ (1 ⊗ t^{m})M]θ

is also a cocategory homomorphism. Indeed,

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1} ⊗ ⋯ ⊗ t^{m})LΔ

= (p^{1} ⊗ ⋯ ⊗ p^{n}).[(1 ⊗ t^{1})M ⊗ ⋯ ⊗ (1 ⊗ t^{m})M]θΔ

= (p^{1} ⊗ ⋯ ⊗ p^{n})Δ ∑_{k=0}^{m}[(1 ⊗ t^{1})M ⊗ ⋯ ⊗ (1 ⊗ t^{k})M]θ

⊗ [(1 ⊗ t^{k+1})M ⊗ ⋯ ⊗ (1 ⊗ t^{m})M]θ

= ∑_{l=0}^{n} ∑_{k=0}^{m}(−)^{(p^{l+1}+···+p^{n})(t^{1}+···+t^{k})}(p^{1} ⊗ ⋯ ⊗ p^{l}).[(1 ⊗ t^{1})M ⊗ ⋯ ⊗ (1 ⊗ t^{k})M]θ

⊗ (p^{l+1} ⊗ ⋯ ⊗ p^{n}).[(1 ⊗ t^{k+1})M
⊗ ⋯ ⊗ (1 ⊗ t^{m})M]θ

= [(p^{1} ⊗ ⋯ ⊗ p^{n})Δ ⊗ (t^{1} ⊗ ⋯ ⊗ t^{m})Δ](1 ⊗ c ⊗ 1)(L ⊗ L)

by Proposition 3.1.

Let us prove that the components of M and L coincide. For m = 0 and any n > 0 we have

(p^{1} ⊗ ⋯ ⊗ p^{n} | g^{0})L_{n0} = (p^{1} ⊗ ⋯ ⊗ p^{n}).(1 ⊗ g^{0})M pr_{1} = (p^{1} ⊗ ⋯ ⊗ p^{n} | g^{0})M_{n0},

hence, L_{n0} = M_{n0}. For m = 1 and any n ⩾ 0 we have

(p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1})L_{n1} = (p^{1} ⊗ ⋯ ⊗ p^{n}).(1 ⊗ t^{1})M pr_{1} = (p^{1} ⊗ ⋯ ⊗ p^{n} ⊗ t^{1})M_{n1},

hence, L_{n1} = M_{n1}. For m > 1 and any n ⩾ 0 we have L_{nm} = 0 and M_{nm} = 0. Therefore, L = M and the proposition is proved.

6.3 Corollary. For all m > 0 and all t^{1} ⊗ ⋯ ⊗ t^{m} as in (6.2.1) we have

[(1 ⊗ t^{1})M ⊗ ⋯ ⊗ (1 ⊗ t^{m})M]B̃_{m} = [1 ⊗ (t^{1} ⊗ ⋯ ⊗ t^{m})B_{m}]M,  (6.3.1)

where B̃ denotes the differential in T sA_{∞}(A_{∞}(𝒜, ℬ), A_{∞}(𝒜, 𝒞)).

Indeed, the general property of an A_{∞}-functor A_{∞}(𝒜, )B̃ = BA_{∞}(𝒜, ) reduces to the above formula, since A_{∞}(𝒜, ) is strict.

In the following definition we introduce A_{∞}-analogs of natural transformations.

6.4 Definition (ω-globular set of A_{∞}-categories). A natural A_{∞}-transforma-tion r : f → g : 𝒜 → ℬ (natural transformation in terms of [Fuk]) is an A_{∞}-trans-formation of degree −1 such that rb + br = 0 (that is, (r)B_{1} = 0). The ω-globular set [Bat98] A_{ω} of A_{∞}-categories is defined as follows: objects (0-morphisms) are A_{∞}-categories 𝒜; 1-morphisms are A_{∞}-functors f : 𝒜 → ℬ; 2-morphisms are natural A_{∞}-transformations r : f → g : 𝒜 → ℬ; 3-morphisms λ : r → s : f → g : 𝒜 → ℬ are (f, g)-coderivations of degree −2, such that r − s = [λ, b]; for n ⩾ 3 n-mor-phisms λ_{n} : λ_{n−1} → μ_{n−1} : ⋯: r → s : f → g : 𝒜 → ℬ are (f, g)-coderivations of degree 1 − n, such that λ_{n−1} − μ_{n−1} = [λ_{n}, b] (notice that the both sides are (f, g)-coderivations of degree 2 − n).

6.5 Remark. Let us notice that the A_{∞}-functor A_{∞}(𝒜, ) from Proposition 6.2 defines a map of the ω-globular set A_{ω} into itself. Indeed, objects ℬ of A_{ω} are mapped into objects A_{∞}(𝒜, ℬ), 1-morphisms g : ℬ → 𝒞 are mapped into 1-mor-phisms (1 ⊗ g)M : A_{∞}(𝒜, ℬ) → A_{∞}(𝒜, 𝒞) and the first component A_{∞}(𝒜, )_{1} maps (g, h)-coderivations into ((1 ⊗ g)M, (1 ⊗ h)M)-coderivations. Moreover, if the equation λ_{n−1} − μ_{n−1} = λ_{n}B_{1} holds for (g, h)-coderivations, then (1 ⊗ λ_{n−1})M − (1 ⊗ μ_{n−1})M = (1 ⊗ λ_{n})MB̃_{1} by (6.3.1), so the sources and the targets are preserved.

It might be useful to turn the ω-globular set A_{ω} into a weak non-unital ω-category in the sense of some of the existing definitions of the latter. Plenty of such definitions including [Bat98] are listed in Leinster’s survey [Lei02]. We do not try to proceed in this direction. Instead we truncate the ω-globular set to a 2-globular set (that is, we deal with 0-, 1- and 2-morphisms) and we make a 2-category out of it.

7. 2-categories of A_{∞}-categories

Let 𝒦 denote the category K(𝕜 -mod) = H^{0}(C(𝕜 -mod)) of differential graded complexes of 𝕜-modules, whose morphisms are chain maps modulo homotopy. Equipped with the usual tensor product, the unit object 𝕜 and the standard symmetry, 𝒦 becomes a 𝕜-linear closed monoidal symmetric category. The inner hom-object is the usual Hom𝕜^{•}(-, -). There is a notion of a category 𝒞 enriched in 𝒦 (𝒦-categories, 𝒦-functors, 𝒦-natural transformations), see Kelly [Kel82]: for all objects X, Y of 𝒞 𝒞(X, Y) is an object of 𝒦. There is a similar notion of a 2-category enriched in 𝒦, or a 𝒦-2-category: it consist of a class of objects Ob 𝒞, a class of 1-morphisms 𝒞(X, Y) for each pair of objects X, Y of 𝒜, an object of 2-morphisms 𝒞(X, Y)(f, g) ∈ Ob 𝒦 for each pair of 1-morphisms f, g ∈ 𝒞(X, Y) and other data. We shall consider 1-uni-tal, non-2-unital 𝒦-2-categories. They are equipped with the following operations: associative composition of 1-morphisms, commuting left and right associative actions of 1-morphisms on 2-morphisms (these actions are morphisms in 𝒦), 1-units, associative vertical composition of 2-morphisms (a morphism in 𝒦) compatible with the left and right actions of 1-morphisms on 2-morphisms and such that the both ways to obtain the horizontal composition coincide. Precise definitions are given in Appendix A.

7.1 Proposition. The following data define a 1-unital, non-2-unital 𝒦-2-category 𝒦A_{∞}:

• objects are A_{∞}-categories;

• 1-morphisms are A_{∞}-functors;

• an object of 2-morphisms between f, g : 𝒜 → ℬ is (A_{∞}(𝒜, ℬ)(f, g), m_{1}) ∈ Ob 𝒦, m_{1} = sB_{1}s^{−1};

• the composition of 1-morphisms is the composition of A_{∞}-functors;

• unit 1-morphisms are identity A_{∞}-functors;

• the right action of a 1-morphism k : ℬ → 𝒞 on 2-morphisms is the chain map (A_{∞}(𝒜, ℬ)(f, g), m_{1}) → (A_{∞}(𝒜, 𝒞)(fk, gk), m_{1}), rs^{−1} ↦ (rs^{−1})·k = (rk)s^{−1}, where r is an (f, g)-coderivation; • the left action of a 1-morphism e : 𝒟 → 𝒜 on 2-morphisms is the chain map (A_{∞}(𝒜, ℬ)(f, g), m_{1}) → (A_{∞}(𝒟, ℬ)(ef, eg), m_{1}), rs^{−1} ↦ e · (rs^{−1}) = (er)s^{−1}, where r is an (f, g)-coderivation;

• the vertical composition is the chain map m_{2} = (s⊗s)B_{2}s^{−1} : A_{∞}(𝒜, ℬ)(f, g)⊗ A_{∞}(𝒜, ℬ)(g, h) → A_{∞}(𝒜, ℬ)(f, h).

Proof. Clearly, the composition of 1-morphisms and the actions of 1-morphisms on 2-morphisms are associative. The right and the left actions are unital, and commute with each other. The equation −(1 ⊗ m_{1} + m_{1} ⊗ 1)m_{2} + m_{2}m_{1} = 0 (see (2.3.2) for k = 2) shows that m_{2} is a chain map. The equation

m_{3}m_{1}+(1⊗1⊗m_{1}+1⊗m_{1}⊗1+m_{1}⊗1⊗1)m_{3}−(m_{2}⊗1)m_{2}+(1⊗m_{2})m_{2} = 0 (7.1.1)

(see (2.3.2) for k = 3) shows that m_{2} is associative in 𝒦.

Let us check that the vertical composition is compatible with the actions of 1-mor-phisms on 2-morphisms. Applying equation (6.1.1) to r ⊗p⊗1 ∈ sA_{∞}(𝒜, ℬ)(f, g)⊗ sA_{∞}(𝒜, ℬ)(g, h) ⊗ 𝕜 ⊂ T^{2}sA_{∞}(𝒜, ℬ)(f, h) ⊗ T^{0}sA_{∞}(ℬ, 𝒞)(k, k) we find that

(r⊗p | k)M_{20}B_{1}+(rk⊗pk)B_{2} = [(r⊗p)(1⊗B_{1}+B_{1}⊗1) | k]M_{20}+(r⊗p)B_{2}k. (7.1.2)

One deduces that (rs^{−1} · k ⊗ ps^{−1} · k)m_{2} = (rs^{−1} ⊗ ps^{−1})m_{2} · k in 𝒦. Applying equation (6.1.1) to 1 ⊗ r ⊗ p ∈ 𝕜 ⊗ sA_{∞}(𝒜, ℬ)(f, g) ⊗ sA_{∞}(𝒜, ℬ)(g, h) ⊂ T^{0}sA_{∞}(𝒟, 𝒜)(e, e) ⊗ T^{2}sA_{∞}(𝒜, ℬ)(f, h) we find that

(er ⊗ ep)B_{2} = (e | r ⊗ p)M_{02}B_{1} + (er ⊗ ep)B_{2} = [e | (r ⊗ p)(1 ⊗ B_{1} + B_{1} ⊗ 1)]M_{02} + e(r ⊗ p)B_{2} = e(r ⊗ p)B_{2} (7.1.3)

(notice that M_{02} = 0). Therefore, (e · rs^{−1} ⊗ e · ps^{−1})m_{2} = e · (rs^{−1} ⊗ ps^{−1})m_{2}.

Now let us prove distributivity. Applying equation (6.1.1) to r ⊗ p ∈ sA_{∞}(𝒜, ℬ)(f, g) ⊗ sA_{∞}(ℬ, 𝒞)(h, k) we find that

(rh ⊗ gp)B_{2} + (−)^{rp}(fp ⊗ rk)B_{2} + (r ⊗ p)M_{11}B_{1} = (r ⊗ p)(1 ⊗ B_{1} + B_{1} ⊗ 1)M_{11}.

Thus, (rh ⊗ gp)B_{2} + (−)^{rp}(fp ⊗ rk)B_{2} = 0 in 𝒦. We deduce that modulo homotopy

(rs^{−1} · h ⊗ g · ps^{−1})m_{2}s = (rhs^{−1} ⊗ gps^{−1})(s ⊗ s)B_{2} = (−)^{p+1}(rh ⊗ gp)B_{2}
= (−)^{rp+p}(fp ⊗ rk)B_{2} = (−)^{rp+p}(fps^{−1}s ⊗ rks^{−1}s)B_{2}
= (−)^{rp+p+r+1}(f · ps^{−1} ⊗ rs^{−1} · k)m_{2}s.

Therefore, (rs^{−1} · h ⊗ g · ps^{−1})m_{2} = (−)^{(r+1)(p+1)}(f · ps^{−1} ⊗ rs^{−1} · k)m_{2} in 𝒦, as stated.

