Theory and Applications of Categories, Vol. 16, No. 9, 2006, pp. 174–205.

FREE A_{∞}-CATEGORIES

VOLODYMYR LYUBASHENKO AND OLEKSANDR MANZYUK

Abstract. For a differential graded 𝕜-quiver 𝒬 we define the free A_{∞}-category ℱ𝒬 generated by 𝒬. The main result is that the restriction A_{∞}-functor A_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜) is an equivalence, where objects of the last A_{∞}-category are morphisms of differential graded 𝕜-quivers 𝒬 → 𝒜.

A_{∞}-categories defined by Fukaya [Fuk93] and Kontsevich [Kon95] are generalizations of differential graded categories for which the binary composition is associative only up to a homotopy. They also generalize A_{∞}-algebras introduced by Stasheff [Sta63, II]. A_{∞}-func-tors are the corresponding generalizations of usual functors, see e.g. [Fuk93, Kel01]. Homomorphisms of A_{∞}-algebras (e.g. [Kad82]) are particular cases of A_{∞}-functors. A_{∞}-transformations are certain coderivations. Examples of such structures are encountered in studies of mirror symmetry (e.g. [Kon95, Fuk02]) and in homological algebra.

For an A_{∞}-category there is a notion of units up to a homotopy (homotopy identity morphisms) [Lyu03]. 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 A_{∞}-transformations between such functors (Fukaya [Fuk02], Kontsevich and Soibelman [KS02, KS], Lefèvre-Hasegawa [LH03], as well as [Lyu03]). This allows to define a 2-cat-egory, whose objects are unital A_{∞}-categories, 1-morphisms are unital A_{∞}-functors and 2-morphisms are equivalence classes of natural A_{∞}-transformations [Lyu03]. We continue to study this 2-category.

The notations and conventions are explained in the first section. We also describe A_{N}-categories, A_{N}-functors and A_{N}-transformations – truncated at N < ∞ versions of A_{∞}-categories. For instance, A_{1}-categories and A_{1}-functors are differential graded 𝕜-quiv-ers and their morphisms. However, A_{1}-transformations bring new 2-categorical features to the theory. In particular, for any differential graded 𝕜-quiver 𝒬 and any A_{∞}-category 𝒜 there is an A_{∞}-category A_{1}(𝒬, 𝒜), whose objects are morphisms of differential graded 𝕜-quivers 𝒬 → 𝒜, and morphisms are A_{1}-transformations. We recall the terminology related to trees in Section 1.7.

FREE A_{∞}-CATEGORIES  175

In the second section we define the free A_{∞}-category ℱ𝒬 generated by a differential graded 𝕜-quiver 𝒬. We classify functors from a free A_{∞}-category ℱ𝒬 to an arbitrary A_{∞}-category 𝒜 in Proposition 2.3. In particular, the restriction map gives a bijection between the set of strict A_{∞}-functors ℱ𝒬 → 𝒜 and the set of morphisms of differential graded 𝕜-quivers 𝒬 → (𝒜, m_{1}) (Corollary 2.4). We classify chain maps into complexes of transformations whose source is a free A_{∞}-category in Proposition 2.8. Description of homotopies between such chain maps is given in Corollary 2.10. Assuming in addition that 𝒜 is unital, we obtain our main result: the restriction A_{∞}-functor restr : A_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜) is an equivalence (Theorem 2.12).

Received by the editors 2003-10-30 and, in revised form, :2006-03-14.

Transmitted by J. Stasheff. Published on 2006-04-09.

2000 Mathematics Subject Classification: 18D05, 18D20, 18G55, 55U15.

Key words and phrases: A_{∞}-categories, A_{∞}-functors, A_{∞}-transformations, 2-category, free A_{∞}-cat-egory.

\begin{matrix}c \\ ◯ \end{matrix} Volodymyr Lyubashenko and Oleksandr Manzyuk, 2006. Permission to copy for private use granted.

In the third section we interpret A_{∞}(ℱ𝒬, ) and A_{1}(𝒬, ) as strict A_{∞}^{u}-2-functors A_{∞}^{u} → A_{∞}^{u}. Moreover, we interpret restr : A_{∞}(ℱ𝒬, ) → A_{1}(𝒬, ) as an A_{∞}^{u}-2-equivalence. In this sense the A_{∞}-category ℱ𝒬 represents the A_{∞}^{u}-2-functor A_{1}(𝒬, ). This is the 2-categorical meaning of freeness of ℱ𝒬.

1. Conventions and preliminaries

We keep the notations and conventions of [Lyu03, LO02], sometimes without explicit mentioning. Some of the conventions are recalled here.

We assume as in [Lyu03, LO02] that most quivers, A_{∞}-categories, etc. are small with respect to some universe 𝒰 .

The ground ring 𝕜 ∈ 𝒰 is a unital associative commutative ring. A 𝕜-module means a 𝒰 -small 𝕜-module.

We use the right operators: the composition of two maps (or morphisms) f : X → Y and g : Y → Z is denoted 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.

ℤ-graded 𝕜-modules are functions X : ℤ ∋ d ↦ X^{d} ∈ 𝕜 -mod. A simple computation shows that the product X = ∏_{ι∈I} X_{ι} in the category of ℤ-graded 𝕜-modules of a family (X_{ι})_{ι∈I} of objects X_{ι} : d ↦ X_{ι}^{d} is given by X : ℤ ∋ d ↦ X^{d} = ∏_{ι∈I} X_{ι}^{d}. Everywhere in this article the product of graded 𝕜-modules means the above product.

If P is a ℤ-graded 𝕜-module, then sP = P [1] denotes the same 𝕜-module with the grading (sP)^{d} = P^{d+1}. The “identity” map P → sP of degree −1 is also denoted s. The map s commutes with the components of the differential in an A_{∞}-category (A_{∞}-algebra) in the following sense: s^{⊗n}b_{n} = m_{n}s.

Let C = C(𝕜 -mod) denote the differential graded category of complexes of 𝕜-modules. Actually, it is a symmetric closed monoidal category.

The cone of a chain of a chain map α : P → Q of complexes of 𝕜-modules is the graded 𝕜-module Cone(α) = Q⊕P [1] with the differential (q, ps)d = (qd^{Q} +pα, psd^{P [1]}) = (qd^{Q} + pα, −pd^{P} s).

1.1. A_{N}-categories. For a positive integer N we define some A_{N}-notions similarly to the case N = ∞. We may say that all data, equations and constructions for A_{N}-case are the same as in A_{∞}-case (e.g. [Lyu03]), however, taken only up to level N.

A differential graded 𝕜-quiver 𝒬 is the following data: a 𝒰 -small set of objects Ob 𝒬; a chain complex of 𝕜-modules 𝒬(X, Y) for each pair of objects X, Y. A morphism of differential graded 𝕜-quivers f : 𝒬 → 𝒜 is given by a map f : Ob 𝒬 → Ob 𝒜, X ↦ Xf and by a chain map 𝒬(X, Y) → 𝒜(Xf, Y f) for each pair of objects X, Y of 𝒬. The category of differential graded 𝕜-quivers is denoted A_{1}.

The category of 𝒰 -small graded 𝕜-linear quivers, whose set of objects is S, admits a monoidal structure with the tensor product 𝒜 × ℬ ↦ 𝒜 ⊗ ℬ, (𝒜 ⊗ ℬ)(X, Y) = ⊕_{Z∈S}𝒜(X, Z) ⊗_{𝕜} ℬ(Z, Y). In particular, we have tensor powers T^{n}𝒜 = 𝒜^{⊗n} of a given graded 𝕜-quiver 𝒜, such that Ob T^{n}𝒜 = Ob 𝒜. Explicitly,

T^{n}𝒜(X, Y) = ⨁_{X=X_{0},X_{1},...,X_{n}=Y ∈Ob 𝒜} 𝒜(X_{0}, X_{1}) ⊗_{𝕜} 𝒜(X_{1}, X_{2}) ⊗_{𝕜} ⋯ ⊗_{𝕜} 𝒜(X_{n−1}, X_{n}).

In particular, T^{0}𝒜(X, Y) = 𝕜 if X = Y and vanishes otherwise. The graded 𝕜-quiver T^{⩽N}𝒜 = ⊕_{n⩾0}^{N}T^{n}𝒜 is called the restricted tensor coalgebra of 𝒜. It is equipped with the cut comultiplication

Δ : T^{⩽N}𝒜(X, Y) → ⨁_{Z∈Ob 𝒜} T^{⩽N}𝒜(X, Z) ⨂_{𝕜}T^{⩽N}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^{⩽N}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). We write T 𝒜 instead of T^{⩽∞}𝒜. If g : T 𝒜 → T ℬ is a map of 𝕜-quivers, then g_{ac} denotes its matrix coefficient T^{a}𝒜^{⊂}^{in_{a}}→ T 𝒜 ^{g}→ T ℬ ^{pr_{c}}▷ T^{c}ℬ. The matrix coefficient g_{a1} is abbreviated to g_{a}.

1.2. Definition. An A_{N}-category 𝒜 consists of the following data: a graded 𝕜-quiver 𝒜; a system of 𝕜-linear maps 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}), 1 ⩽ n ⩽ N,

such that for all 1 ⩽ k ⩽ N

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

The system b_{n} is interpreted as a (1,1)-coderivation b : T^{⩽N}s𝒜 → T^{⩽N}s𝒜 of degree 1 determined by

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},  k, l ⩽ N,

which is a differential.

1.3. Definition. A pointed cocategory homomorphism consists of the following data: A_{N}-categories 𝒜 and ℬ, a map f : Ob 𝒜 → Ob ℬ and a system of 𝕜-linear maps 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), 1 ⩽ n ⩽ N.

The above data are equivalent to a cocategory homomorphism f : T^{⩽N}s𝒜 → T^{⩽N}sℬ of degree 0 such that

f_{01} = (f|_{T^{0}s𝒜} ) pr_{1} = 0 : T^{0}s𝒜 → T^{1}sℬ,  (2)

(this condition was implicitly assumed in [Lyu03, Definition 2.4]). The components of f are

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}},  (3)

where k, l ⩽ N. Indeed, the claim follows from the following diagram, commutative for all l ⩾ 0:

T^{⩽N}s𝒜 \overline{^{f} →} T^{⩽N}sℬ \overline{^{pr_{l}} →} T^{l}sℬ
=  =
(T^{⩽N}s𝒜)^{⊗l} ↓^{Δ^{(l)}}  →^{f^{⊗l}} (T^{⩽N}sℬ)^{⊗l} ↓^{Δ^{(l)}}  \overline{→^{pr_{1}^{⊗l}}} (sℬ)^{⊗l} ‖

where Δ^{(0)} = ε, Δ^{(1)} = id, Δ^{(2)} = Δ and Δ^{(l)} means the cut comultiplication, iterated l − 1 times. Notice that condition (2) can be written as f_{0} = 0.

1.4. Definition. An A_{N}-functor f : 𝒜 → ℬ is a pointed cocategory homomorphism, which commutes with the differential b, that is, for all 1 ⩽ k ⩽ N

∑_{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.

We are interested mostly in the case N = 1. Clearly, A_{1}-categories are differential graded quivers and A_{1}-functors are their morphisms. In the case of one object these reduce to chain complexes and chain maps. The following notion seems interesting even in this case.

1.5. Definition. An A_{N}-transformation r : f → g : 𝒜 → ℬ of degree d consists of the following data: A_{N}-categories 𝒜 and ℬ; pointed cocategory homomorphisms f, g : T^{⩽N}s𝒜 → T^{⩽N}sℬ (or A_{N}-functors f, g : 𝒜 → ℬ); a system of 𝕜-linear maps 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), 0 ⩽ n ⩽ N.

To give a system r_{n} is equivalent to specifying an (f, g)-coderivation r : T^{⩽N}s𝒜 → T^{⩽N+1}sℬ of degree d

r_{kl} = (r|_{T^{k}s𝒜} ) pr_{l} : T^{k}s𝒜 → T^{l}sℬ,  k ⩽ N, l ⩽ N + 1
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}},  (4)

that is, a 𝕜-quiver morphism r, satisfying rΔ = Δ(f ⊗ r + r ⊗ g). This follows from the commutative diagram

T^{⩽N}s𝒜  \overline{^{r}  →} T^{⩽N+1}sℬ \overline{^{pr_{l}} →} T^{l}sℬ
=  =
(T^{⩽N}s𝒜)^{⊗l} ↓^{Δ^{(l)}}  \overline{\underrightarrow{∑_{q+1+t=l} f^{⊗q}⊗r⊗g^{⊗t}}} (T^{⩽N+1}sℬ)^{⊗l} ↓^{Δ^{(l)}}  \overline{→^{pr_{1}^{⊗l}}} (sℬ)^{⊗l} ‖

Let 𝒜, ℬ be A_{N}-categories, and let f^{0}, f^{1}, …, f^{n} : T^{⩽N}s𝒜 → T^{⩽N}sℬ be pointed cocategory homomorphisms. Consider coderivations r_{1}, …, r_{n} as in

f^{0} ^{r^{1}}→ f^{1} ^{r^{2}}→ … f^{n−1} ^{r^{n}}→ f^{n} : T^{⩽N}s𝒜 → T^{⩽N}sℬ.

We construct the following system of 𝕜-linear maps θ_{kl} : T^{k}s𝒜 → T^{l}sℬ, k ⩽ N, l ⩽ N +n of degree deg r^{1} + ⋯ + deg r^{n} from these data:

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}, (5)

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.