The 0-th cohomology functor H^{0} = 𝒦(𝕜, ) : 𝒦 → 𝕜 -mod, X ↦ H^{0}(X) = 𝒦(𝕜, X) is lax monoidal symmetric, since the complex 𝕜 concentrated in degree 0 is the unit object of 𝒦. It determines a functor H^{0} : 𝒦-𝒞at → 𝕜-𝒞at. To a 𝒦-category 𝒞 it assigns a 𝕜-linear category H^{0}(𝒞) with the same class of objects, and for each pair X, Y of objects of 𝒞 the 𝕜-module H^{0}(𝒞)(X, Y) = H^{0}(𝒞(X, Y)). The functor H^{0} : 𝒦-𝒞at → 𝕜-𝒞at is also lax monoidal symmetric. Therefore, there is a functor 𝒦-𝒞at-𝒞at → 𝒞at-𝒞at, again denoted H^{0} by abuse of notation, and the corresponding functor 𝒦-2-𝒞at^{nu} → 2-𝒞at^{nu}. See Appendix A for the definition of 1-unital, non-2-unital 𝒦- or 𝕜- 2-categories. To 𝒦A_{∞} the functor assigns a 1-unital, non-2-unital 𝕜-linear 2-category A_{∞}. Let us describe it in detail. Objects of A_{∞} are A_{∞}-categories, 1-morphisms are A_{∞}-functors, and 2-morphisms are elements of

H^{0}(A_{∞}(𝒜, ℬ)(f, g), m_{1}) \begin{matrix}s \\ ∼ \end{matrix}→ H^{−1}(sA_{∞}(𝒜, ℬ)(f, g), B_{1}),

that is, equivalence classes of natural A_{∞}-transformations r : f → g : 𝒜 → ℬ. Natural A_{∞}-transformations r, t : f → g : 𝒜 → ℬ are equivalent, if they are connected by a 3-morphism λ : r → t, that is, r − t = λB_{1}. Both compositions of 1-morphisms with 2-morphisms Mor_{2}(𝒜, ℬ) × Mor_{1}(ℬ, 𝒞) → Mor_{2}(𝒜, 𝒞), (r, h) ↦ rh and Mor_{1}(𝒜, ℬ) × Mor_{2}(ℬ, 𝒞) → Mor_{2}(𝒜, 𝒞), (f, p) ↦ fp are compositions of 𝕜-linear maps T s𝒜 → T sℬ → T s𝒞. The vertical composition m_{2} of 2-morphisms, translated to H^{−1}(sA_{∞}) assigns the natural A_{∞}-transformation r ·p = (r ⊗p)B_{2} to natural A_{∞}-transformations r : f → g and p : g → h. Indeed, (rs^{−1} ⊗ ps^{−1})m_{2}s = (rs^{−1} ⊗ ps^{−1})(s ⊗ s)B_{2} = (r ⊗ p)B_{2}. Compatibility of these constructions with the equivalence relation is obvious from the construction, and can be verified directly.

7.2 Remark. Let 𝒜 be an A_{∞}-category. It determines a map A_{ω} → A_{ω}, described in Remark 6.5. This map restricts to a map A_{∞}(𝒜, ) : A_{∞} → A_{∞}. It takes an A_{∞}-category ℬ to the A_{∞}-category A_{∞}(𝒜, ℬ), an A_{∞}-functor g : ℬ → 𝒞 to the A_{∞}-functor (1⊗g)M : A_{∞}(𝒜, ℬ) → A_{∞}(𝒜, 𝒞), and an equivalence class of a natural A_{∞}-transformation t : g → h : 𝒜 → ℬ to the equivalence class of the natural A_{∞}-transformation (1 ⊗ t)M : (1 ⊗ g)M → (1 ⊗ h)M : A_{∞}(𝒜, ℬ) → A_{∞}(𝒜, 𝒞), see Remark 6.5. Let us prove that A_{∞}(𝒜, ) : A_{∞} → A_{∞} is a strict 2-functor. Indeed, it preserves the composition of 1-morphisms, (1⊗f)M(1⊗g)M = (1⊗fg)M, and the both compositions of 1-morphisms and 2-morphisms, (1⊗f)M(1⊗t)M = (1⊗ft)M, (1⊗t)M(1⊗f)M = (1⊗tf)M, due to associativity of M. The vertical composition of 2-morphisms is preserved due to (6.3.1) for m = 2:

[(1 ⊗ t^{1})M ⊗ (1 ⊗ t^{2})M]B̃_{2} = [1 ⊗ (t^{1} ⊗ t^{2})B_{2}]M.

7.3 Definition (Unital A_{∞}-categories). Let 𝒞 be an A_{∞}-category. It is called unital if for each object X of 𝒞 there is a unit element – a 𝕜-linear map _{X}i_{0}^{𝒞} : 𝕜 → (s𝒞)^{−1}(X, X) such that _{X}i_{0}^{𝒞} b_{1} = 0, (_{X}i_{0}^{𝒞} ⊗_{X}i_{0}^{𝒞})b_{2} −_{X}i_{0}^{𝒞} ∈ Im b_{1}, and for all pairs X, Y of objects of 𝒞 the chain maps (1⊗_{Y} i_{0}^{𝒞})b_{2}, (_{X}i_{0}^{𝒞} ⊗1)b_{2} : s𝒞(X, Y) → s𝒞(X, Y) are homotopy invertible.

In particular, an A_{∞}-algebra 𝒞 is unital if it has an element i_{0}^{𝒞} ∈ (s𝒞)^{−1} such that i_{0}^{𝒞} b_{1} = 0, (i_{0}^{𝒞} ⊗i_{0}^{𝒞})b_{2} −i_{0}^{𝒞} ∈ Im b_{1}, and the chain maps (1⊗i_{0}^{𝒞})b_{2}, (i_{0}^{𝒞} ⊗1)b_{2} : s𝒞 → s𝒞 are homotopy invertible. Our definition differs from a that of a homological unit (e.g. [LH02]) by the last invertibility condition. It produces rather a homotopical unit:

7.4 Lemma. Let _{X}i_{0}^{𝒞} be as in Definition 7.3 of a unital A_{∞}-category 𝒞, then for each pair X, Y of objects of 𝒞 we have

(1 ⊗_{Y} i_{0}^{𝒞})b_{2} ∼ 1 : s𝒞(X, Y) → s𝒞(X, Y),
(_{X}i_{0}^{𝒞} ⊗ 1)b_{2} ∼ −1 : s𝒞(X, Y) → s𝒞(X, Y).

Proof. For each object X of 𝒞 there is a 𝕜-linear map _{X}v_{0} : 𝕜 → (s𝒞)^{−2}(X, X) such that (_{X}i_{0}^{𝒞} ⊗_{X}i_{0}^{𝒞})b_{2} −_{X}i_{0}^{𝒞} = _{X}v_{0}b_{1}. Hence,

(_{X}i_{0}^{𝒞} ⊗ 1)b_{2}(_{X}i_{0}^{𝒞} ⊗ 1)b_{2} = (_{X}i_{0}^{𝒞} ⊗_{X}i_{0}^{𝒞} ⊗ 1)(1 ⊗ b_{2})b_{2} ∼_{k=3}^{(2.3.1)} −(_{X}i_{0}^{𝒞} ⊗_{X}i_{0}^{𝒞} ⊗ 1)(b_{2} ⊗ 1)b_{2}

= −[(_{X}i_{0}^{𝒞} ⊗_{X}i_{0}^{𝒞})b_{2} ⊗ 1]b_{2} = −(_{X}i_{0}^{𝒞} ⊗ 1)b_{2} − (_{X}v_{0}b_{1} ⊗ 1)b_{2}

= −(_{X}i_{0}^{𝒞} ⊗ 1)b_{2} + b_{1}(_{X}v_{0} ⊗ 1)b_{2} + (_{X}v_{0} ⊗ 1)b_{2}b_{1} ∼ −(_{X}i_{0}^{𝒞} ⊗ 1)b_{2},

(1 ⊗_{Y} i_{0}^{𝒞})b_{2}(1 ⊗_{Y} i_{0}^{𝒞})b_{2} = −(1 ⊗_{Y} i_{0}^{𝒞} ⊗_{Y} i_{0}^{𝒞})(b_{2} ⊗ 1)b_{2} ∼_{k=3}^{(2.3.1)} (1 ⊗_{Y} i_{0}^{𝒞} ⊗_{Y} i_{0}^{𝒞})(1 ⊗ b_{2})b_{2}

= [1 ⊗ (_{Y} i_{0}^{𝒞} ⊗_{Y} i_{0}^{𝒞})b_{2}]b_{2} = (1 ⊗_{Y} i_{0}^{𝒞})b_{2} + (1 ⊗_{Y} v_{0}b_{1})b_{2}

= (1 ⊗_{Y} i_{0}^{𝒞})b_{2} − b_{1}(1 ⊗_{Y} v_{0})b_{2} − (1 ⊗_{Y} v_{0})b_{2}b_{1} ∼ (1 ⊗_{Y} i_{0}^{𝒞})b_{2}.

We see that −(_{X}i_{0}^{𝒞} ⊗1)b_{2} and (1⊗_{Y} i_{0}^{𝒞})b_{2} are invertible idempotents in 𝒦. Therefore, these maps are both homotopic to the identity map.

This lemma shows that a unital A_{∞}-algebra may be defined as an A_{∞}-algebra 𝒞, such that the graded associative 𝕜-algebra H^{•}(𝒞, m_{1}) has a unit 1 ∈ H^{0}(𝒞, m_{1}) and for some/any representative 1^{𝒞} ∈ 𝒞^{0} of the class 1 ∈ H^{0}(𝒞, m_{1}) the chain maps (id ⊗1^{𝒞})m_{2}, (1^{𝒞} ⊗ id)m_{2} : 𝒞 → 𝒞 are homotopic to id_{𝒞}. A unit element i_{0}^{𝒞} ∈ (s𝒞)^{−1} corresponds to a unit 1^{𝒞} ∈ 𝒞^{0} via 1^{𝒞} s = i_{0}^{𝒞}.

7.5 Proposition. Let 𝒞 be a unital A_{∞}-category. Then the collection _{X}i_{0}^{𝒞} extends to a natural A_{∞}-transformation i^{𝒞} : id_{𝒞} → id_{𝒞} : 𝒞 → 𝒞 such that (i^{𝒞} ⊗ i^{𝒞})B_{2} ≡ i^{𝒞}.

Proof. Let 𝕜-linear maps _{X}v_{0} : 𝕜 → (s𝒞)^{−2}(X, X) satisfy the equations (_{X}i_{0}^{𝒞} ⊗ Xi_{0}^{𝒞})b_{2} −_{X}i_{0}^{𝒞} = _{X}v_{0}b_{1}. We will prove that given _{X}i_{0}^{𝒞}, _{X}v_{0} (with _{X}i_{0}^{𝒞} b_{1} = 0) are 0-th components of a natural A_{∞}-transformation i^{𝒞} and a 3-morphism v as follows:

i^{𝒞} : id_{𝒞} → id_{𝒞} : 𝒞 → 𝒞,
v : (i^{𝒞} ⊗ i^{𝒞})B_{2} → i^{𝒞} : id_{𝒞} → id_{𝒞} : 𝒞 → 𝒞.

That is, we will prove the existence of (1, 1)-coderivations i^{𝒞} , v : T s𝒞 → T s𝒞 of degree −1 (resp. −2) such that

i^{𝒞} b + bi^{𝒞} = 0,
(i^{𝒞} ⊗ i^{𝒞})B_{2} − i^{𝒞} = vb − bv.

We already have the 0-th components i_{0}^{𝒞}, v_{0}. Let us construct the other components of i^{𝒞} and v by induction. Given a positive n, assume that we have already found components i_{m}^{𝒞}, v_{m} of the sought i^{𝒞}, v for m < n, such that the equations

(i^{𝒞} b)_{m} + (bi^{𝒞})_{m} = 0 : s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{m−1}, X_{m}) → s𝒞(X_{0}, X_{m}), (7.5.1)

[(i^{𝒞} ⊗ i^{𝒞})B_{2}]_{m} − i_{m}^{𝒞} = (vb)_{m} − (bv)_{m} : s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{m−1}, X_{m}) → s𝒞(X_{0}, X_{m}) (7.5.2)

are satisfied for all m < n. Here (f)_{m} = (T^{m}s𝒞 ^{⊂} → T s𝒞 ^{f}→ T s𝒞 ^{pr_{1}}→ s𝒞) for an arbitrary morphism of quivers f : T s𝒞 → T s𝒞.

Introduce (1, 1)-coderivations ĩ, v : T s𝒞 → T s𝒞 of degree −1 and −2 by their components (i_{0}^{𝒞} , i_{1}^{𝒞} , …,i_{n−1}^{𝒞}, 0, 0, …) (resp. (v_{0}, v_{1}, …, v_{n−1}, 0, 0, …)). Define (1, 1)-coderivations λ = ĩb + bĩ of degree 0 and ν = (ĩ⊗ ĩ)B_{2} − ĩ− ṽB_{1} of degree −1. Then equations (7.5.1), (7.5.2) imply that λ_{m} = 0, ν_{m} = 0 for m < n. The identity λb − bλ = 0 implies that

λ_{n}d = λ_{n}b_{1} − ∑_{q+1+t=n}(1^{⊗q} ⊗ b_{1} ⊗ 1^{⊗t})λ_{n} = 0.

The n-th component of the identity

νB_{1} = (ĩ⊗ĩ)B_{2}B_{1}−ĩB_{1} = −(ĩ⊗ĩ)(1⊗B_{1}+B_{1}⊗1)B_{2}−λ = −(ĩ⊗λ)B_{2}+(λ⊗ĩ)B_{2}−λ

gives

ν_{n}d = ν_{n}b_{1} + ∑_{q+1+t=n}(1^{⊗q} ⊗ b_{1} ⊗ 1^{⊗t})ν_{n} = −(i_{0}^{𝒞} ⊗ λ_{n})b_{2} + (λ_{n} ⊗ i_{0}^{𝒞})b_{2} − λ_{n}
= −λ_{n}(i_{0}^{𝒞} ⊗ 1)b_{2} + λ_{n}(1 ⊗ i_{0}^{𝒞})b_{2} − λ_{n} = −λ_{n}u′.

Here the chain map

u′ = (_{X_{0}}i_{0}^{𝒞} ⊗ 1)b_{2} − (1 ⊗_{X_{n}}i_{0}^{𝒞})b_{2} + 1 : s𝒞(X_{0}, X_{n}) → s𝒞(X_{0}, X_{n})

is homotopic to −1 by Lemma 7.4. Hence, the map

u = Hom(N, u′) : Hom^{•}(N, s𝒞(X_{0}, X_{n})) → Hom^{•}(N, s𝒞(X_{0}, X_{n})), λ_{n} ↦ λ_{n}u′

is also homotopic to −1 for each complex of 𝕜-modules N, in particular, for N = s𝒞(X_{0}, X_{1}) ⊗_{𝕜} ⋯ ⊗_{𝕜} s𝒞(X_{n−1}, X_{n}). Therefore, the complex Cone(u) is contractible by Lemma Appendix B.1. Since −λ_{n}d = 0 and ν_{n}d + λ_{n}u = 0, the element

(ν_{n}, λ_{n}) ∈ Hom_{𝕜}^{−1}(N, s𝒞(X_{0}, X_{n})) ⊕ Hom_{𝕜}^{0}(N, s𝒞(X_{0}, X_{n})) = Cone^{−1}(u)

is a cycle. Due to acyclicity of Cone(u) this element is a boundary of some element

(v_{n}, i_{n}^{𝒞}) ∈ Hom_{𝕜}^{−2}(N, s𝒞(X_{0}, X_{n})) ⊕ Hom_{𝕜}^{−1}(N, s𝒞(X_{0}, X_{n})) = Cone^{−2}(u),

that is, v_{n}d+i_{n}^{𝒞}u = ν_{n} and −i_{n}^{𝒞}d = λ_{n}. These equations can be rewritten as follows:

−i_{n}^{𝒞}b_{1} − ∑_{q+1+t=n}(1^{⊗q} ⊗ b_{1} ⊗ 1^{⊗t})i_{n}^{𝒞} = (ĩb)_{n} + (bĩ)_{n},

v_{n}b_{1} − ∑_{q+1+t=n}(1^{⊗q} ⊗ b_{1} ⊗ 1^{⊗t})v_{n} − (i_{0}^{𝒞} ⊗ i_{n}^{𝒞})b_{2} − (i_{n}^{𝒞} ⊗ i_{0}^{𝒞})b_{2} + i_{n}^{𝒞}

= [(ĩ ⊗ ĩ)B_{2}]_{n} − (ṽb)_{n} + (bṽ)_{n}.

In other words, (1, 1)-coderivations with components (i_{0}^{𝒞} , …, i_{n−1}^{𝒞}, i_{n}^{𝒞}, 0, …), (v_{0}, …, v_{n−1}, v_{n}, 0, …) satisfy equations (7.5.1), (7.5.2) for m ⩽ n. The construction of i^{𝒞}, v goes on inductively.

7.6 Definition. A unit transformation of an A_{∞}-category 𝒞 is a natural A_{∞}-trans-formation i^{𝒞} : id_{𝒞} → id_{𝒞} : 𝒞 → 𝒞 such that (i^{𝒞} ⊗ i^{𝒞})B_{2} ≡ i^{𝒞}, and for each pair X, Y of objects of 𝒞 the chain maps (1 ⊗_{Y} i_{0}^{𝒞})b_{2}, (_{X}i_{0}^{𝒞} ⊗ 1)b_{2} : s𝒞(X, Y) → s𝒞(X, Y) are homotopy invertible.

We have shown in Proposition 7.5 that an A_{∞}-category 𝒞 is unital if and only if it has a unit transformation. Similar (although not identical to our) definitions of units and unital A_{∞}-categories are proposed by Kontsevich and Soibelman [KS02] and Lefèvre-Hasegawa [LH02].

7.7 Proposition. Let 𝒜, 𝒞 be A_{∞}-categories. If 𝒞 is unital, then A_{∞}(𝒜, 𝒞) is unital as well.

Proof. We claim that

(1 ⊗ i^{𝒞})M : (1 ⊗ id_{𝒞})M → (1 ⊗ id_{𝒞})M : A_{∞}(𝒜, 𝒞) → A_{∞}(𝒜, 𝒞)

is a unit of A_{∞}(𝒜, 𝒞). Indeed, (1 ⊗ id_{𝒞})M = id_{A_{∞}(𝒜,𝒞)} , and there is a 3-morphism v : (i^{𝒞} ⊗ i^{𝒞})B_{2} → i^{𝒞} : id_{𝒞} → id_{𝒞} : 𝒞 → 𝒞. Hence, (i^{𝒞} ⊗ i^{𝒞})B_{2} = i^{𝒞} + vB_{1}, and by (6.3.1)

[(1 ⊗ i^{𝒞})M ⊗ (1 ⊗ i^{𝒞})M]B̃_{2} = [1 ⊗ (i^{𝒞} ⊗ i^{𝒞})B_{2}]M

= (1 ⊗ i^{𝒞})M + (1 ⊗ vB_{1})M = (1 ⊗ i^{𝒞})M + (1 ⊗ v)MB̃_{1} ≡ (1 ⊗ i^{𝒞})M.

Let f : 𝒜 → 𝒞 be an A_{∞}-functor. The 0-th component of (1 ⊗ i^{𝒞})M is _{f}[(1 ⊗ i^{𝒞})M]_{0} : 𝕜 → sA_{∞}(𝒜, 𝒞)(f, f), 1 ↦ fi^{𝒞}. It remains to prove that for each pair of A_{∞}-functors f, g : 𝒜 → 𝒞 the maps

(1 ⊗_{g}[(1 ⊗ i^{𝒞})M]_{0})B_{2} : sA_{∞}(𝒜, 𝒞)(f, g) → sA_{∞}(𝒜, 𝒞)(f, g), r ↦ (r ⊗ gi^{𝒞})B_{2},
(_{f}[(1 ⊗ i^{𝒞})M]_{0} ⊗ 1)B_{2} : sA_{∞}(𝒜, 𝒞)(f, g) → sA_{∞}(𝒜, 𝒞)(f, g), r ↦ r(fi^{𝒞} ⊗ 1)B_{2},

are homotopy invertible.

Let us define a decreasing filtration of the complex (sA_{∞}(𝒜, 𝒞)(f, g), B_{1}). For n ∈ ℤ_{⩾0}, we set

Φ_{n} = {r ∈ sA_{∞}(𝒜, 𝒞)(f, g) | ∀l < n r_{l} = 0}
= {r ∈ sA_{∞}(𝒜, 𝒞)(f, g) | ∀l < n (T^{l}s𝒜)r = 0}.

Clearly, Φ_{n} is stable under B_{1} = [ , b], and we have

sA_{∞}(𝒜, 𝒞)(f, g) = Φ_{0} ⊃ Φ_{1} ⊃ ⋯ ⊃ Φ_{n} ⊃ Φ_{n+1} ⊃ ….

Due to (5.1.3) and (3.0.1) the chain maps a = (1⊗gi^{𝒞})B_{2}, c = (fi^{𝒞} ⊗1)B_{2} preserve the subcomplex Φ_{n}. By Definition 2.6 sA_{∞}(𝒜, 𝒞)(f, g) = ∏_{n=0}^{∞} V_{n}, where V_{n} is the 𝕜-module (2.7.1) of n-th components r_{n} of (f, g)-coderivations r. The filtration consists of 𝕜-submodules Φ_{n} = 0 × ⋯ × 0 × ∏_{m=n}^{∞} V_{m}.

The graded complex associated with this filtration is ⊕_{n=0}^{∞}V_{n}, and the differential d : V_{n} → V_{n} induced by B_{1} is given by formula (2.7.2). The associated endomorphisms gr a, gr c of ⊕_{n=0}^{∞}V_{n} are given by the formulas

(r_{n}) gr a = (r_{n} ⊗_{g}i_{0}^{𝒞})b_{2} = ∏_{X_{0},...,X_{n}∈Ob 𝒜} (_{X_{0},...,X_{n}}r_{n} ⊗_{X_{n}g}i_{0}^{𝒞})b_{2},
(r_{n}) gr c = r_{n}(_{f}i_{0}^{𝒞} ⊗ 1)b_{2} = ∏_{X_{0},...,X_{n}∈Ob 𝒜} _{X_{0},...,X_{n}}r_{n}(_{X_{0}f}i_{0}^{𝒞} ⊗ 1)b_{2},

r_{n} ∈ V_{n}, as formulas (3.0.1), (5.1.3) show. Due to Lemma 7.4 for each pair X, Y of objects of 𝒞 the chain maps (1 ⊗_{Y} i_{0}^{𝒞})b_{2}, −(_{X}i_{0}^{𝒞} ⊗ 1)b_{2} : s𝒞(X, Y) → s𝒞(X, Y) are homotopic to the identity map, that is, (1 ⊗_{Y} i_{0}^{𝒞})b_{2} = 1 + hd + dh, (_{X}i_{0}^{𝒞} ⊗ 1)b_{2} = −1 + h′d + dh′ for some 𝕜-linear maps h, h′ : s𝒞(X, Y) → s𝒞(X, Y) of degree −1. Let us choose such homotopies _{X_{0},X_{n}}h, _{X_{0},X_{n}}h′ : s𝒞(X_{0}f, X_{n}g) → s𝒞(X_{0}f, X_{n}g) for each pair X_{0}, X_{n} of objects of 𝒜. Denote by H, H′ : ∏_{n=0}^{∞} V_{n} → ∏_{n=0}^{∞} V_{n} the diagonal maps_{X_{0},...,X_{n}}r_{n} ↦ _{X_{0},...,X_{n}}r_{n}_{X_{0},X_{n}}h,_{X_{0},...,X_{n}}r_{n} ↦ _{X_{0},...,X_{n}}r_{n}_{X_{0},X_{n}}h′. Then gr a = 1+Hd+dH, gr c = −1+H′d+dH′. The chain maps a−HB_{1}−B_{1}H, c− H′B_{1} −B_{1}H′, being restricted to maps ⊕_{m=0}^{∞}V_{m} → ∏_{m=0}^{∞} V_{m} give upper triangular ℕ × ℕ matrices which, in turn, determine the whole map. Thus, a − HB_{1} − B_{1}H = 1 + N, c − H′B_{1} − B_{1}H′ = −1 + N′, where the ℕ × ℕ matrices N, N′ are strictly upper triangular. Therefore, 1 + N and −1 + N′ are invertible (since their inverse maps ∑_{i=0}^{∞}(−N)^{i} and − ∑_{i=0}^{∞}(N′)^{i} make sense). Hence, a = (1 ⊗ gi^{𝒞})B_{2} and c = (fi^{𝒞} ⊗ 1)B_{2} are invertible in 𝒦.

7.8 Corollary. Let f, g : 𝒜 → 𝒞 be A_{∞}-functors. If 𝒞 is unital, then

(1 ⊗ gi^{𝒞})B_{2} ∼ 1 : sA_{∞}(𝒜, 𝒞)(f, g) → sA_{∞}(𝒜, 𝒞)(f, g), and
(fi^{𝒞} ⊗ 1)B_{2} ∼ −1 : sA_{∞}(𝒜, 𝒞)(f, g) → sA_{∞}(𝒜, 𝒞)(f, g).

Proof. Follows from Proposition 7.7 and Lemma 7.4.

7.9 Corollary. Let r : f → g : 𝒜 → 𝒞 be a natural A_{∞}-transformation. If 𝒞 is unital, then

(r ⊗ gi^{𝒞})B_{2} ≡ r, (fi^{𝒞} ⊗ r)B_{2} ≡ r.

Proof. By Corollary 7.8 there are homotopies h, h′ : sA_{∞}(𝒜, 𝒞)(f, g) → sA_{∞}(𝒜, 𝒞)(f, g), which give

(r ⊗ gi^{𝒞})B_{2} = r(1 ⊗ gi^{𝒞})B_{2} = r + rB_{1}h + rhB_{1} = r + (rh)B_{1} ≡ r,
(fi^{𝒞} ⊗ r)B_{2} = −r(fi^{𝒞} ⊗ 1)B_{2} = r + rB_{1}h′ + rh′B_{1} = r + (rh′)B_{1} ≡ r.

7.10 Corollary. The unit transformation of a unital category is determined uniquely up to equivalence.

Indeed, take f = id_{𝒞} and notice that any two unit transformations i^{𝒞} and ′i^{𝒞} of 𝒞 satisfy ′i^{𝒞} ≡ ′i^{𝒞} · i^{𝒞} ≡ i^{𝒞}.

7.11 Corollary. The full 𝒦-2-subcategory 𝒦^{u}A_{∞} of non-2-unital 𝒦-2-category 𝒦A_{∞}, whose objects are unital A_{∞}-categories and the other data are as in 𝒦A_{∞}, is a 1-2-unital 𝒦-2-category. The unit 2-endomorphism of an A_{∞}-functor f : 𝒜 → 𝒞 is the homotopy class of the chain map

1_{f} : 𝕜 → (A_{∞}(𝒜, 𝒞)(f, f), m_{1}), 1 ↦ (fi^{𝒞})s^{−1}.

Proof. The composition

A_{∞}(𝒜, 𝒞)(f, g) ^{1⊗1_{g}} → A_{∞}(𝒜, 𝒞)(f, g) ⊗ A_{∞}(𝒜, 𝒞)(g, g) ^{m_{2}}→ A_{∞}(𝒜, 𝒞)(f, g),

rs^{−1} ↦ rs^{−1} ⊗ (gi^{𝒞})s^{−1} ↦ (rs^{−1} ⊗ (gi^{𝒞})s^{−1})(s ⊗ s)B_{2}s^{−1} = (r ⊗ gi^{𝒞})B_{2}s^{−1}, is homotopic to the identity map by Corollary 7.8. Similarly, the composition

A_{∞}(𝒜, 𝒞)(f, g) ^{1_{f} ⊗1} → A_{∞}(𝒜, 𝒞)(f, f) ⊗ A_{∞}(𝒜, 𝒞)(f, g) ^{m_{2}}→ A_{∞}(𝒜, 𝒞)(f, g),

rs^{−1} ↦ (fi^{𝒞})s^{−1} ⊗ rs^{−1} ↦ (−)^{r−1}(fi^{𝒞} ⊗ r)B_{2}s^{−1} = −r(fi^{𝒞} ⊗ 1)B_{2}s^{−1},

is homotopic to the identity map.