Equivalently, we write

θ_{kl} = ∑_{\begin{matrix}m_{0}+m_{1}+···+m_{n}+n=l \\ p_{0}+j_{1}+p_{1}+···+j_{n}+p_{n}=k \end{matrix}} f_{p_{0}m_{0}}^{0} ⊗ r_{j_{1}}^{1} ⊗ f_{p_{1}m_{1}}^{1} ⊗ ⋯ ⊗ r_{j_{n}}^{n} ⊗ f_{p_{n}m_{n}}^{n}.

The component θ_{kl} vanishes unless n ⩽ l ⩽ k + n. If n = 0, then θ_{kl} is expansion (3) of f^{0}. If n = 1, then θ_{kl} is expansion (4) of r^{1}.

Given an A_{K}-category 𝒜 and an A_{K+N}-category ℬ, 1 ⩽ K, N ⩽ ∞, we construct an A_{N}-category A_{K}(𝒜, ℬ) out of these. The objects of A_{K}(𝒜, ℬ) are A_{K}-functors f : 𝒜 → ℬ. Given two such functors f, g : 𝒜 → ℬ we define the graded 𝕜-module A_{K}(𝒜, ℬ)(f, g) as the space of all A_{K}-transformations r : f → g, namely,

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

The system of differentials B_{n}, n ⩽ N, is defined as follows:

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

[(r)B_{1}]_{k} = ∑_{i_{1}+···+i_{q}+n+j_{1}+···+j_{t}=k}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{q}} ⊗ r_{n} ⊗ g_{j_{1}} ⊗ ⋯ ⊗ g_{j_{t}})b_{q+1+t}

−(−)^{r} ∑_{α+n+β=k}(1^{⊗α} ⊗ b_{n} ⊗ 1^{⊗β})r_{α+1+β}, k ⩽ K,

B_{n} : A_{K}(𝒜, ℬ)(f^{0}, f^{1}) ⊗ ⋯ ⊗ A_{K}(𝒜, ℬ)(f^{n−1}, f^{n}) → A_{K}(𝒜, ℬ)(f^{0}, f^{n}),

r^{1} ⊗ ⋯ ⊗ r^{n} ↦ (r^{1} ⊗ ⋯ ⊗ r^{n})B_{n}, for 1 < n ⩽ N,

where the last A_{K}-transformation is defined by its components:

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

The category of graded 𝕜-linear quivers admits a symmetric monoidal structure with the tensor product 𝒜 × ℬ ↦ 𝒜 ⊠ ℬ, where Ob 𝒜 ⊠ ℬ = Ob 𝒜 × Ob ℬ and (𝒜 ⊠ ℬ)((X, U), (Y, V)) = 𝒜(X, Y) ⊗_{𝕜} ℬ(U, V). The same tensor product was denoted ⊗ in [Lyu03], but we will keep notation 𝒜 ⊗ ℬ only for tensor product from Section 1.1, defined when Ob 𝒜 = Ob ℬ. The two tensor products obey

Distributivity law. Let 𝒜, ℬ, 𝒞, 𝒟 be graded 𝕜-linear quivers, such that Ob 𝒜 = Ob ℬ and Ob 𝒞 = Ob 𝒟. Then the middle four interchange map 1⊗c⊗1 is an isomorphism of quivers

(𝒜 ⊗ ℬ) ⊠ (𝒞 ⊗ 𝒟) ^{∼}→ (𝒜 ⊠ 𝒞) ⊗ (ℬ ⊠ 𝒟),  (6)

identity on objects.

Indeed, the both quivers in (6) have the same set of objects R ×S, where R = Ob 𝒜 = Ob ℬ and S = Ob 𝒞 = Ob 𝒟. Let X, Z ∈ R and U, W ∈ S. The sets of morphisms from (X, U) to (Z, W) are isomorphic via

((𝒜 ⊗ ℬ) ⊠ (𝒞 ⊗ 𝒟))((X, U), (Z, W)) =
(⊕_{Y ∈R}𝒜(X, Y) ⊗_{𝕜} ℬ(Y, Z)) ⊗_{𝕜} (⊕_{V ∈S}𝒞(U, V) ⊗_{𝕜} 𝒟(V, W))

⊕_{(Y,V)∈R×S}𝒜(X, Y) ⊗_{𝕜} ℬ(Y, Z) ⊗_{𝕜} 𝒞(U, V) ⊗_{𝕜} 𝒟(V, W) ↓^{≀}

⊕_{(Y,V)∈R×S}𝒜(X, Y) ⊗_{𝕜} 𝒞(U, V) ⊗_{𝕜} ℬ(Y, Z) ⊗_{𝕜} 𝒟(V, W) ↓^{1⊗c⊗1}
= ((𝒜 ⊠ 𝒞) ⊗ (ℬ ⊠ 𝒟))((X, U), (Z, W)).

The notion of a pointed cocategory homomorphism extends to the case of several arguments, that is, to degree 0 cocategory homomorphisms ψ : T^{⩽L^{1}}s𝒞^{1}⊠⋯⊠T^{⩽L^{q}}s𝒞^{q} → T^{⩽N}sℬ, where N ⩾ L^{1}+⋯+L^{q}. We always assume that ψ_{00...0} : T^{0}s𝒞^{1}⊠⋯⊠T^{0}s𝒞^{q} → sℬ vanishes. We call ψ an A-functor if it commutes with the differential, that is,

(b ⊠ 1 ⊠ ⋯ ⊠ 1 + 1 ⊠ b ⊠ ⋯ ⊠ 1 + ⋯ + 1 ⊠ 1 ⊠ ⋯ ⊠ b)ψ = ψb.

For example, the map α : T^{⩽K}s𝒜 ⊠ T^{⩽N}sA_{K}(𝒜, ℬ) → T^{⩽K+N}sℬ, a ⊠ r^{1} ⊗ ⋯ ⊗ r^{n} ↦ a.[(r^{1} ⊗ ⋯ ⊗ r^{n})θ], is an A-functor.

1.6. Proposition. [cf. Proposition 5.5 of [Lyu03]] Let 𝒜 be an A_{K}-category, let 𝒞^{t} be an A_{L^{t}}-category for 1 ⩽ t ⩽ q, and let ℬ be an A_{N}-category, where N ⩾ K + L^{1} + ⋯ + L^{q}. For any A-functor ϕ : T^{⩽K}s𝒜 ⊠ T^{⩽L^{1}}s𝒞^{1} ⊠ ⋯ ⊠ T^{⩽L^{q}}s𝒞^{q} → T^{⩽N}sℬ there is a unique A-functor ψ : T^{⩽L^{1}}s𝒞^{1} ⊠ ⋯ ⊠ T^{⩽L^{q}}s𝒞^{q} → T^{⩽N−K}sA_{K}(𝒜, ℬ), such that

ϕ = (T^{⩽K}s𝒜⊠T^{⩽L^{1}}s𝒞^{1} ⊠⋯⊠T^{⩽L^{q}}s𝒞^{q} ^{1⊠ψ} → T^{⩽K}s𝒜⊠T^{⩽N−K}sA_{K}(𝒜, ℬ) ^{α}→ T^{⩽N}sℬ).

Let 𝒜 be an A_{N}-category, let ℬ be an A_{N+K}-category, and let 𝒞 be an A_{N+K+L}-cat-egory. The above proposition implies the existence of an A-functor (cf. [Lyu03, Proposition 4.1])

M : T^{⩽K}sA_{N}(𝒜, ℬ) ⊠ T^{⩽L}sA_{N+K}(ℬ, 𝒞) → T^{⩽K+L}sA_{N}(𝒜, 𝒞),

in particular, (1 ⊠ B + B ⊠ 1)M = MB. It has the components

M_{nm} = M|_{T^{n}⊠T^{m}} pr_{1} : T^{n}sA_{N}(𝒜, ℬ) ⊠ T^{m}sA_{N+K}(ℬ, 𝒞) → sA_{N}(𝒜, 𝒞),

n ⩽ K, m ⩽ L. We have M_{00} = 0 and M_{nm} = 0 for m > 1. If m = 0 and n is positive, M_{n0} is given by the formula:

M_{n0} : sA_{N}(𝒜, ℬ)(f^{0}, f^{1}) ⊗ ⋯ ⊗ sA_{N}(𝒜, ℬ)(f^{n−1}, f^{n}) ⊠ 𝕜_{g^{0}} → sA_{N}(𝒜, 𝒞)(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}]_{k} = ∑_{l=n}^{n+k}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}g_{l}^{0}, k ⩽ N,

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

M_{n1} : sA_{N}(𝒜, ℬ)(f^{0}, f^{1}) ⊗ ⋯ ⊗ sA_{N}(𝒜, ℬ)(f^{n−1}, f^{n}) ⊠ sA_{N+K}(ℬ, 𝒞)(g^{0}, g^{1}) → sA_{N}(𝒜, 𝒞)(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}]_{k} = ∑_{l=n}^{n+k}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}t_{l}^{1}, k ⩽ N.

Note that equations

[(r^{1} ⊗ ⋯ ⊗ r^{n})B_{n}]_{k} = [(r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ b)M_{n1}]_{k} − (−)^{r^{1}+···+r^{n}}[(b ⊠ r^{1} ⊗ ⋯ ⊗ r^{n})M_{1n}]_{k} imply that

(r^{1} ⊗ ⋯ ⊗ r^{n})B_{n} = (r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ b)M_{n1} − (−)^{r^{1}+···+r^{n}}(b ⊠ r^{1} ⊗ ⋯ ⊗ r^{n})M_{1n},
B = (1 ⊠ b)M − (b ⊠ 1)M : id → id : A_{N}(𝒜, ℬ) → A_{N}(𝒜, ℬ).

Proposition 1.6 implies the existence of a unique A_{L}-functor

A_{N}(𝒜, ) : A_{N+K}(ℬ, 𝒞) → A_{K}(A_{N}(𝒜, ℬ), A_{N}(𝒜, 𝒞)),

such that

M = [T^{⩽K}sA_{N}(𝒜, ℬ) ⊠ T^{⩽L}sA_{N+K}(ℬ, 𝒞) ^{1⊠A_{N} (𝒜, )} →

T^{⩽K}sA_{N}(𝒜, ℬ) ⊠ T^{⩽L}sA_{K}(A_{N}(𝒜, ℬ), A_{N}(𝒜, 𝒞)) ^{α}→ T^{⩽K+L}sA_{N}(𝒜, 𝒞)].

The A_{L}-functor A_{N}(𝒜, ) is strict, cf. [Lyu03, Proposition 6.2].

Let 𝒜 be an A_{N}-category, and let ℬ be a unital A_{∞}-category with a unit transformation i^{ℬ} . Then A_{N}(𝒜, ℬ) is a unital A_{∞}-category with the unit transformation (1 ⊠ i^{ℬ} )M (cf. [Lyu03, Proposition 7.7]). The unit element for an object f ∈ Ob A_{N}(𝒜, ℬ) is fi_{0}^{A_{N} (𝒜,ℬ)} : 𝕜 → (sA_{N})^{−1}(𝒜, ℬ), 1 ↦ fi^{ℬ} .

When 𝒜 is an A_{K}-category and N < K, we may forget part of its structure and view 𝒜 as an A_{N}-category. If furthermore, ℬ is an A_{K+L}-category, we have the restriction strict A_{L}-functor restr_{K,N} : A_{K}(𝒜, ℬ) → A_{N}(𝒜, ℬ). To prove the results mentioned above, we notice that they are restrictions of their A_{∞}-analogs to finite N. Since the proofs of A_{∞}-results are obtained in [Lyu03] by induction, an inspection shows that the proofs of the above A_{N}-statements are obtained as a byproduct.

1.7. Trees. Since the notions related to trees might be interpreted with some variations, we give precise definitions and fix notation. A tree is a non-empty connected graph without cycles. A vertex which belongs to only one edge is called external, other vertices are internal. A plane tree is a tree equipped for each internal vertex v with a cyclic ordering of the set E_{v} of edges, adjacent to v. Plane trees can be drawn on an oriented plane in a unique way (up to an ambient isotopy) so that the cyclic ordering of each E_{v} agrees with the orientation of the plane. An external vertex distinct from the root is called input vertex.

A rooted tree is a tree with a distinguished external vertex, called root. The set of vertices V (t) of a rooted tree t has a canonical ordering: x ≼ y iff the minimal path connecting the root with y contains x. A linearly ordered tree is a rooted tree t equipped with a linear order ⩽ of the set of internal vertices IV (t), such that for all internal vertices x, y the relation x ≼ y implies x ⩽ y. For each vertex v ∈ V (t) − {root} of a rooted tree, the set E_{v} has a distinguished element e_{v} – the beginning of a minimal path from v to the root. Therefore, for each vertex v ∈ V (t) − {root} of a rooted plane tree, the set E_{v} admits a unique linear order <, for which e_{v} is minimal and the induced cyclic order is the given one. An internal vertex v has degree d, if Card(E_{v}) = d + 1.

For any y ∈ V (t) let P_{y} = {x ∈ V (t) | x ≼ y}. With each plane rooted tree t is associated a linearly ordered tree t_{<} = (t, ⩽) as follows. If x, y ∈ IV (t) are such that x ⋠ y and y ⋠ x, then P_{x} ∩ P_{y} = P_{z} for a unique z ∈ IV (t), distinct from x and y. Let a ∈ E_{z} − {e_{z}} (resp. b ∈ E_{z} − {e_{z}}) be the beginning of the minimal path connecting z and x (resp. y). If a < b, we set x < y. Graphically we <-order the internal vertices by height. Thus, an internal vertex x on the left is depicted lower than a ≼-incomparable internal vertex y on the right:

[figure: x; a; z; y; b; e_{z}; root] , a < b ⟹ x < y.