7.12 Corollary. The full 2-subcategory^{u}A_{∞} of non-2-unital 2-category A_{∞}, which consists of unital A_{∞}-categories, all A_{∞}-functors between them, and equivalence classes of all natural A_{∞}-transformations is a (usual 1-2-unital) 2-category. The unit 2-endomorphism of an A_{∞}-functor f : 𝒜 → 𝒞 is fi^{𝒞}. In this 2-category the notions of an isomorphism between A_{∞}-functors, an equivalence between A_{∞}-categories, etc. make sense. For instance, r : f → g : 𝒜 → ℬ is an isomorphism if there is a natural A_{∞}-transformation p : g → f, such that (r ⊗ p)B_{2} ≡ fi^{ℬ} and (p ⊗ r)B_{2} ≡ gi^{ℬ} . An A_{∞}-functor f : 𝒜 → ℬ is an equivalence if there exists an A_{∞}-functor g : ℬ → 𝒜 and isomorphisms id_{𝒜} → fg and id_{ℬ} → gf.

Proof. Follows from Corollary 7.9 or 7.11.

7.13. Invertible transformations. Let ℬ, 𝒞 be A_{∞}-categories, and let f, g : Ob 𝒞 → Ob ℬ be maps. Assume that ℬ is unital and that for each object X of 𝒞 there are 𝕜-linear maps

Xr_{0} : 𝕜 → (sℬ)^{−1}(Xf, Xg),  _{X}p_{0} : 𝕜 → (sℬ)^{−1}(Xg, Xf),
Xw_{0} : 𝕜 → (sℬ)^{−2}(Xf, Xf),  _{X}v_{0} : 𝕜 → (sℬ)^{−2}(Xg, Xg),

such that

Xr_{0}b_{1} = 0, _{X}p_{0}b_{1} = 0,
(_{X}r_{0} ⊗_{X}p_{0})b_{2} −_{Xf}i_{0}^{ℬ} = _{X}w_{0}b_{1},  (7.13.1)
(_{X}p_{0} ⊗_{X}r_{0})b_{2} −_{Xg}i_{0}^{ℬ} = _{X}v_{0}b_{1}.

7.14 Lemma. Let the above assumptions hold. Then for all objects X of 𝒞 and Y of ℬ the chain maps

(r_{0} ⊗ 1)b_{2} : sℬ(Xg, Y) → sℬ(Xf, Y) and (p_{0} ⊗ 1)b_{2} : sℬ(Xf, Y) → sℬ(Xg, Y),
(1 ⊗ r_{0})b_{2} : sℬ(Y, Xf) → sℬ(Y, Xg) and (1 ⊗ p_{0})b_{2} : sℬ(Y, Xg) → sℬ(Y, Xf)

are homotopy inverse to each other.

Proof. We have

(r_{0} ⊗ 1)b_{2}(p_{0} ⊗ 1)b_{2} = (p_{0} ⊗ r_{0} ⊗ 1)(1 ⊗ b_{2})b_{2}
= −(p_{0} ⊗ r_{0} ⊗ 1)[(b_{2} ⊗ 1)b_{2} + b_{3}b_{1} + (1 ⊗ 1 ⊗ b_{1})b_{3}]  (7.14.1)
= −(_{Xg}i_{0}^{ℬ} ⊗ 1)b_{2} − (v_{0}b_{1} ⊗ 1)b_{2} − (p_{0} ⊗ r_{0} ⊗ 1)b_{3}b_{1} − b_{1}(p_{0} ⊗ r_{0} ⊗ 1)b_{3}
∼ 1 + b_{1}(v_{0} ⊗ 1)b_{2} + (v_{0} ⊗ 1)b_{2}b_{1} ∼ 1 : sℬ(Xg, Y) → sℬ(Xg, Y).

For symmetry reasons also

(p_{0} ⊗ 1)b_{2}(r_{0} ⊗ 1)b_{2} ∼ 1 : sℬ(Xf, Y) → sℬ(Xf, Y).

Therefore, (r_{0} ⊗ 1)b_{2} and (p_{0} ⊗ 1)b_{2} are homotopy inverse to each other. Similarly, (1 ⊗ r_{0})b_{2} and (1 ⊗ p_{0})b_{2} are homotopy inverse to each other.

7.15 Proposition. Let r : f → g : 𝒞 → ℬ be a natural A_{∞}-transformation, where ℬ is unital, and let p_{0}, v_{0}, w_{0} be as in Section 7.13 so that equations (7.13.1) hold. Then p_{0}, w_{0} extend to a natural A_{∞}-transformation p : g → f : 𝒞 → ℬ, a 3-morphism w, and there is a 3-morphism t as follows:

w : (r ⊗ p)B_{2} → fi^{ℬ} : f → f : 𝒞 → ℬ,  (7.15.1)
t : (p ⊗ r)B_{2} → gi^{ℬ} : g → g : 𝒞 → ℬ.  (7.15.2)

In particular, r is invertible and p = r^{−1} in A_{∞}.

Proof. Let us drop equation (7.15.2) and prove the existence of p and w, satisfying (7.15.1). We have to satisfy the equations

pb + bp = 0,  (7.15.3)
(r ⊗ p)B_{2} − fi^{ℬ} = [w, b].  (7.15.4)

Let us construct the components of p and w by induction. Given a positive n, assume that we have already found components p_{m}, w_{m} of the sought p, w for m < n, such that the equations

(pb)_{m} + (bp)_{m} = 0 : s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{m−1}, X_{m}) → sℬ(X_{0}g, X_{m}f), (7.15.5)

[(r ⊗ p)B_{2}]_{m} − (fi^{ℬ} )_{m} = (wb)_{m} − (bw)_{m} :

s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{m−1}, X_{m}) → sℬ(X_{0}f, X_{m}f) (7.15.6)

are satisfied for all m < n. Introduce a (g, f)-coderivation p̃ : T s𝒞 → T sℬ of degree −1 by its components (p_{0}, p_{1}, …, p_{n−1}, 0, 0, …) and an (f, f)-coderivation w̃ : T s𝒞 → T sℬ of degree −2 by its components (w_{0}, w_{1}, …, w_{n−1}, 0, 0, …). Define a (g, f)-coderivation λ = p̃b + bp̃ of degree 0 and an (f, f)-coderivation ν = (r ⊗ p̃)B_{2}−fi^{ℬ} −[w̃, b] of degree −1. Then equations (7.15.3), (7.15.4) imply that λ_{m} = 0, ν_{m} = 0 for m < n. The identity λb − bλ = 0 implies that

λ_{n}d = λ_{n}b_{1} − ∑_{q+1+t=n}(1^{⊗q} ⊗ b_{1} ⊗ 1^{⊗t})λ_{n} = 0.

The identity

[ν, b] = νB_{1} = (r ⊗ p̃)B_{2}B_{1} − (fi^{ℬ} )B_{1} − w̃B_{1}B_{1} = −(r ⊗ p̃)(1 ⊗ B_{1} + B_{1} ⊗ 1)B_{2} = −(r ⊗ p̃B_{1})B_{2} = −(r ⊗ λ)B_{2}

implies that

ν_{n}b_{1} + ∑_{q+1+t=n}(1^{⊗q} ⊗ b_{1} ⊗ 1^{⊗t})ν_{n} = −(r_{0} ⊗ λ_{n})b_{2},

that is, ν_{n}d = −λ_{n}u. Here the map u = Hom(N, (r_{0} ⊗ 1)b_{2}) for N = s𝒞(X_{0}, X_{1}) ⊗_{𝕜} ⋯⊗_{𝕜}s𝒞(X_{n−1}, X_{n}) is homotopy invertible, and the complex Cone(u) is contractible by Lemma Appendix B.1. Hence, the cycle

(ν_{n}, λ_{n}) ∈ Hom_{𝕜}^{−1}(N, sℬ(X_{0}f, X_{n}f)) ⊕ Hom_{𝕜}^{0}(N, sℬ(X_{0}g, X_{n}f)) = Cone^{−1}(u) is the boundary of some element

(w_{n}, p_{n}) ∈ Hom_{𝕜}^{−2}(N, sℬ(X_{0}f, X_{n}f)) ⊕ Hom_{𝕜}^{−1}(N, sℬ(X_{0}g, X_{n}f)) = Cone^{−2}(u),

that is, w_{n}d+p_{n}u = ν_{n} and −p_{n}d = λ_{n}. In other words, equations (7.15.5), (7.15.6) are satisfied for m = n, and we prove the statement by induction.

For similar reasons using Lemma 7.14 there exists a natural A_{∞}-transformation q : g → f : 𝒞 → ℬ with q_{0} = p_{0} and a 3-morphism

v : (q ⊗ r)B_{2} → gi^{ℬ} : g → g : 𝒞 → ℬ

with given v_{0}. Since r has a left inverse and a right inverse, it is invertible in A_{∞} and p is equivalent to q. Hence, (p⊗r)B_{2} is equivalent to (q ⊗r)B_{2}, and there exists t of (7.15.2).

8. Unital A_{∞}-functors

8.1 Definition. Let 𝒜, ℬ be unital A_{∞}-categories. An A_{∞}-functor f : 𝒜 → ℬ is called unital if for all objects X of 𝒜 we have _{X}i_{0}^{𝒜} f_{1} −_{Xf}i_{0}^{ℬ} ∈ Im b_{1}.

For instance, an A_{∞}-homomorphism f : 𝒜 → ℬ of A_{∞}-algebras is unital if the cycles i_{0}^{𝒜} f_{1}, i_{0}^{ℬ} ∈ (sℬ)^{−1} are cohomologous in (sℬ, b_{1}). We may say that a unital A_{∞}-functor (or A_{∞}-homomorphism) preserves the cohomology classes of unit elements.

8.2 Proposition. Let 𝒜, ℬ be unital A_{∞}-categories. An A_{∞}-functor f : 𝒜 → ℬ is unital if and only if i^{𝒜} f ≡ fi^{ℬ} .

Proof. If i^{𝒜} f = fi^{ℬ} + vB_{1}, then _{X}i_{0}^{𝒜} f_{1} = _{X}(i^{𝒜} f)_{0} = _{X}(fi^{ℬ} + vB_{1})_{0} = _{Xf}i_{0}^{ℬ} + Xv_{0}b_{1}, hence, f is unital.

Assume now that f is unital. We want to find a 3-morphism

v : i^{𝒜} f → fi^{ℬ} : f → f : 𝒜 → ℬ,

that is, an (f, f)-coderivation v of degree −2 such that

vB_{1} = i^{𝒜} f − fi^{ℬ} .  (8.2.1)

We subject it to an additional condition described below. Consider 3-morphisms

x : (i^{𝒜} ⊗ i^{𝒜} )B_{2} → i^{𝒜} : id_{𝒜} → id_{𝒜} : 𝒜 → 𝒜,
y : (i^{ℬ} ⊗ i^{ℬ} )B_{2} → i^{ℬ} : id_{ℬ} → id_{ℬ} : ℬ → ℬ,

so that

xB_{1} = (i^{𝒜} ⊗ i^{𝒜} )B_{2} − i^{𝒜} , yB_{1} = (i^{ℬ} ⊗ i^{ℬ} )B_{2} − i^{ℬ} .

The following equations between (f, f)-coderivations of degree −2 are due to (7.1.2), (7.1.3):

(xf)B_{1} = (i^{𝒜} ⊗ i^{𝒜} )B_{2}f − i^{𝒜} f = (i^{𝒜} f ⊗ i^{𝒜} f)B_{2} + (i^{𝒜} ⊗ i^{𝒜} | f)M_{20}B_{1} − i^{𝒜} f,

(fy)B_{1} = f(i^{ℬ} ⊗ i^{ℬ} )B_{2} − fi^{ℬ} = (fi^{ℬ} ⊗ fi^{ℬ} )B_{2} − fi^{ℬ} .

Combining them with (8.2.1) we find that

(xf)B_{1} − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20}B_{1} + vB_{1}
= (i^{𝒜} f ⊗ i^{𝒜} f)B_{2} − fi^{ℬ}
= (i^{𝒜} f ⊗ vB_{1})B_{2} + (i^{𝒜} f ⊗ fi^{ℬ} )B_{2} − fi^{ℬ}
= −(i^{𝒜} f ⊗ v)B_{2}B_{1} + (vB_{1} ⊗ fi^{ℬ} )B_{2} + (fi^{ℬ} ⊗ fi^{ℬ} )B_{2} − fi^{ℬ}
= −(i^{𝒜} f ⊗ v)B_{2}B_{1} + (v ⊗ fi^{ℬ} )B_{2}B_{1} + (fy)B_{1}.

Now we may formulate the problem: we are looking for v as above and an (f, f)- coderivation w of degree −3, such that

wB_{1} = xf − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20} + v + (i^{𝒜} f ⊗ v)B_{2} − (v ⊗ fi^{ℬ} )B_{2} − fy,

in other terms, a 4-morphism

w : xf − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20} + v → fy − (i^{𝒜} f ⊗ v)B_{2} + (v ⊗ fi^{ℬ} )B_{2} :

(i^{𝒜} f ⊗ i^{𝒜} f)B_{2} → fi^{ℬ} : f → f : 𝒜 → ℬ.

Using the chain map

u = (1⊗fi^{ℬ} )B_{2}−1−(i^{𝒜} f ⊗1)B_{2} : (sA_{∞}(𝒜, ℬ)(f, f), B_{1}) → (sA_{∞}(𝒜, ℬ)(f, f), B_{1}),

we may rewrite our system of equations as follows:

−vB_{1} = fi^{ℬ} − i^{𝒜} f,
wB_{1} + vu = xf − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20} − fy.  (8.2.2)

In other words, we look for an element

(w, v) ∈ [sA_{∞}(𝒜, ℬ)(f, f)]^{−3} ⊕ [sA_{∞}(𝒜, ℬ)(f, f)]^{−2} = Cone^{−3}(u),

whose boundary is

(xf − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20} − fy, fi^{ℬ} − i^{𝒜} f)

∈ [sA_{∞}(𝒜, ℬ)(f, f)]^{−2} ⊕ [sA_{∞}(𝒜, ℬ)(f, f)]^{−1} = Cone^{−2}(u).