A forest is a sequence of plane rooted trees. Concatenation of forests is denoted ⊔. The vertical composition F_{1} · F_{2} of forests F_{1}, F_{2} is well-defined if the sum of lengths of sequences F_{1} and F_{2} equals the number of external vertices of F_{2}. These operations allow to construct any tree from elementary ones

1 = ,  and 𝔱_{k} = ^{^{···}}  (k input vertices).

Namely, any linearly ordered tree (t, ⩽) has a unique presentation of the form

(t, ⩽) = (1^{⊔α_{1}} ⊔ 𝔱_{k_{1}} ⊔ 1^{⊔β_{1}}) · (1^{⊔α_{2}} ⊔ 𝔱_{k_{2}} ⊔ 1^{⊔β_{2}}) · . . . · 𝔱_{k_{N}} ,  (7)

where N = |t| =^{def} Card(IV (t)) is the number of internal vertices. Here

1^{⊔α} ⊔ 𝔱_{k} ⊔ 1^{⊔β} = ^{^{...}  ^{^{···}}  ^{...}}^{^{α} ^{k} ^{β}} .

In (7) the highest vertex is indexed by 1, the lowest – by N.

2. Properties of free A_{∞}-categories

2.1. Construction of a free A_{∞}-category. The category strictA_{∞} has A_{∞}-cate-gories as objects and strict A_{∞}-functors as morphisms. There is a functor 𝒰 : strictA_{∞} → A_{1}, 𝒜 ↦ (𝒜, m_{1}) which sends an A_{∞}-category to the underlying differential graded 𝕜-quiver, forgetting all higher multiplications. Following Kontsevich and Soibelman [KS02] we are going to prove that 𝒰 has a left adjoint functor ℱ : A_{1} → strictA_{∞}, 𝒬 ↦ ℱ𝒬. The A_{∞}-category ℱ𝒬 is called free. Below we describe its structure for an arbitrary differential graded 𝕜-quiver 𝒬. We shall work with its shift (s𝒬, d).

Let us define an A_{∞}-category ℱ𝒬 via the following data. The class of objects Ob ℱ𝒬 is Ob 𝒬. The ℤ-graded 𝕜-modules of morphisms between X, Y ∈ Ob 𝒬 are

sℱ𝒬(X, Y) = ⊕_{n⩾1} ⊕_{t∈𝒯_{⩾2}^{n}} sℱ_{t}𝒬(X, Y),
sℱ_{t}𝒬(X, Y) = ⊕_{X_{0},...,X_{n}∈Ob 𝒬}^{X_{0}=X, X_{n}=Y}s𝒬(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒬(X_{n−1}, X_{n})[−|t|],

where 𝒯_{⩾2}^{n} is the class of plane rooted trees with n + 1 external vertices, such that Card(E_{v}) ⩾ 3 for all v ∈ IV (t). We use the following convention: if M, N are (differential) graded 𝕜-modules, then^{1}

(M ⊗ N)[k] = M ⊗ (N[k]),
(M ⊗ N ^{s^{k}}→ (M ⊗ N)[k]) = (M ⊗ N ^{1⊗s^{k}} → M ⊗ (N[k])).

The quiver ℱ𝒬 is equipped with the following operations. For k > 1 the operation b_{k} is a direct sum of maps

b_{k} = s^{|t_{1}|}⊗⋯⊗s^{|t_{k−1}|}⊗s^{|t_{k}|−|t|} : sℱ_{t_{1}}𝒬(Y_{0}, Y_{1})⊗⋯⊗sℱ_{t_{k}}𝒬(Y_{k−1}, Y_{k}) → sℱ_{t}𝒬(Y_{0}, Y_{k}), (8)

where t = (t_{1} ⊔ ⋯ ⊔ t_{k}) · 𝔱_{k}. In particular, |t| = |t_{1}| + ⋯ + |t_{k}| + 1. The operation b_{1} restricted to sℱ_{t}𝒬 is

b_{1} = d ⊕ (−1)^{β(t′)}s^{−1} : sℱ_{t}𝒬(X, Y) → sℱ_{t}𝒬(X, Y) ⊕ ⨁_{t′=t+edge} sℱ_{t′}𝒬(X, Y),  (9)

where the sum extends over all trees t′ ∈ 𝒯_{⩾2}^{n} with a distinguished edge e, such that contracting e we get t from t′. The sign is determined by

β(t′) = β(t′, e) = 1 + h(highest vertex of e),

where an isomorphism of ordered sets

h : IV (t′_{<}) ^{∼}→ [1, |t′|] ∩ ℤ

is simply the height of a vertex in the linearly ordered tree t′_{<}, canonically associated with t′. In (9) d means d ⊗ 1 ⊗ ⋯ ⊗ 1 + ⋯ + 1 ⊗ ⋯ ⊗ d ⊗ 1 + 1 ⊗ ⋯ ⊗ 1 ⊗ d, where the last d is d_{s𝒬[−|t|]} = (−)^{|t|}s^{|t|} · d_{s𝒬} · s^{−|t|}, as usual. According to our conventions, s^{−1} in (9) means 1^{⊗n−1} ⊗ s^{−1}.

2.2. Proposition. ℱ𝒬 is an A_{∞}-category.

1Another gauge choice (M ⊗ N)[1] = M[1] ⊗ N, s = s ⊗ 1 seems less convenient.

Proof. First we prove that b_{1}^{2} = 0 on sℱ_{t}𝒬. Indeed,

b_{1}^{2} = d^{2} ⊕ (−)^{β(t′,e)}(s^{−1}d + ds^{−1}) ⊕ [(−1)^{β(t′_{1},e_{1})+β(t′′,e_{2})} + (−1)^{β(t′_{2},e_{2})+β(t′′,e_{1})}]s^{−2} :

sℱ_{t}𝒬 → sℱ_{t}𝒬 ⊕ ⨁_{t′=t+e} sℱ_{t′}𝒬 ⊕ ⨁_{t′′=t+e_{1}+e_{2}} sℱ_{t′′}𝒬,

where t′′ contains two distinguished edges e_{1}, e_{2}, contraction along which gives t; t′_{2} is t′′ contracted along e_{1}, and t′_{1} is t′′ contracted along e_{2}. We may assume that highest vertex of e_{1} is lower than highest vertex of e_{2} in t′′_{<}. Then β(t′_{1}, e_{1}) = β(t′′, e_{1}) and β(t′′, e_{2}) = β(t′_{2}, e_{2}) + 1, hence,

(−1)^{β(t′_{1},e_{1})+β(t′′,e_{2})} + (−1)^{β(t′_{2},e_{2})+β(t′′,e_{1})} = 0.

Obviously, d^{2} = 0 and s^{−1}d + ds^{−1} = 0, hence, b_{1}^{2} = 0.

Let us prove for each n > 1 that

b_{n}b_{1} + ∑_{p=1}^{n}(1^{⊗p−1} ⊗ b_{1} ⊗ 1^{⊗n−p})b_{n} + ∑_{\begin{matrix}α+k+β=n \\ α+β>0 \end{matrix}}^{k>1}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β})b_{α+1+β} = 0 :

sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 → sℱ_{t}𝒬 ⊕ ⨁_{p,t′_{p}} sℱ_{t′}𝒬 ⊕ ⨁_{\begin{matrix}α+k+β=n \\ α+β>0 \end{matrix}}^{k>1, t′′} sℱ_{t′′}𝒬,

where

t = (t_{1} ⊔ ⋯ ⊔ t_{n}) · 𝔱_{n} = ^{···}^{t_{1}  t_{2}  t_{n−1}  t_{n}} ,  (10)
t′ = (t_{1} ⊔ ⋯ ⊔ t′_{p} ⊔ ⋯ ⊔ t_{n}) · 𝔱_{n} = ^{t_{1}  t′_{p}  t_{n}} ,

t′′ = (t_{1} ⊔ ⋯ ⊔ t_{n}) · (1^{⊔α} ⊔ 𝔱_{k} ⊔ 1^{⊔β}) · 𝔱_{α+1+β}

= [figure: t_{1}  t_{α}  t_{α+1}  t_{α+k}  t_{α+k+1}  t_{n}] ,

and contraction of t′_{p} along distinguished edge e_{p} gives t_{p}. According to the three types of summands in the target, the required equation follows from anticommutativity of the following three diagrams:

sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 \overline{→^{\begin{matrix}s^{|t_{1}|}⊗···⊗s^{|t_{n−1}|}⊗s^{|tn|−|t|} \\ b_{n} \end{matrix}}} sℱ_{t}𝒬
−
sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 ↓^{1^{⊗n−1}⊗d+···+d⊗1^{⊗n−1}}  \overline{→^{\begin{matrix}s^{|t_{1}|}⊗···⊗s^{|t_{n−1}|}⊗s^{|tn|−|t|} \\ b_{n} \end{matrix}}} sℱ_{t}𝒬 ↓^{d}

that is,

(1^{⊗n−1} ⊗ d + ⋯ + d ⊗ 1^{⊗n−1})(s^{|t_{1}|} ⊗ ⋯ ⊗ s^{|t_{n−1}|} ⊗ s^{|t_{n}|−|t|})

+ (s^{|t_{1}|} ⊗ ⋯ ⊗ s^{|t_{n−1}|} ⊗ s^{|t_{n}|−|t|})(1^{⊗n−1} ⊗ d + ⋯ + d ⊗ 1^{⊗n−1}) = 0;
sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{p}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬  \overline{^{s^{|t_{1}|}⊗···⊗s^{|t_{n−1}|}⊗s^{|tn|−|t|}}  →} sℱ_{t}𝒬
−
sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t′_{p}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 ↓^{(−)^{β(t′_{p})}1^{⊗p−1}⊗s^{−1}⊗1^{⊗n−p}} \overline{_{_{s^{|t_{1}|}⊗···⊗s^{|t_{p−1}|}⊗s^{|tp|+1}⊗s^{|t_{p+1}|}⊗···⊗s^{|t_{n−1}|}⊗s^{|tn|−|t|−1}} →}} sℱ_{t′}𝒬 ↓^{(−)^{1+|t_{1}|+···+|t_{p−1}|+β(t′_{p})}s^{−1}}

that is,

(−1)^{β(t′_{p})}(1^{⊗p−1} ⊗s^{−1} ⊗1^{⊗n−p})(s^{|t_{1}|} ⊗⋯⊗s^{|t_{p−1}|} ⊗s^{|t_{p}|+1} ⊗s^{|t_{p+1}|} ⊗⋯⊗s^{|t_{n−1}|} ⊗s^{|t_{n}|−|t|−1})

+ (s^{|t_{1}|} ⊗ ⋯ ⊗ s^{|t_{n−1}|} ⊗ s^{|t_{n}|−|t|})(−1)^{1+|t_{1}|+···+|t_{p−1}|+β(t′_{p})}(1^{⊗n−1} ⊗ s^{−1}) = 0,

in the particular case p = n it holds as well;

sℱ_{t1}𝒬⊗···⊗sℱ_{tn}𝒬  \overline{^{s^{|t1|}⊗···⊗s^{|tn−1|}⊗s^{|tn|−|t|}}  →} sℱ_{t}𝒬
−
sℱ_{t1}𝒬⊗···⊗sℱ_{tα}𝒬⊗sℱt̂𝒬⊗sℱ_{tα+k+1}𝒬⊗···⊗sℱ_{tn}𝒬 ↓^{1^{⊗α}⊗s^{|tα+1|}⊗... ⊗s^{|tα+k−1|}⊗s^{−|tα+1|−···−|tα+k−1|−1}⊗1^{⊗β}} \overline{_{_{s^{|t1|}⊗···⊗s^{|tα|}⊗s^{|t̂|}⊗s^{|tα+k+1|}⊗···⊗s^{|tn−1|}⊗s^{|tn|−|t|−1}} →}} sℱ_{t′′}𝒬 ↓^{(−)^{1+|t1|+···+|tα|}s^{−1}}

where t̂= (t_{α+1} ⊔ ⋯ ⊔ t_{α+k}) · 𝔱_{k}, that is,

(1^{⊗α} ⊗ s^{|t_{α+1}|} ⊗ ⋯ ⊗ s^{|t_{α+k−1}|} ⊗ s^{−|t_{α+1}|−···−|t_{α+k−1}|−1} ⊗ 1^{⊗β})·

(s^{|t_{1}|} ⊗ ⋯ ⊗ s^{|t_{α}|} ⊗ 1^{⊗k−1} ⊗ s^{|t_{α+1}|+···+|t_{α+k}|+1} ⊗ s^{|t_{α+k+1}|} ⊗ ⋯ ⊗ s^{|t_{n−1}|} ⊗ s^{|t_{n}|−|t|−1})

+ (s^{|t_{1}|} ⊗ ⋯ ⊗ s^{|t_{n−1}|} ⊗ s^{|t_{n}|−|t|})(−1)^{1+|t_{1}|+···+|t_{α}|}(1^{⊗n−1} ⊗ s^{−1}) = 0,

in the particular case β = 0 it holds as well.

Therefore, ℱ𝒬 is an A_{∞}-category.

Let us establish a property of free A_{∞}-categories, which explains why they are called free.

2.3. Proposition. [A_{∞}-functors from a free A_{∞}-category] Let 𝒬 be a differential graded quiver, and let 𝒜 be an A_{∞}-category. Let f_{1} : s𝒬 → (s𝒜, b_{1}) be a chain morphism of differential graded quivers with the underlying mapping of objects Ob f : Ob 𝒬 → Ob 𝒜. Suppose given 𝕜-quiver morphisms f_{k} : T^{k}sℱ𝒬 → s𝒜 of degree 0 with the same underlying map Ob f for all k > 1. Then there exists a unique extension of f_{1} to a quiver morphism f_{1} : sℱ𝒬 → s𝒜 such that (f_{1}, f_{2}, …) are components of an A_{∞}-functor f : ℱ𝒬 → 𝒜.