Let us prove that u is homotopy invertible. Since

Xi_{0}^{𝒜} f_{1} = _{Xf}i_{0}^{ℬ} +_{X}zb_{1} : 𝕜 → (sℬ)^{−1}(Xf, Xf),

for some _{X}z, the cycles _{X}r_{0} = _{X}i_{0}^{𝒜} f_{1} and _{X}p_{0} = _{Xf}i_{0}^{ℬ} satisfy conditions (7.13.1) for g = f : Ob 𝒜 → Ob ℬ, that is,

(_{X}i_{0}^{𝒜} f_{1} ⊗_{Xf}i_{0}^{ℬ} )b_{2} −_{Xf}i_{0}^{ℬ} = (_{Xf}i_{0}^{ℬ} ⊗_{Xf}i_{0}^{ℬ} )b_{2} −_{Xf}i_{0}^{ℬ} + (_{X}z ⊗_{Xf}i_{0}^{ℬ} )b_{2}b_{1} ∈ Im b_{1},

(_{Xf}i_{0}^{ℬ} ⊗_{X}i_{0}^{𝒜} f_{1})b_{2} −_{Xf}i_{0}^{ℬ} = (_{Xf}i_{0}^{ℬ} ⊗_{Xf}i_{0}^{ℬ} )b_{2} −_{Xf}i_{0}^{ℬ} − (_{Xf}i_{0}^{ℬ} ⊗_{X}z)b_{2}b_{1} ∈ Im b_{1}.

Hence, the natural A_{∞}-transformation r = i^{𝒜} f : f → f : 𝒜 → ℬ is invertible by Proposition 7.15. In detail, there exists a natural A_{∞}-transformation p : f → f : 𝒜 → ℬ and 3-morphisms q, t such that

(i^{𝒜} f ⊗ p)B_{2} − fi^{ℬ} = qB_{1},  (p ⊗ i^{𝒜} f)B_{2} − fi^{ℬ} = tB_{1}.  (8.2.3)

These equations are interpreted as equations (7.13.1) for the following data. Let 𝒞 = 𝟙 be a 1-object-0-morphisms A_{∞}-category, Ob 𝒞 = {∗}, 𝒞(∗, ∗) = 0. Consider a map Ob 𝒞 → Ob A_{∞}(𝒜, ℬ), ∗ ↦ f, and elements i^{𝒜} f, p ∈ [sA_{∞}(𝒜, ℬ)(f, f)]^{−1}, q, t ∈ [sA_{∞}(𝒜, ℬ)(f, f)]^{−2}. Equations (7.13.1) for these data are precisely (8.2.3), since _{f}i_{0}^{A_{∞}(𝒜,ℬ)} = fi^{ℬ} . By Lemma 7.14 we deduce that

(i^{𝒜} f ⊗ 1)B_{2} : (sA_{∞}(𝒜, ℬ)(f, f), B_{1}) → (sA_{∞}(𝒜, ℬ)(f, f), B_{1}),

is homotopy invertible. Since the map (1 ⊗ fi^{ℬ} )B_{2} − 1 is homotopic to 0 by Corollary 7.8, we deduce that u is homotopy invertible. Therefore, Cone(u) is contractible by Lemma Appendix B.1.

To prove the existence of (w, v) satisfying (8.2.2) it suffices to show that (xf − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20} − fy, fi^{ℬ} − i^{𝒜} f) ∈ Cone^{−2}(u) is a cycle. And indeed,

[xf − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20} − fy]B_{1} + (fi^{ℬ} − i^{𝒜} f)u
= (xB_{1})f − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20}B_{1} − f(yB_{1})
+ [(fi^{ℬ} − i^{𝒜} f) ⊗ fi^{ℬ} ]B_{2} − fi^{ℬ} + i^{𝒜} f + [i^{𝒜} f ⊗ (fi^{ℬ} − i^{𝒜} f)]B_{2}
= (i^{𝒜} ⊗ i^{𝒜} )B_{2}f − i^{𝒜} f − (i^{𝒜} ⊗ i^{𝒜} | f)M_{20}B_{1} − f(i^{ℬ} ⊗ i^{ℬ} )B_{2} + fi^{ℬ}
+ (fi^{ℬ} ⊗ fi^{ℬ} )B_{2} − (i^{𝒜} f ⊗ fi^{ℬ} )B_{2} − fi^{ℬ} + i^{𝒜} f + (i^{𝒜} f ⊗ fi^{ℬ} )B_{2}
− (i^{𝒜} f ⊗ i^{𝒜} f)B_{2}
= −[(i^{𝒜} ⊗ i^{𝒜} )(1 ⊗ B_{1} + B_{1} ⊗ 1) | f]M_{20} = 0

due to (7.1.2) and (7.1.3). Clearly, (fi^{ℬ} − i^{𝒜} f)B_{1} = 0, so the proposition is proven.

Clearly, the composition of unital functors is unital. If ℬ, 𝒞 are unital A_{∞}-cate-gories, r : f → g : ℬ → 𝒞 is an isomorphism of A_{∞}-functors and f is unital, then g is unital as well. Indeed, distributivity law in A_{∞} implies

ℬ →^{\begin{matrix}f \\ r⇓ \end{matrix}} ; _{_{g}→} ; →^{\begin{matrix}i^{ℬ} g⇓ \\ g \end{matrix}} 𝒞 = (ℬ →^{\begin{matrix}id & ℬ \\ i^{ℬ} ⇓ \end{matrix}} ; _{_{id_{ℬ}}→} ℬ →^{\begin{matrix}f \\ r⇓ \end{matrix}} ; _{_{g}→} 𝒞) = ℬ →^{\begin{matrix}f \\ i^{ℬ} f⇓≡fi^{𝒞} \end{matrix}} ; _{_{f} →} ; \begin{matrix}r⇓ \\ g \end{matrix} → 𝒞

= (ℬ →^{\begin{matrix}f \\ r⇓ \end{matrix}} ; _{_{g}→} 𝒞 →^{\begin{matrix}id & 𝒞 \\ i^{𝒞} ⇓ \end{matrix}} ; _{_{id_{𝒞}}→} 𝒞) = ℬ →^{\begin{matrix}f \\ r⇓ \end{matrix}} ; _{_{g}→} ; →^{\begin{matrix}gi^{𝒞} ⇓ \\ g \end{matrix}} 𝒞,

or  r · (i^{ℬ} g) = i^{ℬ} ∘_{h} r = (i^{ℬ} f) · r = (fi^{𝒞}) · r = r ∘_{h} i^{𝒞} = r · (gi^{𝒞}), (8.2.4)

where · and ∘_{h} denote the vertical and the horizontal compositions of 2-morphisms, hence, i^{ℬ} g ≡ gi^{𝒞}.

8.3 Definition. The 2-category A_{∞}^{u} is a 2-subcategory of ^{u}A_{∞}, whose class of objects consists of all unital A_{∞}-categories, 1-morphisms are all unital A_{∞}-functors, and 2-morphisms are equivalence classes of all natural A_{∞}-transformations between such functors.

8.4 Proposition. Let 𝒜 be an A_{∞}-category. The strict 2-functor A_{∞}(𝒜, ) maps a unital A_{∞}-category 𝒞 to the unital A_{∞}-category A_{∞}(𝒜, 𝒞), and a unital functor to a unital functor. Its restrictions A_{∞}(𝒜, ) : ^{u}A_{∞} → ^{u}A_{∞}, A_{∞}(𝒜, ) : A_{∞}^{u} → A_{∞}^{u} preserve 1-units and 2-units, thus, they are 2-functors in the usual sense.

Proof. Proposition 7.7 shows that A_{∞}(𝒜, 𝒞) is unital, if 𝒞 is unital. If g : ℬ → 𝒞 is a unital A_{∞}-functor between unital A_{∞}-categories ℬ and 𝒞, then i^{ℬ} g ≡ gi^{𝒞} implies

(1 ⊗ i^{ℬ} )M(1 ⊗ g)M = (1 ⊗ i^{ℬ} g)M ≡ (1 ⊗ gi^{𝒞})M = (1 ⊗ g)M(1 ⊗ i^{𝒞})M, (8.4.1)

hence, (1 ⊗ g)M is unital. The fact, that A_{∞}(𝒜, ) preserves 1-units and 2-units is already proven in Proposition 7.7.

8.5. Categories modulo homotopy. 𝒦-categories form a 2-category 𝒦-𝒞at. We consider also non-unital 𝒦-categories. They form a 2-category 𝒦-𝒞at^{nu} without 2-units (but with 1-units – identity functors).

8.6 Proposition. There is a strict 2-functor k : A_{∞} → 𝒦-𝒞at^{nu} of non-2-unital 2-categories, which assigns to an A_{∞}-category 𝒞 the 𝒦-category k𝒞 with the same class of objects Ob k𝒞 = Ob 𝒞, the same graded 𝕜-module of morphisms k𝒞(X, Y) = 𝒞(X, Y), equipped with the differential m_{1}. Composition in k𝒞 is given by (the homotopy equivalence class of) m_{2} : 𝒞(X, Y) ⊗ 𝒞(Y, Z) → 𝒞(X, Z). To an A_{∞}-functor f : 𝒜 → ℬ is assigned kf : k𝒜 → kℬ such that Ob kf = f : Ob 𝒜 → Ob ℬ, and for each pair of objects X, Y of 𝒜 we have kf = sf_{1}s^{−1} : 𝒜(X, Y) → ℬ(Xf, Y f). To a natural A_{∞}-transformation r : f → g : 𝒜 → ℬ is assigned kr = r_{0}s^{−1} : kf → kg, that is, for each object X of 𝒜 the component_{X}kr is the homotopy equivalence class of chain map _{X}r_{0}s^{−1} : 𝕜 → ℬ(Xf, Xg). Unital A_{∞}-categories and unital A_{∞}-func-tors are mapped by k to unital 𝒦-categories and unital 𝒦-functors. The restriction k : A_{∞}^{u} → 𝒦-𝒞at is a 2-functor, which preserves 1-units and 2-units.

Proof. The identity (7.1.1) shows that m_{2} is associative in 𝒦. The identity

(kf ⊗ kf)m_{2} + (s ⊗ s)f_{2}s^{−1}m_{1} + (1 ⊗ m_{1} + m_{1} ⊗ 1)(s ⊗ s)f_{2}s^{−1} + m_{2}kf = 0

shows that kf preserves the multiplication in 𝒦.

The map _{X}kr is a chain map since _{X}r_{0}s^{−1}m_{1} = _{X}r_{0}b_{1}s^{−1} = 0. If r ≡ p : f → g : 𝒜 → ℬ, then _{X}r_{0} = _{X}p_{0} +_{X}v_{0}b_{1} for some _{X}v_{0} ∈ (sℬ)^{−2}(Xf, Xg), therefore, Xr_{0}s^{−1} = _{X}p_{0}s^{−1}+(_{X}v_{0}s^{−1})m_{1} and chain maps_{X}r_{0}s^{−1} and_{X}p_{0}s^{−1} are homotopic to each other, that is, kr = kp. The identity

0 = s[(f_{1} ⊗_{Y} r_{0})b_{2} + (_{X}r_{0} ⊗ g_{1})b_{2} + r_{1}b_{1} + b_{1}r_{1}]s^{−1}

= (kf ⊗_{Y} kr)m_{2} −(_{X}kr⊗kg)m_{2} +sr_{1}s^{−1}m_{1} +m_{1}sr_{1}s^{−1} : 𝒜(X, Y) → ℬ(Xf, Y g)

shows that the following diagram commutes in 𝒦 for all objects X, Y of 𝒜

\begin{matrix}𝒜(X, Y) & \overline{^{kf} →} & ℬ(Xf, Y f) \\ ℬ(Xg, Y g) & ↓^{kg} & \overline{→^{(_{X}kr⊗1)m_{2}}} & ℬ(Xf, Y g) & ↓^{(1⊗_{Y} kr)m_{2}} \end{matrix}

Thus kr is, indeed, a 𝒦-natural transformation.

One checks easily that the composition of functors is preserved, and the both compositions of 1-morphisms and 2-morphisms are preserved. The vertical composition of 2-morphisms is preserved due to the property

Xk[(r ⊗ p)B_{2}] = _{X}[(r ⊗ p)B_{2}]_{0}s^{−1} = (_{X}r_{0} ⊗_{X}p_{0})b_{2}s^{−1} = (_{X}kr ⊗_{X}kp)m_{2}.

Let ℬ be a unital category. Then kℬ is a unital 𝒦-category. Indeed, for each object X of ℬ consider the corresponding element 1_{X} = _{X}i_{0}^{ℬ} s^{−1} = _{X}ki^{ℬ} : 𝕜 → ℬ^{0}(X, X). Then for each pair X, Y of objects of 𝒞 the following equations hold in 𝒦

(1 ⊗ 1_{Y})m_{2} = s(1 ⊗ 1_{Y} s)b_{2}s^{−1} = s(1 ⊗_{Y} i_{0}^{ℬ} )b_{2}s^{−1} = ss^{−1}
= 1 : ℬ(X, Y) → ℬ(X, Y),
(1_{X} ⊗ 1)m_{2} = −s(1_{X}s ⊗ 1)b_{2}s^{−1} = −s(_{X}i_{0}^{ℬ} ⊗ 1)b_{2}s^{−1}
= ss^{−1} = 1 : ℬ(X, Y) → ℬ(X, Y).