Proof. For each n > 1 we have to satisfy the equation

b_{n}f_{1} = ∑_{i_{1}+···+i_{l}=n}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}})b_{l} − ∑_{α+k+β=n}^{α+β>0}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β})f_{α+1+β} : T^{n}sℱ𝒬 → s𝒜. (11)

It is used to define recursively f_{1} on sℱ𝒬. Suppose that t_{1}, …, t_{n} are trees, n > 1, and f_{1} : sℱ_{t_{i}}𝒬 → s𝒜 is already defined for all 1 ⩽ i ⩽ n. Since

b_{n} = s^{|t_{1}|} ⊗ ⋯ ⊗ s^{|t_{n−1}|} ⊗ s^{|t_{n}|−|t|} : sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 → sℱ_{t}𝒬

is invertible for t = (t_{1} ⊔ ⋯ ⊔ t_{n}) · 𝔱_{n}, formula (11) determines f_{1} : sℱ_{t}𝒬 → s𝒜 uniquely as

f_{1} = (sℱ_{t}𝒬 ^{b_{n}^{−1}}→ sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 ^{∑(f_{i_{1}}⊗···⊗f_{i_{l}})b_{l}−∑_{α+k+β=n}^{α+β>0}(1^{⊗α}⊗b_{k}⊗1^{⊗β})f_{α+1+β}} → s𝒜).

This proves uniqueness of the extension of f_{1}.

Let us prove that the cocategory homomorphism f with so defined components (f_{1}, f_{2}, …) is an A_{∞}-functor. Equations (11) are satisfied by construction of f_{1}. So it remains to prove that f_{1} is a chain map. Equation f_{1}b_{1} = b_{1}f_{1} holds on sℱ_{|}𝒬 by assumption. We are going to prove by induction on |t| that it holds on sℱ_{t}𝒬. Considering t = (t_{1} ⊔⋯⊔t_{n})·𝔱_{n}, n > 1, we assume that f_{1}b_{1} = b_{1}f_{1} : sℱ_{t′}𝒬 → s𝒜 for all trees t′ with |t′| < |t|. To prove that f_{1}b_{1} = b_{1}f_{1} : sℱ_{t}𝒬 → s𝒜 it suffices to show that b_{n}f_{1}b_{1} = b_{n}b_{1}f_{1} for all n > 1 due to invertibility of b_{n}. Using (11) and the equation b^{2} pr_{1} = 0 we find

b_{n}f_{1}b_{1} − b_{n}b_{1}f_{1} = ∑_{i_{1}+···+i_{l}=n}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}})b_{l}b_{1} − ∑_{α+k+β=n}^{α+β>0}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β})f_{α+1+β}b_{1}

+ ∑_{α+k+β=n}^{α+β>0}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β})b_{α+1+β}f_{1}

= − ∑_{i_{1}+···+i_{l}=n}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}}) ∑_{γ+p+δ=l}^{γ+δ>0}(1^{⊗γ} ⊗ b_{p} ⊗ 1^{⊗δ})b_{γ+1+δ}

+ ∑_{α+k+β=n}^{α+β>0}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β})[ ∑_{j_{1}+···+j_{r}=α+1+β}^{r>1}(f_{j_{1}} ⊗ ⋯ ⊗ f_{j_{r}})b_{r}

− ∑_{γ+p+δ=α+1+β}^{γ+δ>0}(1^{⊗γ} ⊗ b_{p} ⊗ 1^{⊗δ})f_{γ+1+δ}]

= ∑_{r>1}[ ∑_{α+k+β=n}^{α+β>0}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β}) ∑_{j_{1}+···+j_{r}=α+1+β} f_{j_{1}} ⊗ ⋯ ⊗ f_{j_{r}}

− ∑_{i_{1}+···+i_{l}=n}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}}) ∑_{\begin{matrix}γ+p+δ=l \\ γ+1+δ=r \end{matrix}} 1^{⊗γ} ⊗ b_{p} ⊗ 1^{⊗δ}]b_{r}

− ∑_{r>1}[ ∑_{\begin{matrix}α+k+β=n \\ α+β>0 \end{matrix}}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β}) ∑_{\begin{matrix}γ+p+δ=α+1+β \\ γ+1+δ=r \end{matrix}} 1^{⊗γ} ⊗ b_{p} ⊗ 1^{⊗δ}]f_{r}.

Let us show that the expressions in square brackets vanish. The first one is the matrix coefficient bf − fb : sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 → T^{r}s𝒜. Indeed, for r > 1 the inequality r ⩽ j_{1} + ⋯ + j_{r} = α + 1 + β automatically implies that α + β > 0, so this condition can be omitted. Using the induction hypothesis one can transform the left hand side of equation

∑_{α+k+β=n}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β}) ∑_{j_{1}+···+j_{r}=α+1+β} f_{j_{1}} ⊗ ⋯ ⊗ f_{j_{r}}

= ∑_{i_{1}+···+i_{l}=n}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}}) ∑_{\begin{matrix}γ+p+δ=l \\ γ+1+δ=r \end{matrix}} 1^{⊗γ} ⊗ b_{p} ⊗ 1^{⊗δ} : sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 → T^{r}s𝒜

into the right hand side for all n, r ⩾ 1.

The second expression

∑_{\begin{matrix}α+k+β=n \\ α+β>0 \end{matrix}}(1^{⊗α} ⊗ b_{k} ⊗ 1^{⊗β}) ∑_{\begin{matrix}γ+p+δ=α+1+β \\ γ+1+δ=r \end{matrix}} 1^{⊗γ} ⊗ b_{p} ⊗ 1^{⊗δ}  (12)

is the matrix coefficient

(b − b pr_{1})b pr_{r} : T^{n}sℱ𝒬 → T^{r}sℱ𝒬

of the endomorphism (b − b pr_{1})b : T sℱ𝒬 → T sℱ𝒬. However,

(b − b pr_{1})b pr_{r} = b^{2} pr_{r} −b pr_{1} b pr_{r} = −b pr_{1} b pr_{1} pr_{r} = 0

for r > 1, because pr_{1} b = pr_{1} b pr_{1}. Therefore, (12) vanishes and equation b_{n}f_{1}b_{1} = b_{n}b_{1}f_{1} is proven.

Let strictA_{∞}(ℱ𝒬, 𝒜) ⊂ A_{∞}(ℱ𝒬, 𝒜) be a full A_{∞}-subcategory, whose objects are strict A_{∞}-functors. Recall that Ob A_{1}(𝒬, 𝒜) is the set of chain morphisms 𝒬 → 𝒜 of differential graded quivers.

2.4. Corollary. A chain morphism f : 𝒬 → 𝒜 admits a unique extension to a strict A_{∞}-functor f̂: ℱ𝒬 → 𝒜. The maps f ↦ f̂ and

restr : Ob strictA_{∞}(ℱ𝒬, 𝒜) → Ob A_{1}(𝒬, 𝒜), g ↦ (Ob g, g_{1}|_{s𝒬})

are inverse to each other.

Indeed, strict A_{∞}-functors g are distinguished by conditions g_{k} = 0 for k > 1.

We may view strictA_{∞} as a category, whose objects are A_{∞}-categories and morphisms are strict A_{∞}-functors. We may also view A_{1} as a category consisting of differential graded quivers and their morphisms. There is a functor 𝒰 : strictA_{∞} → A_{1}, 𝒜 ↦ (𝒜, m_{1}), which sends an A_{∞}-category to the underlying differential graded 𝕜-quiver, forgetting all higher multiplications. The restriction map

restr : strictA_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒰𝒜),  g ↦ (Ob g, g_{1}|_{s𝒬})  (13)

is functorial in 𝒜.

2.5. Corollary. There is a functor ℱ : A_{1} → strictA_{∞}, 𝒬 ↦ ℱ𝒬, left adjoint to 𝒰.

2.6. Explicit formula for the constructed strict A_{∞}-functor. Let us obtain a more explicit formula for f̂_{1}|_{sℱ_{t}𝒬} . We define f̂_{1} for t = (t_{1} ⊔ ⋯ ⊔ t_{n}) · 𝔱_{n} recursively by a commutative diagram

sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{n}}𝒬 \overline{→^{\begin{matrix}s^{|t_{1}|}⊗···⊗s^{|t_{n−1}|}⊗s^{|tn|−|t|} \\ b_{n} \end{matrix}}} sℱ_{t}𝒬

=^{def}:

s𝒜 ⊗ ⋯ ⊗ s𝒜 f̂_{1}^{⊗n} ; ↓  \overline{^{b_{n}}  →} s𝒜 f̂_{1} ; ↓

Notice that the top map is invertible. Here n, t_{1}, …, t_{n} are uniquely determined by decomposition (10) of t.

Let (t, ⩽) be a linearly ordered tree with the underlying given plane rooted tree t. Decompose (t, ⩽) into a vertical composition of forests as in (7). Then the following diagram commutes

\begin{matrix}s𝒬^{⊗n} & \overline{→^{1^{⊗α_{1}}⊗b_{k_{1}}⊗1^{⊗β_{1}}}} & s𝒬^{⊗α_{1}} ⊗ sℱ_{𝔱_{k_{1}}}𝒬 ⊗ s𝒬^{⊗β_{1}} & \overline{^{1^{⊗α_{2}}⊗b_{k_{2}}⊗1^{⊗β_{2}}} →} & … & \overline{^{b_{k_{N}}} →} & sℱ_{t}𝒬 \\ s𝒜^{⊗n} & ↓^{f_{1}^{⊗n}} & ^{1^{⊗α_{1}}⊗b_{k_{1}}⊗1^{⊗β_{1}}}→ & s𝒜^{⊗α_{1}} ⊗ s𝒜 ⊗ s𝒜^{⊗β_{1}} & ↓^{f̂_{1}^{⊗α_{1}+1+β_{1}}} & \overline{^{1^{⊗α_{2}}⊗b_{k_{2}}⊗1^{⊗β_{2}}} →} & … & ↓^{f̂_{1}^{⊗α_{2}+1+β_{2}}}^{…↓^{f̂_{1}^{⊗k_{N}}}} & \overline{^{b_{k_{N}}} →} & s𝒜 & ↓^{f̂_{1}} \end{matrix}

The upper row consists of invertible maps. One can prove by induction that the composition of maps in the upper row equals ±s^{−|t|}. When (t, ⩽) = t_{<} is the linearly ordered tree, canonically associated with t, then the composition of maps in the upper row equals s^{−|t|}. This is also proved by induction: if t is presented as t = (t_{1} ⊔ ⋯ ⊔ t_{k}) · 𝔱_{k}, then the composition of maps in the upper row is

(1^{⊗k−1} ⊗ s^{−|t_{k}|})(1^{⊗k−2} ⊗ s^{−|t_{k−1}|} ⊗ 1) … (s^{−|t_{1}|} ⊗ 1^{⊗k−1})(s^{|t_{1}|} ⊗ ⋯ ⊗ s^{|t_{k−1}|} ⊗ s^{|t_{k}|−|t|})

= 1^{⊗k−1} ⊗ s^{−|t|} = s^{−|t|}.

Therefore, for an arbitrary tree t ∈ 𝒯_{⩾2}^{n} the map f̂_{1}|_{sℱ_{t}𝒬} is

f̂_{1} = (sℱ_{t}𝒬 ^{s^{|t|}}→ s𝒬^{⊗n} ^{f_{1}^{⊗n}} → s𝒜^{⊗n} ^{1^{⊗α_{1}}⊗b_{k_{1}}⊗1^{⊗β_{1}}} → s𝒜^{⊗α_{1}+1+β_{1}} ^{1^{⊗α_{2}}⊗b_{k_{2}}⊗1^{⊗β_{2}}} → … ^{b_{k_{N}}}→ s𝒜),

where the factors correspond to decomposition (7) of t_{<}.

2.7. Transformations between functors from a free A_{∞}-category. Let 𝒬 be a differential graded quiver, and let 𝒜 be an A_{∞}-category. Then A_{1}(𝒬, 𝒜) is an A_{∞}-category as well. The differential graded quiver (sA_{1}(𝒬, 𝒜), B_{1}) is described as follows. Objects are chain quiver maps ϕ : (s𝒬, b_{1}) → (s𝒜, b_{1}), the graded 𝕜-module of morphisms ϕ → ψ is the product of graded 𝕜-modules

sA_{1}(𝒬, 𝒜)(ϕ, ψ) = ∏_{X∈Ob 𝒬} s𝒜(Xϕ, Xψ)× ∏_{X,Y ∈Ob 𝒬} C(s𝒬(X, Y), s𝒜(Xϕ, Y ψ)), r = (r_{0}, r_{1}).

The differential B_{1} is given by

(rB_{1})_{0} = r_{0}b_{1},
(rB_{1})_{1} = r_{1}b_{1} + (ϕ_{1} ⊗ r_{0})b_{2} + (r_{0} ⊗ ψ_{1})b_{2} − (−)^{r}b_{1}r_{1}.  (14)

Restrictions ϕ, ψ : 𝒬 → 𝒜 of arbitrary A_{∞}-functors ϕ, ψ : ℱ𝒬 → 𝒜 to 𝒬 are A_{1}-functors (chain quiver maps).

2.8. Proposition. Let ϕ, ψ : ℱ𝒬 → 𝒜 be A_{∞}-functors. For an arbitrary complex P of 𝕜-modules chain maps u : P → sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) are in bijection with the following data: (u′, u_{k})_{k>1}

1. a chain map u′ : P → sA_{1}(𝒬, 𝒜)(ϕ, ψ),

2. 𝕜-linear maps

u_{k} : P → ∏_{X,Y ∈Ob 𝒬} C((sℱ𝒬)^{⊗k}(X, Y), s𝒜(Xϕ, Y ψ))

of degree 0 for all k > 1.

The bijection maps u to (u′, u_{k})_{k>1}, where u_{k} = u · pr_{k} and

u′ = (P ^{u}→ sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr}▷ sA_{1}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr}▷ sA_{1}(𝒬, 𝒜)(ϕ, ψ)). (15)

The inverse bijection can be recovered from the recurrent formula

(−)^{p}b_{k}^{ℱ𝒬} (pu_{1}) = −(pd)u_{k} + ∑_{a+q+c=k}^{α,β}(ϕ_{aα} ⊗ pu_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}

− (−)^{p} ∑_{α+q+β=k}^{α+β>0}(1^{⊗α} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗β})(pu_{α+1+β}) : (sℱ𝒬)^{⊗k} → s𝒜,

where k > 1, p ∈ P, and ϕ_{aα}, ψ_{cβ} are matrix elements of ϕ, ψ.

Proof. Since the 𝕜-module of (ϕ, ψ)-coderivations sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) is a product, 𝕜-lin-ear maps u : P → sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) of degree 0 are in bijection with sequences of 𝕜-linear maps (u_{k})_{k⩾0} of degree 0:

u_{0} : P → ∏_{X∈Ob 𝒬} s𝒜(Xϕ, Xψ),  p ↦ pu_{0},
u_{k} : P → ∏_{X,Y ∈Ob 𝒬} C((sℱ𝒬)^{⊗k}(X, Y), s𝒜(Xϕ, Y ψ)), p ↦ pu_{k},

for k ⩾ 1. The complex Φ_{0} = (sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ), B_{1}) admits a filtration by subcomplexes

Φ_{n} = 0 × ⋯ × 0 × ∏_{k=n}^{∞} ∏_{X,Y ∈Ob 𝒬} C((sℱ𝒬)^{⊗k}(X, Y), s𝒜(Xϕ, Y ψ)).

In particular, Φ_{2} is a subcomplex, and

Φ_{0}/Φ_{2} = ∏_{X∈Ob 𝒬} s𝒜(Xϕ, Xψ) × ∏_{X,Y ∈Ob 𝒬} C(sℱ𝒬(X, Y), s𝒜(Xϕ, Y ψ))

is the quotient complex with differential (14). Since (sℱ𝒬, b_{1}) splits into a direct sum of two subcomplexes s𝒬 ⊕ (⊕_{|t|>0}sℱ_{t}𝒬), the complex Φ_{0}/Φ_{2} has a subcomplex

(0 × ∏_{X,Y ∈Ob 𝒬} C(⊕_{|t|>0}sℱ_{t}𝒬(X, Y), s𝒜(Xϕ, Y ψ)), [ , b_{1}]).

The corresponding quotient complex is sA_{1}(𝒬, 𝒜)(ϕ, ψ). The resulting quotient map restr_{1} : sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) → sA_{1}(𝒬, 𝒜)(ϕ, ψ) is the restriction map. Denoting u′ = u · restr_{1}, we get the discussed assignment u ↦ (u′, u_{n})_{n>1}. The claim is that if u is a chain map, then the missing part

u′′_{1} : P → ∏_{X,Y ∈Ob 𝒬} C(⊕_{|t|>0}sℱ_{t}𝒬(X, Y), s𝒜(Xϕ, Y ψ)),

of u_{1} = u′_{1} × u′′_{1} is recovered in a unique way.

Let us prove that the map u ↦ (u′, u_{n})_{n>1} is injective. The chain map u satisfies pdu = puB_{1} for all p ∈ P. That is, pdu_{k} = (puB_{1})_{k} for all k ⩾ 0. Since puB_{1} = (pu)b^{𝒜} − (−)^{p}b^{ℱ𝒬} (pu), these conditions can be rewritten as

pdu_{k} = ∑_{a+q+c=k}^{α,β}(ϕ_{aα} ⊗ pu_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜} − (−)^{p} ∑_{α+q+β=k}(1^{⊗α} ⊗ b_{q} ⊗ 1^{⊗β})(pu_{α+1+β}), (16)

where ϕ_{aα} : T^{a}sℱ𝒬(X, Y) → T^{α}s𝒜(Xϕ, Y ϕ) are matrix elements of ϕ, and ψ_{cβ} are matrix elements of ψ. The same formula can be rewritten as

(−)^{p}b_{k}^{ℱ𝒬} (pu_{1}) = −(pd)u_{k} + ∑_{a+q+c=k}^{α,β}(ϕ_{aα} ⊗ pu_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}

− (−)^{p} ∑_{α+q+β=k}^{α+β>0}(1^{⊗α} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗β})(pu_{α+1+β}) : sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{k}}𝒬 → s𝒜. (17)

When k > 1, the map b_{k}^{ℱ𝒬} : sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{k}}𝒬 → sℱ_{t}𝒬, t = (t_{1} ⊔ ⋯ ⊔ t_{k})𝔱_{k} is invertible, thus, pu_{1} : sℱ_{t}𝒬 → s𝒜 in the left hand side is determined in a unique way by u_{0}, u_{n} for n > 1 and by pu_{1} : sℱ_{t_{i}}𝒬 → s𝒜, 1 ⩽ i ⩽ k, occurring in the right hand side. Since the restriction u′_{1} of u_{1} to sℱ_{|}𝒬 = s𝒬 is known by 1), the map u′′_{1} is recursively recovered from (u_{0}, u′_{1}, u_{n})_{n>1}.

Let us prove that the map u ↦ (u′, u_{n})_{n>1} is surjective. Given (u_{0}, u′_{1}, u_{n})_{n>1} we define maps u′′_{1} of degree 0 recursively by (17). This implies equation (16) for k > 1. For k = 0 this equation in the form pdu_{0} = pu_{0}b_{1} holds due to condition 1). It remains to prove equation (16) for k = 1:

(pd)u_{1} = (pu_{1})b_{1}^{𝒜} + (ϕ_{1} ⊗ pu_{0})b_{2}^{𝒜} + (pu_{0} ⊗ ψ_{1})b_{2}^{𝒜} − (−)^{p}b_{1}(pu_{1}) : sℱ_{t}𝒬(X, Y) → s𝒜(Xϕ, Y ψ) (18)

for all trees t ∈ 𝒯_{⩾2}. For t = | it holds due to assumption 1). Let N > 1 be an integer. Assume that equation (18) holds for all trees t ∈ 𝒯_{⩾2} with the number of input leaves in(t) < N. Let t ∈ 𝒯_{⩾2}^{N} be a tree (with in(t) = N). Then t = (t_{1} ⊔ ⋯ ⊔ t_{k})𝔱_{k} for some k > 1 and some trees t_{i} ∈ 𝒯_{⩾2}, in(t_{i}) < N. For such t equation (18) is equivalent to

(−)^{p}b_{k}(pd)u_{1} = (−)^{p}b_{k}(pu_{1})b_{1}^{𝒜} + (−)^{p}b_{k}(ϕ_{1} ⊗ pu_{0})b_{2}^{𝒜} + (−)^{p}b_{k}(pu_{0} ⊗ ψ_{1})b_{2}^{𝒜}

+ ∑_{γ+j+δ=k}^{γ+δ>0}(1^{⊗γ} ⊗ b_{j} ⊗ 1^{⊗δ})b_{γ+1+δ}(pu_{1}) : sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{k}}𝒬 → s𝒜.

Substituting definition (17) of u_{1} we turn the above equation into an identity

− ∑_{a+q+c=k}^{α,β}(ϕ_{aα} ⊗ pdu_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}  (19)

− (−)^{p} ∑_{α+q+β=k}^{α+β>0}(1^{⊗α} ⊗ b_{q} ⊗ 1^{⊗β})(pdu_{α+1+β})  (20)
= −(pdu_{k})b_{1}  (21)

+ ∑_{a+q+c=k}^{α,β}(ϕ_{aα} ⊗ pu_{q} ⊗ ψ_{cβ})b_{α+1+β}b_{1}

− (−)^{p} ∑_{α+q+β=k}^{α+β>0}(1^{⊗α} ⊗ b_{q} ⊗ 1^{⊗β})(pu_{α+1+β})b_{1}  (22)
+ (−)^{p}b_{k}(ϕ_{1} ⊗ pu_{0})b_{2} + (−)^{p}b_{k}(pu_{0} ⊗ ψ_{1})b_{2}  (23)
+ (−)^{p} ∑_{γ+j+δ=k}^{γ+δ>0}(1^{⊗γ} ⊗ b_{j} ⊗ 1^{⊗δ})[−pdu_{γ+1+δ}  (24)

+ ∑_{a+q+c=γ+1+δ}^{α,β}(ϕ_{aα} ⊗ pu_{q} ⊗ ψ_{cβ})b_{α+1+β}  (25)

− (−)^{p} ∑_{α+q+β=γ+1+δ}^{α+β>0}(1^{⊗α} ⊗ b_{q} ⊗ 1^{⊗β})(pu_{α+1+β})],  (26)

whose validity we are going to prove now. First of all, terms (20) and (24) cancel each other. Term (26) vanishes because for an arbitrary integer g the sum

∑_{\begin{matrix}γ+j+δ=k \\ α+q+β=γ+1+δ \end{matrix}}^{α+1+β=g}(1^{⊗γ} ⊗ b_{j} ⊗ 1^{⊗δ})(1^{⊗α} ⊗ b_{q} ⊗ 1^{⊗β})  (27)

is the matrix coefficient b^{2} = 0 : T^{k}sℱ𝒬 → T^{g}sℱ𝒬, thus, it vanishes. Notice that condition α + β > 0 in (26) automatically implies γ + δ > 0. Furthermore, term (21) cancels one of the terms of sum (19). In the remaining terms of (19) we may use the induction assumptions and replace pdu_{q} with the right hand side of (16). We also absorb terms (23) into sum (25), allowing γ = δ = 0 in it and allowing simultaneously α = β = 0 in (22) to compensate for the missing term b_{k}(pu_{1})b_{1}:

− ∑_{a+q+c=k}^{α+β>0} ∑_{e+j+f=q}^{γ,δ}[ϕ_{aα} ⊗ (ϕ_{eγ} ⊗ pu_{j} ⊗ ψ_{fδ})b_{γ+1+δ}^{𝒜} ⊗ ψ_{cβ}]b_{α+1+β}^{𝒜}  (28)

+ (−)^{p} ∑_{a+q+c=k}^{α+β>0} ∑_{γ+j+δ=q}[ϕ_{aα} ⊗ (1^{⊗γ} ⊗ b_{j} ⊗ 1^{⊗δ})(pu_{γ+1+δ}) ⊗ ψ_{cβ}]b_{α+1+β}^{𝒜}  (29)

= ∑_{a+q+c=k}^{α,β}(ϕ_{aα} ⊗ pu_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}b_{1}^{𝒜}  (30)
− (−)^{p} ∑_{α+q+β=k}(1^{⊗α} ⊗ b_{q} ⊗ 1^{⊗β})(pu_{α+1+β})b_{1}^{𝒜}  (31)

+ (−)^{p} ∑_{γ+j+δ=k} ∑_{a+q+c=γ+1+δ}^{α,β}(1^{⊗γ} ⊗ b_{j} ⊗ 1^{⊗δ})(ϕ_{aα} ⊗ pu_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}.

Recall that ϕ_{a0} vanish for all a except a = 0. Therefore, we may absorb term (30) into sum (28) and term (31) into sum (29), allowing terms with α = β = 0 in them. Denote r = pu ∈ sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ). The proposition follows immediately form the following

2.9. Lemma. For all r ∈ sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) and all k ⩾ 0 we have

− ∑_{a+q+c=k}^{α,β} ∑_{e+j+f=q}^{γ,δ}[ϕ_{aα} ⊗ (ϕ_{eγ} ⊗ r_{j} ⊗ ψ_{fδ})b_{γ+1+δ}^{𝒜} ⊗ ψ_{cβ}]b_{α+1+β}^{𝒜}

+ (−)^{r} ∑_{a+q+c=k}^{α,β} ∑_{γ+j+δ=q}[ϕ_{aα} ⊗ (1^{⊗γ} ⊗ b_{j} ⊗ 1^{⊗δ})r_{γ+1+δ} ⊗ ψ_{cβ}]b_{α+1+β}^{𝒜}

= (−)^{r} ∑_{γ+j+δ=k} ∑_{a+q+c=γ+1+δ}^{α,β}(1^{⊗γ} ⊗ b_{j} ⊗ 1^{⊗δ})(ϕ_{aα} ⊗ r_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}.  (32)

Proof. Sum (32) is split into three sums accordingly to output of b_{j} being an input of ϕ_{aα} or r_{q} or ψ_{cβ}:

− ∑_{a+e+j+f+c=k}^{α,β,γ,δ}(ϕ_{aα} ⊗ ϕ_{eγ} ⊗ r_{j} ⊗ ψ_{fδ} ⊗ ψ_{cβ})(1^{⊗α} ⊗ b_{γ+1+δ}^{𝒜} ⊗ 1^{⊗β})b_{α+1+β}^{𝒜}  (33)

+ (−)^{r} ∑_{a+γ+j+δ+c=k}^{α,β}(1^{⊗a+γ} ⊗ b_{j} ⊗ 1^{⊗δ+c})(ϕ_{aα} ⊗ r_{γ+1+δ} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}  (34)

= (−)^{r} ∑_{x+q+c=k}^{a,α,β}(b_{xa}ϕ_{aα} ⊗ r_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}  (35)

+ (−)^{r} ∑_{a+y+c=k}^{α,q,β}(ϕ_{aα} ⊗ b_{yq}r_{q} ⊗ ψ_{cβ})b_{α+1+β}^{𝒜}  (36)

+ ∑_{a+q+z=k}^{α,β,c}(ϕ_{aα} ⊗ r_{q} ⊗ b_{zc}ψ_{cβ})b_{α+1+β}^{𝒜}.  (37)

Here b_{xa} : T^{x}sℱ𝒬 → T^{a}sℱ𝒬 is a matrix element of b^{ℱ𝒬} . Terms (34) and (36) cancel each other. We shall use A_{∞}-functor identities bϕ = ϕb, bψ = ψb for terms (35) and (37). Being a cocategory homomorphism, ϕ satisfies the identity

∑_{a+e=h} ϕ_{aα} ⊗ ϕ_{eγ} = [Δ(ϕ ⊗ ϕ)]_{h;α,γ} = ϕ_{h,α+γ}Δ_{α+γ;α,γ}

for all non-negative integers h, where Δ is the cut comultiplication. Similarly for ψ. Using this identity in (33) we get the equation to verify:

− ∑_{x+q+z=k}^{v,w}(ϕ_{xv} ⊗ r_{q} ⊗ ψ_{zw}) ∑_{α+y+β=v+1+w}^{α⩽v,β⩽w}(1^{⊗α} ⊗ b_{y}^{𝒜} ⊗ 1^{⊗β})b_{α+1+β}^{𝒜}

= ∑_{x+q+z=k}^{v,w,α}(ϕ_{xv} ⊗ r_{q} ⊗ ψ_{zw})(b_{vα}^{𝒜} ⊗ 1^{⊗1+w})b_{α+1+w}^{𝒜}

+ ∑_{x+q+z=k}^{v,w,β}(ϕ_{xv} ⊗ r_{q} ⊗ ψ_{zw})(1^{⊗v+1} ⊗ b_{wβ}^{𝒜})b_{v+1+β}^{𝒜}.