That is, 1_{X} is the unit endomorphism of X.

If f : 𝒜 → 𝒞 is unital then, applying k to the equivalence i^{𝒜} f ≡ fi^{𝒞}, we find that (1_{id_{k𝒜}} )kf = (kf)1_{id_{k𝒞}} = 1_{kf}, that is, kf is unital (it maps units into units).

8.7 Lemma (Cancellation). Let ϕ : 𝒞 → ℬ be an A_{∞}-functor, such that for all objects X, Y of 𝒞 the chain map ϕ_{1} : (s𝒞(X, Y), b_{1}) → (sℬ(Xϕ, Y ϕ), b_{1}) is invertible in 𝒦. Let f, g : 𝒜 → 𝒞 be A_{∞}-functors. Let y : fϕ → gϕ : 𝒜 → ℬ be a natural A_{∞}-transformation. Then there is a unique up to equivalence natural A_{∞}-transformation t : f → g : 𝒜 → 𝒞 such that y ≡ tϕ.

Proof. First we prove the existence. We are looking for a 2-morphism t : f → g : 𝒜 → 𝒞 and a 3-morphism v : y → tϕ : fϕ → gϕ : 𝒜 → ℬ. We have to satisfy the equations

tb + bt = 0, y − tϕ = vb − bv.

Let us construct the components of t and v by induction. Given a non-negative integer n, assume that we have already found components t_{m}, v_{m} of the sought t, v for m < n, such that the equations

(tb)_{m} + (bt)_{m} = 0 : s𝒜(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒜(X_{m−1}, X_{m}) → s𝒞(X_{0}f, X_{m}g), (8.7.1)

y_{m} − (tϕ)_{m} = (vb − bv)_{m} : s𝒜(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒜(X_{m−1}, X_{m}) → sℬ(X_{0}fϕ, X_{m}gϕ), (8.7.2)

are satisfied for all m < n. Introduce an (f, g)-coderivation t̃ : T s𝒜 → T s𝒞 of degree −1 by its components (t_{0}, …, t_{n−1}, 0, 0, …) and an (fϕ, gϕ)-coderivation ṽ : T s𝒜 → T sℬ of degree −2 by its components (v_{0}, …, v_{n−1}, 0, 0, …). Define an (f, g)-coderivation λ = t̃b + bt̃ of degree 0 and an (fϕ, gϕ)-coderivation ν = y − t̃ϕ − ṽb + bṽ of degree −1. Then equations (8.7.1), (8.7.2) imply that λ_{m} = 0, ν_{m} = 0 for m < n. The identity λb − bλ = 0 implies that

λ_{n}d = λ_{n}b_{1} − ∑_{α+1+β=n}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β})λ_{n} = 0.  (8.7.3)

The identity

νb + bν = −t̃ϕb − bt̃ϕ = −λϕ

implies that

ν_{n}d = ν_{n}b_{1} + ∑_{α+1+β=n}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β})ν_{n} = −λ_{n}ϕ_{1}.  (8.7.4)

Denote N = s𝒜(X_{0}, X_{1}) ⊗_{𝕜} ⋯ ⊗_{𝕜} s𝒜(X_{n−1}, X_{n}), and consider the chain map

u = Hom(N, ϕ_{1}) : Hom^{•}(N, s𝒞(X_{0}f, X_{n}g)) → Hom^{•}(N, s𝒞(X_{0}fϕ, X_{n}gϕ)).

Since ϕ_{1} is homotopy invertible, the map u is homotopy invertible as well. Therefore, the complex Cone(u) is acyclic. Moreover, it is contractible by Lemma Appendix B.1. Equations (8.7.3) and (8.7.4) in the form −λ_{n}d = 0, ν_{n}d + λ_{n}ϕ_{1} = 0 imply that

(ν_{n}, λ_{n}) ∈ Hom_{𝕜}^{−1}(N, sℬ(X_{0}fϕ, X_{n}gϕ)) ⊕ Hom_{𝕜}^{0}(N, s𝒞(X_{0}f, X_{n}g)) = Cone^{−1}(u)

is a boundary of some element

(v_{n}, t_{n}) ∈ Hom_{𝕜}^{−2}(N, sℬ(X_{0}fϕ, X_{n}gϕ)) ⊕ Hom_{𝕜}^{−1}(N, s𝒞(X_{0}f, X_{n}g)) = Cone^{−2}(u),

that is, v_{n}d + t_{n}u = ν_{n} and −t_{n}d = λ_{n}. In other words, equations (8.7.1), (8.7.2) are satisfied for m = n, and we prove the existence of t with the required properties by induction.

Now we prove the uniqueness of t. Assume that we have 2-morphisms t, t′ : f → g : 𝒜 → 𝒞 and 3-morphisms v : y → tϕ : fϕ → gϕ : 𝒜 → ℬ, v′ : y → t′ϕ : fϕ → gϕ : 𝒜 → ℬ. We look for a 3-morphism w and a 4-morphism x:

w : t → t′ : f → g : 𝒜 → 𝒞,
x : v′ → v + wϕ : y → t′ϕ : fϕ → gϕ : 𝒜 → ℬ.

They have to satisfy equations

t − t′ = wb − bw, v′ − v − wϕ = xb + bx.

Let us construct the components of w and x by induction. Given a non-negative integer n, assume that we have already found components w_{m} and x_{m} of the sought w, x for m < n, such that the equations

t_{m} − t′_{m} = (wb − bw)_{m} : s𝒜(X_{0}, X_{1})
⊗ ⋯ ⊗ s𝒜(X_{m−1}, X_{m}) → s𝒞(X_{0}f, X_{m}g),  (8.7.5)

v′_{m} − v_{m} − (wϕ)_{m} = (xb + bx)_{m} : s𝒜(X_{0}, X_{1})
⊗ ⋯ ⊗ s𝒜(X_{m−1}, X_{m}) → sℬ(X_{0}fϕ, X_{m}gϕ),  (8.7.6)

are satisfied for all m < n. Introduce an (f, g)-coderivation w̃ : T s𝒜 → T s𝒞 of degree −2 by its components (w_{0}, …, w_{n−1}, 0, 0, …) and an (fϕ, gϕ)-coderivation x̃ : T s𝒜 → T sℬ of degree −3 by its components (x_{0}, …, x_{n−1}, 0, 0, …). Define an (f, g)-coderivation λ = t′ − t + w̃b + bw̃ of degree −1 and an (fϕ, gϕ)-coderivation ν = v′ − v − w̃ϕ − x̃b − bx̃ of degree −2. Then equations (8.7.5), (8.7.6) imply that λ_{m} = 0, ν_{m} = 0 for m < n. The identity λb + bλ = 0 implies that

λ_{n}d = λ_{n}b_{1} + ∑_{α+1+β=n}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β})λ_{n} = 0.

The identity

νb − bν = v′B_{1} − vB_{1} − w̃ϕb + bw̃ϕ = y − t′ϕ − y + tϕ − w̃bϕ + bw̃ϕ = −λϕ

implies that

ν_{n}d = ν_{n}b_{1} − ∑_{α+1+β=n}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β})ν_{n} = −λ_{n}ϕ_{1}.

Hence,

(ν_{n}, λ_{n}) ∈ Hom_{𝕜}^{−2}(N, sℬ(X_{0}fϕ, X_{n}gϕ)) ⊕ Hom_{𝕜}^{−1}(N, s𝒞(X_{0}f, X_{n}g)) = Cone^{−2}(u)

is a cycle, therefore, it is a boundary of an element

(x_{n}, w_{n}) ∈ Hom_{𝕜}^{−3}(N, sℬ(X_{0}fϕ, X_{n}gϕ)) ⊕ Hom_{𝕜}^{−2}(N, s𝒞(X_{0}f, X_{n}g)) = Cone^{−3}(u),

that is, x_{n}d+w_{n}ϕ_{1} = ν_{n} and −w_{n}d = λ_{n}. In other words, equations (8.7.5), (8.7.6), are satisfied for m = n, and we prove the uniqueness of t, using induction.

A version of the following theorem is proved by Fukaya [Fuk, Theorem 8.6] with a different notion of unitality and under the additional assumption that the 𝕜-modules ℬ(W, Z), 𝒞(X, Y) are free.

8.8 Theorem. Let 𝒞 be an A_{∞}-category and let ℬ be a unital A_{∞}-category. Let ϕ : 𝒞 → ℬ be an A_{∞}-functor such that for all objects X, Y of 𝒞 the chain map ϕ_{1} : (s𝒞(X, Y), b_{1}) → (sℬ(Xϕ, Y ϕ), b_{1}) is invertible in 𝒦. Let h : Ob ℬ → Ob 𝒞 be a mapping. Assume that for each object U of ℬ the 𝕜-linear maps

Ur_{0} : 𝕜 → (sℬ)^{−1}(U, Uhϕ),  _{U}p_{0} : 𝕜 → (sℬ)^{−1}(Uhϕ, U),
Uw_{0} : 𝕜 → (sℬ)^{−2}(U, Uhϕ),  _{U}v_{0} : 𝕜 → (sℬ)^{−2}(Uhϕ, U)

are given such that

Ur_{0}b_{1} = 0, _{U}p_{0}b_{1} = 0,
(_{U}r_{0} ⊗_{U}p_{0})b_{2} −_{U}i_{0}^{ℬ} = _{U}w_{0}b_{1},  (8.8.1)
(_{U}p_{0} ⊗_{U}r_{0})b_{2} −_{Uhϕ}i_{0}^{ℬ} = _{U}v_{0}b_{1}.

Then there is an A_{∞}-functor ψ : ℬ → 𝒞 such that Ob ψ = h, there are natural A_{∞}-transformations r : id_{ℬ} → ψϕ, p : ψϕ → id_{ℬ} such that their 0-th components are the given _{U}r_{0}, _{U}p_{0}. Moreover, r and p are inverse to each other in the sense that

(r ⊗ p)B_{2} ≡ i^{ℬ} , (p ⊗ r)B_{2} ≡ ψϕi^{ℬ} .

There exist unique up to equivalence natural A_{∞}-transformations t : id_{𝒞} → ϕψ, q : ϕψ → id_{𝒞} such that tϕ ≡ ϕr : ϕ → ϕψϕ and qϕ ≡ ϕp : ϕψϕ → ϕ.

Finally, 𝒞 is unital with the unit

i^{𝒞} = (t ⊗ q)B_{2} : id_{𝒞} → id_{𝒞} : 𝒞 → 𝒞,

ϕ and ψ are unital A_{∞}-equivalences, quasi-inverse to each other via mutually inverse isomorphisms r and p, t and q (in particular, (q ⊗ t)B_{2} ≡ ϕψi^{𝒞}).

Proof. We have to satisfy the equations

ψb = bψ, rb + br = 0.

We already know the map Ob ψ and the component r_{0}. Let us construct the remaining components of ψ and r by induction. Given a positive integer n, assume that we have already found components ψ_{m}, r_{m} of the sought ψ, r for m < n, such that the equations

(ψb)_{m} + (bψ)_{m} = 0 : sℬ(X_{0}, X_{1}) ⊗ ⋯ ⊗ sℬ(X_{m−1}, X_{m}) → s𝒞(X_{0}h, X_{m}h), (8.8.2)

(rb + br)_{m} = 0 : sℬ(X_{0}, X_{1}) ⊗ ⋯ ⊗ sℬ(X_{m−1}, X_{m}) → sℬ(X_{0}, X_{m}hϕ) (8.8.3)

are satisfied for all m < n. Introduce a cocategory homomorphism ψ̃ : T sℬ → T s𝒞 of degree 0 by its components (ψ_{1}, …, ψ_{n−1}, 0, 0, …) and a (id_{ℬ} , ψ̃ϕ)-coderivation r̃: T sℬ → T sℬ of degree −1 by its components (r_{0}, r_{1}, …, r_{n−1}, 0, 0, …). Define a (ψ̃, ψ̃)-coderivation λ = ψ̃b − bψ̃ of degree 1 and a map ν = −r̃b − br̃ + (r̃ ⊗ λϕ)θ : T sℬ → T sℬ of degree 0. The commutator r̃b + br̃ has the following property:

(r̃b + br̃)Δ = Δ[1 ⊗ (r̃b + br̃) + (r̃b + br̃) ⊗ ψ̃ϕ + r̃ ⊗ λϕ].

By Proposition 3.1 the map (r̃ ⊗ λϕ)θ has a similar property

(r̃ ⊗ λϕ)θΔ = Δ[1 ⊗ (r̃ ⊗ λϕ)θ + (r̃ ⊗ λϕ)θ ⊗ ψ̃ϕ + r̃ ⊗ λϕ].

Taking the difference we find that ν is an (id_{ℬ} , ψ̃ϕ)-coderivation. Equations (8.8.2), (8.8.3) imply that λ_{m} = 0, ν_{m} = 0 for m < n (the image of (r̃ ⊗ λϕ)θ is contained in T^{⩾2}sℬ).

The identity λb + bλ = 0 implies that

λ_{n}d = λ_{n}b_{1} + ∑_{α+1+β=n}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β})λ_{n} = 0.  (8.8.4)

The identity

νb − bν = (r̃ ⊗ λϕ)θb − b(r̃ ⊗ λϕ)θ

implies that

ν_{n}b_{1} − ∑_{α+1+β=n}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β})ν_{n} = −(r_{0} ⊗ λ_{n}ϕ_{1})b_{2} = −λ_{n}ϕ_{1}(r_{0} ⊗ 1)b_{2}. (8.8.5)

Set N = sℬ(X_{0}, X_{1}) ⊗_{𝕜} ⋯ ⊗_{𝕜} sℬ(X_{n−1}, X_{n}), and introduce a chain map

u = Hom(N, ϕ_{1}(r_{0} ⊗ 1)b_{2}) : Hom^{•}(N, s𝒞(X_{0}h, X_{n}h)) → Hom^{•}(N, sℬ(X_{0}, X_{n}hϕ)).

Since ϕ_{1} and (r_{0} ⊗ 1)b_{2} are homotopy invertible by Lemma 7.14, the map u is homotopy invertible as well. Therefore, the complex Cone(u) is contractible by Lemma Appendix B.1. Equations (8.8.4) and (8.8.5) in the form −λ_{n}d = 0, ν_{n}d + λ_{n}u = 0 imply that

(ν_{n}, λ_{n}) ∈ Hom_{𝕜}^{0}(N, sℬ(X_{0}, X_{n}hϕ)) ⊕ Hom_{𝕜}^{1}(N, s𝒞(X_{0}h, X_{n}h)) = Cone^{0}(u)

is a cycle. Hence, it is a boundary of some element

(r_{n}, ψ_{n}) ∈ Hom_{𝕜}^{−1}(N, sℬ(X_{0}, X_{n}hϕ)) ⊕ Hom_{𝕜}^{0}(N, s𝒞(X_{0}h, X_{n}h)) = Cone^{−1}(u),

that is, r_{n}d + ψ_{n}ϕ_{1}(r_{0} ⊗ 1)b_{2} = ν_{n} and −ψ_{n}d = λ_{n}. In other words, equations (8.8.2), (8.8.3) are satisfied for m = n, and we prove the existence of ψ and r by induction.

Since r_{0} and p_{0} are homotopy inverse to each other in the sense of (8.8.1), we find by Proposition 7.15 that there exists a natural A_{∞}-transformation p : ψϕ → id_{ℬ} such that r and p are inverse to each other.

The existence of t, q such that tϕ ≡ ϕr and qϕ ≡ ϕp follows by Lemma 8.7. Let us prove that i^{𝒞} = (t ⊗ q)B_{2} is a unit of 𝒞. Due to Lemma 7.14 the maps (r_{0} ⊗ 1)b_{2}, (1 ⊗ r_{0})b_{2}, (p_{0} ⊗ 1)b_{2}, (1 ⊗ p_{0})b_{2} are homotopy invertible. Let f denote a homotopy inverse map of ϕ_{1} : s𝒞(X, Y) → sℬ(Xϕ, Y ϕ). The identity tϕ ≡ ϕr implies that Xt_{0}ϕ_{1} = _{Xϕ}r_{0} + κb_{1}. Hence,

(_{Xϕ}r_{0} ⊗ 1)b_{2} ∼ (_{X}t_{0}ϕ_{1} ⊗ 1)b_{2} ∼ f(_{X}t_{0} ⊗ 1)b_{2}ϕ_{1}.

Therefore, (_{X}t_{0} ⊗ 1)b_{2} ∼ ϕ_{1}(_{Xϕ}r_{0} ⊗ 1)b_{2}f is homotopy invertible. Similarly,

(1 ⊗_{Y ϕ}r_{0})b_{2} ∼ (1 ⊗_{Y} t_{0}ϕ_{1})b_{2} ∼ f(1 ⊗_{Y} t_{0})b_{2}ϕ_{1}

implies that (1⊗_{Y} t_{0})b_{2} ∼ ϕ_{1}(1⊗_{Y ϕ}r_{0})b_{2}f is homotopy invertible. Similarly, (_{X}q_{0} ⊗ 1)b_{2} and (1 ⊗_{Y} q_{0})b_{2} are homotopy invertible.

The computation made in (7.14.1) shows that the product of the above homotopy invertible maps

(q_{0} ⊗ 1)b_{2}(t_{0} ⊗ 1)b_{2} ∼ −(t_{0} ⊗ q_{0} ⊗ 1)(b_{2} ⊗ 1)b_{2} = −(i_{0}^{𝒞} ⊗ 1)b_{2}

is the map we are studying. Similarly,

(1 ⊗ t_{0})b_{2}(1 ⊗ q_{0})b_{2} ∼ (1 ⊗ t_{0} ⊗ q_{0})(1 ⊗ b_{2})b_{2} = (1 ⊗ i_{0}^{𝒞})b_{2}.

We conclude that both (i_{0}^{𝒞} ⊗ 1)b_{2} and (1 ⊗ i_{0}^{𝒞})b_{2} are homotopy invertible. Let us prove that (i^{𝒞} ⊗ i^{𝒞})B_{2} ≡ i^{𝒞}. Due to Proposition 7.1 we have

i^{𝒞} ϕ = (t ⊗ q)B_{2}ϕ ≡ (tϕ ⊗ qϕ)B_{2} ≡ (ϕr ⊗ ϕp)B_{2} = ϕ(r ⊗ p)B_{2} ≡ ϕi^{ℬ} . (8.8.6)

Using Proposition 7.1 again we get

(i^{𝒞} ⊗ i^{𝒞})B_{2}ϕ ≡ (i^{𝒞} ϕ ⊗ i^{𝒞} ϕ)B_{2} ≡ (ϕi^{ℬ} ⊗ ϕi^{ℬ} )B_{2} = ϕ(i^{ℬ} ⊗ i^{ℬ} )B_{2} ≡ ϕi^{ℬ} ≡ i^{𝒞} ϕ.

By Lemma 8.7 we deduce that (i^{𝒞} ⊗ i^{𝒞})B_{2} ≡ i^{𝒞}, therefore, i^{𝒞} is a unit of 𝒞.

Let us prove that t and q are inverse to each other. By definition, (t ⊗ q)B_{2} = i^{𝒞}. Due to Proposition 7.1

(q ⊗ t)B_{2}ϕ ≡ (qϕ ⊗ tϕ)B_{2} ≡ (ϕp ⊗ ϕr)B_{2} = ϕ(p ⊗ r)B_{2} ≡ ϕψϕi^{ℬ} ≡ ϕψi^{𝒞} ϕ.

By Lemma 8.7 (q ⊗ t)B_{2} ≡ ϕψi^{𝒞}. Hence, t and q are inverse to each other, as well as r and p. Therefore, ϕ and ψ are equivalences, quasi-inverse to each other.

Relation (8.8.6) shows that ϕ is unital. Let us prove that ψ is unital. We know that ψϕ is isomorphic to the identity functor. Thus, ψϕ is unital by (8.2.4). Hence, i^{ℬ} ψϕ ≡ ψϕi^{ℬ} ≡ ψi^{𝒞} ϕ. By Lemma 8.7 we have i^{ℬ} ψ ≡ ψi^{𝒞}, and ψ is unital. The theorem is proven.

8.9 Corollary. Let 𝒞, ℬ be unital A_{∞}-categories, and let ϕ : 𝒞 → ℬ be an equivalence. Then ϕ is unital.

Proof. Since ϕ is an equivalence, kϕ is an equivalence as well. Hence, ϕ_{1} is invertible in 𝒦. There exists an A_{∞}-functor ψ : ℬ → 𝒞 quasi-inverse to ϕ, and mutually inverse isomorphisms r : id_{ℬ} → ψϕ, p : ψϕ → id_{ℬ} . In particular, the assumptions of Theorem 8.8 are satisfied by ϕ, Ob ψ : Ob ℬ → Ob 𝒞, r_{0} and p_{0}. The theorem implies that ϕ is unital.

8.10 Corollary. Let 𝒞 be an A_{∞}-algebra and let ℬ be a unital A_{∞}-algebra (viewed as A_{∞}-categories with one object). Let ϕ : 𝒞 → ℬ be an A_{∞}-homomorphism such that ϕ_{1} : (s𝒞, b_{1}) → (sℬ, b_{1}) is homotopy invertible. Then 𝒞 and ϕ are unital, and ϕ is an A_{∞}-equivalence.

Existence of ϕ with the above property might be taken as an equivalence relation on the class of unital A_{∞}-algebras.

8.11. Strictly unital A_{∞}-categories. A strict unit of an object X of an A_{∞}-category 𝒜 is an element 1_{X} ∈ 𝒜^{0}(X, X), such that (f ⊗ 1_{X})m_{2} = f, (1_{X} ⊗ g)m_{2} = g, whenever these make sense, and (⋯ ⊗ 1_{X} ⊗ …)m_{n} = 0 if n ≠ 2 (see e.g. [FOOO, Fuk, Kel01]). We may write it as a map 1_{X} : 𝕜 → 𝒜(X, X), 1 ↦ 1_{X}. Assume that 𝒜 has a strict unit for each object X. For example, a differential graded category 𝒜 has strict units. Then we introduce a coderivation i^{𝒜} : id_{𝒜} → id_{𝒜} : 𝒜 → 𝒜, whose components are i_{0}^{𝒜} : 𝕜 → s𝒜(X, X), 1 ↦ 1_{X}s = _{X}i_{0}^{𝒜} , and i_{k}^{𝒜} = 0 for k > 0. The conditions on 1_{X} imply that (1⊗i_{0}^{𝒜} )b_{2} = 1 : s𝒜(Y, X) → s𝒜(Y, X) and (i_{0}^{𝒜} ⊗1)b_{2} = −1 : s𝒜(X, Z) → s𝒜(X, Z). One deduces that i^{𝒜} is a natural A_{∞}-transformation. If an A_{∞}-category 𝒜 has two such transformations – strict units i and i′, then they must coincide because of the above equations. We call 𝒜 strictly unital if it has a strict unit i^{𝒜} . Naturally, a strictly unital A_{∞}-category is unital.

For any A_{∞}-functor f : 𝒞 → 𝒜 the natural A_{∞}-transformation 1_{f}s = fi^{𝒜} : f → f : 𝒞 → 𝒜 has the components _{X}(fi^{𝒜} )_{0} = _{Xf}i_{0}^{𝒜} : 𝕜 → s𝒜(Xf, Xf) and (fi^{𝒜} )_{k} = 0 for k > 0. It is the unit 2-endomorphism of f.

If A_{∞}-category ℬ is strictly unital, then so is 𝒞 = A_{∞}(𝒜, ℬ) for an arbitrary A_{∞}-category 𝒜. Indeed, for an arbitrary A_{∞}-functor f : 𝒜 → ℬ there is a unit 2-endomorphism 1_{f}s = fi^{ℬ} : f → f. We set i_{0}^{𝒞} : 𝕜 → [sA_{∞}(𝒜, ℬ)]^{−1}(f, f), 1 ↦ 1_{f}s, and i_{k}^{𝒞} = 0 for k > 0. For any element r ∈ 𝒞(g, f) we have (r ⊗ 1_{f}s)B_{2} = r. For any element p ∈ 𝒞(f, h) we have p(1_{f}s ⊗ 1)B_{2} = p((fi^{ℬ} )_{0} ⊗ 1)b_{2} = −p. We have also i^{𝒞} B_{1} = 0 and (⋯ ⊗ i^{𝒞} ⊗ …)B_{n} = 0 if n > 2, due to (5.1.3). Therefore, i^{𝒞} satisfies the required conditions.

Another approach to i^{𝒞} uses the A_{∞}-functor M : T sA_{∞}(𝒜, ℬ) ⊗ T sA_{∞}(ℬ, ℬ) → T sA_{∞}(𝒜, ℬ) = 𝒞. We have (1⊗id_{ℬ} )M = id_{𝒞} by (4.1.3), and the natural A_{∞}-trans-formations (1⊗i^{ℬ} )M and i^{𝒞} of id_{𝒞} coincide. Indeed, [(1⊗i^{ℬ} )M]_{0} : 𝕜 → (s𝒞)^{−1}(f, f), 1 ↦ (f | i^{ℬ} )M_{01} = fi^{ℬ} = _{f}i_{0}^{𝒞}. For all n ⩾ 0 we have [(1 ⊗ i^{ℬ} )M]_{n} : r^{1} ⊗ ⋯ ⊗ r^{n} ↦ (r^{1} ⊗ ⋯ ⊗ r^{n} ⊗ i^{ℬ} )M_{n1}. By (4.1.4) the components

[(r^{1} ⊗ ⋯ ⊗ r^{n} ⊗ i^{ℬ} )M_{n1}]_{k} = ∑_{l}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}i_{l}^{ℬ} = (r^{1} ⊗ ⋯ ⊗ r^{n})θ_{k0}i_{0}^{ℬ}

vanish for n > 0.

8.12. Other examples of unital A_{∞}-categories. More examples of unital categories might be obtained via Theorem 8.8. An A_{∞}-category with a homotopy unit in the sense of Fukaya, Oh, Ohta and Ono [FOOO, Definition 20.1] clarified by Fukaya [Fuk, Definition 5.11] is also a unital category in our sense. Indeed, these authors enlarge given A_{∞}-category 𝒞 to a strictly unital A_{∞}-cat-egory ℬ by adding extra elements to 𝒞(X, X), so that the natural embedding (𝒞(X, Y), m_{1}) ↪ (ℬ(X, Y), m_{1}) were a homotopy equivalence. Setting r_{0} = p_{0} = i_{0}^{ℬ} we view the above situation as a particular case of Theorem 8.8.

If an A_{∞}-functor ϕ : 𝒞 → ℬ to a unital A_{∞}-category ℬ is invertible, then 𝒞 is unital. Indeed, since there exists an A_{∞}-functor ψ : ℬ → 𝒞 such that ϕψ = id_{𝒞} and ψϕ = id_{ℬ} , then the map ϕ_{1} is invertible with inverse ψ_{1}. The remaining data are Ob ψ : Ob ℬ → Ob 𝒞 and_{X}r_{0} = _{X}p_{0} = _{X}i_{0}^{ℬ} : 𝕜 → sℬ(X, X). Since (i^{ℬ} ⊗i^{ℬ} )B_{2} ≡ i^{ℬ} we have (_{X}i_{0}^{ℬ} ⊗_{X}i_{0}^{ℬ} )b_{2} −_{X}i_{0}^{ℬ} ∈ Im b_{1} and conditions (8.8.1) are satisfied. The data constructed in Theorem 8.8 will be precisely ψ : ℬ → 𝒞 and r = p = i^{ℬ} . Since ϕ is unital by Theorem 8.8, we may choose i^{𝒞} = ϕi^{ℬ} ψ as a unit of 𝒞.