It follows from the identity b^{2} pr_{1} = 0 : T^{v+1+w}s𝒜 → s𝒜 valid for arbitrary non-negative integers v, w, which we may rewrite like this:

∑_{α+y+β=v+1+w}^{α⩽v,β⩽w}(1^{⊗α}⊗b_{y}^{𝒜} ⊗1^{⊗β})b_{α+1+β}^{𝒜} +∑_{α}(b_{vα}^{𝒜}⊗1^{⊗1+w})b_{α+1+w}^{𝒜}+∑_{β}(1^{⊗v+1}⊗b_{wβ}^{𝒜})b_{v+1+β}^{𝒜} = 0.

So the lemma is proved.

The proposition follows.

Let us consider now the question, when the discussed chain map is null-homotopic.

2.10. Corollary. Let ϕ, ψ : ℱ𝒬 → 𝒜 be A_{∞}-functors. Let P be a complex of 𝕜-modules. Let u : P → sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) be a chain map. The set (possibly empty) of homotopies h : P → sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ), deg h = −1, such that u = dh + hB_{1} is in bijection with the set of data (h′, h_{k})_{k>1}, consisting of

1. a homotopy h′ : P → sA_{1}(𝒬, 𝒜)(ϕ, ψ), deg h′ = −1, such that dh′ +h′B_{1} = u′, where u′ is given by (15);

2. 𝕜-linear maps

h_{k} : P → ∏_{X,Y ∈Ob 𝒬} C((sℱ𝒬)^{⊗k}(X, Y), s𝒜(Xϕ, Y ψ))

of degree −1 for all k > 1. The bijection maps h to (h′, h_{k})_{k>1}, where h_{k} = h · pr_{k} and

h′ = (P ^{h}→ sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr}▷ sA_{1}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr}▷ sA_{1}(𝒬, 𝒜)(ϕ, ψ)).

The inverse bijection can be recovered from the recurrent formula

(−)^{p}b_{k}(ph_{1}) = pu_{k} − (pd)h_{k} − ∑_{a+q+c=k}^{α,β}(ϕ_{aα} ⊗ ph_{q} ⊗ ψ_{cβ})b_{α+1+β}

− (−)^{p} ∑_{a+q+c=k}^{a+c>0}(1^{⊗a} ⊗ b_{q} ⊗ 1^{⊗c})(ph_{a+1+c}) : (sℱ𝒬)^{⊗k} → s𝒜,

where k > 1, p ∈ P, and ϕ_{aα}, ψ_{cβ} are matrix elements of ϕ, ψ.

Proof. We shall apply Proposition 2.8 to the complex Cone(id : P → P) instead of P. The graded 𝕜-module Cone(id_{P}) = P ⊕ P [1] is equipped with the differential (q, ps)d = (qd + p, −pds), p, q ∈ P. The chain maps \overline{u} : Cone(id_{P}) → C to an arbitrary complex C are in bijection with pairs (u : P → C, h : P → C), where u = dh + hd and deg h = −1. The pair (u, h) = (in_{1} \overline{u}, s in_{2} \overline{u}) is assigned to \overline{u}, and the map \overline{u} : P ⊕ P [1] → C, (q, ps) ↦ qu + ph is assigned to a pair (u, h). Indeed, \overline{u} being chain map is equivalent to

(q, ps)d\overline{u} = qdu + pu − pdh = qud + phd = (q, ps)\overline{u}d,

that is, to conditions du = ud, u = dh + hd.

Thus, for a fixed chain map u : P → C the set of homotopies h : P → C, such that u = dh + hd, is in bijection with the set of chain maps \overline{u} : Cone(id_{P}) → C such that in_{1} \overline{u} = u : P → C. Applying this statement to u : P → C = sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) we find by Proposition 2.8 that the set of homotopies h : P → sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) such that u = dh + hB_{1} is in bijection with the set of data (\overline{u}′, \overline{u}_{k})_{k>1}, such that

\overline{u}′ : Cone(id_{P}) → sA_{1}(𝒬, 𝒜)(ϕ, ψ) is a chain map,  in_{1} \overline{u}′ = u′,
\overline{u}_{k} : Cone(id_{P}) → ∏_{X,Y ∈Ob 𝒬} C((sℱ𝒬)^{⊗k}(X, Y), s𝒜(Xϕ, Y ψ)), deg \overline{u}_{k} = 0, in_{1} \overline{u}_{k} = u_{k},

therefore, in bijection with the set of data (h′, h_{k})_{k>1} = (s in_{2} \overline{u}′, s in_{2} \overline{u}_{k})_{k>1}, as stated in corollary.

2.11. Restriction as an A_{∞}-functor. Let 𝒬 be a (𝒰 -small) differential graded 𝕜-quiver. Denote by ℱ𝒬 the free A_{∞}-category generated by 𝒬. Let 𝒜 be a (𝒰 -small) unital A_{∞}-category. There is the restriction strict A_{∞}-functor

restr : A_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜), (f : ℱ𝒬 → 𝒜) ↦ (\overline{f} = (f_{1}|_{𝒬}) : 𝒬 → 𝒜).

In fact, it is the composition of two strict A_{∞}-functors: A_{∞}(ℱ𝒬, 𝒜) ^{restr_{∞,1}} → A_{1}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜), where the second comes from the full embedding 𝒬 ↪ ℱ𝒬. Its first component is

restr_{1} : sA_{∞}(ℱ𝒬, 𝒜)(f, g) → sA_{1}(𝒬, 𝒜)(\overline{f}, \overline{g}),  (38)
r = (r_{0}, r_{1}, …, r_{n}, …) ↦ (r_{0}, r_{1}|_{𝒬}) = \overline{r}.

2.12. Theorem. The A_{∞}-functor restr : A_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜) is an equivalence.

Proof. Let us prove that restriction map (38) is homotopy invertible. We construct a chain map going in the opposite direction

u : sA_{1}(𝒬, 𝒜)(\overline{f}, \overline{g}) → sA_{∞}(ℱ𝒬, 𝒜)(f, g)

via Proposition 2.8 taking P = sA_{1}(𝒬, 𝒜)(\overline{f}, \overline{g}). We choose

u′ : sA_{1}(𝒬, 𝒜)(\overline{f}, \overline{g}) → sA_{1}(𝒬, 𝒜)(\overline{f}, \overline{g})

to be the identity map and u_{k} = 0 for k > 1. Therefore,

u · restr_{1} = u′ = id_{sA_{1}(𝒬,𝒜)(\overline{f},\overline{g})} .

Denote

v = id_{sA_{∞}(ℱ𝒬,𝒜)(f,g)} −[sA_{∞}(ℱ𝒬, 𝒜)(f, g) ^{restr_{1}} → sA_{1}(𝒬, 𝒜)(\overline{f}, \overline{g}) ^{u}→ sA_{∞}(ℱ𝒬, 𝒜)(f, g)].

Let us prove that v is null-homotopic via Corollary 2.10, taking P = sA_{∞}(ℱ𝒬, 𝒜)(f, g). A homotopy h : sA_{∞}(ℱ𝒬, 𝒜)(f, g) → sA_{∞}(ℱ𝒬, 𝒜)(f, g), deg h = −1, such that v = B_{1}h + hB_{1} is specified by h′ = 0 : sA_{∞}(ℱ𝒬, 𝒜)(f, g) → sA_{1}(𝒬, 𝒜)(\overline{f}, \overline{g}) and h_{k} = 0 for k > 1. Indeed,

v′ = v · restr_{1} = restr_{1} − restr_{1} ·u · restr_{1} = restr_{1} − restr_{1} = 0,

so v′ = B_{1}h′+h′B_{1} and condition 1 of Corollary 2.10 is satisfied^{2}. Therefore, u is homotopy inverse to restr_{1}.

Let i^{𝒜} be a unit transformation of the unital A_{∞}-category 𝒜. Then A_{1}(𝒬, 𝒜) is a unital A_{∞}-category with the unit transformation (1⊗i^{𝒜} )M (cf. [Lyu03, Proposition 7.7]). The unit element for an object ϕ ∈ Ob A_{1}(𝒬, 𝒜) is _{ϕ}i_{0}^{A_{1}(𝒬,𝒜)} : 𝕜 → sA_{1}(𝒬, 𝒜), 1 ↦ ϕi^{𝒜} . The A_{∞}-category A_{∞}(ℱ𝒬, 𝒜) is also unital. To establish equivalence of these two A_{∞}-categories via restr : A_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜) we verify the conditions of Theorem 8.8 from [Lyu03].

Consider the mapping Ob A_{1}(𝒬, 𝒜) → Ob A_{∞}(ℱ𝒬, 𝒜), ϕ ↦ ϕ̂, which extends a given chain map to a strict A_{∞}-functor, constructed in Corollary 2.4. Clearly, \overline{ϕ̂} = ϕ. It remains to give two mutually inverse cycles, which we choose as follows:

ϕr_{0} : 𝕜 → sA_{1}(𝒬, 𝒜)(ϕ, \overline{ϕ̂}),  1 ↦ ϕi^{𝒜} ,
ϕp_{0} : 𝕜 → sA_{1}(𝒬, 𝒜)(\overline{ϕ̂}, ϕ),  1 ↦ ϕi^{𝒜} .

Clearly, _{ϕ}r_{0}B_{1} = 0, _{ϕ}p_{0}B_{1} = 0,

(_{ϕ}r_{0} ⊗_{ϕ}p_{0})B_{2} −_{ϕ}i_{0}^{A_{1}(𝒬,𝒜)} : 1 ↦ (ϕi^{𝒜} ⊗ ϕi^{𝒜} )B_{2} − ϕi^{𝒜} ∈ Im B_{1},
(_{ϕ}p_{0} ⊗_{ϕ}r_{0})B_{2} −_{ϕ}i_{0}^{A_{1}(𝒬,𝒜)} : 1 ↦ (ϕi^{𝒜} ⊗ ϕi^{𝒜} )B_{2} − ϕi^{𝒜} ∈ Im B_{1}.

Therefore, all assumptions of Theorem 8.8 [Lyu03] are satisfied. Thus, restr : A_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜) is an A_{∞}-equivalence.

2By the way, the only non-vanishing component of h is h_{1}.

2.13. Corollary. Every A_{∞}-functor f : ℱ𝒬 → 𝒜 is isomorphic to the strict A_{∞}-functor ˆ\overline{f} : ℱ𝒬 → 𝒜.

Proof. Note that \overline{f} = \overline{ˆ\overline{f}}. The A_{1}-transformation \overline{f}i^{𝒜} : \overline{f} → \overline{ˆ\overline{f}} : 𝒬 → 𝒜 with the components (_{Xf}i_{0}^{𝒜} , \overline{f}_{1}i_{1}^{𝒜}) is natural. It is mapped by u into a natural A_{∞}-transformation (\overline{f}i^{𝒜} )u : f → ˆ \overline{f} : ℱ𝒬 → 𝒜. Its zero component _{Xf}i_{0}^{𝒜} is invertible, therefore (\overline{f}i^{𝒜} )u is invertible by [Lyu03, Proposition 7.15].

3. Representable 2-functors A_{∞}^{u} → A_{∞}^{u}

Recall that unital A_{∞}-categories, unital A_{∞}-functors and equivalence classes of natural A_{∞}-transformations form a 2-category [Lyu03]. In order to distinguish between the A_{∞}-category A_{∞}^{u}(𝒞, 𝒟) and the ordinary category, whose morphisms are equivalence classes of natural A_{∞}-transformations, we denote the latter by

\overline{A_{∞}^{u}}(𝒞, 𝒟) = H^{0}(A_{∞}^{u}(𝒞, 𝒟), m_{1}).

The corresponding notation for the 2-category is \overline{A_{∞}^{u}}. We will see that arbitrary A_{N}-cat-egories can be viewed as 2-functors \overline{A_{∞}^{u}} → \overline{A_{∞}^{u}}. Moreover, they come from certain generalizations called A_{∞}^{u}-2-functors. There is a notion of representability of such 2-functors, which explains some constructions of A_{∞}-categories. For instance, a differential graded 𝕜-quiver 𝒬 will be represented by the free A_{∞}-category ℱ𝒬 generated by it.

3.1. Definition. A (strict) A_{∞}^{u}-2-functor F : A_{∞}^{u} → A_{∞}^{u} consists of

1. a map F : Ob A_{∞}^{u} → Ob A_{∞}^{u};

2. a unital A_{∞}-functor F = F_{𝒞,𝒟} : A_{∞}^{u}(𝒞, 𝒟) → A_{∞}^{u}(F 𝒞, F 𝒟) for each pair 𝒞, 𝒟 of unital A_{∞}-categories;

such that

3. id_{F 𝒞} = F (id_{𝒞}) for any unital A_{∞}-category 𝒞;

4. the equation

\begin{aligned}T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}^{u}(𝒟, ℰ) & \overline{^{M} →} & T sA_{∞}^{u}(𝒞, ℰ) \\ & = \\ T sA_{∞}^{u}(F 𝒞, F 𝒟) ⊠ T sA_{∞}^{u}(F 𝒟, F ℰ) & ↓^{F_{𝒞,𝒟} ⊠F_{𝒟,ℰ}} & →^{M} & T sA_{∞}^{u}(F 𝒞, F ℰ) & ↓^{F_{𝒞,ℰ}} \end{aligned}  (39)

holds strictly for each triple 𝒞, 𝒟, ℰ of unital A_{∞}-categories.

The A_{∞}-functor F : A_{∞}^{u}(𝒞, 𝒟) → A_{∞}^{u}(F 𝒞, F 𝒟) consists of the mapping of objects

Ob F : Ob A_{∞}^{u}(𝒞, 𝒟) → Ob A_{∞}^{u}(F 𝒞, F 𝒟), f ↦ F f,

and the components F_{k}, k ⩾ 1:

F_{1} : sA_{∞}^{u}(𝒞, 𝒟)(f, g) → sA_{∞}^{u}(F 𝒞, F 𝒟)(F f, F g),
F_{2} : sA_{∞}^{u}(𝒞, 𝒟)(f, g) ⊗ sA_{∞}^{u}(𝒞, 𝒟)(g, h) → sA_{∞}^{u}(F 𝒞, F 𝒟)(F f, F h),

and so on.

Weak versions of A_{∞}^{u}-2-functors and 2-transformations between them might be considered elsewhere.

3.2. Definition. A (strict) A_{∞}^{u}-2-transformation λ : F → G : A_{∞}^{u} → A_{∞}^{u} of strict A_{∞}^{u}-2-functors is

1. a family of unital A_{∞}-functors λ_{𝒞} : F 𝒞 → G𝒞, 𝒞 ∈ Ob A_{∞}^{u};

such that

2. the diagram of A_{∞}-functors

\begin{aligned}A_{∞}^{u}(𝒞, 𝒟) & \overline{^{F} →} & A_{∞}^{u}(F 𝒞, F 𝒟) \\ & = \\ A_{∞}^{u}(G𝒞, G𝒟) & ↓^{G} & \overline{→^{(λ_{𝒞} ⊠1)M}} & A_{∞}^{u}(F 𝒞, G𝒟) & ↓^{(1⊠λ_{𝒟} )M} \end{aligned}  (40)

strictly commutes.

An A_{∞}^{u}-2-transformation λ = (λ_{𝒞}) for which λ_{𝒞} are A_{∞}-equivalences is called a natural A_{∞}^{u}-2-equivalence.

Let us show now that the above notions induce ordinary strict 2-functors and strict 2-transformations in 0-th cohomology. Recall that the strict 2-category \overline{A_{∞}^{u}} consists of objects – unital A_{∞}-categories, the category \overline{A_{∞}^{u}}(𝒞, 𝒟) for any pair of objects 𝒞, 𝒟, the identity functor id_{𝒞} for any unital A_{∞}-category 𝒞, and the composition functor [Lyu03]

\overline{A_{∞}^{u}}(𝒞, 𝒟)(f, g) × \overline{A_{∞}^{u}}(𝒟, ℰ)(h, k) ^{•^{2}}→ \overline{A_{∞}^{u}}(𝒞, ℰ)(fh, gk),
(rs^{−1}, ps^{−1}) → (rhs^{−1} ⊗ gps^{−1})m_{2}.

Given a strict A_{∞}^{u}-2-functor F as in Definition 3.1 we construct from it an ordinary strict 2-functor \overline{F} = F : Ob \overline{A_{∞}^{u}} → Ob \overline{A_{∞}^{u}}, \overline{F} = H^{0}(sF_{1}s^{−1}) : \overline{A_{∞}^{u}}(𝒞, 𝒟) → \overline{A_{∞}^{u}}(F 𝒞, F 𝒟) as follows.

Denote

M_{10} ⊙ M_{01} = \{sA_{∞}^{u}(𝒞, 𝒟) ⊠ sA_{∞}^{u}(𝒟, ℰ) \begin{matrix}Δ_{10}⊠Δ_{01} \\ ∼ \end{matrix} →

[sA_{∞}^{u}(𝒞, 𝒟) ⊗ T^{0}sA_{∞}^{u}(𝒞, 𝒟)] ⊠ [T^{0}sA_{∞}^{u}(𝒟, ℰ) ⊗ sA_{∞}^{u}(𝒟, ℰ)]
∼→ [sA_{∞}^{u}(𝒞, 𝒟) ⊠ T^{0}sA_{∞}^{u}(𝒟, ℰ)] ⊗ [T^{0}sA_{∞}^{u}(𝒞, 𝒟) ⊠ sA_{∞}^{u}(𝒟, ℰ)]

M_{10}⊗M_{01} → sA_{∞}^{u}(𝒞, ℰ) ⊗ sA_{∞}^{u}(𝒞, ℰ)}, (41)

where the obvious isomorphisms Δ_{10} and Δ_{01} are components of the comultiplication Δ, the middle isomorphism is that of distributivity law (6), and the components M_{10} and M_{01} of M are the composition maps.

Property (39) of F implies that

(M_{10} ⊙ M_{01})(F_{1} ⊗ F_{1}) = (F_{1} ⊠ F_{1})(M_{10} ⊙ M_{01}).  (42)

Indeed, the following diagram commutes

\begin{matrix}sA_{∞}^{u}(𝒞, 𝒟) ⊠ T^{0}sA_{∞}^{u}(𝒟, ℰ) & \overline{^{M_{10}}→} & sA_{∞}^{u}(𝒞, ℰ) \\ sA_{∞}^{u}(F 𝒞, F 𝒟) ⊠ T^{0}sA_{∞}^{u}(F 𝒟, F ℰ) & ↓^{F_{1}⊠Ob F} & →^{M_{10}} & sA_{∞}^{u}(F 𝒞, F ℰ) & ↓^{F_{1}} \end{matrix}

due to (39). ⊗-tensoring it with one more similar diagram we get

(M_{10} ⊗ M_{01})(F_{1} ⊗ F_{1}) = [(F_{1} ⊠ Ob F) ⊗ (Ob F ⊠ F_{1})](M_{10} ⊗ M_{01}).

The isomorphisms in (41) commute with F in expected way, so (42) follows. We claim that the diagram

\begin{matrix}sA_{∞}^{u}(𝒞, 𝒟) ⊠ sA_{∞}^{u}(𝒟, ℰ) & \overline{^{(M_{10}⊙M_{01})B_{2}}→} & sA_{∞}^{u}(𝒞, ℰ) \\ sA_{∞}^{u}(F 𝒞, F 𝒟) ⊠ sA_{∞}^{u}(F 𝒟, F ℰ) & ↓^{F_{1}⊠F_{1}} & \overline{→^{(M_{10}⊙M_{01})B_{2}}} & sA_{∞}^{u}(F 𝒞, F ℰ) & ↓^{F_{1}} \end{matrix}  (43)

homotopically commutes. Indeed, since

(1 ⊗ B_{1} + B_{1} ⊗ 1)F_{2} + B_{2}F_{1} = (F_{1} ⊗ F_{1})B_{2} + F_{2}B_{1},

we get

(M_{10} ⊙ M_{01})B_{2}F_{1}

= (M_{10} ⊙ M_{01})(F_{1} ⊗ F_{1})B_{2} + (M_{10} ⊙ M_{01})F_{2}B_{1} − (M_{10} ⊙ M_{01})(1 ⊗ B_{1} + B_{1} ⊗ 1)F_{2} = (F_{1} ⊠ F_{1})(M_{10} ⊙ M_{01})B_{2} + (M_{10} ⊙ M_{01})F_{2}B_{1} − (1 ⊠ B_{1} + B_{1} ⊠ 1)(M_{10} ⊙ M_{01})F_{2}.

We have used equations

(M_{10} ⊙ M_{01})(1 ⊗ B_{1}) = (1 ⊠ B_{1})(M_{10} ⊙ M_{01}),
(M_{10} ⊙ M_{01})(B_{1} ⊗ 1) = (B_{1} ⊠ 1)(M_{10} ⊙ M_{01}),

which can be proved similarly to (42) due to M being an A_{∞}-functor. Passing to cohomology we get from (43) a strictly commutative diagram of functors

H^{0}(A_{∞}^{u}(𝒞, 𝒟) ⊠ A_{∞}^{u}(𝒟, ℰ))  \overline{^{•^{2}} →} H^{0}(A_{∞}^{u}(𝒞, ℰ))

=

H^{0}(A_{∞}^{u}(F 𝒞, F 𝒟) ⊠ A_{∞}^{u}(F 𝒟, F ℰ)) ↓^{H^{0}(sF_{1}s^{−1}⊠sF_{1}s^{−1})}  →^{•^{2}} H^{0}(A_{∞}^{u}(F 𝒞, F ℰ)) ↓^{H^{0}(sF_{1}s^{−1})}

since •^{2} = H^{0}((s ⊠ s)(M_{10} ⊙ M_{01})B_{2}s^{−1}). Using the Künneth map we come to strictly commutative diagram of functors

\overline{A_{∞}^{u}}(𝒞, 𝒟) × \overline{A_{∞}^{u}}(𝒟, ℰ)  \overline{^{•^{2}} →} \overline{A_{∞}^{u}}(𝒞, ℰ)

=

\overline{A_{∞}^{u}}(F 𝒞, F 𝒟) × \overline{A_{∞}^{u}}(F 𝒟, F ℰ) ↓^{H^{0}(sF_{1}s^{−1})×H^{0}(sF_{1}s^{−1})}  →^{•^{2}} \overline{A_{∞}^{u}}(F 𝒞, F ℰ) ↓^{H^{0}(sF_{1}s^{−1})}

that is, to a usual strict 2-functor \overline{F} : \overline{A_{∞}^{u}} → \overline{A_{∞}^{u}}.

Let us show that an A_{∞}^{u}-2-transformation λ : F → G : A_{∞}^{u} → A_{∞}^{u} as in Definition 3.2 induces an ordinary strict 2-transformation \overline{λ} : \overline{F} → \overline{G} : \overline{A_{∞}^{u}} → \overline{A_{∞}^{u}} in cohomology. Indeed, diagram (40) implies commutativity of diagram

sA_{∞}^{u}(𝒞, 𝒟)  \overline{^{F_{1}}  →} sA_{∞}^{u}(F 𝒞, F 𝒟)

=

sA_{∞}^{u}(G𝒞, G𝒟) ↓^{G_{1}}  \overline{^{(λ_{𝒞} ⊠1)M}  →} sA_{∞}^{u}(F 𝒞, G𝒟) ↓^{(1⊠λ_{𝒟} )M_{10}}
r : f → g : G𝒞 → G𝒟  → λ_{𝒞}r : λ_{𝒞}f → λ_{𝒞}g : F 𝒞 → G𝒟.

Passing to cohomology we get

\overline{A_{∞}^{u}}(𝒞, 𝒟) \overline{→^{H^{0}(sF_{1}s^{−1})}} \overline{A_{∞}^{u}}(F 𝒞, F 𝒟)

=

\overline{A_{∞}^{u}}(G𝒞, G𝒟) ↓^{H^{0}(sG_{1}s^{−1})}  \overline{\begin{matrix}λ_{𝒞} · \\ \overline{A_{∞}^{u}}(λ_{𝒞} ,G𝒟) \end{matrix}→} \overline{A_{∞}^{u}}(F 𝒞, G𝒟) ↓^{·λ_{𝒟} =\overline{A_{∞}^{u}}(F 𝒞,λ_{𝒟} )}

Therefore, \overline{λ_{𝒞}} ∈ Ob \overline{A_{∞}^{u}}(F 𝒞, G𝒞) form a strict 2-transformation \overline{λ} : \overline{F} → \overline{G} : \overline{A_{∞}^{u}} → \overline{A_{∞}^{u}}.

3.3. Examples of A_{∞}^{u}-2-functors. Let 𝒜 be an A_{N}-category, 1 ⩽ N ⩽ ∞. It determines an A_{∞}^{u}-2-functor F = A_{N}(𝒜, ) : A_{∞}^{u} → A_{∞}^{u}, given by the following data:

1. the map F : Ob A_{∞}^{u} → Ob A_{∞}^{u}, 𝒞 ↦ A_{N}(𝒜, 𝒞) (the category A_{N}(𝒜, 𝒞) is unital by [Lyu03, Proposition 7.7]);

2. the unital strict A_{∞}-functor F = A_{N}(𝒜, ) : A_{∞}^{u}(𝒞, 𝒟) → A_{∞}^{u}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, 𝒟)) for each pair 𝒞, 𝒟 of unital A_{∞}-categories (cf. [Lyu03, Propositions 6.2, 8.4]).

Clearly, id_{A_{N} (𝒜,𝒞)} = (1⊠id_{𝒞})M = A_{N}(𝒜, id_{𝒞}). We want to prove now that the equation

[T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}^{u}(𝒟, ℰ) ^{M}→ T sA_{∞}^{u}(𝒞, ℰ) ^{A_{N} (𝒜, )} → T sA_{∞}^{u}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, ℰ))]

= [T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}^{u}(𝒟, ℰ) ^{A_{N} (𝒜, )⊠A_{N} (𝒜, )}→
T sA_{∞}^{u}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, 𝒟)) ⊠ T sA_{∞}^{u}(A_{N}(𝒜, 𝒟), A_{N}(𝒜, ℰ))

M→ T sA_{∞}^{u}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, ℰ))] (44)

holds strictly for each triple 𝒞, 𝒟, ℰ of unital A_{∞}-categories. In fact, this F is a restriction of an A_{∞}-2-functor F : Ob A_{∞} → Ob A_{∞}, 𝒞 ↦ A_{N}(𝒜, 𝒞), which is defined just as in Definition 3.1 without mentioning the unitality. Equation (44) follows from a similar equation without the unitality index u. To prove it we consider the compositions

[T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒟, ℰ) ^{1⊠M} → T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, ℰ)

1⊠A_{N} (𝒜, ) → T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, ℰ)) ^{α}→ T sA_{N}(𝒜, ℰ)]
= [T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒟, ℰ)
1⊠M → T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, ℰ) ^{M}→ T sA_{N}(𝒜, ℰ)]
= [T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒟, ℰ)
M⊠1 → T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, ℰ) ^{M}→ T sA_{N}(𝒜, ℰ)]
= [T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒟, ℰ) ^{1⊠A_{N} (𝒜, )⊠1} →
T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, 𝒟)) ⊠ T sA_{∞}(𝒟, ℰ)
α⊠1 → T sA_{N}(𝒜, 𝒟) ⊠ T sA_{∞}(𝒟, ℰ)
1⊠A_{N} (𝒜, ) → T sA_{N}(𝒜, 𝒟) ⊠ T sA_{∞}(A_{N}(𝒜, 𝒟), A_{N}(𝒜, ℰ)) ^{α}→ T sA_{N}(𝒜, ℰ)]
= [T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒟, ℰ) ^{1⊠A_{N} (𝒜, )⊠A_{N} (𝒜, )}→
T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, 𝒟)) ⊠ T sA_{∞}(A_{N}(𝒜, 𝒟), A_{N}(𝒜, ℰ))

1⊠M → T sA_{N}(𝒜, 𝒞) ⊠ T sA_{∞}(A_{N}(𝒜, 𝒞), A_{N}(𝒜, ℰ)) ^{α}→ T sA_{N}(𝒜, ℰ)].

By Proposition 1.6 we deduce equation (44) (see also [Lyu03, Proposition 5.5]).

Let now 𝒜 be a unital A_{∞}-category. It determines an A_{∞}^{u}-2-functor G = A_{∞}^{u}(𝒜, ) : A_{∞}^{u} → A_{∞}^{u}, given by the following data:

1. the map G : Ob A_{∞}^{u} → Ob A_{∞}^{u}, 𝒞 ↦ A_{∞}^{u}(𝒜, 𝒞) (the category A_{∞}^{u}(𝒜, 𝒞) is unital by [Lyu03, Proposition 7.7]);

2. the unital strict A_{∞}-functor G = A_{∞}^{u}(𝒜, ) : A_{∞}^{u}(𝒞, 𝒟) → A_{∞}^{u}(A_{∞}^{u}(𝒜, 𝒞), A_{∞}^{u}(𝒜, 𝒟)) for each pair 𝒞, 𝒟 of unital A_{∞}-categories, determined from

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

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

(cf. [Lyu03, Propositions 6.2, 8.4]).

Clearly, G𝒞 are full A_{∞}-subcategories of F 𝒞 for the A_{∞}^{u}-2-functor F = A_{∞}(𝒜, ). Furthermore, A_{∞}-functors G_{𝒞,𝒟} (f) are restrictions of A_{∞}-functors F_{𝒞,𝒟} (f), so G is a full A_{∞}^{u}-2-subfunctor of F. In particular, G satisfies equation (39). Another way to prove that G is an A_{∞}^{u}-2-functor is to repeat the reasoning concerning F.

3.4. Example of an A_{∞}^{u}-2-equivalence. Assume that 𝒬 is a differential graded 𝕜-quiver. As usual, ℱ𝒬 denotes the free A_{∞}-category generated by it. We claim that restr : A_{∞}(ℱ𝒬, ) → A_{1}(𝒬, ) : A_{∞}^{u} → A_{∞}^{u} is a strict 2-natural A_{∞}-equivalence. Indeed, it is given by the family of unital A_{∞}-functors restr_{𝒞} : A_{∞}(ℱ𝒬, 𝒞) → A_{1}(𝒬, 𝒞), 𝒞 ∈ Ob A_{∞}^{u}, which are equivalences by Theorem 2.12. We have to prove that the diagram of A_{∞}-func-tors

A_{∞}^{u}(𝒞, 𝒟)  \overline{^{A_{∞}(ℱ𝒬, )}→} A_{∞}^{u}(A_{∞}(ℱ𝒬, 𝒞), A_{∞}(ℱ𝒬, 𝒟))

=

A_{∞}^{u}(A_{1}(𝒬, 𝒞), A_{1}(𝒬, 𝒟)) ↓^{A_{1}(𝒬, )}  \overline{→^{(restr_{𝒞} ⊠1)M}} A_{∞}^{u}(A_{∞}(ℱ𝒬, 𝒞), A_{1}(𝒬, 𝒟)) ↓^{(1⊠restr_{𝒟} )M}

(45)

commutes. Notice that all arrows in this diagram are strict A_{∞}-functors. Indeed, A_{∞}(ℱ𝒬, ) and A_{1}(𝒬, ) are strict by [Lyu03, Proposition 6.2]. For an arbitrary A_{∞}-functor f the components [(f ⊠ 1)M]_{n} = (f ⊠ 1)M_{0n} vanish for all n except for n = 1, thus, (f ⊠ 1)M is strict. The A_{∞}-functor g = restr_{𝒟} is strict, hence, the n-th component

[(1 ⊠ g)M]_{n} : r^{1} ⊗ ⋯ ⊗ r^{n} ↦ (r^{1} ⊗ ⋯ ⊗ r^{n} | g)M_{n0}

of the A_{∞}-functor (1 ⊠ g)M satisfies the equation

[(r^{1} ⊗ ⋯ ⊗ r^{n} | g)M_{n0}]_{k} = (r^{1} ⊗ ⋯ ⊗ r^{n})θ_{k1}g_{1}.

If the right hand side does not vanish, then n ⩽ 1 ⩽ k + n, so n = 1 and (1 ⊠ g)M is strict.

Given an A_{∞}-transformation t : g → h : 𝒞 → 𝒟 between unital A_{∞}-functors we find that

A_{∞}(ℱ𝒬, )(t) = [(1 ⊠ t)M : (1 ⊠ g)M → (1 ⊠ h)M : A_{∞}(ℱ𝒬, 𝒞) → A_{∞}(ℱ𝒬, 𝒟)],
A_{1}(𝒬, )(t) = [(1 ⊠ t)M : (1 ⊠ g)M → (1 ⊠ h)M : A_{1}(𝒬, 𝒞) → A_{1}(𝒬, 𝒟)],
[(1 ⊠ restr_{𝒟} )M]A_{∞}(ℱ𝒬, )(t) = [((1 ⊠ t)M) · restr_{𝒟} : ((1 ⊠ g)M) · restr_{𝒟}
→ ((1 ⊠ h)M) · restr_{𝒟} : A_{∞}(ℱ𝒬, 𝒞) → A_{1}(𝒬, 𝒟)],
[(restr_{𝒞} ⊠1)M]A_{1}(𝒬, )(t) = [restr_{𝒞} ·((1 ⊠ t)M) : restr_{𝒞} ·((1 ⊠ g)M)
→ restr_{𝒞} ·((1 ⊠ h)M) : A_{∞}(ℱ𝒬, 𝒞) → A_{1}(𝒬, 𝒟)].

We have to verify that the last two A_{∞}-transformations are equal. First of all, let us show that mappings of objects in (45) commute. Given a unital A_{∞}-functor g : 𝒞 → 𝒟, we are going to check that

[(1 ⊠ g)M]_{n} · restr_{1} = restr_{1}^{⊗n} ·[(1 ⊠ g)M]_{n}  (46)

for any n ⩾ 1. Indeed, for any n-tuple of composable A_{∞}-transformations

f^{0} ^{r^{1}}→ f^{1} → … ^{r^{n}}→ f^{n} : ℱ𝒬 → 𝒞,

we have in both cases

{(r^{1} ⊗ ⋯ ⊗ r^{n})[(1 ⊠ g)M]_{n}}_{0} = [(r^{1} ⊗ ⋯ ⊗ r^{n}|g)M_{n0}]_{0} = (r_{0}^{1} ⊗ ⋯ ⊗ r_{0}^{n})g_{n},
{(r^{1} ⊗ ⋯ ⊗ r^{n})[(1 ⊠ g)M]_{n}}_{1} = [(r^{1} ⊗ ⋯ ⊗ r^{n}|g)M_{n0}]_{1}

= ∑_{i=1}^{n}(r_{0}^{1} ⊗ ⋯ ⊗ r_{0}^{i−1} ⊗ r_{1}^{i} ⊗ r_{0}^{i+1} ⊗ ⋯ ⊗ r_{0}^{n})g_{n}

+ ∑_{i=1}^{n}(r_{0}^{1} ⊗ ⋯ ⊗ r_{0}^{i−1} ⊗ f_{1}^{i} ⊗ r_{0}^{i+1} ⊗ ⋯ ⊗ r_{0}^{n})g_{n+1}.

Note that the right hand sides depend only on 0-th and 1-st components of r^{i}, f^{i}. This is precisely what is claimed by equation (46).

The coincidence of A_{∞}-transformations ((1 ⊠ t)M) · restr_{𝒟} = restr_{𝒞} ·((1 ⊠ t)M) follows similarly from the computation:

{(r^{1} ⊗ ⋯ ⊗ r^{n})[(1 ⊠ t)M]_{n}}_{0} = [(r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ t)M_{n1}]_{0} = (r_{0}^{1} ⊗ ⋯ ⊗ r_{0}^{n})t_{n},
{(r^{1} ⊗ ⋯ ⊗ r^{n})[(1 ⊠ t)M]_{n}}_{1} = [(r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ t)M_{n1}]_{1}

= ∑_{i=1}^{n}(r_{0}^{1} ⊗ ⋯ ⊗ r_{0}^{i−1} ⊗ r_{1}^{i} ⊗ r_{0}^{i+1} ⊗ ⋯ ⊗ r_{0}^{n})t_{n}

+ ∑_{i=1}^{n}(r_{0}^{1} ⊗ ⋯ ⊗ r_{0}^{i−1} ⊗ f_{1}^{i} ⊗ r_{0}^{i+1} ⊗ ⋯ ⊗ r_{0}^{n})t_{n+1}.

3.5. Representability. An A_{∞}^{u}-2-functor F : A_{∞}^{u} → A_{∞}^{u} is called representable, if it is naturally A_{∞}^{u}-2-equivalent to the A_{∞}^{u}-2-functor A_{∞}(𝒜, ) : A_{∞}^{u} → A_{∞}^{u} for some A_{∞}-category 𝒜. The above results imply that the A_{∞}^{u}-2-functor A_{1}(𝒬, ) corresponding to a differential graded 𝕜-quiver 𝒬 is represented by the free A_{∞}-category ℱ𝒬 generated by 𝒬.

This definition of representability has a disadvantage: many different A_{∞}-categories can represent the same A_{∞}^{u}-2-functor. More attractive notion is the following. An A_{∞}^{u}-2-functor F : A_{∞}^{u} → A_{∞}^{u} is called unitally representable, if it is naturally A_{∞}^{u}-2-equivalent to the A_{∞}^{u}-2-functor A_{∞}^{u}(𝒜, ) : A_{∞}^{u} → A_{∞}^{u} for some unital A_{∞}-category 𝒜. Such 𝒜 is unique up to an A_{∞}-equivalence. Indeed, composing a natural 2-equivalence \overline{λ} : \overline{A_{∞}^{u}(𝒜, )} → \overline{A_{∞}^{u}(ℬ, )} : \overline{A_{∞}^{u}} → \overline{A_{∞}^{u}} with the 0-th cohomology 2-functor H^{0} : \overline{A_{∞}^{u}} → 𝒞at, we get a natural 2-equivalence H^{0}\overline{λ} : H^{0}\overline{A_{∞}^{u}(𝒜, )} → H^{0}\overline{A_{∞}^{u}(ℬ, )} : \overline{A_{∞}^{u}} → 𝒞at. However, H^{0}\overline{A_{∞}^{u}(𝒜, )} = \overline{A_{∞}^{u}}(𝒜, ), so using a 2-category version of Yoneda lemma one can deduce that 𝒜 and ℬ are equivalent in \overline{A_{∞}^{u}}. We shall present an example of unital representability in subsequent publication [LM04].

3.6. Acknowledgements. We are grateful to all the participants of the A_{∞}-category seminar at the Institute of Mathematics, Kyiv, for attention and fruitful discussions, especially to Yu. Bespalov and S. Ovsienko. One of us (V.L.) is grateful to Max-Planck-Institut für Mathematik for warm hospitality and support at the final stage of this research.

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

Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany

Email: lub@imath.kiev.ua
      manzyuk@mathematik.uni-kl.de

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