If a unital A_{∞}-category 𝒞 is equivalent to a strictly unital A_{∞}-category ℬ via an A_{∞}-functor ϕ : 𝒞 → ℬ, then (1 ⊗ ϕ)M : A_{∞}(𝒜, 𝒞) → A_{∞}(𝒜, 𝒞) is also an equivalence for an arbitrary A_{∞}-category 𝒜 as Proposition 8.4 shows. Thus, a unital A_{∞}-category A_{∞}(𝒜, 𝒞) is equivalent to a strictly unital A_{∞}-category A_{∞}(𝒜, ℬ). In particular, if ϕ is invertible, then (1 ⊗ ϕ)M is invertible as well.

8.13. Cohomology of A_{∞}-categories. Using a lax monoidal functor from 𝒦 to some monoidal category we get another 2-functor, which can be composed with k. For instance, there is a cohomology functor H^{•} : 𝒦 → ℤ -grad -𝕜 -mod, which induces a 2-functor H^{•} : 𝒦-𝒞at → ℤ -grad -𝕜-𝒞at. In practice we will use the 0-th cohomology functor H^{0} : 𝒦 → 𝕜 -mod, the corresponding 2-functor H^{0} : 𝒦-𝒞at → 𝕜-𝒞at, and the composite 2-functor

A_{∞}^{u} ^{k}→ 𝒦-𝒞at ^{H^{0}}→ 𝕜-𝒞at,

which is also denoted by H^{0}. It takes a unital A_{∞}-category 𝒞 into a 𝕜-linear category H^{0}(𝒞) with the same class of objects Ob H^{0}(𝒞) = Ob 𝒞. Its morphism space between objects X and Y is H^{0}(𝒞)(X, Y) = H^{0}(𝒞(X, Y), m_{1}), the 0-th cohomology with respect to the differential m_{1} = sb_{1}s^{−1}. The composition in H^{0}(𝒞) is induced by m_{2}, and the units by i_{0}^{𝒞} s^{−1}.

For example, the homotopy category K(𝒜) of complexes of objects of an abelian category 𝒜 is the 0-th cohomology H^{0}(C(𝒜)) of the differential graded category of complexes C(𝒜).

Appendix A. Enriched 2-categories

Recall that 𝒰 is a fixed universe. Let 𝒱 = (𝒱, ⊗, c, 𝟙) be a symmetric monoidal 𝒰 -category (that is, all 𝒱(X, Y) are 𝒰 -small sets). In this article we shall use (𝒱, ⊗, c, 𝟙) = (𝕜 -mod, ⊗_{𝕜}, σ, 𝕜), where σ is the permutation isomorphism, or (𝒱, ⊗, c, 𝟙) = (𝒦, ⊗_{𝕜}, c, 𝕜), where 𝒦 is the category of differential graded 𝕜-modules, whose morphisms are chain maps modulo homotopy, and c is its standard symmetry. There is a notion of a category 𝒞 enriched in 𝒱 (𝒱-categories, 𝒱-functors, 𝒱-natural transformations), see Kelly [Kel82], summarized e.g. in [KL01]: for all objects X, Y of 𝒞 𝒞(X, Y) is an object of 𝒱. Denote by 𝒱-𝒞at the category, whose objects are 𝒱-categories and morphisms are 𝒱-functors. Since 𝒱 is symmetric, the category 𝒱-𝒞at is symmetric monoidal with the tensor product 𝒜 ⊗ ℬ of 𝒱-categories 𝒜, ℬ defined via Ob(𝒜 ⊗ ℬ) = Ob 𝒜 × Ob ℬ, 𝒜 ⊗ ℬ(X × Y, U × V) = 𝒜(X, U) ⊗ ℬ(Y, V). Thus, we may consider the 1-category 𝒱-𝒞at-𝒞at of 𝒱-𝒞at-categories and 𝒱-𝒞at-functors. We may interpret it in the same way, as 𝒞at-𝒞at is interpreted as the category of 2-categories. So we say that objects of 𝒱-𝒞at-𝒞at are 𝒱-2-categories 𝔄, as defined below. To restore the definition of a usual 2-category, it suffices to take 𝒱 = (𝒰 -Sets, ×, ∅).

Appendix A.1 Definition (𝒱-2-category). A 1-unital 2-unital 𝒱-2-category 𝔄 consists of

• a class of objects Ob 𝔄;

• for any pair of objects 𝒜, ℬ ∈ Ob 𝔄 a 𝒱-category 𝔄(𝒜, ℬ);

• for any object 𝒜 ∈ Ob 𝔄 a 𝒱-functor \underline{𝟙} → 𝔄(𝒜, 𝒜), 1 ↦ id_{𝒜} ;

• for any triple 𝒜, ℬ, 𝒞 of objects of 𝔄 a 𝒱-functor

𝔄(𝒜, ℬ) ⊗ 𝔄(ℬ, 𝒞) → 𝔄(𝒜, 𝒞), (f, g) ↦ fg,

such that the following 𝒱-functors are equal (modulo the associativity isomorphism in 𝒱): f id = f = id f, f(gh) = (fg)h.

Here the unit 𝒱-category \underline{𝟙} has the set of objects Ob \underline{𝟙} = \{1\}, and \underline{𝟙}(1, 1) = 𝟙 is the unit object of 𝒱. The above definition has an equivalent unpacked form, namely, Definition Appendix A.3. We also need generalizations of the above 𝒱-2-categories – 1-unital non-2-unital 𝒱-2-categories, which contain unit 1-morphisms, but do not contain unit 2-morphisms. An expanded definition of the latter follows. It seems that it does not have a concise version.

Appendix A.2 Definition (Non-2-unital 𝒱-2-category). A 1-unital non-2-unital 𝒱-2-category 𝔄 consists of

• a class of objects Ob 𝔄;

• a class of 1-morphisms 𝔄(𝒜, ℬ) for any pair 𝒜, ℬ of objects of 𝔄;

• an object of 2-morphisms 𝔄(𝒜, ℬ)(f, g) ∈ Ob 𝒱 for any pair of 1-morphisms f, g ∈ 𝔄(𝒜, ℬ);

• a strictly associative composition of 1-morphisms 𝔄(𝒜, ℬ) × 𝔄(ℬ, 𝒞) → 𝔄(𝒜, 𝒞), (f, g) ↦ fg; • a strict two-sided unit 1-morphism id_{𝒜} ∈ 𝔄(𝒜, 𝒜) for each object 𝒜 of 𝔄;

• a right action of a 1-morphism k : ℬ → 𝒞 on 2-morphisms ·k : 𝔄(𝒜, ℬ)(f, g) → 𝔄(𝒜, 𝒞)(fk, gk) ∈ Mor 𝒱;

• a left action of a 1-morphism e : 𝒟 → 𝒜 on 2-morphisms e· : 𝔄(𝒜, ℬ)(f, g) → 𝔄(𝒟, ℬ)(ef, eg) ∈ Mor 𝒱;

• a vertical composition of 2-morphisms m_{2} : 𝔄(𝒜, ℬ)(f, g) ⊗ 𝔄(𝒜, ℬ)(g, h) → 𝔄(𝒜, ℬ)(f, h) ∈ Mor 𝒱,

such that • m_{2} is associative (in monoidal category 𝒱);

• the right and the left actions (a) commute with each other:

(e·)(·k) = (·k)(e·), for 𝒟 ^{e}→ 𝒜 \begin{aligned}f & → \\ g & → \end{aligned} ℬ ^{k}→ 𝒞,

(b) are associative:

(·k)(·k′) = ·(kk′),  for 𝒜 \begin{aligned}f & → \\ g & → \end{aligned} ℬ ^{k}→ 𝒞 ^{k′}→ 𝒟,
(e′·)(e·) = (e′e)·,  for 𝒞 ^{e′}→ 𝒟 ^{e}→ 𝒜 \begin{aligned}f & → \\ g & → \end{aligned} ℬ,

(c) and unital: (· id_{ℬ} ) = id, (id_{𝒜} ·) = id;

• the right and the left actions of 1-morphisms on 2-morphisms preserve the vertical composition:

\begin{aligned}𝔄(𝒜, ℬ)(f, g) ⊗ 𝔄(𝒜, ℬ)(g, h) & \overline{→^{m_{2}}} & 𝔄(𝒜, ℬ)(f, h) \\ & = \\ 𝔄(𝒜, 𝒞)(fk, gk) ⊗ 𝔄(𝒜, 𝒞)(gk, hk) & ↓^{(·k)⊗(·k)} & →^{m_{2}} & 𝔄(𝒜, 𝒞)(fk, hk) & ↓^{·k} \end{aligned} for 𝒜 \begin{aligned}f & → \\ g & → \\ h & → \end{aligned} ℬ ^{k}→ 𝒞,

\begin{aligned}𝔄(𝒜, ℬ)(f, g) ⊗ 𝔄(𝒜, ℬ)(g, h) & \overline{→^{m_{2}}} & 𝔄(𝒜, ℬ)(f, h) \\ & = \\ 𝔄(𝒟, ℬ)(ef, eg) ⊗ 𝔄(𝒟, ℬ)(eg, eh) & ↓^{(e·)⊗(e·)} & →^{m_{2}} & 𝔄(𝒟, ℬ)(ef, eh) & ↓^{e·} \end{aligned} for 𝒟 ^{e}→ 𝒜 \begin{aligned}f & → \\ g & → \\ h & → \end{aligned} ℬ;

• the distributivity law holds:

[figure: 𝔄(ℬ, 𝒞)(h, k) ⊗ 𝔄(𝒜, ℬ)(f, g); 𝔄(𝒜, ℬ)(f, g) ⊗ 𝔄(ℬ, 𝒞)(h, k); 𝔄(𝒜, 𝒞)(fh, fk) ⊗ 𝔄(𝒜, 𝒞)(fk, gk); 𝔄(𝒜, 𝒞)(fh, gh) ⊗ 𝔄(𝒜, 𝒞)(gh, gk); 𝔄(𝒜, 𝒞)(fh, gk); =; (f·)⊗(·k); c; (·h)⊗(g·); m_{2}; m_{2}]

for 𝒜 \begin{aligned}f & → \\ g & → \end{aligned} ℬ →^{h} ; _{_{k}→} 𝒞.

The following definition is equivalent to Definition Appendix A.1.

Appendix A.3 Definition (2-unital 𝒱-2-category). A 1-unital 2-unital 𝒱-2-category 𝔄 consists of the same data as in Definition Appendix A.2 plus a morphism 1_{f} : 𝟙 → 𝔄(𝒜, ℬ)(f, f) for any 1-morphism f, which is a two-sided unit with respect to m_{2}, such that homomorphisms ·k, e· preserve the units 1_{−}.

Appendix B. Contractibility

One can avoid using the following lemma in this article. However, it might be used in order to replace inductive constructions with recurrent formulas.

Appendix B.1 Lemma. Let a chain map u : A → C be homotopically invertible. Then Cone(u) is contractible.

Proof. The homotopy category 𝒦 = K(𝕜 -mod) is triangulated and it has a distinguished triangle A ^{u}→ C ^{p}→ Cone(u) ^{q}→ A[1] ^{u[1]}→ (e.g. [Gri87, Corollaire 5.13]). Since up = 0, qu[1] = 0 (e.g. [Gri87, Proposition 2.8]), and u is invertible in 𝒦, we deduce that p = 0 and q = 0 in 𝒦. Since 𝒦(Cone(u), ) is a homological functor (e.g. [Gri87, Proposition 2.10]), we have 𝒦(Cone(u), Cone(u)) = 0, that is, Cone(u) ≃ 0 in 𝒦.

Let us construct an explicit homotopy between id_{Cone(u)} and 0_{Cone(u)}. There exists a chain map v : C → A homotopically inverse to u. That is, there are maps h′ : A → A, h′′ : C → C of degree −1 such that uv = 1 + h′d^{A} + d^{A}h′ : A → A, vu = 1+h′′d^{C}+d^{C}h′′ : C → C. Using the notation at the end of Section 1 we define a map h : Cone(u) → Cone(u) of degree −1 by the formula (c, a)h = (−ch′′, cv +ah′), (c, a) ∈ C^{k} ⊕ A^{k+1} = Cone^{k}(u). Let us compute the boundary of h:

(c, a)(hd + dh) = (−ch′′, cv + ah′)d + (cd^{C} + au, −ad^{A})h
= (−ch′′d^{C} + cvu + ah′u, −cvd^{A} − ah′d^{A})
+ (−cd^{C}h′′ − auh′′, cd^{C}v + auv − ad^{A}h′)
= (c + ah′u − auh′′, a).

Hence, hd + dh = 1 − f : Cone(u) → Cone(u), where the map f : Cone(u) → Cone(u) is defined via (c, a)f = (auh′′ − ah′u, 0). We conclude that f is a chain map homotopic to the identity map. A sequence of equivalences id_{Cone(u)} ∼ f = id_{Cone(u)} f ∼ f^{2} = 0 proves that Cone(u) is contractible. It gives also an explicit homotopy – the map \underline{h} = h + hf : Cone(u) → Cone(u) of degree −1, which satisfies id_{Cone(u)} = \underline{h}d+d\underline{h}. This homotopy might be used to replace inductive constructions in this paper with recurrent formulas.

Acknowledgements. I am grateful to all the participants of the A_{∞}-category seminar at the Institute of Mathematics, Kyiv, especially to Yu. Bespalov, O. Manzyuk, S. Ovsienko.

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Volodymyr Lyubashenko lub@imath.kiev.ua

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine