Quotients of unital A_{∞}-categories

Volodymyr Lyubashenko^{∗} and Oleksandr Manzyuk^{†}

April 1, 2004

Abstract Assuming that ℬ is a full A_{∞}-subcategory of a unital A_{∞}-category 𝒞 we construct the quotient unital A_{∞}-category 𝒟 =‘𝒞/ℬ’. It represents the A_{∞}-2-functor 𝒜 ↦ A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} , which associates with a given unital A_{∞}-category 𝒜 the A_{∞}-category of unital A_{∞}-functors 𝒞 → 𝒜, whose restriction to ℬ is contractible. Namely, there is a unital A_{∞}-functor e : 𝒞 → 𝒟 such that the composition ℬ ↪ 𝒞 ⟶^{e} 𝒟 is contractible, and for an arbitrary unital A_{∞}-category 𝒜 the restriction A_{∞}-functor (e ⊠ 1)M : A_{∞}^{u}(𝒟, 𝒜) → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} is an equivalence.

Let C be the differential graded category of differential graded 𝕜-modules. We prove that the Yoneda A_{∞}-functor Y : 𝒜 → A_{∞}^{u}(𝒜^{op}, C) is a full embedding for an arbitrary unital A_{∞}-category 𝒜. In particular, such 𝒜 is A_{∞}-equivalent to a differential graded category with the same set of objects.

Let 𝒜 be an Abelian category. The question: what is the quotient

{category of complexes in 𝒜}/{category of acyclic complexes}?

admits several answers. The first answer – the derived category of 𝒜 – was given by Grothendieck and Verdier [Ver77].

The second answer – a differential graded category 𝒟 – is given by Drinfeld [Dri02]. His article is based on the work of Bondal and Kapranov [BK90] and of Keller [Kel99]. The derived category D(𝒜) can be obtained as H^{0}(𝒟^{pre-tr}).

The third answer – an A_{∞}-category of bar-construction type – is given by Lyubashenko and Ovsienko [LO02]. This A_{∞}-category might turn out useful in additional assumptions, e.g. when the basic ring 𝕜 is a field. It is an A_{∞}-version of one of the constructions of Drinfeld [Dri02].

The fourth answer – an A_{∞}-category freely generated over the category of complexes in 𝒜 – is given in this article. It is likely to be related to the first three answers, but the precise relationship is not yet established.

∗Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine; lub@imath.kiev.ua

†Department of Algebra, Faculty of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 64 Volodymyrska st., Kyiv, 01033, Ukraine; manzyuk@univ.kiev.ua; Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany; manzyuk@mathematik.uni-kl.de

1. Introduction

Since A_{∞}-algebras were introduced by Stasheff [Sta63, II] there existed a possibility to consider A_{∞}-generalizations of categories. It did not happen until A_{∞}-categories were encountered in studies of mirror symmetry by Fukaya [Fuk93] and Kontsevich [Kon95]. A_{∞}-categories may be viewed as generalizations of differential graded categories for which the binary composition is associative only up to a homotopy. The possibility to define A_{∞}-functors was mentioned by Smirnov [Smi89], who reformulated one of his results in the language of A_{∞}-functors between differential graded categories. The definition of A_{∞}-functors between A_{∞}-categories was published by Keller [Kel01], who studied their applications to homological algebra. Homomorphisms of A_{∞}-algebras (e.g. [Kad82]) are particular cases of A_{∞}-functors.

A_{∞}-transformations between A_{∞}-functors are certain coderivations. Given two A_{∞}-cat-egories 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 (Fukaya [Fuk02], Kontsevich and Soibelman [KS02, KS99], Lefèvre-Hasegawa [LH03], as well as [Lyu03]). For an A_{∞}-category 𝒞 there is a homotopy invariant notion of unit elements (identity morphisms) [Lyu03]. They are cycles _{X}i_{0}^{𝒞} ∈ s𝒞(X, X) of degree −1 such that the maps (1 ⊗ i_{0}^{𝒞})b_{2}, −(i_{0}^{𝒞} ⊗ 1)b_{2} : s𝒞(X, Y) → s𝒞(X, Y) are homotopic to the identity map. This allows to define the 2-category \overline{A_{∞}^{u}}, whose objects are unital A_{∞}-categories (those which have units), 1-morphisms are unital A_{∞}-functors (their first components preserve the units up to a boundary) and 2-morphisms are equivalence classes of natural A_{∞}-transforma-tions [Lyu03]. We continue to study this 2-category. Notations and terminology follow [Lyu03], complemented by [LO02] and [LM03].

Unital A_{∞}-categories and unital A_{∞}-functors can be considered as strong homotopy generalizations of differential graded categories and functors. Let us illustrate the notion of A_{∞}-transformations in a familiar context.

1.1. Differential for A_{∞}-transformations compared with the Hochschild differential. Let 𝒜, ℬ be ordinary 𝕜-linear categories. We consider 𝒜( , ) and ℬ( , ) as complexes of 𝕜-modules concentrated in degree 0. This turns 𝒜 and ℬ into differential graded categories and, thereby, into unital A_{∞}-categories. An A_{∞}-functor between 𝒜 and ℬ is necessarily strict, for (s𝒜)^{⊗k} = 𝒜^{⊗k}[k] and sℬ = ℬ[1] are concentrated in different degrees if k > 1. Thus, a unital A_{∞}-functor f : 𝒜 → ℬ is the same as an ordinary 𝕜-linear functor f. Let f, g : 𝒜 → ℬ be 𝕜-linear functors. All complexes C((s𝒜)^{⊗k}(X, Y), sℬ(Xf, Y g)) are concentrated in degree k − 1. Their direct product

Ψ_{k} = ∏_{X,Y ∈Ob 𝒜} C((s𝒜)^{⊗k}(X, Y), sℬ(Xf, Y g))

is the same, whether taken in the category of 𝕜-modules or graded 𝕜-modules or complexes of 𝕜-modules. It is the module of k-th components of A_{∞}-transformations. The graded 𝕜-module of A_{∞}-transformations sA_{∞}(𝒜, ℬ)(f, g) is isomorphic to the direct product ∏_{k=0}^{∞} Ψ_{k} taken in the category of graded 𝕜-modules [Lyu03, Section 2.7]. That is, [sA_{∞}(𝒜, ℬ)(f, g)]^{n} = ∏_{k=0}^{∞} Ψ_{k}^{n}, where Ψ_{k}^{n} is the degree n part of Ψ_{k}. Therefore, in our case it simply coincides with the graded 𝕜-module Ψ[1] : ℤ ∋ n ↦ Ψ_{n+1} ∈ 𝕜 -mod. The graded 𝕜-module sA_{∞}(𝒜, ℬ)(f, g) is equipped with the differential B_{1}, rB_{1} = rb^{ℬ} − (−)^{r}b^{𝒜} r [Lyu03, Proposition 5.1]. Since the only non-vanishing component of b (resp. f) is b_{2} (resp. f_{1}), the explicit formula for components of rB_{1} is the following:

(rB_{1})_{k+1} = (f_{1} ⊗ r_{k})b_{2} + (r_{k} ⊗ g_{1})b_{2} − (−)^{r_{k}} ∑_{a+c=k−1}(1^{⊗a} ⊗ b_{2} ⊗ 1^{⊗c})r_{k}.

Recalling that deg r_{k} = k − 1, we get the differential in Ψ[1] also denoted B_{1}:

r_{k}B_{1} = (f_{1} ⊗ r_{k})b_{2} + (r_{k} ⊗ g_{1})b_{2} + (−)^{k} ∑_{a+c=k−1}(1^{⊗a} ⊗ b_{2} ⊗ 1^{⊗c})r_{k},

where r_{k} ∈ Ψ_{k}, r_{k}B_{1} ∈ Ψ_{k+1}. We consider an isomorphism of graded 𝕜-modules Ψ → Ψ′ : ℤ ∋ k ↦ Ψ′_{k} given by

Ψ_{k} ∋ r_{k} ↦ (s ⊗ ⋯ ⊗ s)r_{k}s^{−1} = s^{⊗k}r_{k}s^{−1} ∈ Ψ′_{k} =^{def} ∏_{X,Y ∈Ob 𝒜} C(𝒜^{⊗k}(X, Y), ℬ(Xf, Y g)).

Its inverse is Ψ′_{k} ∋ t_{k} ↦ (s^{⊗k})^{−1}t_{k}s ∈ Ψ_{k}. This isomorphism induces the differential

d : Ψ′_{k} → Ψ′_{k+1}, t_{k}d = s^{⊗k+1} · [(s^{⊗k})^{−1}t_{k}s]B_{1} · s^{−1}.

The explicit formula for d is

t_{k}d = (f ⊗ t_{k})m_{2} + ∑_{a+c=k−1}(−1)^{a+1}(1^{⊗a} ⊗ m_{2} ⊗ 1^{⊗c})t_{k} + (−1)^{k+1}(t_{k} ⊗ g)m_{2}.

Up to an overall sign this coincides with the differential in the Hochschild cochain complex C^{•}(𝒜,_{f}ℬ_{g}) (cf. [Mac63, Section X.3]). The 𝒜-bimodule _{f}ℬ_{g} acquires its left 𝒜-mod-ule structure via f and its right 𝒜-module structure via g. Therefore, in our situation A_{∞}-transformations are nothing else but Hochschild cochains. Natural A_{∞}-trans-formations r : f → g : 𝒜 → ℬ (such that deg r = −1 and rB_{1} = 0) are identified with the Hochschild cocycles of degree 0, that is, with natural transformations t = rs^{−1} : f → g : 𝒜 → ℬ in the ordinary sense.

When 𝒜, ℬ are differential graded categories and f, g : 𝒜 → ℬ are differential graded functors, we may still interpret the complex (sA_{∞}(𝒜, ℬ)(f, g), B_{1}) as the complex of Hochschild cochains C^{•}(𝒜,_{f}ℬ_{g}) for the differential graded category 𝒜 and the differential graded bimodule _{f}ℬ_{g}. Indeed, for a homogeneous element r ∈ sA_{∞}(𝒜, ℬ)(f, g) the components of rB_{1} are

(rB_{1})_{k} = r_{k}b_{1} + (f_{1} ⊗ r_{k−1})b_{2} + (r_{k−1} ⊗ g_{1})b_{2}

− (−)^{r} ∑_{a+1+c=k}(1^{⊗a} ⊗ b_{1} ⊗ 1^{⊗c})r_{k} − (−)^{r} ∑_{a+2+c=k}(1^{⊗a} ⊗ b_{2} ⊗ 1^{⊗c})r_{k−1}.

For A_{∞}-functors between differential graded categories or A_{∞}-categories the differential B_{1} is not interpreted as Hochschild differential any more. But we may view the complex of A_{∞}-transformations as a generalization of the Hochschild cochain complex.

1.2. Main result. By Definition 6.4 of [LO02] an A_{∞}-functor g : ℬ → 𝒜 from a unital A_{∞}-category ℬ is contractible if for all objects X, Y of ℬ the chain map g_{1} : sℬ(X, Y) → s𝒜(Xg, Y g) is null-homotopic. If g : ℬ → 𝒜 is a unital A_{∞}-functor, then it is contractible if and only if for any X ∈ Ob ℬ and any V ∈ Ob 𝒜 the complexes s𝒜(Xg, V) and s𝒜(V, Xg) are contractible. Equivalently, gi^{𝒜} ≡ 0 : g → g : ℬ → 𝒜 [LO02, Proposition 6.1(C5)]. Other equivalent conditions are listed in Propositions 6.1– 6.3 of [LO02].

Let ℬ be a full A_{∞}-subcategory of a unital A_{∞}-category 𝒞. Let 𝒜 be an arbitrary unital A_{∞}-category. Denote by A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} the full A_{∞}-subcategory of A_{∞}^{u}(𝒞, 𝒜), whose objects are unital A_{∞}-functors 𝒞 → 𝒜, whose restriction to ℬ is contractible. We allow to consider A_{∞}-categories with the empty set of objects.

1.3 Main Theorem. In the above assumptions there exists a unital A_{∞}-category 𝒟 and a unital A_{∞}-functor e : 𝒞 → 𝒟 such that

1) the composition ℬ^{⊂} → 𝒞 ^{e}→ 𝒟 is contractible;

2) the strict A_{∞}-functor given by composition with e

(e ⊠ 1)M : A_{∞}^{u}(𝒟, 𝒜) → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} , f ↦ ef,

is an A_{∞}-equivalence for an arbitrary unital A_{∞}-category 𝒜.

Proof. There exists a differential graded category 𝒞̃ with Ob 𝒞̃ = Ob 𝒞, and quasi-inverse to each other A_{∞}-functors Ỹ : 𝒞 → 𝒞̃, Ψ : 𝒞̃ → 𝒞 such that Ob Ỹ = Ob Ψ = id_{Ob 𝒞} (by Remark A.13 it follows from the A_{∞}-version of Yoneda Lemma – Theorem A.11). Let ℬ̃ ⊂ 𝒞̃ be the full differential graded subcategory with Ob ℬ̃ = Ob ℬ. By considerations in Appendix B (Corollaries B.10, B.11) if a unital A_{∞}-category 𝒟 and a unital A_{∞}-functor ẽ: 𝒞̃ → 𝒟 represent the A_{∞}-2-functor 𝒜 ↦ A_{∞}^{u}(𝒞̃, 𝒜)_{mod ℬ̃} in the sense 1), 2), then the

pair (𝒟, e = (𝒞 Ỹ→ 𝒞̃ ẽ→ 𝒟)) represents A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} . Indeed,

(e ⊠ 1)M = (A_{∞}^{u}(𝒟, 𝒜) ^{(ẽ⊠1)M} → A_{∞}^{u}(𝒞̃, 𝒜)_{mod ℬ̃} ^{(Ỹ ⊠1)M} → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} )

is a composition of two A_{∞}-equivalences.

For such ℬ̃ ⊂ 𝒞̃, and, more generally, for a full subcategory ℬ̃ of an arbitrary strictly unital A_{∞}-category 𝒞̃ the representing A_{∞}-category 𝒟 = Q(𝒞̃|ℬ̃) exists. It is constructed in Section 5.2 as an A_{∞}-category, freely generated over 𝒞̃ by application of contracting homotopies H to morphisms, whose source or target is in ℬ̃. The strict A_{∞}-functor ẽ: 𝒞̃ → 𝒟 is identity on objects and ẽ_{1} is an embedding. Theorem 6.5 asserts unitality of 𝒟 = Q(𝒞̃|ℬ̃).

The 2-category \overline{A_{∞}^{u}} = H^{0}A_{∞}^{u} has unital A_{∞}-categories as objects, unital A_{∞}-functors as 1-morphisms and equivalence classes of natural A_{∞}-transformations as 2-morphisms [Lyu03]. Thus, \overline{A_{∞}^{u}}(𝒞, 𝒜)(f, g) = [H^{0}A_{∞}^{u}(𝒞, 𝒜)](f, g) = H^{0}(A_{∞}^{u}(𝒞, 𝒜)(f, g), m_{1}).

A zero object of a category ℰ is an object Z, which is simultaneously initial and terminal. For a linear (Ab-enriched, not necessarily additive) category ℰ this can be formulated as follows: ℰ(Z, X) = 0 and ℰ(X, Z) = 0 for any object X of ℰ. This condition is equivalent to the equation 1_{Z} = 0 ∈ ℰ(Z, Z).

1.4 Corollary. The unital A_{∞}-functor e : 𝒞 → 𝒟 from the main theorem has the following property: composition with e in the sequence of functors

\overline{A_{∞}^{u}}(𝒟, 𝒜) ^{e•}→ \overline{A_{∞}^{u}}(𝒞, 𝒜) → \overline{A_{∞}^{u}}(ℬ, 𝒜)

is an equivalence of the category \overline{A_{∞}^{u}}(𝒟, 𝒜) with the full subcategory

Ker(\overline{A_{∞}^{u}}(𝒞, 𝒜) → \overline{A_{∞}^{u}}(ℬ, 𝒜)) = H^{0}(A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} , m_{1}) ⊂ \overline{A_{∞}^{u}}(𝒞, 𝒜),

consisting of such unital A_{∞}-functors f : 𝒞 → 𝒜 that its restriction f|_{ℬ} : ℬ → 𝒜 is a zero object of \overline{A_{∞}^{u}}(ℬ, 𝒜).

Proof. A unital A_{∞}-functor g : ℬ → 𝒜 is contractible if and only if gi^{𝒜} ≡ 0 : g → g : ℬ → 𝒜, that is, 1_{g} = 0 ∈ \overline{A_{∞}^{u}}(ℬ, 𝒜)(g, g). Thus, for unital g contractibility is equivalent to g being a zero object of \overline{A_{∞}^{u}}(ℬ, 𝒜).

The main theorem asserts that the chain map e• = s(e⊠1)M_{01}s^{−1} : A_{∞}^{u}(𝒟, 𝒜)(f, g) → A_{∞}^{u}(𝒞, 𝒜)(ef, eg) is a homotopy isomorphism, while Corollary 1.4 claims only that it induces isomorphism in 0-th homology.

1.5. Uniqueness of the representing A_{∞}-category. With each strict A_{∞}-2-functor F : A_{∞}^{u} → A_{∞}^{u} is associated an ordinary strict 2-functor \overline{F} : \overline{A_{∞}^{u}} → \overline{A_{∞}^{u}}, \overline{F} 𝒜 = F 𝒜 [LM03, Section 3.2]. With a strict A_{∞}-2-transformation λ = (λ_{𝒜} ) : F → G : A_{∞}^{u} → A_{∞}^{u} is associated an ordinary strict 2-transformation \overline{λ} = (\overline{λ_{𝒜}} ) : \overline{F} → \overline{G} : \overline{A_{∞}^{u}} → \overline{A_{∞}^{u}} in cohomology [ibid]. Assume that λ is a natural A_{∞}-2-equivalence. Since λ_{𝒜} : F 𝒜 → G𝒜 are A_{∞}-equivalences, the 1-morphisms \overline{λ_{𝒜}} : \overline{F} 𝒜 → \overline{G}𝒜 are equivalences in the 2-category \overline{A_{∞}^{u}}. Composing \overline{λ} with the 0-th cohomology 2-functor H^{0} : \overline{A_{∞}^{u}} → 𝒞at, we get a 2-natural equivalence H^{0}\overline{λ} : H^{0}\overline{F} → H^{0}\overline{G} : \overline{A_{∞}^{u}} → 𝒞at, which consists of equivalences of ordinary categories H^{0}(λ_{𝒜} ) : H^{0}(F 𝒜) → H^{0}(G𝒜). In particular, if F = A_{∞}^{u}(𝒟, ) for some unital A_{∞}-category 𝒟, then

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

Indeed, the both categories H^{0}\overline{A_{∞}^{u}(𝒟, 𝒜)} and \overline{A_{∞}^{u}}(𝒟, 𝒜) have unital A_{∞}-functors 𝒟 → 𝒜 as objects and equivalence classes of natural A_{∞}-transformations as morphisms. If λ : A_{∞}^{u}(𝒟, ) → G : A_{∞}^{u} → A_{∞}^{u} is a natural A_{∞}-2-equivalence (G is unitally representable by 𝒟), then H^{0}(λ_{𝒜} ) : \overline{A_{∞}^{u}}(𝒟, 𝒜) → H^{0}(G𝒜) : \overline{A_{∞}^{u}} → 𝒞at is a 2-natural equivalence. Thus, H^{0}\overline{G} is represented by 𝒟 in the 2-category sense and 𝒟 is unique up to an equivalence by Section C.17.

In particular, if G = A_{∞}^{u}(𝒞, )_{mod ℬ} , then with each object e of G𝒟 = A_{∞}^{u}(𝒞, 𝒟)_{mod ℬ} is associated a strict A_{∞}-2-transformation

λ = (e ⊠ 1)M : A_{∞}^{u}(𝒟, 𝒜) → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} : A_{∞}^{u} → A_{∞}^{u}.

We have identified H^{0}\overline{G}(𝒜) with Ker(\overline{A_{∞}^{u}}(𝒞, 𝒜) → \overline{A_{∞}^{u}}(ℬ, 𝒜)) in Corollary 1.4. The strict 2-natural equivalence H^{0}\overline{λ} : H^{0}\overline{F} → H^{0}\overline{G} : \overline{A_{∞}^{u}} → 𝒞at identifies with the strict 2-transformation

H^{0}\overline{G}λ^{𝒟,e} : \overline{A_{∞}^{u}}(𝒟, ) → H^{0}\overline{G} : \overline{A_{∞}^{u}} → 𝒞at

from Proposition C.11, since

(f : 𝒟 → 𝒜) ^{(e⊠1)M} → ef = (e)(H^{0}\overline{G}(f)) =^{def}: (f)^{H^{0}\overline{G}}λ^{𝒟,e} ,
(r : f → g : 𝒟 → 𝒜) ^{(e⊠1)M} → er = (e)(H^{0}\overline{G}(r)) =^{def}: (r)^{H^{0}\overline{G}}λ^{𝒟,e} .

Therefore, the pair (𝒟, e) represents the strict 2-functor H^{0}\overline{G} : \overline{A_{∞}^{u}} → 𝒞at in the sense of Definition C.16.

1.6 Corollary. The pair (𝒟, e : 𝒞 → 𝒟) is unique up to an equivalence in the sense of Section C.17.

The proof immediately follows from results of Section C.17.

The unital A_{∞}-category 𝒟 obtained in the main theorem can be replaced with a differential graded category by the A_{∞}-version of Yoneda Lemma (Theorem A.11). We may restrict Corollary 1.4 to differential graded categories ℬ, 𝒞, 𝒜 for the same reason. Then it becomes parallel to the second half of main Theorem 1.6.2 of Drinfeld’s work [Dri02], which asserts exactness of the sequence of categories

T (𝒟, 𝒜) → T (𝒞, 𝒜) → T (ℬ, 𝒜)

in the same sense as in Corollary 1.4. Here T is a certain 2-category whose objects are differential graded categories. It is not known in general whether categories T (𝒞, 𝒜) and \overline{A_{∞}^{u}}(𝒞, 𝒜) are equivalent. If 𝕜 is a field, then, as B. Keller explained to us, equivalence of \overline{A_{∞}^{u}}(𝒞, 𝒜) and T (𝒞, 𝒜) can be deduced from results of Lefèvre-Hasegawa [LH03, Section 8.2].

1.7. Basic properties of the main construction. The proof of universality of 𝒟 = Q(𝒞|ℬ) is based on the fact that 𝒟 is relatively free over 𝒞, that is, it admits a filtration

𝒞 = 𝒟_{0} ⊂ 𝒬_{1} ⊂ 𝒟_{1} ⊂ 𝒬_{2} ⊂ 𝒟_{2} ⊂ 𝒬_{3} ⊂ ⋯ ⊂ 𝒟

by A_{∞}-subcategories 𝒟_{j} and differential graded subquivers 𝒬_{j}, such that the graded subquiver 𝒟_{j} ⊂ 𝒬_{j+1} has a direct complement 𝒩_{j+1} (a graded subquiver of 𝒬_{j+1}), and such that 𝒟_{j+1} is generated by 𝒩_{j+1} over 𝒟_{j}. The precise conditions are given in Definition 5.1 (see also Proposition 2.1). This filtration allows to write down a sequence of restriction A_{∞}-functors

A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} ← A_{∞1}^{ψu}(𝒟_{0}, 𝒬_{1}; 𝒜) ← A_{∞}^{ψu}(𝒟_{1}, 𝒜)

← A_{∞1}^{ψu}(𝒟_{1}, 𝒬_{2}; 𝒜) ← A_{∞}^{ψu}(𝒟_{2}, 𝒜) ← A_{∞1}^{ψu}(𝒟_{2}, 𝒬_{3}; 𝒜) ← … (5.13.1)

and to prove that each of these A_{∞}-functors is an equivalence, surjective on objects (Theorem 4.7, Propositions 5.7, 5.10). The category A_{∞1}^{ψu}(𝒟_{j}, 𝒬_{j+1}; 𝒜) is defined via pullback square (4.2.1)

\begin{aligned}A_{∞1}^{ψu}(𝒟_{j}, 𝒬_{j+1}; 𝒜) & → & A_{1}(𝒬_{j+1}, 𝒜) \\ A_{∞}^{ψu}(𝒟_{j}, 𝒜) & ↓ & \overline{→} & A_{1}(𝒟_{j}, 𝒜) & ↓ \end{aligned}

The A_{∞}-categories 𝒟_{j} are not unital, but only pseudounital – there are distinguished cycles i_{0}^{𝒞} ∈ (s𝒟_{j})^{−1}, which are not unit elements of 𝒟_{j} if j > 0. The index ψu in A_{∞}^{ψu} indicates that we consider pseudounital A_{∞}-functors – a generalization of unital ones (Definition 4.1). Their first components preserve the distinguished cycles up to a boundary. The A_{∞}-equivalence A_{∞}^{u}(𝒟, 𝒜) → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} is the limit case of (5.13.1) (Theorem 5.13).

The proof of unitality of 𝒟 = Q(𝒞|ℬ) for strictly unital 𝒞 is based on the study of multicategory of A_{∞}-operations and contracting homotopies operating in 𝒟 (Theorem 6.5).

1.8. Description of various results. The proof of Theorem 4.7 is based on description of chain maps P → sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) to the complex of (ϕ, ψ)-coderivations (Proposition 2.6), where ℱ𝒬 is the free A_{∞}-category, generated by a differential graded quiver 𝒬. A similar result for the quotient ℱ𝒬/s^{−1}I over an A_{∞}-ideal I is given in Proposition 2.9. Of course, any A_{∞}-category is a quotient of a free one (Proposition 3.2). We also describe homotopies between chain maps P → sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) (Corollary 2.7), and generalize the result to quotients ℱ𝒬/s^{−1}I (Corollary 2.10).

In Section 7 we consider the example of differential graded category 𝒞 = C(𝒜) of complexes in a 𝕜-linear Abelian category 𝒜, and the full subcategory ℬ ⊂ 𝒞 of acyclic complexes. The functor H^{0}e factors through a functor D(𝒜) → H^{0}(Q(𝒞|ℬ)) by Corollary 7.2.

In Appendix A we define, following Fukaya [Fuk02, Lemma 9.8], the Yoneda A_{∞}-func-tor Y : 𝒜^{op} → A_{∞}(𝒜, C), where C is the differential graded category of complexes of 𝕜-modules. We prove for an arbitrary unital A_{∞}-category 𝒜 that the Yoneda A_{∞}-func-tor Y is an equivalence of 𝒜^{op} with its image – full differential graded subcategory of A_{∞}(𝒜, C) (Theorem A.11). This is already proven by Fukaya in the case of strictly unital A_{∞}-category 𝒜 [Fuk02, Theorem 9.1]. As a corollary we deduce that any 𝒰 -small unital A_{∞}-category 𝒜 is A_{∞}-equivalent to a 𝒰 -small differential graded category (Corollary A.12).

In Appendix C we lift the classical Yoneda Lemma one dimension up – to strict 2-categories, weak 2-functors and weak 2-transformations.

An important result from another paper is recalled in simplified form, in which it is used in the present paper:

1.9 Corollary (to Theorem 8.8 [Lyu03]). 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 homotopy invertible. If Ob ϕ : Ob 𝒞 → Ob ℬ is surjective, then 𝒞 is unital and ϕ is an A_{∞}-equivalence.

Proof. Let h : Ob ℬ → Ob 𝒞 be an arbitrary mapping such that h · Ob ϕ = id_{Ob ℬ} . The remaining data required in Theorem 8.8 of [Lyu03] can be chosen as _{U}r_{0} = _{U}p_{0} = _{U}i_{0}^{ℬ} : 𝕜 → (sℬ)^{−1}(U, U) for all objects U of ℬ. We conclude by this theorem that there exists an A_{∞}-functor ψ : ℬ → 𝒞 with Ob ψ = h, quasi-inverse to ϕ.

Logical dependence of sections is the following. Appendices A and C do not depend on other sections. Appendix B depends on Appendix A. Sections 2–7 depend on appendices and on sections with smaller number (excluding the first section). The first section depends on all the rest of the article.

1.10. Conventions and preliminaries. We keep the notations and conventions of [Lyu03, LO02, LM03], sometimes without explicit mentioning. Some of the conventions are recalled here.

We assume that most quivers, A_{∞}-categories, etc. are small with respect to some universe 𝒰 . It means that the set of objects and the set of morphisms are 𝒰 -small, that is, isomorphic as sets to an element of 𝒰 [GV73, Section 1.0]. The universe 𝒰 is supposed to be an element of a universe 𝒰′, which in its turn is an element of a universe 𝒰′′, and so on. All sets are supposed to be in bijection with some elements of some of the universes. Some differential graded categories in this paper will be 𝒰′-small 𝒰 -categories. A category 𝒞 is a 𝒰 -category if all its sets of morphisms 𝒞(X, Y) are 𝒰 -small [GV73, Definition 1.1].

The 𝒰 -small ground ring 𝕜 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.

The set of non-negative integers is denoted ℕ = ℤ_{⩾0}.

We consider only such A_{∞}-categories 𝒞 that the differential b : T s𝒞 → T s𝒞 vanishes on T^{0}s𝒞, that is, b_{0} = 0. We consider only those A_{∞}-functors f : 𝒜 → ℬ, whose 0-th component f_{0} vanishes.

1.11 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. We thank all the staff of Max-Planck-Institut für Mathematik in Bonn for warm hospitality and support of this research. The main results of this article were obtained during the stage of the first author in MPIM, and a short term visit to MPIM of the second author. We are indebted to M. Jibladze for a valuable advice to look for an operadic approach to the quotient category. We thank B. Keller for the explanation of some results obtained by K. Lefèvre-Hasegawa in his Ph.D. thesis.

2. Quotients of free A_{∞}-categories

Let 𝒬 be a differential graded 𝕜-quiver. There is a free A_{∞}-category ℱ𝒬, generated by 𝒬 [LM03, Section 2.1]. Let R ⊂ sℱ𝒬 be a graded subquiver. Denote by I = (R) ⊂ sℱ𝒬 the graded subquiver spanned by multiplying elements of R with some elements of sℱ𝒬 via several operations b_{k}^{ℱ𝒬} , k > 1. It can be described as follows. Let t ∈ 𝒯_{⩾2}^{n} be a plane rooted tree with i(t) = n input leaves. Each decomposition

(t, ⩽) = (1^{⊔α_{1}} ⊔ 𝔱_{k_{1}} ⊔ 1^{⊔β_{1}}) · (1^{⊔α_{2}} ⊔ 𝔱_{k_{2}} ⊔ 1^{⊔β_{2}}) · . . . · 𝔱_{k_{N}} ,  (2.0.1)

of t into the product of elementary forests gives a linear ordering ⩽ of t. Here α_{1}+k_{1}+β_{1} = n and N = |t| is the number of internal vertices of t. An operation

b_{(t,⩽)}^{ℱ𝒬} = (sℱ𝒬^{⊗n} ^{1^{⊗α_{1}}⊗b_{k_{1}}^{ℱ𝒬} ⊗1^{⊗β_{1}}} → sℱ𝒬^{⊗α_{1}+1+β_{1}} ^{1^{⊗α_{2}}⊗b_{k_{2}}^{ℱ𝒬} ⊗1^{⊗β_{2}}} → … ^{b_{k_{N}}^{ℱ𝒬}}→ sℱ𝒬)  (2.0.2)

is associated with the linearly ordered tree (t, ⩽). Different choices of the ordering of t change only the sign of the above map. In particular, one may consider the canonical linear ordering t_{<} of t [LM03, Section 1.7] and the corresponding map b_{t_{<}}^{ℱ𝒬} . So the subquiver I ⊂ sℱ𝒬 is defined as

I = (R) = ∑_{t∈𝒯_{⩾2}^{α+1+β}} Im(b_{t_{<}}^{ℱ𝒬} : s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β} → sℱ𝒬),

where the summation goes over all α, β ∈ ℤ_{⩾0} and all trees with α + 1 + β input leaves.

2.1 Proposition. Let R ⊂ sℱ𝒬 be a graded subquiver such that Rb_{1}^{ℱ𝒬} ⊂ (R) = I. Then Ib_{1}^{ℱ𝒬} ⊂ I, I is an A_{∞}-ideal of ℱ𝒬, and ℰ = ℱ𝒬/s^{−1}I is an A_{∞}-category.

Proof. Clearly, I is closed under multiplications b_{k}, k > 1, with elements of sℱ𝒬. Let us prove that for all t ∈ 𝒯_{⩾2}

Im(b_{t_{<}}^{ℱ𝒬} : s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β} → sℱ𝒬)b_{1}^{ℱ𝒬} ⊂ I  (2.1.1)

using induction on |t|. This holds for |t| = 0, t = | by assumption. Let |t| = N > 0 and assume that (2.1.1) holds for all t′ ∈ 𝒯_{⩾2} with |t′| < N. The tree t can be presented as t = (t_{1} ⊔ ⋯ ⊔ t_{k})𝔱_{k} for some k > 1. We have b_{t_{<}}^{ℱ𝒬} = ±(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )b_{k}^{ℱ𝒬} , |t_{i}| < N and

b_{k}^{ℱ𝒬} b_{1}^{ℱ𝒬} = − ∑_{a+q+c=k}^{a+c>0}(1^{⊗a} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗c})b_{a+1+c}^{ℱ𝒬}.

One of the a + 1 + c factors of

(s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β})(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )(1^{⊗a} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗c})  (2.1.2)

is contained in I (for q = 1 this is the induction assumption). Hence,

(s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β})(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )(1^{⊗a} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗c})b_{a+1+c}^{ℱ𝒬} ⊂ I,

and the inclusion Ib_{1}^{ℱ𝒬} ⊂ I follows by induction.

Therefore, I is stable under all b_{k}^{ℱ𝒬} , k ⩾ 1, so it is an A_{∞}-ideal, and ℰ = ℱ𝒬/s^{−1}I is an A_{∞}-category.

2.2. A_{∞}-functors from a quotient of a free A_{∞}-category. The following statement is Proposition 2.3 from [LM03].

2.3 Proposition. Let 𝒬 be a differential graded quiver, and let 𝒜 be an A_{∞}-category. A_{∞}-functors f : ℱ𝒬 → 𝒜 are in bijection with sequences (f′_{1}, f_{k})_{k>1}, where f′_{1} : s𝒬 → (s𝒜, b_{1}) is a chain morphism of differential graded quivers with the underlying mapping of objects Ob f : Ob 𝒬 → Ob 𝒜 and f_{k} : T^{k}sℱ𝒬 → s𝒜 are 𝕜-quiver morphisms of degree 0 with the same underlying map Ob f for all k > 1. The morphisms f_{k} are components of f for k > 1. The component f_{1} : sℱ𝒬 → s𝒜 is an extension of f′_{1}.

We are going to extend this description to quotients of free A_{∞}-categories. Let a graded subquiver R ⊂ sℱ𝒬 satisfy the assumption Rb_{1}^{ℱ𝒬} ⊂ (R) = I. Denote by π : ℱ𝒬 ▷ ℰ = ℱ𝒬s^{−1}I the natural projection strict A_{∞}-functor.

2.4 Proposition. An A_{∞}-functor f : ℱ𝒬 → 𝒜 factorizes as f = (ℱ𝒬 ^{π}→ ℰ ^{f̃}→ 𝒜) for some (unique) A_{∞}-functor f̃: ℰ → 𝒜 if and only if the following two conditions are satisfied:

1. Rf_{1} = 0;

2. (sℱ𝒬^{⊗α} ⊗ I ⊗ sℱ𝒬^{⊗β})f_{α+1+β} = 0 for all α, β ∈ ℤ_{⩾0} such that α + β > 0.

Proof. If f = πf̃, then f_{k} = π_{1}^{⊗k}f̃_{k} and the above conditions are necessary.

Assume that the both conditions are satisfied. Let us prove that If_{1} = 0. We are going to prove that for all t ∈ 𝒯_{⩾2}

(s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β} ^{b_{t<}^{ℱ𝒬}}→ sℱ𝒬 ^{f_{1}}→ s𝒜) = 0  (2.4.1)

by induction on |t|. This holds for |t| = 0, t = | by assumption. Let |t| = N > 0 and assume that (2.4.1) holds for all t′ ∈ 𝒯_{⩾2} with |t′| < N. The tree t can be presented as t = (t_{1} ⊔ ⋯ ⊔ t_{k})𝔱_{k} for some k > 1. We have b_{t_{<}}^{ℱ𝒬} = ±(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )b_{k}^{ℱ𝒬} , |t_{i}| < N and

b_{k}^{ℱ𝒬} f_{1} = ∑_{i_{1}+···+i_{l}=k}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}})b_{l}^{𝒜} − ∑_{a+q+c=n}^{a+c>0}(1^{⊗a} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗c})f_{a+1+c}.  (2.4.2)

One of the a + 1 + c factors of (2.1.2) is contained in I and also one of the l factors of

(s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β})(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}})

is contained in I (for i_{j} = 1 this is the induction assumption). Hence, the right hand side of (2.4.2) is contained in I, and If_{1} ⊂ I follows by induction. Therefore,

(sℱ𝒬^{⊗α} ⊗ I ⊗ sℱ𝒬^{⊗β})f ⊂ ∑_{i_{1}+···+i_{l}=α+1+β}(sℱ𝒬^{⊗α} ⊗ I ⊗ sℱ𝒬^{⊗β})(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{l}}) = 0

for all α, β ⩾ 0. Clearly, f factorizes as f = πf̃.

2.5. Transformations from a free A_{∞}-category. The following statement is Proposition 2.8 from [LM03].

2.6 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_{k} = u · pr_{k},

u′ = (P ^{u}→ sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr_{⩽1}}▷ sA_{1}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr}▷ sA_{1}(𝒬, 𝒜)(ϕ, ψ)). (2.6.1)

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, (2.6.2)

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

The following statement is Corollary 2.10 from [LM03].

2.7 Corollary. Let ϕ, ψ : ℱ𝒬 → 𝒜 be A_{∞}-functors. Let P be a complex of 𝕜-modules. Let w : 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 w = 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} = w′, where

w′ = (P ^{w}→ sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr_{⩽1}}▷ sA_{1}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr}▷ sA_{1}(𝒬, 𝒜)(ϕ, ψ));

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_{k} = h · pr_{k},

h′ = (P ^{h}→ sA_{∞}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr_{⩽1}}▷ sA_{1}(ℱ𝒬, 𝒜)(ϕ, ψ) ^{restr}▷ sA_{1}(𝒬, 𝒜)(ϕ, ψ)). (2.7.1)

The inverse bijection can be recovered from the recurrent formula

(−)^{p}b_{k}(ph_{1}) = pw_{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, (2.7.2)

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

2.8. Transformations from a quotient of a free A_{∞}-category. We are going to extend the above description to quotients of free A_{∞}-categories. Assume that a graded subquiver R ⊂ sℱ𝒬 satisfies Rb_{1}^{ℱ𝒬} ⊂ (R) = I. Let 𝒜 be an A_{∞}-category. Composition with the projection A_{∞}-functor π : ℱ𝒬 ▷ ℰ = ℱ𝒬/s^{−1}I gives a strict A_{∞}-functor L^{π} = (π ⊠ 1)M : A_{∞}(ℰ, 𝒜) → A_{∞}(ℱ𝒬, 𝒜). It is injective on objects and morphisms, that is, both maps Ob L^{π} : ϕ ↦ πϕ and L_{1}^{π} : sA_{∞}(ℰ, 𝒜)(ϕ, ψ) → sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ), r ↦ πr are injective. We are going to characterize the subcomplex sA_{∞}(ℰ, 𝒜)(ϕ, ψ)^{⊂} → sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ).

2.9 Proposition. Let P be a complex of 𝕜-modules, and let u : P → sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ) be a chain map. Denote

u_{k} = u · pr_{k} : P → ∏_{X,Y ∈Ob 𝒬} C((sℱ𝒬)^{⊗k}(X, Y), s𝒜(Xϕ, Y ψ)), k ⩾ 0.

Then the image of u is contained in the subcomplex sA_{∞}(ℰ, 𝒜)(ϕ, ψ) if and only if the following two conditions are satisfied:

1. R(pu_{1}) = Im(pu_{1} : R(X, Y) → s𝒜(Xϕ, Y ψ)) = 0 for all p ∈ P;

2. (sℱ𝒬^{⊗α} ⊗ I ⊗ sℱ𝒬^{⊗β})(pu_{k}) = 0 for all p ∈ P and all α, β ⩾ 0 such that α + 1 + β = k > 1.

Proof. The conditions are obviously necessary. Let us prove that they are sufficient. First of all, we are going to show that I(pu_{1}) = 0. Namely, we claim that

(s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β})b_{t_{<}}^{ℱ𝒬} (pu_{1}) = 0  (2.9.1)

for all trees t ∈ 𝒯_{⩾2}^{α+1+β}. We prove it by induction on |t|. For |t| = 0, t = | this holds by assumption 1. Let t ∈ 𝒯_{⩾2} be a tree with |t| = N > 0 internal vertices. Assume that (2.9.1) holds for all t′ ∈ 𝒯_{⩾2} with |t′| < N. The tree t can be presented as t = (t_{1}⊔⋯⊔t_{k})𝔱_{k} for some k > 1, so b_{t_{<}}^{ℱ𝒬} = ±(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )b_{k}^{ℱ𝒬} , and |t_{i}| < N. Formula (2.6.2) in the form

(−)^{p}b_{k}^{ℱ𝒬} (pu_{1}) = −(pd)u_{k} + ∑_{a+q+c=k}^{m,n}(π_{1}^{⊗a}ϕ_{am} ⊗ pu_{q} ⊗ π_{1}^{⊗c}ψ_{cn})b_{m+1+n}^{𝒜}

− (−)^{p} ∑_{a+q+c=k}^{a+c>0}(1^{⊗a} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗c})(pu_{a+1+c}) : sℱ_{t_{1}}𝒬 ⊗ ⋯ ⊗ sℱ_{t_{k}}𝒬 → s𝒜 (2.9.2)

implies that

(s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β})b_{t_{<}}^{ℱ𝒬} (pu_{1}) = (s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β})(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )b_{k}^{ℱ𝒬} (pu_{1})

⊂ (sℱ𝒬^{⊗γ} ⊗ I ⊗ sℱ𝒬^{⊗δ})[(pd)u_{k}]

+ ∑(s𝒬^{⊗α} ⊗ R ⊗ s𝒬^{⊗β})(b_{t_{1}}^{ℱ𝒬} ⊗ ⋯ ⊗ b_{t_{k}}^{ℱ𝒬} )(π_{1}^{⊗a}ϕ_{am} ⊗ pu_{q} ⊗ π_{1}^{⊗c}ψ_{cn})b_{m+1+n}^{𝒜}

+ ∑_{λ+μ>0}(sℱ𝒬^{⊗λ} ⊗ I ⊗ sℱ𝒬^{⊗μ})(pu_{λ+1+μ}) = 0.

Indeed, summands with q = 0 vanish due to Iπ_{1} = 0, summands with q = 1 vanish due to induction assumption (2.9.1), and other summands vanish due to condition 2.

Thus condition 2 holds not only for k > 1 but for k = 1 as well. At last

(sℱ𝒬^{⊗α}⊗I ⊗sℱ𝒬^{⊗β})(pu) ⊂ ∑_{a+q+c=α+1+β}^{m,n}(sℱ𝒬^{⊗α}⊗I ⊗sℱ𝒬^{⊗β})(π_{1}^{⊗a}ϕ_{am}⊗pu_{q} ⊗π_{1}^{⊗c}ψ_{cn}) = 0,

and the proof is finished.

Let us extend the description of homotopies between chain maps to the case of quotient of a free A_{∞}-category. We keep the assumptions of Section 2.8.

2.10 Corollary. Let P be a complex of 𝕜-modules, and let v : P → sA_{∞}(ℰ, 𝒜)(ϕ, ψ) be a chain map. Denote

w = (P ^{v}→ sA_{∞}(ℰ, 𝒜)(ϕ, ψ)^{⊂} → sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ)).

Let h : P → sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ) be a homotopy, deg h = −1, w = dh + hB_{1}. Then the image of h is contained in the subcomplex sA_{∞}(ℰ, 𝒜)(ϕ, ψ) if and only if it factorizes as

h = (P ^{η}→ sA_{∞}(ℰ, 𝒜)(ϕ, ψ)^{⊂} → sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ)),

where deg η = −1, v = dη+ηB_{1}, or if and only if the following two conditions are satisfied:

1. R(ph_{1}) = Im(ph_{1} : R(X, Y) → s𝒜(Xϕ, Y ψ)) = 0 for all p ∈ P;

2. (sℱ𝒬^{⊗α} ⊗ I ⊗ sℱ𝒬^{⊗β})(ph_{k}) = 0 for all p ∈ P and all α, β ⩾ 0 such that α + 1 + β = k > 1.

Proof. Given pair (w, h) defines a chain map \overline{w} : Cone(id_{P}) → sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ), (q, ps) ↦ qw + ph, such that

w = (P ^{in_{1}}→ P ⊕ P [1] = Cone(id_{P}) ^{\overline{w}}→ sA_{∞}(ℱ𝒬, 𝒜)(πϕ, πψ)).

The image of h is contained in sA_{∞}(ℰ, 𝒜)(ϕ, ψ) if and only if the image of \overline{w} is contained in sA_{∞}(ℰ, 𝒜)(ϕ, ψ). By Proposition 2.9 this is equivalent to conditions:

1′. R(qw_{1}) = 0;  1′′. R(ph_{1}) = 0;
2′. (sℱ𝒬^{⊗α} ⊗ I ⊗ sℱ𝒬^{⊗β})(qw_{k}) = 0; 2′′. (sℱ𝒬^{⊗α} ⊗ I ⊗ sℱ𝒬^{⊗β})(ph_{k}) = 0

for all q, p ∈ P and all α, β ⩾ 0 such that α + 1 + β = k > 1. However, the conditions 1′ and 2′ are satisfied automatically by Proposition 2.9 applied to w and v.

3. A simple example

We want to consider an example, which is almost tautological. The non-trivial part of it is the concrete choice of a system of relations R generating an ideal. Let 𝒟 be an A_{∞}-category. We view it as a differential graded quiver and construct the free A_{∞}-category ℱ𝒟 out of it. We choose the following subquiver of relations: R_{𝒟} = ∑_{n⩾2} Im(b_{n}^{ℱ𝒟} − b_{n}^{𝒟} : s𝒟^{⊗n} → sℱ𝒟). More precisely, a map δ_{n} : (s𝒟)^{⊗n} → sℱ𝒟, n ⩾ 2, is defined as the difference

δ_{n} = ((s𝒟)^{⊗n} = (sℱ_{|}𝒟)^{⊗n} \begin{matrix}b_{n}^{ℱ𝒟} \\ s^{−1} \end{matrix} → sℱ_{𝔱_{n}}𝒟^{⊂} → sℱ𝒟)−((s𝒟)^{⊗n} ^{b_{n}^{𝒟}}→ s𝒟 = sℱ_{|}𝒟^{⊂} → sℱ𝒟),

and R_{𝒟} = ∑_{n⩾2} Im(δ_{n}).

3.1 Lemma. The subquiver I = (R_{𝒟} ) ⊂ sℱ𝒟 is an A_{∞}-ideal.

Proof. According to Proposition 2.1 it suffices to check that R_{𝒟} b_{1}^{ℱ𝒟} ⊂ (R_{𝒟} ). As

(b_{n}^{ℱ𝒟} − b_{n}^{𝒟})b_{1}^{ℱ𝒟} = −b_{n}^{𝒟} b_{1}^{ℱ𝒟} − ∑_{a+k+c=n}^{a+c>0}(1^{⊗a} ⊗ b_{k}^{ℱ𝒟} ⊗ 1^{⊗c})b_{a+1+c}^{ℱ𝒟},

we have

Im(b_{n}^{ℱ𝒟} − b_{n}^{𝒟})b_{1}^{ℱ𝒟} ⊂ − Im(b_{n}^{𝒟} b_{1}^{ℱ𝒟} + ∑_{a+k+c=n}^{a+c>0}(1^{⊗a} ⊗ b_{k}^{𝒟} ⊗ 1^{⊗c})b_{a+1+c}^{ℱ𝒟}) + (R_{𝒟} )

⊂ − Im(b_{n}^{𝒟} b_{1}^{𝒟} + ∑_{a+k+c=n}^{a+c>0}(1^{⊗a} ⊗ b_{k}^{𝒟} ⊗ 1^{⊗c})b_{a+1+c}^{𝒟}) + (R_{𝒟} ) = (R_{𝒟} ),

and the lemma is proven.

Let us describe the ideal I = (R_{𝒟} ) ⊂ sℱ𝒟. Let t′ ∈ 𝒯 be a tree, and let t = (1^{⊔α} ⊔ 𝔱_{k} ⊔ 1^{⊔β}) · t′_{<}. Define a map δ_{α,k,t′} as the difference

δ_{α,k,t′} = ((s𝒟)^{⊗n} ^{1^{⊗α}⊗b_{k}^{ℱ𝒟} ⊗1^{⊗β}} → (sℱ_{|}𝒟)^{⊗α} ⊗ sℱ_{𝔱_{k}}𝒟 ⊗ (sℱ_{|}𝒟)^{⊗β} ^{b_{t′_{<}}^{ℱ𝒟}}→ sℱ_{t}𝒟 → sℱ𝒟)

− ((s𝒟)^{⊗n} ^{1^{⊗α}⊗b_{k}^{𝒟} ⊗1^{⊗β}} → (s𝒟)^{⊗α+1+β} = (sℱ_{|}𝒟)^{⊗α+1+β} ^{b_{t′_{<}}^{ℱ𝒟}}→ sℱ_{t′}𝒟 → sℱ𝒟)

for k > 1, α+k +β = n. Clearly, ∑ Im(δ_{α,k,t′}) = I. These compositions can be simplified. Let h be the height of the distinguished vertex 𝔱_{k} in t_{<}. Then the above difference equals

δ_{α,k,t′} = ((s𝒟)^{⊗α+k+β} = (sℱ_{|}𝒟)^{⊗n} ^{(−)^{|t|−h}s^{−|t|}} → sℱ_{t}𝒟 → sℱ𝒟)

− ((s𝒟)^{⊗α+k+β} ^{1^{⊗α}⊗b_{k}^{𝒟} ⊗1^{⊗β}} → (s𝒟)^{⊗α+1+β} = (sℱ_{|}𝒟)^{⊗α+1+β} ^{s^{−|t′|}} → sℱ_{t′}𝒟 → sℱ𝒟).

With the identity map id : 𝒟 → 𝒟 is associated a strict A_{∞}-functor îd : ℱ𝒟 → 𝒟 [LM03, Section 2.6]. Its first component equals

îd_{1} = (sℱ_{t}𝒟 ^{[b_{(t,⩽)}^{ℱ𝒟}]^{−1}} → T^{i(t)}s𝒟 ^{b_{(t,⩽)}^{𝒟}} → s𝒟)  (3.1.1)

for each linear ordering (t, ⩽) of a plane rooted tree t ∈ 𝒯_{⩾2}^{n} with i(t) = n input leaves. Note that in the above formula b_{(t,⩽)}^{ℱ𝒟} = ±s^{−|t|}. In particular, for the canonical linear ordering t_{<} the formula becomes

îd_{1} = (sℱ_{t}𝒟 ^{s^{|t|}}→ T^{i(t)}s𝒟 ^{b_{t>}^{𝒟}}→ s𝒟).

Here

b_{(t,⩽)}^{𝒟} = (s𝒟^{⊗n} ^{1^{⊗α_{1}}⊗b_{k_{1}}^{𝒟}⊗1^{⊗β_{1}}} → s𝒟^{⊗α_{1}+1+β_{1}} ^{1^{⊗α_{2}}⊗b_{k_{2}}^{𝒟}⊗1^{⊗β_{2}}} → … ^{b_{k_{N}}^{𝒟}}→ s𝒟)

corresponds to the ordered decomposition

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

of (t, ⩽) into the product of forests, α_{1} + k_{1} + β_{1} = n, and b_{(t,⩽)}^{ℱ𝒟} has a similar meaning.

3.2 Proposition. The A_{∞}-category ℱ𝒟/s^{−1}(R_{𝒟} ) is isomorphic to 𝒟.

Proof. Let ℰ = ℱ𝒟/s^{−1}(R_{𝒟} ) be the quotient category. The projection map π_{1} : sℱ𝒟 → sℰ with the underlying map of objects Ob π = id_{Ob 𝒟} determines a strict A_{∞}-functor π : ℱ𝒟 → ℰ. The embedding ι_{1} = (s𝒟^{⊂} → sℱ𝒟 ^{π_{1}} ▷ sℰ) with the underlying identity map of objects Ob ι = id_{Ob 𝒟} determines a strict A_{∞}-functor ι : 𝒟 → ℰ. Indeed, ι_{1}^{⊗n}b_{n}^{ℰ} = b_{n}^{𝒟} ι_{1} : s𝒟^{⊗n} → sℰ, for Im(b_{n}^{ℱ𝒟} − b_{n}^{𝒟}) = Im δ_{n} ⊂ I ⊂ sℱ𝒟.

We claim that the A_{∞}-functor îd : ℱ𝒟 → 𝒟 factorizes as îd = (ℱ𝒟 ^{π}→ ℰ ĩd→ 𝒟) for some A_{∞}-functor ĩd : ℰ → 𝒟. Indeed, both conditions of Proposition 2.4 are satisfied. The second is obvious since îd is strict. The first condition R_{𝒟} îd_{1} = 0 follows from the computation (b_{n}^{ℱ𝒟} − b_{n}^{𝒟})îd_{1} = id_{1}^{⊗n} b_{n}^{𝒟} − b_{n}^{𝒟} id_{1} = 0 : s𝒟^{⊗n} → s𝒟. Thus, ĩd : ℰ → 𝒟 is a strict A_{∞}-functor.

Both ι and ĩd induce the identity map on objects. Clearly, ι · ĩd = id_{𝒟} . Furthermore, ι is surjective on morphisms because every element of ℱ_{t}𝒟 reduces to an element of 𝒟 modulo (R_{𝒟} ). Therefore, ι is invertible and A_{∞}-functors ι and ĩd are strictly inverse to each other.

3.3 Corollary. (R_{𝒟} )îd_{1} = 0.

4. A_{∞}-categories and quivers

4.1 Definition (Pseudounital A_{∞}-categories). A pseudounital structure of an A_{∞}-cat-egory 𝒟 is a choice of an element [ι_{X}] ∈ H^{−1}(s𝒟(X, X), b_{1}) for each object X of 𝒟. By ι_{X} we mean a representative of the chosen cohomology class, deg ι_{X} = −1. An A_{∞}-functor f : 𝒟 → 𝒜 between two pseudounital A_{∞}-categories is called pseudounital if it preserves the distinguished cohomology classes, that is, ι_{Xf}^{𝒜} − ι_{X}^{𝒟}f_{1} ∈ Im b_{1} for all objects X of 𝒟.

The composition of pseudounital A_{∞}-functors is pseudounital as well. The full subcategory of pseudounital A_{∞}-functors is denoted A_{∞}^{ψu}(𝒟, 𝒜) ⊂ A_{∞}(𝒟, 𝒜).

A unital A_{∞}-category 𝒜 has a canonical pseudounital structure: ι_{X}^{𝒜} = _{X}i_{0}^{𝒜}. An A_{∞}-functor f : 𝒟 → 𝒜 between unital A_{∞}-categories is unital if and only if it is pseudounital for the canonical pseudounital structures of 𝒟 and 𝒜 [Lyu03, Definition 8.1].

Let 𝒟 be a pseudounital A_{∞}-category, let 𝒬 be a differential graded quiver with Ob 𝒬 = Ob 𝒟, and let in^{𝒟} : 𝒟 → 𝒬 be an A_{1}-functor, such that Ob in^{𝒟} = id_{Ob 𝒟} and in_{1}^{𝒟} : s𝒟 ↪ s𝒬 is an embedding. Let 𝒜 be a pseudounital A_{∞}-category.

4.2. A_{∞1}-functors and transformations. We define A_{∞}-category A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) via pull-back square

\begin{matrix}A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) & \overline{→} & A_{1}(𝒬, 𝒜) \\ A_{∞}^{ψu}(𝒟, 𝒜) & ↓ & \overline{→^{restr_{⩽1}}} & A_{1}(𝒟, 𝒜) & ↓^{A_{1}(in^{𝒟} ,𝒜)} \end{matrix}  (4.2.1)

In details, the objects of A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) are pairs (f, f′), where f : 𝒟 → 𝒜 is a pseudounital A_{∞}-functor, f′ : 𝒬 → 𝒜 is an A_{1}-functor such that Ob f = Ob f′ and f_{1} = f′_{1}|_{s𝒟} . Morphisms of A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) from (f, f′) to (g, g′) are pairs (r, r′), where r ∈ A_{∞}(𝒟, 𝒜)(f, g), r′ ∈ A_{1}(𝒬, 𝒜)(f′, g′) are such that deg r = deg r′, r_{0} = r′_{0} and r_{1} = r′_{1}|_{s𝒟} . For any n-tuple of composable morphisms (r^{j}, p^{j}) ∈ sA_{∞1}(𝒟, 𝒬; 𝒜)((f^{j−1}, g^{j−1}), (f^{j}, g^{j})), 1 ⩽ j ⩽ n, their n-th product is defined as

[(r^{1}, p^{1}) ⊗ ⋯ ⊗ (r^{n}, p^{n})]B_{n} =^{def} ((r^{1} ⊗ ⋯ ⊗ r^{n})B_{n}, (p^{1} ⊗ ⋯ ⊗ p^{n})B_{n}).

It is well-defined, because formulas for B_{k} agree for all A_{N}-categories, 1 ⩽ N ⩽ ∞. The identity B^{2} = 0 for A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) follows from that for A_{∞}^{ψu}(𝒟, 𝒜) and A_{1}(𝒬, 𝒜). Thus, A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) is an A_{∞}-category.

4.3 Proposition. If 𝒜 is unital, then the A_{∞}-category A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) is unital as well.

Proof. Let us denote by (1 ⊠ i^{𝒜} )M the A_{∞}-transformation id → id : A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) → A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜), whose n-th component is

(r^{1}, p^{1}) ⊗ ⋯ ⊗ (r^{n}, p^{n}) ↦ ((r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ i^{𝒜} )M_{n1}, (p^{1} ⊗ ⋯ ⊗ p^{n} ⊠ i^{𝒜} )M_{n1}).

It is well-defined, since formulas for multiplication M agree for all A_{N}-categories, 1 ⩽ N ⩽ ∞. In a sense, (1⊠i^{𝒜} )M for A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) is determined by a pair of A_{∞}-transformations: (1 ⊠ i^{𝒜} )M for A_{∞}^{ψu}(𝒟, 𝒜) and for A_{1}(𝒬, 𝒜). Since the latter two A_{∞}-transformations are natural and satisfy

[(1 ⊠ i^{𝒜} )M ⊗ (1 ⊠ i^{𝒜} )M]B_{2} ≡ (1 ⊠ i^{𝒜} )M,

the A_{∞}-transformation (1 ⊠ i^{𝒜} )M for A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) has the same properties.

We claim that (1 ⊠ i^{𝒜} )M is the unit transformation of A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) as defined in [Lyu03, Definition 7.6]. Indeed, it remains to prove that chain endomorphisms

a =^{def} (1 ⊗_{(g,g′)}[(1 ⊠ i^{𝒜} )M]_{0})B_{2}, c =^{def} (_{(f,f′)}[(1 ⊠ i^{𝒜} )M]_{0} ⊗ 1)B_{2} :

Φ =^{def} (sA_{∞1}(𝒟, 𝒬; 𝒜)((f, f′), (g, g′)), B_{1}) → Φ

are homotopy invertible for all pairs (f, f′), (g, g′) of objects of A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜). We have

(r, r′)a = ((r ⊗ gi^{𝒜} )B_{2}, (r′ ⊗ g′i^{𝒜} )B_{2}),
(r, r′)c = (r(fi^{𝒜} ⊗ 1)B_{2}, r′(f′i^{𝒜} ⊗ 1)B_{2}).

As a graded 𝕜-module Φ = ∏_{n=0}^{∞} V_{n}, where

V_{0} = ∏_{X∈Ob 𝒟} s𝒜(Xf, Xg),
V_{1} = ∏_{X,Y ∈Ob 𝒟} C(s𝒬(X, Y), s𝒜(Xf, Y g)),
V_{n} = ∏_{X,Y ∈Ob 𝒟} C((s𝒟)^{⊗n}(X, Y), s𝒜(Xf, Y g)) for n ⩾ 2.  (4.3.1)

Consider the decreasing filtration Φ = Φ_{0} ⊃ Φ_{1} ⊃ ⋯ ⊃ Φ_{n} ⊃ Φ_{n+1} ⊃ … of the complex Φ, defined by Φ_{n} = 0 × ⋯ × 0 × ∏_{m=n}^{∞} V_{m}. We may write

Φ_{1} = {(r, r′) ∈ Φ | r_{0} = 0},
Φ_{n} = {(r, 0) ∈ Φ | ∀l < n r_{l} = 0} for n ⩾ 2.

The differential B_{1} preserves the submodules Φ_{n}. It induces the differential

d = C(1, b_{1}) − ∑_{α+1+β=n} C(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗β}, 1)

in the quotient V_{n} = Φ_{n}/Φ_{n+1}. Due to

[(r ⊗ gi^{𝒜} )B_{2}]_{k} = ∑_{l}(r ⊗ gi^{𝒜} )θ_{kl}b_{l} = ∑_{a+p+c+q+e=k}^{α,γ,ε}(f_{aα} ⊗ r_{p} ⊗ g_{cγ} ⊗ (gi^{𝒜} )_{q} ⊗ g_{eε})b_{α+γ+ε+2}

and similar formulas, the endomorphisms a, c : Φ → Φ preserve the subcomplexes Φ_{n}. They induce the endomorphisms gr a, gr c : V_{n} → V_{n} in the quotient complex V_{n}:

(r′_{1}) gr a = ∏_{X,Y ∈Ob 𝒟} (_{X,Y} r′_{1} ⊗_{Y g}i_{0}^{𝒜})b_{2}^{𝒜} ,  r′_{1} ∈ V_{1},
(r′_{1}) gr c = ∏_{X,Y ∈Ob 𝒟} _{X,Y} r′_{1}(_{Xf}i_{0}^{𝒜} ⊗ 1)b_{2}^{𝒜} ,
(r_{n}) gr a = ∏_{X,Y ∈Ob 𝒟} (_{X,Y} r_{n} ⊗_{Y g}i_{0}^{𝒜})b_{2}^{𝒜} ,  r_{n} ∈ V_{n}, n = 0 or n ⩾ 2.
(r_{n}) gr c = ∏_{X,Y ∈Ob 𝒟} _{X,Y} r_{n}(_{Xf}i_{0}^{𝒜} ⊗ 1)b_{2}^{𝒜} .

Due to unitality of 𝒜, for each pair X, Y of objects of 𝒟 there exist 𝕜-linear maps X,Y h,_{X,Y} h′ : s𝒜(Xf, Y g) → s𝒜(Xf, Y g) of degree −1 such that

(1 ⊗_{Y g}i_{0}^{𝒜})b_{2}^{𝒜} = 1 +_{X,Y} h · d + d ·_{X,Y} h,  (4.3.2)
(_{Xf}i_{0}^{𝒜} ⊗ 1)b_{2}^{𝒜} = −1 +_{X,Y} h′ · d + d ·_{X,Y} h′.  (4.3.3)

We equip the 𝕜-modules Φ^{d} = ∏_{n=0}^{∞} V_{n}^{d} with the topology of the product of discrete Abelian groups V_{n}^{d}. Thus the 𝕜-submodules Φ_{m}^{d} = 0^{m−1} × ∏_{n=m}^{∞} V_{n}^{d} form a basis of neighborhoods of 0 in Φ^{d}. Continuous maps A : V^{d} → V^{d+p} are identified in the proof of Proposition A.9 with ℕ × ℕ-matrices of linear maps A_{nm} : V_{n}^{d} → V_{m}^{d+p} with finite number of non-vanishing elements in each column.

In particular, the maps B_{1} : Φ^{d} → Φ^{d+1} are continuous for all d ∈ ℤ. Let us introduce continuous 𝕜-linear maps H, H′ : ∏_{n=0}^{∞} V_{n}^{d} → ∏_{n=0}^{∞} V_{n}^{d−1} by diagonal matrices _{X,Y} r_{n} ↦ X,Y r_{n}_{X,Y} h, _{X,Y} r_{n} ↦ _{X,Y} r_{n}_{X,Y} h′. We may view gr a, gr c as diagonal matrices and as the corresponding continuous endomorphisms of ∏_{n=0}^{∞} V_{n}^{d}. Equations (4.3.2), (4.3.3) can be written as

gr a = 1 + Hd + dH, gr c = −1 + H′d + dH′.

The continuous chain maps

N = a − HB_{1} − B_{1}H − 1, N′ = c − H′B_{1} − B_{1}H′ + 1 : ∏_{n=0}^{∞} V_{n}^{d} → ∏_{n=0}^{∞} V_{n}^{d}

have strictly upper triangular matrices. Therefore, the endomorphisms 1 + N, −1 + N′ : ∏_{n=0}^{∞} V_{n}^{d} → ∏_{n=0}^{∞} V_{n}^{d} are invertible. Their inverse maps correspond to well-defined ℕ×ℕmatrices ∑_{i=0}^{∞}(−N)^{i} and − ∑_{i=0}^{∞}(N′)^{i}. Since a, c are homotopic to invertible maps, they are homotopy invertible and the proposition is proven.

4.4. A strict A_{∞}-2-functor. Let us describe a strict A_{∞}-2-functor F : A_{∞}^{u} → A_{∞}^{u}. It maps a unital A_{∞}-category 𝒜 to the unital A_{∞}-category A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜). The strict unital A_{∞}-functor

F_{𝒜,ℬ} = A_{∞1}^{ψu}(𝒟, 𝒬; ) : A_{∞}^{u}(𝒜, ℬ) → A_{∞}^{u}(A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜), A_{∞1}^{ψu}(𝒟, 𝒬; ℬ))

is specified by the following data. It maps an object f : 𝒜 → ℬ (a unital A_{∞}-functor) to the object

A_{∞1}^{ψu}(𝒟, 𝒬; f) = (A_{∞}^{ψu}(𝒟, f), A_{1}(𝒬, f)) = ((1 ⊠ f)M, (1 ⊠ f)M),

a unital A_{∞}-functor A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) → A_{∞1}^{ψu}(𝒟, 𝒬; ℬ), which sends (g, g′) ∈ Ob A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) to (gf, g′f) ∈ Ob A_{∞1}^{ψu}(𝒟, 𝒬; ℬ). Its n-th component is

[A_{∞1}^{ψu}(𝒟, 𝒬; f)]_{n} : (r^{1}, p^{1})⊗⋯⊗(r^{n}, p^{n}) ↦ ((r^{1} ⊗⋯⊗r^{n} ⊠f)M_{n0}, (p^{1} ⊗⋯⊗p^{n} ⊠f)M_{n0}).

An A_{∞}-transformation q : f → g : 𝒜 → ℬ is mapped by [A_{∞1}^{ψu}(𝒟, 𝒬; )]_{1} to the A_{∞}-trans-formation

A_{∞1}^{ψu}(𝒟, 𝒬; q) = (A_{∞}^{ψu}(𝒟, q), A_{1}(𝒬, q)) = ((1 ⊠ q)M, (1 ⊠ q)M),

whose n-th component is

[A_{∞1}^{ψu}(𝒟, 𝒬; q)]_{n} : (r^{1}, p^{1})⊗⋯⊗(r^{n}, p^{n}) ↦ ((r^{1} ⊗⋯⊗r^{n} ⊠q)M_{n1}, (p^{1} ⊗⋯⊗p^{n} ⊠q)M_{n1}).

Thus, a strict A_{∞}-functor F_{𝒜,ℬ} = A_{∞1}^{ψu}(𝒟, 𝒬; ) is constructed. It is unital, because the unit element fi^{ℬ} of f ∈ Ob A_{∞}^{u}(𝒜, ℬ) is mapped to the unit element A_{∞1}^{ψu}(𝒟, 𝒬; fi^{ℬ} ) = ((1 ⊠ fi^{ℬ} )M, (1 ⊠ fi^{ℬ} )M) of A_{∞1}^{ψu}(𝒟, 𝒬; f) ∈ Ob A_{∞}^{u}(F 𝒜, F ℬ).

Necessary equation, given by diagram (3.1.1) of [LM03] follows from the same equation written for A_{∞}^{ψu}(𝒟, ) and for A_{1}(𝒬, ). Therefore, the strict A_{∞}-2-functor F is constructed.

4.5. An A_{∞}-category freely generated over an A_{∞}-category. Let 𝒟 be a pseudounital A_{∞}-category, let 𝒬 be a differential graded quiver with Ob 𝒬 = Ob 𝒟, equipped with a chain embedding s𝒟^{⊂} → s𝒬, identity on objects. Assume that there exists a graded 𝕜-subquiver 𝒩 ⊂ 𝒬, Ob 𝒩 = Ob 𝒟, which is a direct complement of 𝒟. Thus, 𝒟 ⊕ 𝒩 = 𝒬 is an isomorphism of graded 𝕜-linear quivers. Then there exists a differential in the graded quiver sℳ = 𝒩 and a chain map α : sℳ → s𝒟 such that Ob α = id_{Ob 𝒟} and 𝒬 = Cone α. Indeed, the embedding in^{𝒟} in the exact sequence

0 → s𝒟 ^{in^{𝒟}}→ s𝒬 ^{pr^{𝒩}}→ ℳ[2] → 0

is a chain map. Thus the graded 𝕜-quiver ℳ[2] = s𝒩 = Coker(in^{𝒟} ) acquires a differential d^{ℳ[2]} , such that pr^{𝒩} is a chain map. The differential in s𝒬 has the form

(t, ms)b_{1}^{𝒬} = (tb_{1}^{𝒟} + mα, msd^{ℳ[2]} ) = (tb_{1}^{𝒟} + mα, −md^{ℳ[1]} s)  (4.5.1)

for t ∈ s𝒟(X, Y), m ∈ ℳ[1](X, Y), where α : sℳ(X, Y) → s𝒟(X, Y) are 𝕜-linear maps of degree 0. The condition (b_{1}^{𝒬})^{2} = 0 is equivalent to α being a chain map. Therefore, s𝒬 = Cone(α : sℳ → s𝒟).

Define a pseudounital A_{∞}-category ℰ = ℱ𝒬/s^{−1}(R_{𝒟} ), where R_{𝒟} = ∑_{n⩾2} Im(δ_{n}) for

δ_{n} = ((s𝒟)^{⊗n} ^{⊂} → (sℱ𝒬)^{⊗n} ^{b_{n}^{ℱ𝒬}}→ sℱ𝒬) − ((s𝒟)^{⊗n} ^{b_{n}^{𝒟}}→ s𝒟^{⊂} → sℱ𝒬).

Repeating word by word the proof of Lemma 3.1 we deduce that J = (R_{𝒟} ) ⊂ sℱ𝒬 is an A_{∞}-ideal. The distinguished elements ι_{X} ∈ (sℰ)^{−1}(X, X) are those of 𝒟 ⊂ ℰ. Let 𝒜 be a pseudounital A_{∞}-category. There is the restriction strict A_{∞}-functor

restr : A_{∞}^{ψu}(ℰ, 𝒜) → A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜), x ↦ (x|_{𝒟} , x|_{𝒬}).

If 𝒜 is unital, then the above A_{∞}-functor is unital, because the unit element fi^{𝒜} of f ∈ Ob A_{∞}^{ψu}(ℰ, 𝒜) is mapped to the unit element ((fi^{𝒜} )|_{𝒟} , (fi^{𝒜} )|_{𝒬}) = (f|_{𝒟} i^{𝒜} , f|_{𝒬}i^{𝒜} ) of (f|_{𝒟} , f|_{𝒬}) ∈ Ob A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜).

4.6 Proposition. The map Ob restr : Ob A_{∞}^{ψu}(ℰ, 𝒜) → Ob A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) is surjective. An object (f, f′) of A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) is the restriction of a unique pseudounital A_{∞}-functor f̃ : ℰ → 𝒜 such that Ob f̃ = Ob f, f̃|_{𝒬} = f′, f̃_{k}|_{s𝒟^{⊗k}} = f_{k}, and f̃_{k} vanishes on all summands of T^{k}sℰ containing a factor s𝒩 for k > 1.

Proof. Let us define an A_{∞}-functor f̂ : ℱ𝒬 → 𝒜 via Proposition 2.3 by the following data. On objects it is Ob f̂ = Ob f = Ob f′, the restriction to s𝒬 of the first component is f̂_{1}|_{s𝒬} = f′. On each direct summand of T^{k}sℱ𝒬, k > 1, containing the factor s𝒩 we set f̂_{k} = 0. On the direct summand T^{k}sℱ𝒟 of T^{k}sℱ𝒬 we define

f̂_{k} = (T^{k}sℱ𝒟 ^{îd_{1}^{⊗k}} → T^{k}s𝒟 ^{f_{k}}→ s𝒜)

for k > 1, where îd_{1} is defined by (3.1.1). These requirements specify f̂ completely. It is pseudounital, since f̂_{1}|_{s𝒟} = f_{1} and f is pseudounital.

Let us prove that the A_{∞}-functor f̂factors as f̂ = (ℱ𝒬 ^{π} ▷ ℰ f̃→ 𝒜) for some unique A_{∞}-functor f̃. Denote J = (R_{𝒟} ) ⊂ sℱ𝒬. We have to check conditions of Proposition 2.4. The second condition, (sℱ𝒬^{⊗α} ⊗ J ⊗ sℱ𝒬^{⊗β})f̂_{α+1+β} = 0 if α + β > 0, holds on direct summands of sℱ𝒬^{⊗α} ⊗ J ⊗ sℱ𝒬^{⊗β}, which contain a factor s𝒩 in some of ℱ𝒬. It holds also on summands of J of the form Im(b_{t}^{ℱ𝒬} : ⋯ ⊗ s𝒩 ⊗ ⋯ ⊗ R_{𝒟} ⊗ ⋯ → sℱ𝒬) or Im(b_{t}^{ℱ𝒬} : ⋯⊗R_{𝒟} ⊗⋯⊗s𝒩⊗⋯ → sℱ𝒬). We have to verify that (sℱ𝒟^{⊗α} ⊗I ⊗sℱ𝒟^{⊗β})f̂_{α+1+β} = 0 if α + β > 0, where I ⊂ sℱ𝒟 is the ideal described in Lemma 3.1. This equation holds true because

(sℱ𝒟^{⊗α} ⊗ I ⊗ sℱ𝒟^{⊗β})f̂_{α+1+β} = (sℱ𝒟^{⊗α} ⊗ I ⊗ sℱ𝒟^{⊗β})îd_{1}^{α+1+β}f_{α+1+β} = 0

due to equation Iîd_{1} = 0, obtained in Corollary 3.3. Therefore, the second condition of Proposition 2.4 is verified.

Let us observe that the restriction of f̂ to ℱ𝒟 coincides with ℱ𝒟 îd→ 𝒟 ^{f}→ 𝒜. Indeed, their components are equal, f̂_{k}|_{ℱ𝒟} = îd_{1}^{⊗k} · f_{k} for k > 1, and f̂_{1}|_{sℱ_{|}𝒟} = f_{1} =

îd_{1} · f_{1} : s𝒟 → s𝒜. Hence, f̂|_{ℱ𝒟} = îd · f by Proposition 2.3.

In particular, f̂_{1}|_{sℱ𝒟} = (sℱ𝒟 ^{îd_{1}}→ s𝒟 ^{f_{1}}→ s𝒜). Hence, R_{𝒟} f̂_{1} = R_{𝒟} îd_{1}f_{1} = 0 due to Corollary 3.3. Therefore, the first condition of Proposition 2.4 is also verified and f̃ exists. Uniqueness of such f̃ is obvious.

The projection map π_{1} : sℱ𝒬 → sℰ with the underlying map of objects Ob π = id_{Ob 𝒟} determines a strict A_{∞}-functor π : ℱ𝒬 → ℰ. The embedding ι_{1} = (s𝒟^{⊂} ^{i}→ sℱ𝒬 ^{π_{1}}▷ sℰ) with the underlying identity map of objects Ob ι = id_{Ob 𝒟} determines a strict A_{∞}-functor ι : 𝒟 → ℰ. Indeed, ι_{1}^{⊗n}b_{n}^{ℰ} = b_{n}^{𝒟} ι_{1} : s𝒟^{⊗n} → sℰ, for Im(b_{n}^{ℱ𝒟} − b_{n}^{𝒟}) = Im δ_{n} ⊂ J ⊂ sℱ𝒬. These A_{∞}-functors produce other strict A_{∞}-functors for an arbitrary A_{∞}-category 𝒜. For instance, the functor

(π ⊠ 1)M : A_{∞}(ℰ, 𝒜) → A_{∞}(ℱ𝒬, 𝒜), x ↦ πx,

injective on objects and morphisms, and the restriction A_{∞}-functor

restr = (ι ⊠ 1)M : A_{∞}(ℰ, 𝒜) → A_{∞}(𝒟, 𝒜), y ↦ ιy = \overline{y}.

4.7 Theorem. Let ℰ = ℱ𝒬/s^{−1}(R_{𝒟} ), where 𝒟, 𝒬 satisfy assumptions of Section 4.5. Then the restriction A_{∞}-functor

restr : A_{∞}^{ψu}(ℰ, 𝒜) → A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)  (4.7.1)

is an A_{∞}-equivalence, surjective on objects. The chain surjections restr_{1} admit a chain splitting.

Proof. Let us prove that the chain map

restr_{1} : sA_{∞}^{ψu}(ℰ, 𝒜)(f, g) → sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬}))  (4.7.2)

is homotopy invertible for all pairs of pseudounital A_{∞}-functors f, g : ℰ → 𝒜. This will be achieved in a sequence of Lemmata.

The graded 𝕜-quiver decomposition 𝒬 = 𝒟 ⊕ 𝒩 implies that the graded 𝕜-quiver ℱ𝒟 is a direct summand of ℱ𝒬. The projection pr^{ℱ𝒟} : sℱ𝒬 → sℱ𝒟 annihilates all summands with factors s𝒩. Define a degree 0 map

ϖ = (sℱ𝒬 ^{pr^{ℱ𝒟}}→ sℱ𝒟 ^{îd_{1}}→ s𝒟).

4.8 Lemma. There exists a unique chain map

u : sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg)

such that u′ obtained from u via formula (2.6.1) equals

u′ = restr_{𝒬} : sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) → sA_{1}(𝒬, 𝒜)(f|_{𝒬}, g|_{𝒬}), (p, p′) ↦ p′,

and for k > 1 the maps u_{k} are

u_{k} = [sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) ^{pr_{k}}→ ∏_{X,Y ∈Ob 𝒟} C(s𝒟^{⊗k}(X, Y), s𝒜(Xf, Y g))

∏ C(ϖ^{⊗k},1) → ∏_{X,Y ∈Ob 𝒟} C(sℱ𝒬^{⊗k}(X, Y), s𝒜(Xf, Y g))], (p, p′) ↦ p_{k} ↦ ϖ^{⊗k}p_{k}.

Proof. Apply Proposition 2.6 to P = sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})).

4.9 Lemma. The map u from Lemma 4.8 takes values in

sA_{∞}(ℰ, 𝒜)(f, g) ⊂ sA_{∞}(ℱ𝒬, 𝒜)(πf, πg).

Proof. Let us verify conditions of Proposition 2.9. We have Jϖ = 0. Indeed, ϖ vanishes on summands of J = (R_{𝒟} ) of the form Im(b_{t}^{ℱ𝒬} : ⋯ ⊗ s𝒩 ⊗ ⋯ ⊗ R_{𝒟} ⊗ ⋯ → sℱ𝒬) or Im(b_{t}^{ℱ𝒬} : ⋯ ⊗ R_{𝒟} ⊗ ⋯ ⊗ s𝒩 ⊗ ⋯ → sℱ𝒬). Looking at I = J ∩ ℱ𝒟 we find that Jϖ = Iϖ = Iîd_{1} = 0 by Corollary 3.3. Therefore,

(sℱ𝒬^{⊗α} ⊗ J ⊗ sℱ𝒬^{⊗β})((p, p′)u_{k}) = (sℱ𝒬^{⊗α} ⊗ J ⊗ sℱ𝒬^{⊗β})ϖ^{⊗k}p_{k} = 0,  (4.9.1)

and the second condition of Proposition 2.9 is verified.

Let us check now that R_{𝒟} ((p, p′)u_{1}) = 0 for any element

(p, p′) ∈ sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})).

That is,

(−)^{p}(i^{⊗k}b_{k}^{ℱ𝒬} − b_{k}^{𝒟} i)((p, p′)u_{1}) = 0 : s𝒟^{⊗k}(X, Y) → s𝒜(Xf, Y g)  (4.9.2)

for k > 1, where i : s𝒟^{⊂} → sℱ𝒬 is the embedding of differential graded 𝕜-quivers. By definition (2.6.2)

(−)^{p}i^{⊗k}b_{k}^{ℱ𝒬} ((p, p′)u_{1}) = −i^{⊗k}[(pB_{1}, p′B_{1})u_{k}]

+ ∑_{a+q+c=k}^{m,n}(i^{⊗a}π_{1}^{⊗a}f_{am} ⊗ i^{⊗q}((p, p′)u_{q}) ⊗ i^{⊗c}π_{1}^{⊗c}g_{cn})b_{m+1+n}^{𝒜}

− (−)^{p}i^{⊗k} ∑_{a+q+c=k}^{a+c>0}(1^{⊗a} ⊗ b_{q}^{ℱ𝒬} ⊗ 1^{⊗c})((p, p′)u_{a+1+c}) : s𝒟^{⊗k}(X, Y) → s𝒜(Xf, Y g).

For an arbitrary (t, t′) ∈ sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})), in particular for (p, p′) or (pB_{1}, p′B_{1}), we have for k > 1

i^{⊗k}((t, t′)u_{k}) = (t, t′)u_{k}C(i^{⊗k}, 1) = t_{k}C(ϖ^{⊗k}, 1)C(i^{⊗k}, 1) = t_{k}C(i^{⊗k}ϖ^{⊗k}, 1) = t_{k} (4.9.3)

due to relation iϖ = id_{s𝒟} . For k = 0 or 1 we also have i^{⊗k}((t, t′)u_{k}) = t_{k}. Indeed,

(t, t′)u_{0} = (t, t′)u′ pr_{0} = t′_{0} = t_{0},
i((t, t′)u_{1}) = in^{𝒟} ((t, t′)u′) pr_{1} = in^{𝒟} t′_{1} = t_{1}.

Notice also that (s𝒟^{⊂} ^{i}→ sℱ𝒬 ^{π_{1}} ▷ sℰ) = ι_{1}, hence, i^{⊗a}π_{1}^{⊗a}f_{am} = ι_{1}^{⊗a}f_{am} = \overline{f}_{am} =^{def}

(f|_{𝒟} )_{am}. Due to already proven property (4.9.1) we may replace i^{⊗q}b_{q}^{ℱ𝒬} in the last sum with b_{q}^{𝒟} i. Therefore,

(−)^{p}i^{⊗k}b_{k}^{ℱ𝒬} ((p, p′)u_{1}) = −(pB_{1})_{k} + ∑_{a+q+c=k}^{m,n}(\overline{f}_{am} ⊗ p_{q} ⊗ \overline{g}_{cn})b_{m+1+n}^{𝒜}

− (−)^{p} ∑_{a+q+c=k}^{a+c>0}(1^{⊗a} ⊗ b_{q}^{𝒟} ⊗ 1^{⊗c})i^{⊗a+1+c}((p, p′)u_{a+1+c})

= −(pB_{1})_{k} + (pb^{𝒜} )_{k} − (−)^{p} ∑_{a+q+c=k}(1^{⊗a} ⊗ b_{q}^{𝒟} ⊗ 1^{⊗c})p_{a+1+c} + (−)^{p}b_{k}^{𝒟} p_{1}

= −(pb^{𝒜} − (−)^{p}b^{𝒟} p)_{k} + (pb^{𝒜} )_{k} − (−)^{p}(b^{𝒟} p)_{k} + (−)^{p}b_{k}^{𝒟} i((p, p′)u_{1})

= (−)^{p}b_{k}^{𝒟} i((p, p′)u_{1}) : s𝒟^{⊗k}(X, Y) → s𝒜(Xf, Y g) and (4.9.2) is proven. We conclude by Proposition 2.9 that there is a chain map Φ such that

u = (sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) ^{Φ}→ sA_{∞}^{ψu}(ℰ, 𝒜)(f, g)

⊂^{(π⊠1)M_{01}} → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg)), (4.9.4)

so the lemma is proven.

4.10 Lemma. The map Φ from (4.9.4) is a one-sided inverse to restr_{1}:

(sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) ^{Φ}→ sA_{∞}^{ψu}(ℰ, 𝒜)(f, g)

restr_{1} → sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬}))) = id .

Proof. Recall that restr_{1} is the componentwise map

((ι ⊠ 1)M_{01}, (j ⊠ 1)M_{01}) : sA_{∞}^{ψu}(ℰ, 𝒜)(f, g) → sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬}))
r = (r_{k})_{k} ↦ (ιr, jr) = ((ι_{1}^{⊗k}r_{k})_{k⩾0}, (j_{1}^{⊗k}r_{k})_{k=0,1})

Also (π ⊠ 1)M_{01} : sA_{∞}^{ψu}(ℰ, 𝒜)(f, g)^{⊂} → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg), r = (r_{k})_{k} ↦ πr = (π_{1}^{⊗k}r_{k})_{k} is componentwise. Introduce another componentwise map of degree 0

L^{i} : sA_{∞}(ℱ𝒬, 𝒜)(πf, πg) → sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})),
q = (q_{k})_{k} ↦ ((i^{⊗k}q_{k})_{k⩾0}, (in^{𝒬 ⊗k} q_{k})_{k=0,1}).

As ι_{1} = (s𝒟^{⊂} ^{i}→ sℱ𝒬 ^{π_{1}} ▷ sℰ) and j_{1} = (s𝒬^{⊂}^{in^{𝒬}}→ sℱ𝒬 ^{π_{1}} ▷ sℰ), the lower triangle in the following diagram commutes:

sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬}))

=

sA_{∞}^{ψu}(ℰ, 𝒜)(f, g)^{^{Φ}  \overline{→}}

=

sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) ↓^{Φ·restr_{1}  =}  ← ^{L^{i}} \underline{←}^{restr^{1}}  sA_{∞}(ℱ𝒬, 𝒜)(πf, πg) ↓^{u}^{^{(π⊠1)M_{01}} \frac{⊂}{→}}

(4.10.1)

Thus the whole diagram is commutative and Φ · restr_{1} = uL^{i}. We have proved in (4.9.3) and (4.5) that for all (p, p′) ∈ sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) and all k ∈ ℤ_{⩾0} we have i^{⊗k}((p, p′)u_{k}) = p_{k}. Similarly,

in^{𝒬}[(p, p′)u_{1}] = (p, p′)u′ pr_{1} = p′_{1}.

Therefore,

(p, p′)Φ restr_{1} = ((i^{⊗k}(p, p′)u_{k})_{k⩾0}, (in^{𝒬 ⊗k} (p, p′)u_{k})_{k=0,1}) = ((p_{k})_{k⩾0}, (p′_{k})_{k=0,1}) = (p, p′),

and the equation Φ · restr_{1} = id_{sA_{∞1}(𝒟,𝒬;𝒜)((f|_{𝒟} ,f|_{𝒬} ),(g|_{𝒟} ,g|_{𝒬} ))} is proven.

4.11 Lemma. Denote by v the chain map

v = id − restr_{1} ·Φ : sA_{∞}(ℰ, 𝒜)(f, g) → sA_{∞}(ℰ, 𝒜)(f, g).

Denote by w the chain map

w = [sA_{∞}(ℰ, 𝒜)(f, g) ^{v}→ sA_{∞}(ℰ, 𝒜)(f, g)^{⊂}^{(π⊠1)M_{01}} → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg)].

There exists a unique homotopy h : sA_{∞}(ℰ, 𝒜)(f, g) → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg) of degree −1 such that w = B_{1}h + hB_{1},

h′ = (sA_{∞}(ℰ, 𝒜)(f, g) ^{h}→ sA_{∞}(ℱ𝒬, 𝒜)(πf, πg)

restr_{⩽1}▷ sA_{1}(ℱ𝒬, 𝒜)(πf, πg) ^{restr}▷ sA_{1}(𝒬, 𝒜)(jf, jg)) = 0,

h_{k} = 0 : sA_{∞}(ℰ, 𝒜)(f, g) → ∏_{X,Y ∈Ob 𝒟} C(sℱ𝒬^{⊗k}(X, Y), s𝒜(Xf, Y g)), for k > 1.

Proof. We use Corollary 2.7, setting P = sA_{∞}(ℰ, 𝒜)(f, g). We have

w = [sA_{∞}(ℰ, 𝒜)(f, g)^{⊂}^{(π⊠1)M_{01}} → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg)]

−[sA_{∞}(ℰ, 𝒜)(f, g) ^{restr_{1}} → sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) ^{u}→ sA_{∞}(ℱ𝒬, 𝒜)(πf, πg)]

due to (4.9.4). Due to (2.6.1), w′ defined in condition 1 of Corollary 2.7 is

w′ = [sA_{∞}(ℰ, 𝒜)(f, g)^{⊂}^{(π⊠1)M_{01}} → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg)

restr_{⩽1}▷ sA_{1}(ℱ𝒬, 𝒜)(πf, πg) ^{restr}▷ sA_{1}(𝒬, 𝒜)(jf, jg)]

−[sA_{∞}(ℰ, 𝒜)(f, g) ^{restr_{1}} → sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) \begin{matrix}u′ \\ restr & 𝒬 \end{matrix} → sA_{1}(𝒬, 𝒜)(jf, jg)],

where j : 𝒬 ↪ ℰ is the embedding A_{1}-functor, j_{1} = (s𝒬 ↪ sℱ𝒬 ^{π_{1}}→ sℰ). We get

w′ = [sA_{∞}(ℰ, 𝒜)(f, g) ^{restr_{⩽1}}▷ sA_{1}(ℰ, 𝒜)(f, g)

⊂^{(π⊠1)M_{01}} → sA_{1}(ℱ𝒬, 𝒜)(πf, πg) ^{(in^{𝒬} ⊠1)M_{01}}▷ sA_{1}(𝒬, 𝒜)(jf, jg)]
− [sA_{∞}(ℰ, 𝒜)(f, g) ^{((ι⊠1)M_{01},(j⊠1)M_{01})} → sA_{∞1}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬}))
restr_{𝒬} → sA_{1}(𝒬, 𝒜)(jf, jg)]
= [sA_{∞}(ℰ, 𝒜)(f, g) ^{restr_{⩽1}}▷ sA_{1}(ℰ, 𝒜)(f, g) ^{(j⊠1)M_{01}}▷ sA_{1}(𝒬, 𝒜)(jf, jg)]

− [sA_{∞}(ℰ, 𝒜)(f, g) ^{restr_{⩽1}}▷ sA_{1}(ℰ, 𝒜)(f, g) ^{(j⊠1)M_{01}}▷ sA_{1}(𝒬, 𝒜)(jf, jg)] = 0.

Therefore, h′ = 0 satisfies B_{1}h′ + h′B_{1} = 0 = w′. Hence, the unique homotopy h is constructed by Corollary 2.7.

4.12 Remark. In the case of Lemma 4.11 h · pr_{k} = h_{k} = 0 if k > 1 or if k = 0. Indeed, (2.7.1) together with h′ = 0 implies that h_{0} = h pr_{0} = h′ pr_{0} = 0, moreover,

rh_{1}|_{s𝒬} = rh′ pr_{1} = 0 : s𝒬(X, Y) = sℱ_{|}𝒬(X, Y) → s𝒜(Xf, Y g),  (4.12.1)

where r ∈ sA_{∞}(ℰ, 𝒜)(f, g). Therefore, recurrent formula (2.7.2) simplifies here to

(−)^{r}b_{k}^{ℱ𝒬} (rh_{1}) = rw_{k} − ∑_{a+1+c=k}^{m,n}(π_{1}^{⊗a}f_{am} ⊗ rh_{1} ⊗ π_{1}^{⊗c}g_{cn})b_{m+1+n}^{𝒜} :

(sℱ𝒬)^{⊗k}(X, Y) → s𝒜(Xf, Y g). (4.12.2)

4.13 Lemma. The homotopy h constructed in Lemma 4.11 factorizes as

h = (sA_{∞}(ℰ, 𝒜)(f, g) ^{η}→ sA_{∞}(ℰ, 𝒜)(f, g)^{⊂}^{(π⊠1)M_{01}} → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg))

for a unique homotopy η of degree −1 such that v = B_{1}η + ηB_{1}.

Proof. Let us show that h satisfies conditions of Corollary 2.10. The second is obvious. The first is R_{𝒟} (rh_{1}) = 0 for any r ∈ sA_{∞}(ℰ, 𝒜)(f, g), that is,

(−)^{r}(i^{⊗k}b_{k}^{ℱ𝒬} − b_{k}^{𝒟} i)(rh_{1}) = 0 : s𝒟^{⊗k}(X, Y) → s𝒜(Xf, Y g).  (4.13.1)

From (4.12.2) we find the formula for k > 1

(−)^{r}i^{⊗k}b_{k}^{ℱ𝒬} (rh_{1}) = i^{⊗k}(rw_{k}) − ∑_{a+1+c=k}^{m,n}(ι_{1}^{⊗a}f_{am} ⊗ i(rh_{1}) ⊗ ι_{1}^{⊗c}g_{cn})b_{m+1+n}^{𝒜}

= ι_{1}^{⊗k}r_{k}−i^{⊗k}[(ιr, jr)u_{k}]− ∑_{a+1+c=k}^{m,n}(\overline{f}_{am}⊗i(rh_{1})⊗\overline{g}_{cn})b_{m+1+n}^{𝒜} : s𝒟^{⊗k}(X, Y) → s𝒜(Xf, Y g).

A particular case of (4.12.1) is

i(rh_{1}) = [in^{𝒟} (rh′)] pr_{1} = 0,  (4.13.2)

due to h′ = 0, where the A_{1}-functor in^{𝒟} : 𝒟 ↪ 𝒬 is the natural embedding. For our concrete choice of u_{k} we get

(−)^{r}i^{⊗k}b_{k}^{ℱ𝒬} (rh_{1}) = ι_{1}^{⊗k}r_{k} − i^{⊗k}ϖ^{⊗k}ι_{1}^{⊗k}r_{k} = 0 : s𝒟^{⊗k}(X, Y) → s𝒜(Xf, Y g),

since (s𝒟 ^{i}→ sℱ𝒬 ^{ϖ}→ s𝒟) = id. Therefore, i^{⊗k}b_{k}^{ℱ𝒬} (rh_{1}) = 0 and b_{k}^{𝒟} i(rh_{1}) = 0 due to (4.13.2). We conclude that (4.13.1) is satisfied, and by Corollary 2.10 there exists a homotopy

η : sA_{∞}(ℰ, 𝒜)(f, g) → sA_{∞}(ℰ, 𝒜)(f, g),

such that deg η = −1, h = η · [(π ⊠ 1)M_{01}] and v = B_{1}η + ηB_{1}.

Lemmata 4.10 and 4.13 show that the maps restr_{1} and Φ given by (4.7.2) and (4.9.4) are homotopy inverse to each other.

The A_{∞}-functor restr is surjective on objects by Proposition 4.6, and its first component is a homotopy isomorphism. Therefore, it is an A_{∞}-equivalence by Corollary 1.9, and Theorem 4.7 is proven.

4.14 Corollary. The collection of A_{∞}-functors (4.7.1) is natural A_{∞}-2-equivalence.

Proof. The restriction A_{∞}-functors A_{∞}^{ψu}(ℰ, 𝒜) → A_{∞}^{ψu}(𝒟, 𝒜) and A_{∞}^{ψu}(ℰ, 𝒜) → A_{1}(𝒬, 𝒜) are strict A_{∞}-2-transformations. By (4.2.1) the restriction A_{∞}-functor A_{∞}^{ψu}(ℰ, 𝒜) → A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) is also a strict A_{∞}-2-transformation. It is an A_{∞}-equivalence by Theorem 4.7.

4.15 Remark. The maps Φ, η constructed in the proof of Theorem 4.7 satisfy

Φ · η = 0 : sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) → sA_{∞}^{ψu}(ℰ, 𝒜)(f, g).

Indeed, Φ · η composed with an embedding,

Φ · η · (π ⊠ 1)M_{01} = Φ · h : sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬})) → sA_{∞}(ℱ𝒬, 𝒜)(πf, πg),

is a degree −1 homotopy such that

B_{1}(Φh) + (Φh)B_{1} = Φ(B_{1}h + hB_{1}) = Φw = Φ(id − restr_{1} Φ)(π ⊠ 1)M_{01} = (id −Φ restr_{1})Φ(π ⊠ 1)M_{01} = 0.

We have Φh pr_{k} = Φ · h_{k} = 0 for k > 1 and

(Φh)′ = Φh restr_{⩽1} restr = Φ · h′ = 0.

The 0 homotopy for 0 chain map also has these properties, and by Corollary 2.7 we conclude that Φh = 0.

4.16 Remark. The equation

η · restr_{1} = 0 : sA_{∞}^{ψu}(ℰ, 𝒜)(f, g) → sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬}))

also holds. Indeed, the decomposition restr_{1} = (π ⊠ 1)M_{01} · L^{i} from diagram (4.10.1) implies that

η · restr_{1} = η · (π ⊠ 1)M_{01} · L^{i} = h · L^{i}.

For any r ∈ sA_{∞}^{ψu}(ℰ, 𝒜)(f, g) all components of the element

rhL^{i} = ((i^{⊗k}(rh_{k}))_{k⩾0}, (in^{𝒬 ⊗k} (rh_{k}))_{k=0,1}) ∈ sA_{∞1}^{ψu}(𝒟, 𝒬; 𝒜)((f|_{𝒟} , f|_{𝒬}), (g|_{𝒟} , g|_{𝒬}))

vanish except, possibly, those indexed by k = 1 by Remark 4.12. However, i(rh_{1}) = 0 by (4.13.2), and, moreover, in^{𝒬}(rh_{1}) = 0 by (4.12.1), thus, all the components of rhL^{i} vanish.

4.17 Remark. We have not used in the proof of Theorem 4.7 the assumption of pseudounitality of 𝒟 and 𝒜. Its assertion holds without this property. The restriction A_{∞}-functor

restr : A_{∞}(ℰ, 𝒜) → A_{∞1}(𝒟, 𝒬; 𝒜)

is an A_{∞}-equivalence, surjective on objects. Its first component maps admit a chain splitting. In the particular case 𝒟(X, Y) = 0 for all X, Y ∈ Ob 𝒬 we get Theorem 2.12 of [LM03]: the A_{∞}-functor

restr : A_{∞}(ℱ𝒬, 𝒜) → A_{1}(𝒬, 𝒜)

is an A_{∞}-equivalence.

5. Relatively free A_{∞}-categories

Hinich [Hin97] defines standard cofibrations of differential graded algebras. This notion is generalized by Drinfeld to semi-free differential graded categories [Dri02]. We give a definition in the spirit of these two definitions in the framework of A_{∞}-categories.

5.1 Definition. Let e : 𝒞 → 𝒟 be a strict A_{∞}-functor such that Ob e is an isomorphism, and e_{1} : s𝒞 → s𝒟 is an embedding. The A_{∞}-category 𝒟 is relatively free over 𝒞, if it can be represented as the union of an increasing sequence of its A_{∞}-subcategories 𝒟_{j} and differential graded subquivers 𝒬_{j}

𝒟_{0} ⊂ 𝒬_{1} ⊂ 𝒟_{1} ⊂ 𝒬_{2} ⊂ 𝒟_{2} ⊂ 𝒬_{3} ⊂ ⋯ ⊂ 𝒟  (5.1.1)

with the same set of objects Ob 𝒟, such that

1. s𝒟_{0} = (s𝒞)e_{1};

2. for each j ⩾ 0 the embedding of graded quivers 𝒟_{j} ^{⊂} → 𝒬_{j+1} admits a splitting map 𝒬_{j+1} ▷ 𝒟_{j} of degree 0;

3. for each j > 0 the unique strict A_{∞}-functor ℱ𝒬_{j} → 𝒟_{j} extending the embedding 𝒬_{j} ^{⊂} → 𝒟_{j} factors into the natural projection and an isomorphism

ℱ𝒬_{j} ▷ ℱ𝒬_{j}/s^{−1}(R_{j}) ^{∼}→ 𝒟_{j},

where the system of relations R_{j} = R_{𝒟_{j−1}} ⊂ sℱ𝒬_{j} is R_{j} = ∑_{n⩾2} Im(δ_{n}) for

δ_{n} = ((s𝒟_{j−1})^{⊗n} ^{⊂} → (sℱ𝒬_{j})^{⊗n} ^{b_{n}^{ℱ𝒬_{j}}}→ sℱ𝒬_{j}) − ((s𝒟_{j−1})^{⊗n} ^{b_{n}^{𝒟_{j−1}}}→ s𝒟_{j−1} ^{⊂} → sℱ𝒬_{j}).

When all differential graded quivers 𝒩_{j} = Coker(𝒟_{j−1} ^{⊂} → 𝒬_{j}) have zero differential and the 𝕜-modules 𝒩_{j}^{k}(X, Y) are free for all j ⩾ 1, k ∈ ℤ, we say that 𝒟 is semi-free over 𝒞 in accordance with terminology of Drinfeld. In fact, if in Definition 5.1 one replaces A_{∞}-categories with differential graded categories and adds the above assumption on 𝒩_{j}, then one recovers Definition 13.4 from [Dri02] of semi-free differential graded categories.

The system of relations R_{j} is the minimal one that ensures that the natural embedding s𝒟_{j−1} ↪ sℱ𝒬_{j}/(R_{j}) = s𝒟_{j} is the first component of a strict A_{∞}-functor. In semi-free case we may say that 𝒟_{j} is freely generated by 𝒩_{j} over 𝒟_{j−1}.

5.2. The main construction. Let 𝒞 be a unital A_{∞}-category, and let ℬ ⊂ 𝒞 be its full subcategory. The unit i_{0}^{𝒞} is abbreviated to i_{0}.

A vertex of a tree is k-ary if it is adjacent to k + 1 edges. A unary vertex is a 1-ary one.

Define a labeled tree t = (t; X_{0}, X_{1}, …, X_{n}) as a non-empty (non-reduced) plane rooted tree t with n leaves, such that unary vertices are not joined by an edge, equipped with a sequence (X_{0}, X_{1}, …, X_{n}) of objects of 𝒞.

Let e be an edge of t. If i is the smallest number such that i-th leaf is above e, the domain of e is defined as dom(e) = X_{i−1}. If k is the biggest number such that k-th leaf is above e, the codomain of e is defined as codom(e) = X_{k}. An admissible tree is a labeled tree (t; X_{0}, X_{1}, …, X_{n}) such that for each edge e adjacent to a unary vertex dom(e) ∈ Ob ℬ or codom(e) ∈ Ob ℬ (or both).

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 𝒞-admissible tree is an admissible tree (t; X_{0}, X_{1}, …, X_{n}) such that top (maximal with respect to ≼) internal vertices are unary.

Define a graded quiver ℰ with the set of objects Ob ℰ = Ob 𝒞. The ℤ-graded 𝕜-module of morphisms between X, Y ∈ Ob ℰ is defined as

sℰ(X, Y) = ⨁_{n⩾1} ⨁_{admissible (t;X_{0},X_{1},...,X_{n})}^{X_{0}=X, X_{n}=Y} sℰ(t)(X, Y),  (5.2.1)
sℰ(t)(X_{0}, X_{n}) = sℰ(t) = s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n})[|t|_{1} − |t|_{>}],

where |t|_{1} is the number of unary internal vertices of t, and |t|_{>} is the the number of internal vertices of arity > 1 .

The vertices of arity k > 1 are interpreted as k-ary multiplications of degree 1. Unary vertices are interpreted as contracting homotopies H of degree −1. Define an A_{∞}-structure on ℰ in which operations b_{k}, k > 1, are given by grafting. So 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|} : sℰ(t_{1})(Y_{0}, Y_{1}) ⊗ ⋯ ⊗ sℰ(t_{k})(Y_{k−1}, Y_{k}) → sℰ(t)(Y_{0}, Y_{k}),

where |t| = |t|_{>} − |t|_{1} and t = (t_{1} ⊔ ⋯ ⊔ t_{k}) · 𝔱_{k}. In particular, |t| = |t_{1}| + ⋯ + |t_{k}| + 1.

Let t = (t; X_{0}, X_{1}, …, X_{n}) be an admissible tree, whose lowest internal vertex is not unary. In particular, t might be the trivial tree t = (|; X_{0}, X_{1}). Assume that X_{0} ∈ Ob ℬ or X_{n} ∈ Ob ℬ (or both). Denote by H the 𝕜-linear map

H = s : sℰ(t)(X_{0}, X_{n}) = s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n})[|t|_{1} − |t|_{>}] → s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n})[1 + |t|_{1} − |t|_{>}] = sℰ(t · 𝔱_{1})(X_{0}, X_{n})

of degree −1. Here 𝔱_{1} = ( ^{}; X_{0}, X_{n}) is the unary corolla.

The operation b_{1} is determined by the given differential b_{1} : s𝒞 → s𝒞 and by the recursive substitutions

b_{k}b_{1} := − ∑_{α+p+β=k}^{α+β>0}(1^{⊗α} ⊗ b_{p} ⊗ 1^{⊗β})b_{α+1+β},  k > 1,  (5.2.2)
Hb_{1} := 1 − b_{1}H,  (5.2.3)

where H stands for a unary vertex. Identity (5.2.2) is satisfied for k > 1 by definition of b_{1}. We have to prove that b_{1}^{2} = 0. Assume that k > 1, then

b_{k}b_{1}^{2} = − ∑_{α+p+β=k}^{α+β>0}(1^{⊗α} ⊗ b_{p} ⊗ 1^{⊗β})b_{α+1+β}b_{1}

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

= ∑_{α+p+ε+q+δ=k}(1^{⊗α} ⊗ b_{p} ⊗ 1^{⊗ε} ⊗ b_{q} ⊗ 1^{⊗δ})b_{α+ε+δ+2}

− ∑_{γ+q+η+p+β=k}(1^{⊗γ} ⊗ b_{q} ⊗ 1^{⊗η} ⊗ b_{p} ⊗ 1^{⊗β})b_{γ+η+β+2}

+ ∑_{γ+r+δ=k}^{γ+δ>0}{1^{⊗γ} ⊗ [ ∑_{κ+p+λ=r}(1^{⊗κ} ⊗ b_{p} ⊗ 1^{⊗λ})b_{κ+1+λ}] ⊗ 1^{⊗δ}}b_{γ+1+δ}

= ∑_{γ+1+δ=k}(1^{⊗γ} ⊗ b_{1}^{2} ⊗ 1^{⊗δ})b_{k},

because the sum in square brackets vanishes for r > 1 by (5.2.2). We also have

Hb_{1}^{2} = (1 − b_{1}H)b_{1} = b_{1} − b_{1}(1 − b_{1}H) = b_{1}^{2}H.

By induction the equation b_{1}^{2}|_{s𝒞} = 0 implies that b_{1}^{2} = 0 on sℰ. Therefore, ℰ is an A_{∞}-category.

It has an ideal (R_{𝒞})_{+}, generated by the 𝕜-subquiver R_{𝒞} = ∑_{n⩾2} Im(δ_{n}) for

δ_{n} = ((s𝒞)^{⊗n} ^{⊂} → (sℰ)^{⊗n} ^{b_{n}^{ℰ}}→ sℰ) − ((s𝒞)^{⊗n} ^{b_{n}^{𝒞}}→ s𝒞^{⊂} → sℰ)

by application of operations 1^{⊗α}⊗H⊗1^{⊗β}, 1^{⊗α}⊗b_{p}⊗1^{⊗β} for p ⩾ 2. By Lemma 3.1 R_{𝒞}b_{1}^{ℰ} ⊂ (R_{𝒞}) ⊂ (R_{𝒞})_{+}, where (R_{𝒞}) denotes the ideal generated by application of 1^{⊗α} ⊗ b_{p} ⊗ 1^{⊗β} (p ⩾ 2) only. Similarly to Proposition 2.1 this implies that (R_{𝒞})_{+}b_{1}^{ℰ} ⊂ (R_{𝒞})_{+}. Indeed, let t = (t; X_{0}, X_{1}, …, X_{n}) be an admissible tree, whose lowest internal vertex is not unary. Assume that X_{0} ∈ Ob ℬ or X_{n} ∈ Ob ℬ, so that t · 𝔱_{1} is admissible. For an arbitrary z ∈ (R_{𝒞})_{+}(t)(X_{0}, X_{n}) there exists zH ∈ (R_{𝒞})_{+}(t · 𝔱_{1})(X_{0}, X_{n}), and zb_{1} ∈ (R_{𝒞})_{+}(X_{0}, X_{n}) by induction. Due to (5.2.3) zHb_{1} = z−zb_{1}H ∈ (R_{𝒞})_{+}, which proves the claim. Therefore, the ideal (R_{𝒞})_{+} is stable with respect to all A_{∞}-operations, including b_{1}.

Denote by 𝒟 = ℰ/s^{−1}(R_{𝒞})_{+} = Q(𝒞|ℬ) the quotient A_{∞}-category. It has a direct sum decomposition similar to that of ℰ

s𝒟(X, Y) = ⨁_{n⩾1} ⨁_{𝒞-admissible (t;X_{0},X_{1},...,X_{n})}^{X_{0}=X, X_{n}=Y} s𝒟(t)(X, Y),
s𝒟(t)(X_{0}, X_{n}) = sℰ(t)(X_{0}, X_{n}) = s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n})[|t|_{1} − |t|_{>}],

with the only difference that the sum is taken over 𝒞-admissible trees t. We can view 𝒟 as a graded 𝕜-subquiver of ℰ.

The category 𝒞 is embedded in 𝒟 (via a strict A_{∞}-functor) as

s𝒟_{0}(X, Y) = s𝒟(|)(X, Y) = s𝒞(X, Y)

for the trivial tree t = (|; X, Y). Let us show that 𝒟 is relatively free over 𝒞.

Let us define for j ⩾ 0 the A_{∞}-subcategories 𝒟_{j} and differential graded subquivers 𝒬_{j+1} of 𝒟 so that embeddings (5.1.1) hold. Each leaf ℓ and the root of a tree can be connected by the unique minimal path. We say that internal vertices occurring at this path are between the root and the leaf ℓ. Define for j ⩾ 0 the A_{∞}-subcategory 𝒟_{j} = ⊕_{t}𝒟(t) of 𝒟, where the summation goes over all

𝒞-admissible trees t with no more than j unary internal vertices between the root and any leaf.

(C1)

Define for j ⩾ 1 the graded subquiver 𝒩_{j} = ⊕_{t}𝒟(t) of 𝒟, where the summation goes over all trees t satisfying (C1) and such that

there exists a leaf ℓ of t with j unary internal vertices between the root and ℓ; the lowest internal vertex (adjacent to the root) is unary.

(C2)

One can easily see that for j ⩾ 1

s𝒬_{j} = s𝒟_{j−1} ⊕ s𝒩_{j}

is a differential graded subquiver of 𝒟_{j} ⊂ 𝒟. For example, 𝒟_{0} = 𝒟(|) = 𝒞, 𝒩_{1} = 𝒟(^{}), 𝒬_{1} = 𝒟(|) ⊕ 𝒟(^{}), and 𝒟_{1} = 𝒟(|) ⊕ ⊕_{t}𝒟(t), where t runs over admissible trees with the only unary internal vertex v, such that all other internal vertices lie on the minimal path between the root and v.

The inclusion map of differential graded quivers i : s𝒬_{j} ^{⊂} → s𝒟_{j} induces a unique strict A_{∞}-functor î: ℱ𝒬_{j} → 𝒟_{j} [LM03, Corollary 2.4].

5.3 Proposition. The map î_{1} : sℱ𝒬_{j} → s𝒟_{j} is surjective and its kernel is (R_{𝒟_{j−1}} ). Thus it induces an isomorphism ι_{1} : sℱ𝒬_{j}/(R_{𝒟_{j−1}} ) → s𝒟_{j}.

Proof. The strict A_{∞}-functor î: ℱ𝒬_{j} → 𝒟_{j} is described in [LM03, Section 2.6] as follows. Let t be a reduced labeled tree, with n input leaves, and let ⩽ be a linear order on Vert(t), such that x ≼ y implies x ⩽ y for all x, y ∈ Vert(t). The choice of ⩽ is equivalent to the choice of decomposition into product of elementary forests (2.0.1). The linearly ordered tree (t, ⩽) determines the map b_{(t,⩽)}^{ℱ𝒬_{j}} : (sℱ𝒬_{j})^{⊗n} → sℱ𝒬_{j} given by (2.0.2) and a similar

map b_{(t,⩽)}^{𝒟_{j}} : (s𝒟_{j})^{⊗n} → s𝒟_{j}. In the commutative diagram from [LM03, Section 2.6]

\begin{matrix}(s𝒬_{j})^{⊗n} & \overline{→^{\begin{matrix}b_{(t,⩽)}^{ℱ𝒬_{j}} \\ ±s^{−|t|} \end{matrix}}} & sℱ_{t}𝒬_{j} \\ (s𝒟_{j})^{⊗n} & ↓^{i_{1}^{⊗n}} & \overline{^{b_{(t,⩽)}^{𝒟_{j}}}→} & s𝒟_{j} & ↓^{î_{1}} \end{matrix}  (5.3.1)

the top map is invertible, so î_{1} is uniquely determined by this diagram.

Being the first component of a strict A_{∞}-functor î_{1} satisfies, in particular, the equation

((s𝒟_{j−1})^{⊗n} ^{⊂} → (sℱ_{|}𝒬_{j})^{⊗n} ^{b_{n}^{ℱ𝒬_{j}}}→ sℱ𝒬_{j} ^{î_{1}}→ s𝒟_{j}) = ((s𝒟_{j−1})^{⊗n} ^{⊂}^{i_{1}^{⊗n}} → (s𝒟_{j})^{⊗n} ^{b_{n}^{𝒟_{j}}}→ s𝒟_{j})

= ((s𝒟_{j−1})^{⊗n} ^{b_{n}^{𝒟_{j−1}}} → s𝒟_{j−1} ^{⊂} ^{i_{1}}→ s𝒟_{j}) = ((s𝒟_{j−1})^{⊗n} ^{b_{n}^{𝒟_{j−1}}} → s𝒟_{j−1} ^{⊂} → sℱ_{|}𝒬_{j} ^{î_{1}}→ s𝒟_{j}).

It implies R_{𝒟_{j−1}} î_{1} = 0. Since î is strict we have also (R_{𝒟_{j−1}} )î_{1} = 0. Thus there is a strict A_{∞}-functor ι with the first component ι_{1} : sℱ𝒬_{j}/(R_{𝒟_{j−1}} ) → s𝒟_{j}, identity on objects.

Let us construct a degree 0 map ϕ : s𝒟_{j} → sℱ𝒬_{j} for j ⩾ 1. Let t be a tree that satisfies (C1). Denote by UV(t) ⊂ Vert(t) the subset of unary internal vertices. Let minUV(t) be the subset of partially ordered set (UV(t), ≼) consisting of minimal elements. Let L ⊂ Leaf(t) be the subset of leaves ℓ such that between ℓ and the root there are no unary vertices. Let \overline{L} be the set of leaf vertices above leaves from L. Using the canonical linear ordering t_{<} = (t, ⩽) of \overline{Vert}(t) [LM03, Section 1.7] we can write the set \overline{L} ⊔ minUV(t) as {u_{1} < ⋯ < u_{k}}. For any 1 ⩽ p ⩽ k denote by t_{p} the 𝒞-admissible subtree of t with

\overline{Vert}(t_{p}) = \{y ∈ \overline{Vert}(t) | y ≽ u_{p}\} ⊔ \{new root vertex r_{p}\}.

Edges of t_{p} are all edges of t above u_{p} plus a new root edge between u_{p} and r_{p}. In particular, if u_{p} ∈ \overline{L}, then t_{p} = | is the trivial tree. Denote by t′ the reduced labeled tree, which is t with all vertices and edges above minUV(t) removed. It has precisely k leaves. Thus t is the concatenation of a forest and t′:

t = (t_{1} ⊔ t_{2} ⊔ ⋯ ⊔ t_{k}) · t′.  (5.3.2)

We have

Vert(t) = Vert(t′) ⊔ ⨆_{p=1}^{k} Vert(t_{p}),  Leaf(t) = ⨆_{p=1}^{k} Leaf(t_{p}),  Leaf(t′) = ⨆_{p=1}^{k} Out(t_{p}).

Correspondingly, the labels of the p-th leaf of t′ are those of Out(t_{p}).

Being simply a shift, the 𝕜-linear map b_{t′_{<}}^{𝒟} : s𝒟(t_{1})⊗⋯⊗s𝒟(t_{k}) → s𝒟(t) is invertible. Therefore, for any element x ∈ s𝒟(t) there exists a unique tensor ∑_{i} z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i} ∈ s𝒟(t_{1}) ⊗ ⋯ ⊗ s𝒟(t_{k}) such that x = ∑_{i}(z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i})b_{t′_{<}}^{𝒟}. We have z_{p}^{i} ∈ s𝒟(t_{p}) ⊂ s𝒬_{j}, in particular, z_{p}^{i} ∈ s𝒟(|) = s𝒞, if u_{p} ∈ \overline{L}. Define

xϕ = ∑_{i}(z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i})b_{t′_{<}}^{ℱ𝒬_{j}} ∈ sℱ_{t′}𝒬_{j}.

Commutative diagram (5.3.1) implies that xϕî_{1} = ∑_{i}(z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i})b_{t′_{<}}^{𝒟_{j}} = x. Therefore,

[s𝒟_{j} ^{ϕ}→ sℱ𝒬_{j} ^{π_{1}} ▷ sℱ𝒬_{j}/(R_{𝒟_{j−1}} ) ^{ι_{1}}→ s𝒟_{j}] = id .  (5.3.3)

Let us prove that ϕπ_{1} preserves the operations b_{n} for n > 1. Indeed,

b_{n}^{𝒟_{j}} (ϕπ_{1}) − (ϕπ_{1})^{⊗n}b_{n} = [(s𝒟_{j})^{⊗n} ^{b_{n}^{𝒟_{j}} ϕ−ϕ^{⊗n}b_{n}^{ℱ𝒬_{j}}}→ sℱ𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )].

Consider trees τ_{1}, τ_{2}, …, τ_{n} satisfying condition (C1), labeled so that the operation b_{n}^{𝒟_{j}} : s𝒟(τ_{1}) ⊗ ⋯ ⊗ s𝒟(τ_{n}) → s𝒟_{j} makes sense. The quiver (s𝒟_{j})^{⊗n} is a direct sum of such s𝒟(τ_{1})⊗⋯⊗s𝒟(τ_{n}). If some of trees τ_{p} are not trivial, then b_{n}^{𝒟_{j}} ϕ = ϕ^{⊗n}b_{n}^{ℱ𝒬_{j}} , because constructing ϕ for τ = (τ_{1}⊔τ_{2}⊔⋯⊔τ_{n})·𝔱_{n} is equivalent to decomposing each τ_{p} as in (5.3.2), collecting the upper parts, and gluing the lowest parts τ′_{p} into τ′ = (τ′_{1} ⊔ τ′_{2} ⊔ ⋯ ⊔ τ′_{n}) · 𝔱_{n}. If all trees τ_{p} are trivial, then 𝒟(τ_{p}) = 𝒟(|) = 𝒞 and

(b_{n}^{𝒞}ϕ − ϕ^{⊗n}b_{n}^{ℱ𝒬_{j}} )π_{1} = (b_{n}^{𝒞} − b_{n}^{ℱ𝒬_{j}} )π_{1} = 0 : (s𝒞)^{⊗n} → sℱ𝒬_{j}/(R_{𝒟_{j−1}} ),

due to R_{𝒞}π_{1} ⊂ R_{𝒟_{j−1}} π_{1} = 0.

We claim that

[sℱ𝒬_{j} ^{î_{1}}→ s𝒟_{j} ^{ϕ}→ sℱ𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )] = [sℱ𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )]. (5.3.4)

First of all, the restriction of this equation to s𝒬_{j} = sℱ_{|}𝒬_{j} holds true:

[s𝒬_{j} ^{i_{1}}→ s𝒟_{j} ^{ϕ}→ sℱ𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )] = [s𝒬_{j} ^{⊂} → sℱ𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )].

(5.3.5)

Indeed, s𝒬_{j} = s𝒟_{j−1} ⊕ s𝒩_{j}. On the first summand we get for x ∈ s𝒟_{j−1}(t)

x ^{i_{1}}→ x = ∑_{i}(z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i})b_{t′_{<}}^{𝒟_{j}} ^{ϕ}→ ∑_{i}(z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i})b_{t′_{<}}^{ℱ𝒬_{j}}

π_{1}→ ∑_{i}(z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i})b_{t′_{<}}^{ℱ𝒬_{j}} + (R_{𝒟_{j−1}} ) = ∑_{i}(z_{1}^{i} ⊗ ⋯ ⊗ z_{k}^{i})b_{t′_{<}}^{𝒟_{j−1}} + (R_{𝒟_{j−1}} ) = xπ_{1}

by Proposition 3.2 because z_{p}^{i} ∈ s𝒟_{j−1}. On the second summand we get for x ∈ s𝒩_{j}(t)

x ^{i_{1}}→ x ^{ϕ}→ x ^{π_{1}}→ xπ_{1}

since t′ = |. Thus, (5.3.5) is verified.

Now we prove (5.3.4) on the generic summand sℱ_{τ}𝒬_{j} of sℱ𝒬_{j}, where τ is a reduced labeled tree with n leaves. The first map below is an isomorphism:

[(s𝒬_{j})^{⊗n} ^{b_{τ<}^{ℱ𝒬_{j}}}→ sℱ_{τ}𝒬_{j} ^{î_{1}}→ s𝒟_{j} ^{ϕπ_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )]
= [(s𝒬_{j})^{⊗n} ^{⊂}^{i_{1}^{⊗n}}→ (s𝒟_{j})^{⊗n} ^{b_{τ<}^{𝒟_{j}}}→ s𝒟_{j} ^{ϕπ_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )]
= [(s𝒬_{j})^{⊗n} ^{⊂}^{i_{1}^{⊗n}}→ (s𝒟_{j})^{⊗n} ^{ϕπ_{1}}→ (sℱ𝒬_{j}/(R_{𝒟_{j−1}} ))^{⊗n} ^{b_{τ<}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )]
= [(s𝒬_{j})^{⊗n} ^{⊂} → (sℱ𝒬_{j})^{⊗n} ^{π_{1}^{⊗n}} → (sℱ𝒬_{j}/(R_{𝒟_{j−1}} ))^{⊗n} ^{b_{τ<}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )]
= [(s𝒬_{j})^{⊗n} ^{b_{τ<}^{ℱ𝒬_{j}}}→ sℱ_{τ}𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )]

by (5.3.5) and by the fact that the considered maps î_{1}, ϕπ_{1} and π_{1} commute with b_{τ_{<}}. Rewriting (5.3.4) in the form

[sℱ𝒬_{j} ^{π_{1}} ▷ sℱ𝒬_{j}/(R_{𝒟_{j−1}} ) ^{ι_{1}}→ s𝒟_{j} ^{ϕ}→ sℱ𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )]

= [sℱ𝒬_{j} ^{π_{1}} ▷ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )],

we find by surjectivity of π_{1} that

[sℱ𝒬_{j}/(R_{𝒟_{j−1}} ) ^{ι_{1}}→ s𝒟_{j} ^{ϕ}→ sℱ𝒬_{j} ^{π_{1}}→ sℱ𝒬_{j}/(R_{𝒟_{j−1}} )] = id .

Together with (5.3.3) this proves that ϕπ_{1} is an inverse to ι_{1}.

5.4 Corollary. The A_{∞}-category 𝒟 = Q(𝒞|ℬ) is relatively free over 𝒞.

5.5. The first equivalence. Let 𝒜 be pseudounital, then the restriction functor

A_{∞1}^{ψu}(𝒟_{0}, 𝒬_{1}; 𝒜) → A_{∞}^{ψu}(𝒞, 𝒜), (f, f′) ↦ f

takes values in the full subcategory A_{∞}^{ψu}(𝒞, 𝒜)_{mod ℬ} . Indeed, let (f, f′) be an object of A_{∞1}^{ψu}(𝒟_{0}, 𝒬_{1}; 𝒜). For arbitrary objects X, Y of ℬ we have

f_{1} = [s𝒞(X, Y)^{⊂} → s𝒬_{1}(X, Y) ^{f′_{1}}→ s𝒜(Xf, Y f)]
= [s𝒟(|)(X, Y) ^{b_{1}H+Hb_{1}} → s𝒬_{1}(X, Y) ^{f′_{1}}→ s𝒜(Xf, Y f)]
= [s𝒟(|)(X, Y) ^{b_{1}(Hf′_{1})+(Hf′_{1})b_{1}} → s𝒜(Xf, Y f)],

where H is the map H : s𝒟(|) ^{s}→ s𝒟(𝔱_{1})^{⊂} → s𝒬_{1}. Hence, the above f_{1} is null-homo-topic. By Definition 6.4 of [LO02] the A_{∞}-functor f|_{ℬ} is contractible.

A short exact sequence of chain maps of complexes is semisplit (resp. semisplittable) if it is split (resp. splittable) as a sequence of degree 0 maps of graded 𝕜-modules.

5.6 Lemma. Let 0 → C ^{α}→ A ^{β}→ B → 0 be a semisplittable exact sequence of complexes of 𝕜-modules. If C is contractible, then this sequence is splittable, and the splitting chain map ν : B → A can be chosen so that ν is homotopy inverse to β.

Proof. Let ϕ : A → C be a map of degree 0, such that αϕ = 1_{C}. Assume that 1_{C} = Hd + dH for a homotopy H : C → C of degree −1. Then ψ = (ϕH)d = ϕHd + dϕH : A → C is a chain map, and αψ = αϕHd + dαϕH = Hd + dH = 1_{C}. Denote by ν : B → A the unique 𝕜-linear map such that νψ = 0, νβ = 1. The splitting injection ν = ker ψ is a chain map. The sequence looks as follows

0 → C →^{α} ; ←_{ψ} A →^{β} ; ←_{ν} B → 0.

Since A is a direct sum C ⊕ B, we have

id_{A} −βν = ψα = (ϕHd + dϕH)α = (ϕHα)d + d(ϕHα) = γd + dγ,

where γ = (A ^{ϕ}→ C ^{H}→ C ^{α}→ A) is a homotopy.

5.7 Proposition. Let 𝒜 be unital, then the restriction strict A_{∞}-functor

restr : A_{∞1}^{ψu}(𝒟_{0}, 𝒬_{1}; 𝒜) → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ}  (5.7.1)

is an A_{∞}-equivalence, surjective on objects. The chain surjections restr_{1} admit a chain splitting.

Proof. First of all, restr is surjective on objects. Indeed, assume that f : 𝒞 → 𝒜 is unital and ℬ^{⊂} → 𝒞 ^{f}→ 𝒜 is contractible. We have to extend the chain maps f_{1} : s𝒞(X, Y) → s𝒜(Xf, Y f) to chain maps f′_{1} : s𝒬_{1}(X, Y) → s𝒜(Xf, Y f). If X, Y ∉ Ob ℬ, then s𝒬_{1}(X, Y) = s𝒞(X, Y) and f′_{1} = f_{1}. If X ∈ Ob ℬ or Y ∈ Ob ℬ, then s𝒬_{1}(X, Y) = s𝒞(X, Y) ⊕ s𝒟(𝔱_{1})(X, Y) as a graded quiver, and the complex 𝒜(Xf, Y f) is contractible by Proposition 6.1(C1), (C2) of [LO02]. Let χ_{XY} : s𝒜(Xf, Y f) → s𝒜(Xf, Y f) be a contracting homotopy for 𝒜(Xf, Y f). Define

f′_{1} = (s𝒟(𝔱_{1})(X, Y) ^{s^{−1}}→ s𝒞(X, Y) ^{f_{1}}→ s𝒜(Xf, Y f) ^{χ_{XY}} → s𝒜(Xf, Y f)).

Then Hf′_{1} = f_{1}χ_{XY}. Since H = s : s𝒞 → s𝒟(𝔱_{1}) is invertible, the equation

Hf′_{1}b_{1} − Hb_{1}f′_{1} = Hf′_{1}b_{1} + b_{1}Hf′_{1} − f′_{1} = f_{1}χ_{XY} b_{1} + b_{1}f_{1}χ_{XY} − f_{1} = 0 : s𝒞(X, Y) → s𝒜(Xf, Y f)

implies that f′_{1} is a chain map.

Let us prove that the restriction chain map

restr_{1} : sA_{∞1}^{ψu}(𝒞, 𝒬_{1}; 𝒜)((f, f′), (g, g′)) → sA_{∞}(𝒞, 𝒜)(f, g), (r, r′) ↦ r

is homotopy invertible. This map is a product over n ∈ ℤ_{⩾0} of the restriction maps ρ_{n} : V_{n} → V′_{n} of the graded 𝕜-modules of n-th components (compare (4.3.1) with analogous decomposition of sA_{∞}(𝒞, 𝒜)(f, g)). Clearly, the maps ρ_{n} = id for n = 0 or for n > 1. On the other hand, for n = 1

ρ_{1} = ∏ C(in^{𝒞}, 1) : ∏_{X,Y ∈Ob 𝒞} C(s𝒬_{1}(X, Y), s𝒜(Xf, Y g)) → ∏_{X,Y ∈Ob 𝒞} C(s𝒞(X, Y), s𝒜(Xf, Y g))

is surjective with the kernel Ker ρ_{1} = ∏_{X,Y ∈Ob 𝒞} C(s𝒟(𝔱_{1})(X, Y), s𝒜(Xf, Y g)), because the sequence 0 → s𝒞 → s𝒬_{1} → s𝒟(𝔱_{1}) → 0 is semisplit. Since we may write the kernel as

Ker ρ_{1} = ∏_{X∈Ob ℬ,Y ∈Ob 𝒞 or X∈Ob 𝒞,Y ∈Ob ℬ} C(s𝒟(𝔱_{1})(X, Y), s𝒜(Xf, Y g)),

it is contractible, because contractibility of f|_{ℬ} , g|_{ℬ} : ℬ → 𝒜 implies contractibility of complexes 𝒜(Xf, Y g) by Proposition 6.1(C1), (C2) of [LO02].

Summing up, the first term of the semisplit exact sequence

0 → Ker restr_{1} → sA_{∞1}(𝒞, 𝒬_{1}; 𝒜)((f, f′), (g, g′)) ^{restr_{1}} → sA_{∞}(𝒞, 𝒜)(f, g) → 0

is contractible. By Lemma 5.6 this sequence admits a splitting chain map

ν : sA_{∞}(𝒞, 𝒜)(f, g) → sA_{∞1}(𝒞, 𝒬; 𝒜)((f, f′), (g, g′)),

and ν is homotopy inverse to restr_{1}. Applying Corollary 1.9 we conclude that (5.7.1) is an A_{∞}-equivalence.

5.8 Corollary. A_{∞}-2-transformation (5.7.1) is a natural A_{∞}-2-equivalence.

5.9 Proposition. Let 𝒜 be a unital A_{∞}-category, let 𝒟 be a pseudounital A_{∞}-category with distinguished cycles ι_{X}^{𝒟}, let 𝒬 be a differential graded quiver and let 𝒩 be a graded quiver such that Ob 𝒟 = Ob 𝒬 = Ob 𝒩 and 𝒬 = 𝒟⊕𝒩. Suppose that 𝒩(X, Y) ≠ 0 implies that ι_{X}^{𝒟} ∈ Im b_{1} or ι_{Y}^{𝒟} ∈ Im b_{1}. Then an arbitrary pseudounital A_{∞}-functor f : 𝒟 → 𝒜 extends to an object (f, f′) ∈ Ob A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) for some f′.

Proof. Let sℳ = 𝒩 be the differential graded quiver and let α : sℳ → s𝒟 be the chain map defined in Section 4.5. There exists a homotopy \overline{f} : sℳ(X, Y) → s𝒜(Xf, Y f) of degree −1 such that

αf_{1} = \overline{f}b_{1} + d^{ℳ[1]} \overline{f} : sℳ(X, Y) → s𝒜(Xf, Y f).

Indeed, the case of ℳ(X, Y) = 0 being obvious, we may assume that ι_{X} ∈ Im b_{1} or ι_{Y} ∈ Im b_{1}. Then _{Xf}i_{0}^{𝒜} ∈ ι_{X}f_{1} + Im b_{1} ⊂ Im b_{1} or _{Y f}i_{0}^{𝒜} ∈ ι_{Y} f_{1} + Im b_{1} ⊂ Im b_{1}. Since 𝒜 is unital, the complex s𝒜(Xf, Y f) is contractible with some contracting homotopy \overline{h}. We may take \overline{f} = α\overline{h}.

Define a degree 0 map

f′_{1} = (s𝒬(X, Y) = s𝒟(X, Y) ⊕ s𝒩(X, Y) ^{(f_{1},s^{−1}\overline{f})} → s𝒜(Xf, Y f)).

For arbitrary p ∈ s𝒟(X, Y), m ∈ sℳ(X, Y) = 𝒩(X, Y) we have

(p, ms)(f′_{1}b_{1}^{𝒜} − b_{1}^{𝒬} f′_{1}) = pf_{1}b_{1}^{𝒜} + m\overline{f}b_{1}^{𝒜} − (pb_{1}^{𝒟} + mα)f_{1} + (md^{ℳ[1]} s)s^{−1}\overline{f}
= p(f_{1}b_{1}^{𝒜} − b_{1}^{𝒟} f_{1}) + m(\overline{f}b_{1}^{𝒜} + d^{ℳ[1]} \overline{f} − αf_{1}) = 0

by (4.5.1). Therefore, f′_{1} is a chain map and f′_{1}|_{s𝒟} = f_{1}.

5.10 Proposition. In assumptions of Proposition 5.9 the restriction strict A_{∞}-functor

restr : A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) → A_{∞}^{ψu}(𝒟, 𝒜),  (x, x′) ↦ x  (5.10.1)

is an A_{∞}-equivalence, surjective on objects. The chain surjections restr_{1} admit a chain splitting.

Proof. Let us prove that for an arbitrary pair of objects (f, f′), (g, g′) of A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) the restriction chain map

sA_{∞1}(𝒟, 𝒬; 𝒜)((f, f′), (g, g′)) → sA_{∞}(𝒟, 𝒜)(f, g)

is homotopy invertible. This map is a product over n ∈ ℤ_{⩾0} of the restriction maps ρ_{n} : V_{n} → V′_{n} of the graded 𝕜-modules of n-th components (compare (4.3.1) with analogous decomposition of sA_{∞}(𝒟, 𝒜)(f, g)). Clearly, the maps ρ_{n} = id for n = 0 or for n > 1. On the other hand, for n = 1

ρ_{1} = ∏ C(in^{𝒟} , 1) : ∏_{X,Y ∈Ob 𝒟} C(s𝒬(X, Y), s𝒜(Xf, Y g)) → ∏_{X,Y ∈Ob 𝒟} C(s𝒟(X, Y), s𝒜(Xf, Y g))

is surjective with the kernel Ker ρ_{1} = ∏_{X,Y ∈Ob 𝒟} C(s𝒩(X, Y), s𝒜(Xf, Y g)). As in proof of Proposition 5.9 𝒩(X, Y) ≠ 0 implies that _{Xf}i_{0}^{𝒜} ∈ ι_{X}f_{1} + Im b_{1} ⊂ Im b_{1} or _{Y g}i_{0}^{𝒜} ∈ ι_{Y} g_{1} + Im b_{1} ⊂ Im b_{1}, hence, s𝒜(Xf, Y g) is contractible. Therefore, for all objects X, Y of 𝒟 the complex C(s𝒩(X, Y), s𝒜(Xf, Y g)) is contractible. Thus, Ker restr_{1} = Ker ρ_{1} is contractible.

Summing up, the first term of the semisplit exact sequence

0 → Ker restr_{1} → sA_{∞1}(𝒟, 𝒬; 𝒜)((f, f′), (g, g′)) ^{restr_{1}} → sA_{∞}(𝒟, 𝒜)(f, g) → 0

is contractible. By Lemma 5.6 this sequence admits a splitting chain map

ν : sA_{∞}(𝒟, 𝒜)(f, g) → sA_{∞1}(𝒟, 𝒬; 𝒜)((f, f′), (g, g′)),

and ν is homotopy inverse to restr_{1}.

By Proposition 5.9 the surjection

Ob A_{∞1}^{ψu}(𝒟, 𝒬; 𝒜) ∋ (f, f′) ↦ f ∈ Ob A_{∞}^{ψu}(𝒟, 𝒜)

admits a splitting f ↦ f̃ = (f, f′). Applying Corollary 1.9 we conclude that (5.10.1) is an A_{∞}-equivalence.

5.11 Corollary. A_{∞}-2-transformation (5.10.1) is a natural A_{∞}-2-equivalence.

An easy converse to Lemma 5.6 is given by

5.12 Lemma. Let β : A → B, ν : B → A be chain maps of complexes of 𝕜-modules, such that νβ = id_{B}. Denote C = Ker ν, then A ≃ C ⊕ B. If β is a homotopy isomorphism, then the chain complex C is contractible.

Proof. Being one-sided inverse to homotopy isomorphism β, the map ν is homotopy inverse to β. Therefore, id_{A} −βν = hd + dh for some homotopy h : A → A of degree −1. Since C is the image of the idempotent id_{A} −βν = pr^{C} · in^{C}, we have

id_{C} = in^{C} pr^{C} in^{C} pr^{C} = in^{C}(id_{A} −βν) pr^{C} = (in^{C} h pr^{C})d + d(in^{C} h pr^{C}).

Thus, id_{C} = Hd + dH for H = in^{C} h pr^{C} : C → C, and C is contractible.

5.13 Theorem. Let 𝒜 be a unital A_{∞}-category, and let 𝒟 = ∪_{j⩾0}𝒟_{j} = lim_{⟶}_{j}𝒟_{j}. Then the restriction strict A_{∞}-functor

restr : A_{∞}^{ψu}(𝒟, 𝒜) → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ}

is an A_{∞}-equivalence, 2-natural in 𝒜, surjective on objects. The chain surjections restr_{1} admit a chain splitting.

Proof. All restriction strict A_{∞}-functors in the sequence

A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} ← A_{∞1}^{ψu}(𝒟_{0}, 𝒬_{1}; 𝒜) ← A_{∞}^{ψu}(𝒟_{1}, 𝒜)

← A_{∞1}^{ψu}(𝒟_{1}, 𝒬_{2}; 𝒜) ← A_{∞}^{ψu}(𝒟_{2}, 𝒜) ← A_{∞1}^{ψu}(𝒟_{2}, 𝒬_{3}; 𝒜) ← … (5.13.1)

are A_{∞}-equivalences (and natural A_{∞}-2-equivalences). They are surjective on objects. The first components are surjective and admit a chain splitting. For the first functor it follows from Proposition 5.7. For other odd-numbered functors it follows from Proposition 5.10. Indeed, if X ∈ Ob ℬ, then _{X}i_{0}^{𝒞} = _{X}i_{0}^{𝒞}Hb_{1} ∈ Im(b_{1} : s𝒟_{1}(X, X) → s𝒟_{1}(X, X)). For even-numbered functors it follows from Theorem 4.7.

Let us show that A_{∞}^{ψu}(𝒟, 𝒜) is the inverse limit of (5.13.1) on objects and on morphisms. There are restriction strict A_{∞}-functors

restr : A_{∞}^{ψu}(𝒟, 𝒜) → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} ⊂ A_{∞}^{u}(𝒞, 𝒜),  f ↦ f|_{𝒞},
restr : A_{∞}^{ψu}(𝒟, 𝒜) → A_{∞1}^{ψu}(𝒟_{j}, 𝒬_{j+1}; 𝒜),  f ↦ (f|_{𝒟_{j}} , f|_{𝒬_{j+1}} ), j ⩾ 0,
restr : A_{∞}^{ψu}(𝒟, 𝒜) → A_{∞}^{ψu}(𝒟_{j}, 𝒜),  f ↦ f|_{𝒟_{j}} , j ⩾ 1.

They agree with the functors restr from (5.13.1) in the sense restr · restr = restr.

Since 𝒟 = ∪_{j⩾0}𝒟_{j}, pseudounital A_{∞}-functors f : 𝒟 → 𝒜 are in bijection with sequences (f^{j})_{j} of pseudounital A_{∞}-functors f^{j} : 𝒟_{j} → 𝒜 such that f^{j+1}|_{𝒟_{j}} = f^{j}. In other words, Ob A_{∞}^{ψu}(𝒟, 𝒜) is the inverse limit of the sequence of surjections

Ob A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} ◁  Ob A_{∞1}^{ψu}(𝒟_{0}, 𝒬_{1}; 𝒜) ◁  Ob A_{∞}^{ψu}(𝒟_{1}, 𝒜)
◁  Ob A_{∞1}^{ψu}(𝒟_{1}, 𝒬_{2}; 𝒜) ◁  Ob A_{∞}^{ψu}(𝒟_{2}, 𝒜) ◁  Ob A_{∞1}^{ψu}(𝒟_{2}, 𝒬_{3}; 𝒜) ◁  …

In particular, the map

Ob restr : Ob A_{∞}^{ψu}(𝒟, 𝒜) → Ob A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ}

is surjective.

Let f, g : 𝒟 → 𝒜 be pseudounital A_{∞}-functors. Since 𝒟 = (∪_{j⩾0}𝒟_{j}) ⋃(∪_{j⩾1}𝒬_{j}), A_{∞}-transformations p : f → g : 𝒟 → 𝒜 are in bijection with sequences

(p^{0}, p^{′1}, p^{1}, p^{′2}, p^{2}, p^{′3}, …)

of A_{∞}-transformations p^{j} : f|_{𝒟_{j}} → g|_{𝒟_{j}} : 𝒟_{j} → 𝒜, j ⩾ 0, and A_{1}-transformations p^{′j} : f|_{𝒬_{j}} → g|_{𝒬_{j}} : 𝒬_{j} → 𝒜, j ⩾ 1, such that p^{j+1}|_{𝒟_{j}} = p^{j}, p^{j}|_{𝒬_{j}} = p^{′j}. In other words, A_{∞}(𝒟, 𝒜)(f, g) is the inverse limit of the sequence of splittable chain surjections

A_{∞}(𝒞, 𝒜)(f|_{𝒞}, g|_{𝒞}) ◁ A_{∞1}(𝒟_{0}, 𝒬_{1}; 𝒜)((f|_{𝒟_{0}} , f|_{𝒬_{1}} ), (g|_{𝒟_{0}} , g|_{𝒬_{1}} )) ◁ … ◁ A_{∞}(𝒟_{j}, 𝒜)(f|_{𝒟_{j}} , g|_{𝒟_{j}} ) ◁ A_{∞1}(𝒟_{j}, 𝒬_{j+1}; 𝒜)((f|_{𝒟_{j}} , f|_{𝒬_{j+1}} ), (g|_{𝒟_{j}} , g|_{𝒬_{j+1}} )) ◁ …

Since these surjections are splittable, the above sequence is isomorphic to the sequence of natural projections

C_{0} ◁ C_{0} × C_{1} ◁ C_{0} × C_{1} × C_{2} ◁  … ◁  ∏_{m=0}^{n} C_{m} ◁  ∏_{m=0}^{n+1} C_{m} ◁  …

for some complexes C_{m} of 𝕜-modules. Its inverse limit is ∏_{m=0}^{∞} C_{m} ≃ A_{∞}(𝒟, 𝒜)(f, g). By Lemma 5.12 all C_{m} are contractible for m > 0. Therefore, ∏_{m=1}^{∞} C_{m} is contractible. We obtain a split exact sequence

0 → ∏_{m=1}^{∞} C_{m} → A_{∞}(𝒟, 𝒜)(f, g) ^{β}→ A_{∞}(𝒞, 𝒜)(f|_{𝒞}, g|_{𝒞}) → 0

with contractible first term. By Lemma 5.6 β = s restr_{1} s^{−1} is a homotopy isomorphism. Using Corollary 1.9 we prove the theorem.

6. Unitality of 𝒟

We are going to prove that if 𝒞 is strictly unital, then the A_{∞}-category 𝒟 constructed in Section 5.2 is not only pseudounital, but unital with the unit elements_{X}i_{0}^{𝒞} ∈ s𝒟(|)(X, X) ⊂ s𝒟(X, X). Let us describe 𝕜-linear maps h : s𝒟(X, Y) → s𝒟(X, Y) of degree −1 such that

1 − (1 ⊗ i_{0}^{𝒞})b_{2} = b_{1}h + hb_{1} : s𝒟(X, Y) → s𝒟(X, Y).  (6.0.1)

The homotopy h is called the right unit homotopy. Let t be a 𝒞-admissible tree. Let y be its rightmost leaf vertex. Let v_{0} ≺ v_{1} ≺ ⋯ ≺ v_{p} ≺ v_{p+1} = y be the directed path connecting the root v_{0} with y, p ⩾ 0. Vertices v_{i} and v_{i+1} are connected by an edge for all 0 ⩽ i ⩽ p. Let t_{i}^{+}, 1 ⩽ i ⩽ p be the tree t with an extra leaf attached on the right to the vertex v_{i} if v_{i} is n-ary, n > 1 as in

X_{0} X_{1}  X_{n−1} Y_{_{v_{i}}  ^{···}} ↦ X_{0} X_{1}  X_{n−1}  Y _{Y} i_{0}^{𝒞} Y_{_{v_{i}^{+}}  ^{···}}

If v_{i} is unary, we attach an extra leaf on the right to v_{i} and add two more unary vertices above and below it as in

X  Y_{_{v_{i}}}  ↦ [figure: X  Y _{Y} i_{0}^{𝒞} Y; v_{i}^{+}; v_{i}^{0}; v_{i}^{−}; X  Y]

The obtained trees t_{i}^{+} are 𝒞-admissible.

Let x be a homogeneous element of s𝒟(t)(Z, Y). Define x_{i}^{+} = x⊗_{Y} i_{0}^{𝒞} ∈ s𝒟(t_{i}^{+})(Z, Y) = s𝒟(t)(Z, Y) ⊗ s𝒞(Y, Y). We claim that if 𝒞 is strictly unital, then the map

h : x ↦ ∑_{i=1}^{p} ±x_{i}^{+}  (6.0.2)

with an appropriate choice of signs satisfies (6.0.1). To describe the signs and to prove the claim we study the set of operations acting in 𝒟.

6.1. A multicategory operating in 𝒟. Let 𝒜_{∞}^{𝒞/ℬ} be the free graded 𝕜-linear (nonsymmetric) operad, generated by

— a 0-ary operation i_{0} ∈ 𝒜_{∞}^{𝒞/ℬ} (0) of degree −1,

— a unary operation H ∈ 𝒜_{∞}^{𝒞/ℬ} (1) of degree −1,

— an n-ary operation b_{n} ∈ 𝒜_{∞}^{𝒞/ℬ} (n) of degree 1 for each n > 1.

The construction of free non-symmetric operads uses plane trees instead of abstract trees. Otherwise it is similar to the case of symmetric operads, see e.g. [MSS02]. The operad 𝒜_{∞} of operations in A_{∞}-algebras (e.g. [Mar00]) is a suboperad of 𝒜_{∞}^{𝒞/ℬ} .

Actually we need a multicategory [Lam69, Lam89] rather than an operad. A multicategory is a many object version of an operad, a (non-symmetric) operad is a one-object multicategory. So we define a graded 𝕜-linear multicategory 𝒜_{∞}^{𝒞/ℬ} , whose objects are pairs of objects of 𝒞, thus Ob 𝒜_{∞}^{𝒞/ℬ} = Ob 𝒞 × Ob 𝒞. For n > 0 the graded 𝕜-module of morphisms

𝒜_{∞}^{𝒞/ℬ} ((X′_{1}, X_{1}), (X′_{2}, X_{2}), …, (X′_{n}, X_{n}); (Y′, Y))

is 0 unless X′_{1} = Y′, X_{n} = Y and X_{i} = X′_{i+1} for all 1 ⩽ i < n. For n = 0 the graded 𝕜-module of morphisms 𝒜_{∞}^{𝒞/ℬ} (; (Y′, Y)) is 0 unless Y′ = Y. The morphisms of 𝒜_{∞}^{𝒞/ℬ} are freely generated by

— 0-ary operations _{X}i_{0} ∈ 𝒜_{∞}^{𝒞/ℬ} (; (X, X)), X ∈ Ob 𝒞 of degree deg_{X}i_{0} = −1,

— unary operations H = H_{X,Y} ∈ 𝒜_{∞}^{𝒞/ℬ} ((X, Y); (X, Y)), X, Y ∈ Ob 𝒞, where X ∈ Ob ℬ or Y ∈ Ob ℬ, of degree deg H = −1,

— n-ary operations b_{n} ∈ 𝒜_{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), (X_{1}, X_{2}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n})), of degree deg b_{n} = 1 for X_{0}, …, X_{n} ∈ Ob 𝒞, n > 1.

We shall not insist on distinguishing between the operad 𝒜_{∞}^{𝒞/ℬ} and its refinement – the multicategory 𝒜_{∞}^{𝒞/ℬ} , leaving the choice of context to the reader.

Similarly to the operad 𝒜_{∞} [Mar00], governing A_{∞}-algebras, the multicategory 𝒜_{∞}^{𝒞/ℬ} has a differential d – a derivation of degree 1, such that d^{2} = 0. In general, a derivation d of degree p of a graded 𝕜-linear multicategory ℳ is a collection of 𝕜-linear endomorphisms d of ℳ (Z_{1}, …, Z_{n}; Z) of degree p, such that all compositions (which are of degree 0)

μ_{i} : ℳ (Y_{1}, …, Y_{k}; Z_{i})⊗ℳ (Z_{1}, …, Z_{n}; Z) → ℳ (Z_{1}, …, Z_{i−1}, Y_{1}, …, Y_{k}, Z_{i+1}, …, Z_{n}; Z)

(6.1.1)

satisfy the equation

μ_{i}d = (1 ⊗ d + d ⊗ 1)μ_{i}

with the sign conventions of this article. If d^{2} = 0, we may say that μ_{i} are chain maps.

Since 𝒜_{∞}^{𝒞/ℬ} is free, its derivations are uniquely determined by their values on generators. In particular, the derivation d of degree 1 is determined by these values:

Xi_{0}d = 0,
H_{X,Y} d = 1_{(X,Y)} (the unit element of 𝒜_{∞}^{𝒞/ℬ} ((X, Y); (X, Y))),
b_{n}d = − ∑_{a+p+c=n}^{p>1, a+c>0}(1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗c})b_{a+1+c},  n > 1.

For 𝒜_{∞}^{𝒞/ℬ} we use the notation (1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗c})b_{a+1+c} referring to the action of 𝒜_{∞}^{𝒞/ℬ} in A_{∞}-algebras as a synonym of the usual operadic notation (b_{p}⊗b_{a+1+c})μ_{a+1} = b_{p}∘_{a+1}b_{a+1+c}. Since the derivation d has odd degree, the 𝕜-linear map d^{2} is also a derivation. Its value on all generators is 0 (for b_{n} it follows from a similar result for 𝒜_{∞}, see e.g. [Mar00, MSS02]). Therefore, d^{2} = 0, d is a differential, and 𝒜_{∞} ^{⊂} → 𝒜_{∞}^{𝒞/ℬ} is a chain embedding.

The so defined differential d is distinguished by the following property. The action maps

α : sℰ(X_{0}, X_{1}) ⊗ sℰ(X_{1}, X_{2}) ⊗ ⋯ ⊗ sℰ(X_{n−1}, X_{n})

⊗ 𝒜_{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), (X_{1}, X_{2}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n})) → sℰ(X_{0}, X_{n}) (6.1.2)

are chain maps, where sℰ( , ) is equipped with the differential b_{1}. It suffices to check this on generators of 𝒜_{∞}^{𝒞/ℬ} . For them the property follows from (5.2.2), (5.2.3) and the equation _{X}i_{0}^{𝒞}b_{1} = 0.

Let \underline{𝒜} _{∞}^{𝒞/ℬ} be a submulticategory of 𝒜_{∞}^{𝒞/ℬ} , generated by H and b_{n}, n ⩾ 2, with the same set of objects Ob \underline{𝒜} _{∞}^{𝒞/ℬ} = Ob 𝒜_{∞}^{𝒞/ℬ} . It is a differential graded submulticategory without 0-ary operations: \underline{𝒜} _{∞}^{𝒞/ℬ} (; (X, Y)) = 0 for all X, Y ∈ Ob 𝒞. As a 𝕜-linear graded multigraph \underline{𝒜} _{∞}^{𝒞/ℬ} has the following description. For n ⩾ 1

\underline{𝒜} _{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), (X_{1}, X_{2}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n})) = ⨁_{admissible (t;X_{0},X_{1},...,X_{n})} 𝕜[|t|_{1} − |t|_{>}],

(6.1.3)

other 𝕜-modules \underline{𝒜} _{∞}^{𝒞/ℬ} ((X′_{1}, X_{1}), (X′_{2}, X_{2}), …, (X′_{n}, X_{n}); (Y′, Y)) vanish.

The embedding of \underline{𝒜} _{∞}^{𝒞/ℬ} is denoted

ι : \underline{ℳ} =^{def} \underline{𝒜} _{∞}^{𝒞/ℬ} ^{⊂} → ℳ =^{def} 𝒜_{∞}^{𝒞/ℬ} .  (6.1.4)

The following general results are valid for an arbitrary embedding ι : \underline{ℳ} ^{⊂} → ℳ of differential graded 𝕜-linear multicategories, such that Ob ι = id_{Ob ℳ} , and \underline{ℳ} (; Z) = 0 for all Z ∈ Ob \underline{ℳ} = Ob ℳ.

6.2 Definition. A right derivation δ of degree p of the embedding ι : \underline{ℳ} ^{⊂} → ℳ is a collection of 𝕜-linear maps

δ : \underline{ℳ} (Z_{1}, …, Z_{n}; Z) → ℳ (Z_{1}, …, Z_{n}; Z)

of degree p for all Z_{1}, …, Z_{n}, Z ∈ Ob ℳ, such that compositions μ_{i} from (6.1.1) satisfy

\begin{aligned}\underline{ℳ} ⊗ \underline{ℳ} & \overline{^{μ_{n}} →} & \underline{ℳ} \\ & = \\ ℳ ⊗ ℳ & ↓^{ι⊗δ+δ⊗ι} & \overline{^{μ_{n}} →} & ℳ & ↓^{δ} \end{aligned}  ,  \begin{aligned}\underline{ℳ} ⊗ \underline{ℳ} & \overline{^{μ_{i}} →} & \underline{ℳ} \\ & = \\ ℳ ⊗ ℳ & ↓^{ι⊗δ} & \overline{^{μ_{i}} →} & ℳ & ↓^{δ} \end{aligned}

if 1 ⩽ i < n.

Clearly, for n = 0 the map δ : \underline{ℳ} (; Z) = 0 → ℳ (; Z) is 0.

An example of a right derivation is given by an inner right derivation. Let λ_{Z} ∈ ℳ (Z; Z) be a family of morphisms of degree p. For n > 0 define

ad_{λ} : \underline{ℳ} (Z_{1}, …, Z_{n}; Z) → ℳ (Z_{1}, …, Z_{n}; Z)
f ↦ (f ⊗ λ_{Z})μ_{1} − (−)^{fλ}(λ_{Z_{n}} ⊗ f)μ_{n} = f ∘_{1} λ_{Z} − (−)^{fp}λ_{Z_{n}} ∘_{n} f.

One verifies easily that ad_{λ} is a right derivation, which we call inner.

One can show that if δ is a right derivation of ι, and d is a derivation of ℳ such that \underline{ℳ} d ⊂ \underline{ℳ} , then their commutator [δ, d] = δd − (−)^{δ·d}dδ is a right derivation of ι as well. In particular, it applies to a differential d of degree 1. If δ = ad_{λ} is inner, then [δ, d] = ad_{λd} is also inner.

6.3. Examples of right derivations. Consider embedding (6.1.4) and take the family of morphisms

λ_{(X,Y)} = (1 ⊗_{Y} i_{0})b_{2} = _{Y} i_{0} ∘_{2} b_{2} ∈ 𝒜_{∞}^{𝒞/ℬ} ((X, Y); (X, Y)).

Since λ_{(X,Y)}d = 0, we have [ad_{λ}, d] = ad_{λd} = 0. Thus, δ = ad_{(1⊗i_{0})b_{2}} commutes with d.

Let us show that ad_{λ} = [η, d] for some right derivation η : \underline{𝒜} _{∞}^{𝒞/ℬ} → 𝒜_{∞}^{𝒞/ℬ} of ι. Since \underline{𝒜} _{∞}^{𝒞/ℬ} is a free multicategory, any such right derivation is uniquely determined by its value on the generators H and b_{n}, n > 1. We define a right derivation η of ι of degree −1 by the following assignment:

H_{X,Y} η = H_{X,Y} (1 ⊗_{Y} i_{0})b_{2}H_{X,Y} ,
b_{n}η = (1^{⊗n} ⊗_{X_{n}}i_{0})b_{n+1}.  (6.3.1)

Any operation f ∈ \underline{𝒜} _{∞}^{𝒞/ℬ} can be presented as

f = (g ⊗ 1)(1^{⊗a_{1}} ⊗ e_{p_{1}})(1^{⊗a_{2}} ⊗ e_{p_{2}}) … (1^{⊗a_{k−1}} ⊗ e_{p_{k−1}})e_{p_{k}}

for some g ∈ \underline{𝒜} _{∞}^{𝒞/ℬ} , where e_{1} = H and e_{p} = b_{p} if p > 1. Then

fη = ∑_{i=1}^{k}(−)^{k−i}(g ⊗ 1)(1^{⊗a_{1}} ⊗ e_{p_{1}}) … (1^{⊗a_{i}} ⊗ e_{p_{i}}η) … (1^{⊗a_{k−1}} ⊗ e_{p_{k−1}})e_{p_{k}}.  (6.3.2)

Let us prove that

[η, d] = ηd + dη = ad_{(1⊗i_{0})b_{2}} .  (6.3.3)

Since the difference of the both sides is a right derivation of ι, it suffices to prove (6.3.3) on generators. First we find that

H_{X,Y} (ηd + dη) = [H_{X,Y} (1 ⊗_{Y} i_{0})b_{2}H_{X,Y}]d + 1_{(X,Y)}η = H_{X,Y} (1 ⊗_{Y} i_{0})b_{2} − (1 ⊗_{Y} i_{0})b_{2}H_{X,Y} = H_{X,Y} ad_{(1⊗i_{0})b_{2}} .

For n ⩾ 2 we have

b_{n}(ηd + dη) = (1^{⊗n} ⊗ i_{0})(b_{n+1}d) − ∑_{a+p+c=n}^{p>1, c>0}(1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗c})(b_{a+1+c}η)

− ∑_{a+p=n}^{p>1, a>0}[(1^{⊗a} ⊗ b_{p})b_{a+1}]η

= − ∑_{a+p+c=n+1}^{p>1, a+c>0}(1^{⊗n} ⊗ i_{0})(1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗c})b_{a+1+c}

− ∑_{a+p+c=n}^{p>1, c>0}(1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗c})(1^{⊗a+1+c} ⊗ i_{0})b_{a+c+2}

− ∑_{a+p=n}^{p>1, a>0}(1^{⊗a} ⊗ b_{p})(1^{⊗a+1} ⊗ i_{0})b_{a+2} + ∑_{a+p=n}^{p>1, a>0}[1^{⊗a} ⊗ (1^{⊗p} ⊗ i_{0})b_{p+1}]b_{a+1}

= − ∑_{a+p+c=n+1}^{p>1, a+c>0}(1^{⊗n} ⊗ i_{0})(1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗c})b_{a+1+c}

+ ∑_{a+p+e=n+1}^{p>1, e>1}(1^{⊗n} ⊗ i_{0})(1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗e})b_{a+1+e}

+ ∑_{a+p=n}^{p>1, a>0}(1^{⊗n} ⊗ i_{0})(1^{⊗a} ⊗ b_{p} ⊗ 1)b_{a+2} + ∑_{a+q=n+1}^{q>2, a>0}(1^{⊗n} ⊗ i_{0})(1^{⊗a} ⊗ b_{q})b_{a+1}

= − ∑_{0+p+1=n+1}(1^{⊗n} ⊗ i_{0})(b_{p} ⊗ 1)b_{2} − ∑_{a+2+0=n+1}(1^{⊗n} ⊗ i_{0})(1^{⊗a} ⊗ b_{2})b_{a+1}

= b_{n}(1 ⊗ i_{0})b_{2} − [1^{⊗n−1} ⊗ (1 ⊗ i_{0})b_{2}]b_{n}

= b_{n} ad_{(1⊗i_{0})b_{2}} .

Thus equation (6.3.3) is verified.

Notice that the graded quiver sℰ defined by (5.2.1) is a free \underline{𝒜} _{∞}^{𝒞/ℬ} -algebra, generated by the graded quiver s𝒞. Indeed, (5.2.1) can be written as

sℰ(X, Y) = ⨁_{n⩾1} ⨁_{X_{1},...,X_{n−1}∈Ob 𝒞}^{X_{0}=X, X_{n}=Y} s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n})⊗

⊗ \underline{𝒜} _{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), (X_{1}, X_{2}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n}))

due to (6.1.3). Compare with the usual free algebras over an operad, e.g. [MSS02]. The operations H, b_{n} for n ⩾ 2 act in sℰ via multicategory compositions in \underline{𝒜} _{∞}^{𝒞/ℬ} .

The multigraph 𝒜_{∞}^{𝒞/ℬ} is expressed via \underline{𝒜} _{∞}^{𝒞/ℬ} as follows:

𝒜_{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), (X_{1}, X_{2}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n}))

= ⨁_{k_{0},...,k_{n}∈ℤ_{⩾0}} \underline{𝒜} _{∞}^{𝒞/ℬ} ((Y_{0}, Y_{1}), (Y_{1}, Y_{2}), …, (Y_{p−1}, Y_{p}); (X_{0}, X_{n}))[k_{0} + ⋯ + k_{n}], (6.3.4)

(Y_{0}, Y_{1}, …, Y_{p}) = (\underbrace{X_{0}, …, X_{0}}_{k_{0}+1}, \underbrace{X_{1}, …, X_{1}}_{k_{1}+1}, …, \underbrace{X_{n}, …, X_{n}}_{k_{n}+1}),

where p = k_{0} + ⋯ + k_{n} + n. Indeed, 0-ary operations can be performed first. The summand of (6.3.4) corresponds to insertion of k_{0} symbols _{X_{0}}i_{0}, k_{1} symbols _{X_{1}}i_{0}, and so on. In terms of trees such summand is described by concatenation of the forest

X_{0} X_{0} X_{0} X_{1} X_{1} X_{1}  X_{n} X_{n} X_{n}

[figure: ...; ...; ...; ...] _{k_{0}}  _{k_{1}}  _{k_{n}}

with an admissible tree (t; Y_{0}, Y_{1}, …, Y_{p}).

The action (6.1.2) of 𝒜_{∞}^{𝒞/ℬ} in sℰ is described as follows. An element

(_{X_{0}}i_{0}^{⊗k_{0}} ⊗1⊗_{X_{1}}i_{0}^{⊗k_{1}} ⊗1⊗⋯⊗1⊗_{X_{n}}i_{0}^{⊗k_{n}})·f ∈ 𝒜_{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n})),

where f ∈ \underline{𝒜} _{∞}^{𝒞/ℬ} ((Y_{0}, Y_{1}), (Y_{1}, Y_{2}), …, (Y_{p−1}, Y_{p}); (X_{0}, X_{n})), acts by the map

sℰ(X_{0}, X_{1}) ⊗ ⋯ ⊗ sℰ(X_{n−1}, X_{n}) ^{_{X_{0}}^{𝒞}i_{0}^{⊗k_{0}}⊗1⊗_{X_{1}}^{𝒞}i_{0}^{⊗k_{1}}⊗1⊗···⊗1⊗_{Xn}^{𝒞}i_{0}^{⊗kn}} →

s𝒞(X_{0}, X_{0})^{⊗k_{0}} ⊗ sℰ(X_{0}, X_{1}) ⊗ s𝒞(X_{1}, X_{1})^{⊗k_{1}} ⊗ ⋯ ⊗ sℰ(X_{n−1}, X_{n}) ⊗ s𝒞(X_{n}, X_{n})^{⊗k_{n}}

followed by the action of f via multiplication in multicategory \underline{𝒜} _{∞}^{𝒞/ℬ} , that is, via grafting of trees.

Define a 𝕜-linear map h̄ : sℰ(X, Y) → sℰ(X, Y) of degree −1 for all objects X, Y of 𝒞 as follows:

sℰ(X, Y)

⨁_{X_{1},...,X_{n−1}∈Ob 𝒞}^{n, X_{0}=X, X_{n}=Y} s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n}) ⊗ \underline{𝒜} _{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n})) ‖

⨁_{X_{1},...,X_{n−1}∈Ob 𝒞}^{n, X_{0}=X, X_{n}=Y} s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n}) ⊗ 𝒜_{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n})) ↓^{−1^{⊗n}⊗η}

sℰ(X, Y) ↓^{α}

The concrete choice (6.3.1) of value of η on generators shows that on the summand

s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n}) ⊗ \underline{𝒜} _{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n}))

the map h̄ takes values in

s𝒞(X_{0}, X_{1})⊗⋯⊗s𝒞(X_{n−1}, X_{n})⊗s𝒞(X_{n}, X_{n})⊗\underline{𝒜} _{∞}^{𝒞/ℬ} (…, (X_{n−1}, X_{n}), (X_{n}, X_{n}); (X_{0}, X_{n}))

Due to (6.3.2) the explicit formula for h̄ has the form (6.0.2) with the concrete choice of signs.

Let us compute the commutator

−h̄b_{1}^{ℰ} − b_{1}^{ℰ} h̄ = (1^{⊗n} ⊗ η)αb_{1}^{ℰ} + b_{1}^{ℰ}(1^{⊗n} ⊗ η)α

= (1^{⊗n} ⊗ η)( ∑_{a+1+c=n}(1^{⊗a} ⊗ b_{1} ⊗ 1^{⊗c}) ⊗ 1 + 1^{⊗n} ⊗ d)α

+ ( ∑_{a+1+c=n}(1^{⊗a} ⊗ b_{1} ⊗ 1^{⊗c}) ⊗ 1 + 1^{⊗n} ⊗ d)(1^{⊗n} ⊗ η)α

= [1^{⊗n} ⊗ (ηd + dη)]α = [1^{⊗n} ⊗ ad_{(1⊗i_{0})b_{2}}]α :

s𝒞(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒞(X_{n−1}, X_{n}) ⊗ \underline{𝒜} _{∞}^{𝒞/ℬ} ((X_{0}, X_{1}), …, (X_{n−1}, X_{n}); (X_{0}, X_{n})) →

s𝒞(X_{0}, X_{1})⊗⋯⊗s𝒞(X_{n−1}, X_{n})⊗s𝒞(X_{n}, X_{n})⊗\underline{𝒜} _{∞}^{𝒞/ℬ} (…, (X_{n−1}, X_{n}), (X_{n}, X_{n}); (X_{0}, X_{n})).

We write an element z_{1} ⊗ ⋯ ⊗ z_{n} ⊗ f of the source, which is a direct summand of sℰ(X_{0}, X_{n}), in the form (z_{1} ⊗ ⋯ ⊗ z_{n})f, meaning that z_{i} ∈ s𝒞(X_{i−1}, X_{i}) and f is a composition of expressions 1^{⊗a} ⊗ b_{p} ⊗ 1^{⊗c} and 1^{⊗a} ⊗ H ⊗ 1^{⊗c} ending in b_{p} or H. Then

− [(z_{1} ⊗ ⋯ ⊗ z_{n})f](h̄b_{1}^{ℰ} + b_{1}^{ℰ} h̄) = [(z_{1} ⊗ ⋯ ⊗ z_{n})(f. ad_{(1⊗i_{0})b_{2}})]α

= {[(z_{1} ⊗ ⋯ ⊗ z_{n})f] ⊗_{X_{n}}i_{0}^{𝒞}}b_{2}^{ℰ} − (z_{1} ⊗ ⋯ ⊗ z_{n} ⊗_{X_{n}}i_{0}^{𝒞})(1^{⊗n−1} ⊗ b_{2}^{ℰ})f. (6.3.5)

6.4 Proposition. Assume that 𝒞 is a unital A_{∞}-category with the unit elements satisfying equations

(1 ⊗ i_{0}^{𝒞})b_{2} = 1,
(1^{⊗n} ⊗ i_{0}^{𝒞})b_{n+1} = 0,  if n > 1.

Then there exists a map h : s𝒟 → s𝒟 such that

(sℰ ^{h̄}→ sℰ ^{π} ▷ s𝒟) = (sℰ ^{π} ▷ s𝒟 ^{h}→ s𝒟).

The map h : s𝒟 → s𝒟 is a right unit homotopy for 𝒟.

Proof. We have to prove that (R)_{+}h̄ ⊂ (R)_{+}. The left hand side is the sum of images of maps

[(1^{⊗a} ⊗ b_{n}^{ℰ} ⊗ 1^{⊗c})f]η − (1^{⊗a} ⊗ b_{n}^{𝒞} ⊗ 1^{⊗c})(fη) : s𝒞^{⊗k} → sℰ.  (6.4.1)

If c > 0, this map equals

(1^{⊗a} ⊗ b_{n}^{ℰ} ⊗ 1^{⊗c})(fη) − (1^{⊗a} ⊗ b_{n}^{𝒞} ⊗ 1^{⊗c})(fη),

and the claim holds. If c = 0, expression (6.4.1) is

(1^{⊗a} ⊗ b_{n}^{ℰ})(fη) − (1^{⊗a} ⊗ b_{n}^{𝒞})(fη) + (−)^{f}(1^{⊗a} ⊗ (1^{⊗n} ⊗ i_{0}^{𝒞})b_{n+1}^{ℰ})f
= (1^{⊗a} ⊗ b_{n}^{ℰ})(fη) − (1^{⊗a} ⊗ b_{n}^{𝒞})(fη)

+ (−)^{f}(1^{⊗a+n} ⊗ i_{0}^{𝒞})(1^{⊗a} ⊗ b_{n+1}^{ℰ})f − (−)^{f}(1^{⊗a+n} ⊗ i_{0}^{𝒞})(1^{⊗a} ⊗ b_{n+1}^{𝒞})f

+ (−)^{f}(1^{⊗a} ⊗ (1^{⊗n} ⊗ i_{0}^{𝒞})b_{n+1}^{𝒞})f : s𝒞^{⊗k} → sℰ

and the last summand equals 0. So the claim holds, and h exists.

Property (6.3.5) turns into

hb_{1}^{𝒟} + b_{1}^{𝒟} h = 1 − (1 ⊗ i_{0}^{𝒞})b_{2} : s𝒟(X, Y) → s𝒟(X, Y).

Therefore, h is a right unit homotopy for 𝒟.

6.5 Theorem. Assume that 𝒞 is a unital A_{∞}-category with the unit elements satisfying equations

(1 ⊗ i_{0}^{𝒞})b_{2} = 1,  (i_{0}^{𝒞} ⊗ 1)b_{2} = −1,
(1^{⊗n} ⊗ i_{0}^{𝒞})b_{n+1} = 0,  (i_{0}^{𝒞} ⊗ 1^{⊗n})b_{n+1} = 0,  if n > 1.

Then the A_{∞}-category 𝒟 is unital.

Proof. Besides constructing 𝒟 from the pair (𝒞, ℬ), we may apply the construction to the pair (𝒞^{op}, ℬ^{op}), and we get an A_{∞}-category isomorphic to 𝒟^{op}. The opposite A_{∞}-cate-gory 𝒜^{op} to an A_{∞}-category 𝒜 is the opposite quiver, equipped with operations b_{k}^{op}, see Definition A.5. In particular, b_{1}^{op} = b_{1} and (x ⊗ i_{0})b_{2}^{op} = −x(i_{0} ⊗ 1)b_{2}. Thus we may use h_{op} = h for 𝒟^{op} in place of h′ for 𝒟. Thus, A_{∞}-category 𝒟 is unital.

6.6 Corollary. If 𝒞 is strictly unital, then the A_{∞}-category 𝒟 is unital.

6.7 Corollary. The A_{∞}-2-functor 𝒜 ↦ A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} is unitally representable for an arbitrary unital A_{∞}-category 𝒞.

Proof. By Corollary B.11 it suffices to prove unital representability for differential graded categories 𝒞̃ in place of 𝒞. In this case the representing unital A_{∞}-category 𝒟 exists by Corollary 6.6.

7. The example of complexes

Let 𝒜 be a 𝕜-linear Abelian category, let 𝒞 = C(𝒜) be the differential graded category of complexes in 𝒜, and let ℬ be its full subcategory of acyclic complexes. Let 𝒟 = Q(𝒞|ℬ) be the quotient unital A_{∞}-category. The embedding e : 𝒞 ↪ 𝒟 induces a 𝕜-linear functor of homotopy categories

H^{0}e : Ho 𝒞 = H^{0}(𝒞, m_{1}) → H^{0}(𝒟, m_{1}) = Ho 𝒟,

where m_{1} = sb_{1}s^{−1}. Morphisms of Ho 𝒞 are homotopy equivalence classes [q] of chain morphisms q : X → Y.

7.1 Proposition. For a quasi-isomorphism q its image [qe] ∈ Ho 𝒟(X, Y) is invertible.

Proof. If q : X → Y is a quasi-isomorphism, then C = Cone q is acyclic. The complex C is the graded object Y ⊕ X[1] with the differential given by the matrix

d^{C} = (\begin{matrix}d^{Y} & 0 \\ s^{−1}q & d^{X[1]} \end{matrix}) = (\begin{matrix}d^{Y} & 0 \\ s^{−1}q & −s^{−1}d^{X}s \end{matrix}) .

There is a semisplit exact sequence of chain morphisms 0 → Y ^{n}→ C ^{k}→ X[1] → 0, where n = in^{Y} = (1, 0), k = pr^{X[1]} = ( \begin{matrix}0 \\ 1 \end{matrix} ). Thus n ∈ 𝒞(Y, C)^{0}, k ∈ 𝒞(C, X[1])^{0} are cycles, nm_{1} = 0, km_{1} = 0. The morphisms s ∈ 𝒞(X, X[1])^{−1}, s^{−1} ∈ 𝒞(X[1], X)^{1} are also cycles, because in our conventions sm_{1} = sd^{X[1]} + d^{X}s = 0, similarly s^{−1}m_{1} = 0. Hence, ks^{−1} ∈ 𝒞(C, X)^{1} is also a cycle, (ks^{−1})m_{1} = 0.

Since C ∈ Ob ℬ, we have a map

\underline{H} = sHs^{−1} : 𝒞(C, C) → 𝒟(𝔱_{1})(C, C)^{⊂} → 𝒟(C, C), f ↦ fsHs^{−1} = f\underline{H}.

It satisfies the equation \underline{H}m_{1} + m_{1}\underline{H} = e : 𝒞(C, C) → 𝒟(C, C). In particular, there is an element 1_{C} \underline{H} ∈ 𝒟(C, C). Define an element

p = (n ⊗ 1_{C} \underline{H} ⊗ ks^{−1})(1 ⊗ m_{2})m_{2} ∈ 𝒟(Y, X),

where m_{2} = (s ⊗ s)b_{2}s^{−1}. More generally, m_{n} = s^{⊗n}b_{n}s^{−1}. We have deg p = 0 and

pm_{1} = −(n ⊗ 1_{C} ⊗ ks^{−1})(1 ⊗ m_{2})m_{2} = nks^{−1} = 0.

Let us show that [p] ∈ H^{0}𝒟(Y, X) is inverse to [q] ∈ H^{0}𝒟(X, Y). Define h =

(X ^{s}→ X[1] ^{in^{X[1]}} → C), then h ∈ 𝒞(X, C)^{−1}, and

hm_{1} = hd^{C} + d^{X}h = (0, s) (\begin{matrix}d^{Y} & 0 \\ s^{−1}q & −s^{−1}d^{X}s \end{matrix}) + d^{X}(0, s) = (q, 0) = qn.

One can check that

(h ⊗ 1_{C} \underline{H} ⊗ ks^{−1})(1 ⊗ m_{2})m_{2}m_{1} − (q ⊗ n ⊗ 1_{C} \underline{H} ⊗ ks^{−1})(1 ⊗ 1 ⊗ m_{2})m_{3}m_{1} = −hks^{−1} + (q ⊗ n ⊗ 1_{C} \underline{H} ⊗ ks^{−1})(1 ⊗ 1 ⊗ m_{2})(1 ⊗ m_{2})m_{2} = −1_{X} + (q ⊗ p)m_{2},

because hk = s.

Denote by z the morphism z = pr^{Y} : C → Y, then z ∈ 𝒞(C, Y)^{0} and

zm_{1} = zd^{Y} − d^{C}z = \binom{1}{0}d^{Y} − (\begin{matrix}d^{Y} & 0 \\ s^{−1}q & d^{X[1]} \end{matrix}) \binom{1}{0} = −\binom{0}{s^{−1}q} = −ks^{−1}q.

One can check that

(n ⊗ 1_{C} \underline{H} ⊗ z)(1 ⊗ m_{2})m_{2}m_{1} − (n ⊗ 1_{C} \underline{H} ⊗ ks^{−1} ⊗ q)(1 ⊗ m_{3})m_{2}m_{1}

− (n ⊗ 1_{C} \underline{H} ⊗ ks^{−1} ⊗ q)(1 ⊗ m_{2} ⊗ 1)m_{3}m_{1}

= nz − (n ⊗ 1_{C} \underline{H} ⊗ ks^{−1} ⊗ q)(1 ⊗ m_{2} ⊗ 1)(m_{2} ⊗ 1)m_{2} = 1_{Y} − (p ⊗ q)m_{2},

because nz = 1_{Y}. Therefore, the cycles p ∈ 𝒟(Y, X)^{0} and q ∈ 𝒟(X, Y)^{0} are inverse to each other modulo boundaries.

7.2 Corollary. The functor H^{0}e factors through a functor D(𝒜) → H^{0}𝒟, identity on objects.

We would like to know, when this functor is an equivalence.

A. The Yoneda Lemma for unital A_{∞}-categories

A.1. Basic identities in symmetric closed monoidal category of complexes. We want to work out in detail a system of notations suitable for computations in symmetric closed monoidal categories. Actually we need only the category of ℤ-graded 𝕜-modules with a differential of degree 1. The corresponding system of notations was already used in [Lyu03, LO02].

There exists a 𝒰′-small set S of 𝒰 -small 𝕜-modules such that any 𝒰 -small 𝕜-module M is isomorphic to some 𝕜-module N ∈ S (due to presentations 𝕜^{(P)} → 𝕜^{(Q)} → M → 0). We turn S into a category of 𝕜-modules 𝕜 -mod with Ob 𝕜 -mod = S. Thus 𝕜 -mod is an Abelian 𝕜-linear symmetric closed monoidal 𝒰′-small 𝒰 -category. The category C = C(𝕜 -mod) of complexes in 𝕜 -mod inherits all these properties from 𝕜 -mod, except that the symmetry becomes c : X ⊗ Y → Y ⊗ X, x ⊗ y ↦ (−)^{xy}y ⊗ x = (−)^{deg x·deg y}y ⊗ x. Therefore, we may consider the category of complexes as enriched in itself (a differential graded category), and it is denoted by C in this case. The (inner) hom-object between complexes X and Y is the complex

C(X, Y)^{n} = ∏_{i∈ℤ}Hom_{𝕜}(X^{i}, Y^{i+n}), (f^{i})_{i∈ℤ}d = (f^{i}d^{i+deg f} − (−)^{deg f}d^{i}f^{i+1})_{i∈ℤ}.

The product X = ∏_{ι∈I} X_{ι} in the category of complexes of 𝕜-modules of the family of objects (X_{ι})_{ι∈I} coincides with the product in the category of ℤ-graded 𝕜-modules (and differs from the product in the category of 𝕜-modules). It is given by X^{m} = ∏_{ι∈I} X_{ι}^{m}.

Given a complex Z and an element a ∈ C(X, Y), we assign to it elements 1⊗a ∈ C(Z ⊗ X, Z⊗Y), (z⊗x)(1⊗a) = z⊗xa, and a⊗1 ∈ C(X⊗Z, Y ⊗Z), (x⊗z)(a⊗1) = (−)^{za}xa⊗z. Clearly, (1⊗a)c = c(a⊗1) ∈ C(Z ⊗X, Y ⊗Z) and (a⊗1)c = c(1⊗a) ∈ C(X ⊗Z, Z ⊗Y). If g ∈ C(Z, W), then we have (1 ⊗ a)(g ⊗ 1) = (−)^{ag}(g ⊗ 1)(1 ⊗ a) ∈ C(Z ⊗ X, W ⊗ Y) (Koszul’s rule).

For any pair of complexes X, Y ∈ Ob C denote by

ev_{X,Y} : X ⊗ C(X, Y) → Y, coev_{X,Y} : Y → C(X, X ⊗ Y)

the canonical evaluation and coevaluation maps respectively. Then the adjunction isomorphisms are explicitly given as follows:

C(Y, C(X, Z)) ⟷ C(X ⊗ Y, Z),
f → (1_{X} ⊗ f) ev_{X,Z},
coev_{X,Y} C(X, g) ← g.  (A.1.1)

Given a complex Z and an element a ∈ C(X, Y), we assign to it the element C(1, a) = C(Z, a) = aϕ of C(C(Z, X), C(Z, Y)) obtained from the equation

m_{2}^{C} = (C(Z, X) ⊗ C(X, Y) \begin{matrix}1⊗ϕ \\ ∃! \end{matrix} → C(Z, X) ⊗ C(C(Z, X), C(Z, Y)) ^{ev}→ C(Z, Y)),

commutative for a unique chain map ϕ. Despite that the map a is not a chain map we write this element as a : X → Y, and we write aϕ as

C(1, a) : C(Z, X) → C(Z, Y), (f^{i})_{i∈ℤ} ↦ (f^{i}a^{i+deg f})_{i∈ℤ}.

Similarly, given a complex X and an element g ∈ C(W, Z), we assign to it the element C(g, 1) = C(g, X) = gψ ∈ C(C(Z, X), C(W, X)) obtained from the diagram

C(Z, X) ⊗ C(W, Z)  \overline{→^{c}} C(W, Z) ⊗ C(Z, X)

C(Z, X) ⊗ C(C(Z, X), C(W, X)) ↓^{∃! 1⊗ψ}  \overline{^{ev} →} C(W, X) ↓^{m_{2}^{C}}

commutative for a unique chain map ψ. Although the map C(g, 1) is not a chain map we write it as

C(g, 1) : C(Z, X) → C(W, X), (f^{i})_{i∈ℤ} ↦ ((−)^{deg f·deg g}g^{i}f^{i+deg g})_{i∈ℤ}.

For each pair of homogeneous elements a ∈ C(X, Y), g ∈ C(W, Z) we have

(C(Z, X) ^{C(Z,a)} → C(Z, Y) ^{C(g,Y)} → C(W, Y)) = (−)^{ag}(C(Z, X) ^{C(g,X)} → C(W, X) ^{C(W,a)} → C(W, Y)).

This equation follows from one of the standard identities in symmetric closed monoidal categories [EK66], and can be verified directly. We also have C(1, a)C(1, h) = C(1, ah) and C(g, 1)C(e, 1) = (−)^{ge}C(eg, 1), whenever these maps are defined.

One easily sees that m_{1}^{C} = d : C(X, Y) → C(X, Y) coincides with C(1, d_{Y}) − C(d_{X}, 1). Let f : A ⊗ X → B, g : B ⊗ Y → C be two homogeneous 𝕜-linear maps of arbitrary

degrees. Then the following holds:

(X ⊗ Y ^{coev_{A,X} ⊗ coev_{B,Y}} → C(A, A ⊗ X) ⊗ C(B, B ⊗ Y) ^{C(A,f)⊗C(B,g)} → C(A, B) ⊗ C(B, C)

m_{2}→ C(A, C)) = (X⊗Y ^{coev_{A,X⊗Y}} → C(A, A⊗X⊗Y) ^{C(A,f⊗1)} → C(A, B⊗Y) ^{C(A,g)} → C(A, C)). (A.1.2)

Indeed, (coev_{A,X} ⊗ coev_{B,Y})(C(A, f)⊗C(B, g)) = (coev_{A,X} C(A, f)⊗coev_{B,Y} C(B, g)), for coev has degree 0. Denote f̄ = coev_{A,X} C(A, f), ḡ = coev_{B,Y} C(B, g). The morphisms f̄ and ḡ correspond to f and g by adjunction. Further, the morphism m_{2} comes by adjunction from the following map:

A ⊗ C(A, B) ⊗ C(B, C) ^{ev_{A,B} ⊗1} → B ⊗ C(B, C) ^{ev_{B,C}} → C,

in particular the following diagram commutes:

A ⊗ C(A, B) ⊗ C(B, C) \overline{→^{ev_{A,B} ⊗1}} B ⊗ C(B, C)

A ⊗ C(A, C) ↓^{1⊗m_{2}}  \overline{^{ev_{A,C}}  →} C ↓^{ev_{B,C}}

Thus we have a commutative diagram

A ⊗ X ⊗ Y \overline{^{1⊗f̄⊗1} →} A ⊗ C(A, B) ⊗ Y \overline{^{ev_{A,B} ⊗1} →} B ⊗ Y

A ⊗ C(A, B) ⊗ C(B, C) ↓^{1⊗1⊗ḡ}  \overline{→^{ev_{A,B} ⊗1}} B ⊗ C(B, C) ↓^{1⊗ḡ}

A ⊗ C(A, C) ↓^{1⊗m_{2}}  ^{ev_{A,C}}  → C ↓^{ev_{B,C}}

The top row composite coincides with f ⊗ 1 and the right-hand side vertical composite coincides with g (by adjunction). Thus (1 ⊗ f̄ ⊗ 1)(1 ⊗ 1 ⊗ ḡ)(1 ⊗ m_{2}) ev_{A,C} coincides with (f ⊗ 1)g. But the mentioned morphism comes from (f̄ ⊗ ḡ)m_{2} by adjunction, so that (f̄⊗ ḡ)m_{2} = coev_{A,X⊗Y} C(A, (f ⊗ 1)g) (the latter morphism is the image of (f ⊗ 1)g under the adjunction), and we are done.

One verifies similarly the following assertion [EK66]: given f ∈ C(A, B), then the diagram

X  \overline{→^{coev_{A,X}}} C(A, A ⊗ X)

C(B, B ⊗ X) ↓^{coev_{B,X}}  \overline{→^{C(f,B⊗X)}} C(A, B ⊗ X) ↓^{C(A,f⊗1)}

(A.1.3)

commutes.

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. We also define the map σ : P → P [1], p ↦ (−)^{deg p}ps. The map s commutes with the components of the differential b in an A_{∞}-category (A_{∞}-algebra) in the following sense: s^{⊗n}b_{n} = m_{n}s. On the other hand, σ^{⊗n}b_{n} = (−)^{n}m_{n}σ. The main identity b^{2} = 0 written in components takes the form

∑_{r+n+t=k}(1^{⊗r} ⊗ b_{n} ⊗ 1^{⊗t})b_{r+1+t} = 0 : T^{k}s𝒜 → s𝒜.  (A.1.4)

A.2. A_{∞}-functor h^{X}. Let 𝒜 be an A_{∞}-category. Following Fukaya [Fuk02, Definition 7.28] for any object X of 𝒜 we define a cocategory homomorphism h^{X} : T s𝒜 → T sC as follows. It maps an object Z to the complex h^{X}Z = (s𝒜(X, Z), b_{1}). Actually, h^{X}Z is some fixed complex of 𝕜-modules from Ob C, with a fixed isomorphism h^{X}Z ^{∼}→ (s𝒜(X, Z), b_{1}). These isomorphisms enter implicitly into the structure maps of h^{X}, however, we shall pretend that they are identity morphisms. The closed monoidal structure of C gives us the right to omit these isomorphisms in all the formulae.

We require h^{X} to be pointed, that is, (T^{0}s𝒜)h^{X} ⊂ T^{0}sC. Therefore, h^{X} is completely specified by its components h_{k}^{X} for k ⩾ 1:

h_{k}^{X} = [s𝒜(Z_{0}, Z_{1}) ⊗ ⋯ ⊗ s𝒜(Z_{k−1}, Z_{k}) ^{coev} → C(h^{X}Z_{0}, h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k})

C(1,b_{k+1}) → C(s𝒜(X, Z_{0}), s𝒜(X, Z_{k})) ^{σ}→ sC(s𝒜(X, Z_{0}), s𝒜(X, Z_{k}))] . (A.2.1)

We claim that h^{X} is actually an A_{∞}-functor h^{X} : 𝒜 → C. A similar statement is already known from Fukaya’s work [Fuk02, Proposition 7.18] under slightly more restrictive general assumptions which are not needed for this result. Nevertheless, for the sake of completeness, we give a full proof in our own notations. We have to verify the following equation for an arbitrary k ⩾ 1:

∑_{p+t+q=k}(1^{⊗p} ⊗ b_{t} ⊗ 1^{⊗q})h_{p+1+q}^{X} = h_{k}^{X}b_{1}^{C} + ∑_{i+j=k}(h_{i}^{X} ⊗ h_{j}^{X})b_{2}^{C} :

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{X}Z_{k}).

Let us gradually expand each summand. First of all,

h_{k}^{X}b_{1}^{C} = coev C(1, b_{k+1})σb_{1}^{C}
= − coev C(1, b_{k+1})m_{1}^{C}σ
= − coev C(1, b_{k+1})(C(1, b_{1}) − C(b_{1}, 1))σ
= − coev C(1, b_{k+1}b_{1})σ − coev C(b_{1}, 1)C(1, b_{k+1})σ.

Similarly,

(h_{i}^{X} ⊗ h_{j}^{X})b_{2}^{C} = (coev C(1, b_{i+1})σ ⊗ coev C(1, b_{j+1})σ)b_{2}^{C}
= −(coev ⊗ coev)(C(1, b_{i+1}) ⊗ C(1, b_{j+1}))m_{2}σ
= − coev C(1, (b_{i+1} ⊗ 1^{⊗j})b_{j+1})σ.  (A.2.2)

The sign is due to the fact σ and b have odd degree. The last line is due to (A.1.2). Further, given α : Y → U, then the following diagram commutes

X ⊗ Y ⊗ Z  \overline{^{1⊗α⊗1}  →} X ⊗ U ⊗ Z

C(A, A ⊗ X ⊗ Y ⊗ Z) ↓^{coev_{A,X⊗Y ⊗Z}}  \overline{→^{C(A,1⊗1⊗α⊗1)}} C(A, A ⊗ X ⊗ U ⊗ Z), ↓^{coev_{A,X⊗U⊗Z}}

for coev is a functor morphism. Applying this to α = b_{t} we obtain

(1^{⊗p} ⊗ b_{t} ⊗ 1^{⊗q})h_{p+1+q}^{X} = (1^{⊗p} ⊗ b_{t} ⊗ 1^{⊗q}) coev C(1, b_{p+2+q})σ = coev C(1, (1^{⊗p+1} ⊗ b_{t} ⊗ 1^{⊗q})b_{p+2+q})σ. (A.2.3)

Thus,

∑_{p+t+q=k}(1^{⊗p} ⊗ b_{t} ⊗ 1^{⊗q})h_{p+1+q}^{X} = coev C(1, ∑_{\begin{matrix}p+t+q=k+1, \\ p⩾1 \end{matrix}}(1^{⊗p} ⊗ b_{t} ⊗ 1^{⊗q})b_{p+1+q})σ,

so that

∑_{p+t+q=k}(1^{⊗p} ⊗ b_{t} ⊗ 1^{⊗q})h_{p+1+q}^{X} − h_{k}^{X}b_{1}^{C} − ∑_{\begin{matrix}i+j=k, \\ i,j⩾1 \end{matrix}}(h_{i}^{X} ⊗ h_{j}^{X})b_{2}^{C} = coev C(b_{1}, 1)C(1, b_{k+1})σ

+ coev C(1, ∑_{\begin{matrix}p+t+q=k+1, \\ p⩾1 \end{matrix}}(1^{⊗p} ⊗ b_{t} ⊗ 1^{⊗q})b_{p+1+q} + b_{k+1}b_{1} + ∑_{\begin{matrix}i+j=k, \\ i,j⩾1 \end{matrix}}(b_{i+1} ⊗ 1^{⊗j})b_{j+1})σ.

Finally, coev C(b_{1}, 1) = coev C(1, b_{1} ⊗ 1^{⊗k}) by (A.1.3), and the sum in the right-hand side of the above formula vanishes by equation (A.1.4).

A.3 Lemma. Let 𝒜 be a unital A_{∞}-category. Then for each object X of 𝒜 the A_{∞}-functor h^{X} : 𝒜 → C is unital.

Proof. For arbitrary objects Y, Z of 𝒜 the map h_{1}^{X} is given by

h_{1}^{X} = [s𝒜(Y, Z) ^{coev} → C(h^{X}Y, h^{X}Y ⊗ h^{Y} Z) ^{C(1,b_{2})} → C(h^{X}Y, h^{X}Z) ^{σ}→ sC(h^{X}Y, h^{X}Z)].

We need this formula for Z = Y. We have to find such v ∈ (sC)^{−2}(h^{X}Y, h^{X}Y) that

Y i_{0}^{𝒜} h_{1}^{X} −_{h^{X}Y} i_{0}^{C} = vb_{1}^{C} : 𝕜 → (sC)^{−1}(h^{X}Y, h^{X}Y).

We have _{h^{X}Y} i_{0}^{C} = 1_{h^{X}Y} s. We seek for an element v in the form v = h′s, where h′ ∈ C^{−1}(h^{X}Y, h^{X}Y). The required equation is:

[𝕜 ^{_{Y} i_{0}^{𝒜}}→ (s𝒜)^{−1}(Y, Y) ^{coev} → C^{−1}(h^{X}Y, h^{X}Y ⊗ h^{Y} Y) ^{C(1,b_{2})} → C^{0}(h^{X}Y, h^{X}Y)]

= [𝕜 ^{1_{h^{X} Y} +h′m_{1}^{C}} → C^{0}(h^{X}Y, h^{X}Y)].

By functoriality of coev we have _{Y} i_{0}^{𝒜} coev_{h^{X}Y,h^{Y} Y} = coev_{h^{X}Y,𝕜} C(h^{X}Y, 1 ⊗ _{Y} i_{0}^{𝒜}), and the above equation reduces to

(h^{X}Y = h^{X}Y ⊗ 𝕜 ^{1⊗_{Y} i_{0}^{𝒜}} → h^{X}Y ⊗ h^{Y} Y ^{b_{2}}→ h^{X}Y ) = (h^{X}Y ^{1+h′b_{1}+b_{1}h′} → h^{X}Y ).

The existence of such a homotopy h′ immediately follows from the definition of a unital A_{∞}-category, see [Lyu03, Lemma 7.4].

A.4. The opposite A_{∞}-category. Let 𝒜 be a graded 𝕜-quiver. Then its opposite quiver 𝒜^{op} is defined as the quiver with the same class of objects Ob 𝒜^{op} = Ob 𝒜, and with graded 𝕜-modules of morphisms 𝒜^{op}(X, Y) = 𝒜(Y, X).

Let γ : T s𝒜^{op} → T s𝒜 denote the following cocategory anti-isomorphism:

γ = (−1)^{k}ω_{c}^{0} : s𝒜^{op}(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒜^{op}(X_{k−1}, X_{k}) → s𝒜(X_{k}, X_{k−1}) ⊗ ⋯ ⊗ s𝒜(X_{1}, X_{0}),

(A.4.1)

where the permutation ω^{0} = ( \begin{matrix}1 & 2 & ... & k−1 & k \\ k & k−1 & ... & 2 & 1 \end{matrix} ) is the longest element of 𝔖_{k}, and ω_{c}^{0} is the corresponding signed permutation, the action of ω^{0} in tensor products via standard symmetry. Clearly, γΔ = Δ(γ ⊗γ)c = Δc(γ ⊗γ), which is the anti-isomorphism property. Notice also that (𝒜^{op})^{op} = 𝒜 and γ^{2} = id.

When 𝒜 is an A_{∞}-category with the codifferential b : T s𝒜 → T s𝒜, then γbγ : T s𝒜^{op} → T s𝒜^{op} is also a codifferential. Indeed,

γbγΔ = γbΔc(γ ⊗ γ) = γΔ(1 ⊗ b + b ⊗ 1)c(γ ⊗ γ) = Δ(γ ⊗ γ)c(1 ⊗ b + b ⊗ 1)c(γ ⊗ γ)
= Δ(γ ⊗ γ)(b ⊗ 1 + 1 ⊗ b)(γ ⊗ γ) = Δ(γbγ ⊗ 1 + 1 ⊗ γbγ).

A.5 Definition (cf. Fukaya [Fuk02] Definition 7.8). The opposite A_{∞}-category 𝒜^{op} to an A_{∞}-category 𝒜 is the opposite quiver, equipped with the codifferential b^{op} = γbγ : T s𝒜^{op} → T s𝒜^{op}.

The components of b^{op} are computed as follows:

b_{k}^{op} = (−)^{k+1}[s𝒜^{op}(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒜^{op}(X_{k−1}, X_{k})

ω_{c}^{0}→ s𝒜(X_{k}, X_{k−1}) ⊗ ⋯ ⊗ s𝒜(X_{1}, X_{0}) ^{b_{k}}→ s𝒜(X_{k}, X_{0}) = s𝒜^{op}(X_{0}, X_{k})].

The sign (−1)^{k} in (A.4.1) ensures that the above definition agrees with the definition of the opposite usual category.

A.6. The Yoneda A_{∞}-functor. Since 𝒜 is 𝒰 -small, and C is a 𝒰′-small 𝒰 -category, A_{∞}(𝒜, C) is a 𝒰′-small differential graded 𝒰 -category as well. Indeed, every its set of morphisms sA_{∞}(𝒜, C)(f, g) is isomorphic to the product of graded 𝕜-modules

∏_{k=0}^{∞} ∏_{X,Y ∈Ob 𝒜} C(T^{k}s𝒜(X, Y), sC(Xf, Y g)),

that is 𝒰 -small.

Denote by Y a pointed cocategory homomorphism T s𝒜^{op} → T sA_{∞}(𝒜, C) given as follows: on objects X ↦ h^{X}, the components

Y_{n} : s𝒜^{op}(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒜^{op}(X_{n−1}, X_{n}) → sA_{∞}(𝒜, C)(h^{X_{0}}, h^{X_{n}})  (A.6.1)

are determined by the following formulas. The composition of Y_{n} with

pr_{k} : sA_{∞}(𝒜, C)(h^{X_{0}}, h^{X_{n}}) → C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}))

(that is, the k-th component of the coderivation (x_{1} ⊗ ⋯ ⊗ x_{n})Y_{n}) is given by the formula

Y_{nk} = (−)^{n}[s𝒜(X_{1}, X_{0}) ⊗ ⋯ ⊗ s𝒜(X_{n}, X_{n−1}) ^{coev} → C(h^{X_{0}}Z_{0} ⊗h^{Z_{0}}Z_{1} ⊗⋯⊗h^{Z_{k−1}}Z_{k}, h^{X_{0}}Z_{0} ⊗h^{Z_{0}}Z_{1} ⊗⋯⊗h^{Z_{k−1}}Z_{k} ⊗h^{X_{1}}X_{0} ⊗⋯⊗h^{X_{n}}X_{n−1})

C(1,τ_{c}b_{k+n+1}) → C(h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, h^{X_{n}}Z_{k})
∼→ C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}))

C(1,σ) → C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}))],

where the permutation τ ∈ 𝔖_{k+n+1} is τ^{k,n} = ( \begin{matrix}0 & 1 & ... & k & k+1 & ... & k+n \\ n & 1+n & ... & k+n & n−1 & ... & 0 \end{matrix} ).

In other words, the coderivation (x_{1} ⊗ ⋯ ⊗ x_{n})Y_{n} has components s𝒜(Z_{0}, Z_{1}) ⊗ ⋯ ⊗ s𝒜(Z_{k−1}, Z_{k}) → sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}), (z_{1} ⊗ ⋯ ⊗ z_{k}) ↦ (z_{1} ⊗ ⋯ ⊗ z_{k} ⊗ x_{1} ⊗ ⋯ ⊗ x_{n})Y′_{nk}, where

Y′_{nk} = (−)^{n}[s𝒜(Z_{0}, Z_{1}) ⊗ ⋯ ⊗ s𝒜(Z_{k−1}, Z_{k}) ⊗ s𝒜(X_{1}, X_{0}) ⊗ ⋯ ⊗ s𝒜(X_{n}, X_{n−1})

coev → C(h^{X_{0}}Z_{0}, h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1})

C(1,τ_{c}b_{k+n+1}) → C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}) ^{σ}→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})]. (A.6.2)

We claim that Y is an A_{∞}-functor. An equivalent statement is already proved by Fukaya [Fuk02, Lemma 9.8]. Since our conventions (and even definitions) differ from Fukaya’s, it would take some efforts to establish an equivalence. Instead, we give a proof in our notations.

A.7 Proposition. The pointed cocategory homomorphism Y is an A_{∞}-functor 𝒜^{op} → A_{∞}(𝒜, C).

Proof. The equation to prove is the following:

∑_{l>0;i_{1}+···+i_{l}=n}(Y_{i_{1}} ⊗ Y_{i_{2}} ⊗ ⋯ ⊗ Y_{i_{l}})B_{l} = ∑_{α+p+β=n}(1^{⊗α} ⊗ b_{p}^{op} ⊗ 1^{⊗β})Y_{α+1+β} :

s𝒜^{op}(X_{0}, X_{1}) ⊗ ⋯ ⊗ s𝒜^{op}(X_{n−1}, X_{n}) → sA_{∞}(𝒜, C)(h^{X_{0}}, h^{X_{n}}).

We compare the k-th components of the both sides:

[Y_{n}B_{1}]_{k} + ∑_{p+q=n}[(Y_{p} ⊗ Y_{q})B_{2}]_{k} = ∑_{α+p+β=n}[(1^{⊗α} ⊗ b_{p}^{op} ⊗ 1^{⊗β})Y_{α+1+β}]_{k} :

s𝒜(X_{1}, X_{0}) ⊗ ⋯ ⊗ s𝒜(X_{n}, X_{n−1}) → sA_{∞}(𝒜, C)(h^{X_{0}}, h^{X_{n}})

pr_{k}→ C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})).

Let us denote by r̄ an element of s𝒜(X_{1}, X_{0}) ⊗ ⋯ ⊗ s𝒜(X_{n}, X_{n−1}). The terms in the left hand side are expanded to

[(r̄Y_{n})B_{1}]_{k} = (r̄Y_{n})_{k}b_{1}^{C}

+ ∑_{i+j=k}^{i>0}(h_{i}^{X_{0}} ⊗(r̄Y_{n})_{j})b_{2}^{C}+ ∑_{i+j=k}^{j>0}((r̄Y_{n})_{i}⊗h_{j}^{X_{n}})b_{2}^{C}−(−)^{r} ∑_{α+p+β=k}(1^{⊗α}⊗b_{p}⊗1^{⊗β})(r̄Y_{n})_{α+1+β},

[(Y_{p} ⊗ Y_{q})B_{2}]_{k} = ∑_{i+j=k}^{p,q>0}(Y′_{pi} ⊗ Y′_{qj})b_{2}^{C}.

To reduce further the number of terms we introduce the notation Y′_{0k} = h_{k}^{X}. Indeed, when n = 0, formulas (A.6.2) and (A.2.1) coincide. So the left hand side expands to the following expression:

[(r̄Y_{n})B_{1}]_{k} + ∑_{p+q=n}[(Y_{p} ⊗ Y_{q})B_{2}]_{k}

= (r̄Y_{n})_{k}b_{1}^{C} + ∑_{i+j=k}^{p,q⩾0}(Y′_{pi} ⊗ Y′_{qj})b_{2}^{C} − (−)^{r} ∑_{α+p+β=k}(1^{⊗α} ⊗ b_{p} ⊗ 1^{⊗β})(r̄Y_{n})_{α+1+β},

Thus the equation to prove is this one:

[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} ^{Y′_{nk}}→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})

b_{1}^{C}→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})]+ ∑_{\begin{matrix}p+q=n \\ i+j=k \end{matrix}}^{\begin{matrix}p,q,i,j⩾0 \\ p+i>0,q+j>0 \end{matrix}}[h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{k−1}}Z_{k}⊗h^{X_{1}}X_{0}⊗⋯⊗h^{X_{n}}X_{n−1} ^{1⊗c⊗1} →

h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{i−1}}Z_{i}⊗h^{X_{1}}X_{0}⊗⋯⊗h^{X_{p}}X_{p−1}⊗h^{Z_{i}}Z_{i+1}⊗⋯⊗h^{Z_{k−1}}Z_{k}⊗h^{X_{p+1}}X_{p}⊗⋯⊗h^{X_{n}}X_{n−1}

Y′_{pi}⊗Y′_{qj} → sC(h^{X_{0}}Z_{0}, h^{X_{p}}Z_{i}) ⊗ sC(h^{X_{p}}Z_{i}, h^{X_{n}}Z_{k}) ^{b_{2}^{C}}→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})]

− ∑_{α+p+β=k}[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} ^{1^{⊗α}⊗b_{p}⊗1^{⊗β+n}} →

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{α−1}}Z_{α} ⊗ h^{Z_{α}}Z_{α+p} ⊗ h^{Z_{α+p}}Z_{α+p+1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1}

Y′_{n,α+1+β} → sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})]

= ∑_{α+p+β=n}[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} ^{1^{⊗k+α}⊗b_{p}^{op}⊗1^{⊗β}} →

h^{Z_{0}}Z_{1} ⊗⋯⊗h^{Z_{k−1}}Z_{k} ⊗h^{X_{1}}X_{0} ⊗⋯⊗h^{X_{α}}X_{α−1} ⊗h^{X_{α+p}}X_{α} ⊗h^{X_{α+p+1}}X_{α+p} ⊗⋯⊗h^{X_{n}}X_{n−1}

Y′_{α+1+β,k} → sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})].

Substituting explicit expression for Y′_{nk} we get the following equation to verify:

− (−)^{n}[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1}

coev → C(h^{X_{0}}Z_{0}, h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1}) C(1,τ_{c}^{n,k}b_{k+n+1}) → C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}) ^{C(1,b_{1})−C(b_{1},1)} → C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}) ^{σ}→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})]

− ∑_{\begin{matrix}p+q=n \\ i+j=k \end{matrix}}^{\begin{matrix}p,q,i,j⩾0 \\ p+i>0,q+j>0 \end{matrix}}(−)^{p+q}[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} ^{1^{⊗i}⊗c⊗1^{⊗n−p}} →

h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{i−1}}Z_{i}⊗h^{X_{1}}X_{0}⊗⋯⊗h^{X_{p}}X_{p−1}⊗h^{Z_{i}}Z_{i+1}⊗⋯⊗h^{Z_{k−1}}Z_{k}⊗h^{X_{p+1}}X_{p}⊗⋯⊗h^{X_{n}}X_{n−1} coev ⊗ coev → C(h^{X_{0}}Z_{0}, h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{i−1}}Z_{i} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{p}}X_{p−1})

⊗ C(h^{X_{p}}Z_{i}, h^{X_{p}}Z_{i} ⊗ h^{Z_{i}}Z_{i+1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{p+1}}X_{p} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1})

C(1,τ_{c}^{i,p}b_{1+i+p})⊗C(1,τ_{c}^{j,q}b_{1+j+q}) → C(h^{X_{0}}Z_{0}, h^{X_{p}}Z_{i}) ⊗ C(h^{X_{p}}Z_{i}, h^{X_{n}}Z_{k}) ^{m_{2}^{C}}→ C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})

σ→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})]

− (−)^{n} ∑_{α+p+β=k}[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} ^{1^{⊗α}⊗b_{p}⊗1^{⊗β+n}} →

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{α−1}}Z_{α} ⊗ h^{Z_{α}}Z_{α+p} ⊗ h^{Z_{α+p}}Z_{α+p+1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} coev → C(h^{X_{0}}Z_{0}, h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{α−1}}Z_{α} ⊗ h^{Z_{α}}Z_{α+p} ⊗ h^{Z_{α+p}}Z_{α+p+1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}

⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1}) ^{C(1,τ_{c}^{α+1+β,n}b_{n+α+β+2})} → C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}) ^{σ}→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})]

= ∑_{α+p+β=n}(−1)^{α+β+p}[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} ^{1^{⊗k+α}⊗(ω_{c}^{0}b_{p})⊗1^{⊗β}} →

h^{Z_{0}}Z_{1} ⊗⋯⊗h^{Z_{k−1}}Z_{k} ⊗h^{X_{1}}X_{0} ⊗⋯⊗h^{X_{α}}X_{α−1} ⊗h^{X_{α+p}}X_{α} ⊗h^{X_{α+p+1}}X_{α+p} ⊗⋯⊗h^{X_{n}}X_{n−1}

coev → C(h^{X_{0}}Z_{0}, h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{α}}X_{α−1} ⊗ h^{X_{α+p}}X_{α}⊗

h^{X_{α+p+1}}X_{α+p} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1}) ^{C(1,τ_{c}^{k,α+1+β}b_{k+α+β+2})} → C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})

σ→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})].

Using functoriality of coev, identities (A.1.2) and (A.1.3) we reduce all terms of this equation to the form

[h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1}

coev → C(h^{X_{0}}Z_{0}, h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1})

→ … → C(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k}) ^{σ}→ sC(h^{X_{0}}Z_{0}, h^{X_{n}}Z_{k})].

Cancelling the above coev and σ, we obtain an equation between complexes and maps of the form C(h^{X_{0}}Z_{0}, ). Simplifying it we get an equation between maps : h^{X_{0}}Z_{0} ⊗h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1} → h^{X_{n}}Z_{k}, whose corollary is the required identity. Multiplying it with the composite symmetry

( \begin{matrix}1 & ... & n−1 & n & n+1 & ... & n+k+1 \\ k+n+1 & ... & k+3 & k+2 & 1 & ... & k+1 \end{matrix} )_{c} : h^{X_{n}}X_{n−1} ⊗⋯⊗h^{X_{1}}X_{0} ⊗h^{X_{0}}Z_{0} ⊗h^{Z_{0}}Z_{1} ⊗⋯⊗h^{Z_{k−1}}Z_{k}

→ h^{X_{0}}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{X_{1}}X_{0} ⊗ ⋯ ⊗ h^{X_{n}}X_{n−1}

and with the common sign (−1)^{n} we get the equation

b_{n+1+k}b_{1} + (1^{⊗n} ⊗ b_{1} ⊗ 1^{⊗k})b_{n+1+k} + ∑_{\begin{matrix}p+q=n \\ i+j=k \end{matrix}}^{\begin{matrix}p,q,i,j⩾0 \\ p+i>0,q+j>0 \end{matrix}}(1^{⊗q} ⊗ b_{p+1+i} ⊗ 1^{⊗j})b_{q+1+j}

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

This is identity (A.1.4), so the proof is finished.

A.8. The Yoneda embedding. Now our goal is to prove that the Yoneda A_{∞}-functor Y is an equivalence of 𝒜^{op} with its image. This is already proven by Fukaya in the case of strictly unital A_{∞}-category 𝒜 [Fuk02, Theorem 9.1]. We are going to extend this result to arbitrary unital A_{∞}-categories.

A.9 Proposition. Let 𝒜 be a unital A_{∞}-category. Let X, W be objects of 𝒜. Then the map Y_{1} : s𝒜^{op}(X, W) → sA_{∞}(𝒜, C)(h^{X}, h^{W}) is homotopy invertible.

Proof. The composition of Y_{1} with

pr_{k} : sA_{∞}(𝒜, C)(h^{X}, h^{W}) → C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X}Z_{0}, h^{W} Z_{k}))

is given by the formula

Y_{1k} = −[s𝒜(W, X) ^{coev} → C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{W} X)

C(1,τ_{c}^{k,1}b_{k+2}) → C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, h^{W} Z_{k})
∼→ C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}Z_{0}, h^{W} Z_{k}))
C(1,σ) → C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X}Z_{0}, h^{W} Z_{k}))],

where the permutation τ^{k,1} ∈ 𝔖_{k+2} is τ^{k,1} = ( \begin{matrix}0 & 1 & ... & k & k+1 \\ 1 & 2 & ... & k+1 & 0 \end{matrix} ).

Define a degree 0 map α : sA_{∞}(𝒜, C)(h^{X}, h^{W}) → s𝒜^{op}(X, W) as follows:

α = [sA_{∞}(𝒜, C)(h^{X}, h^{W}) ^{pr_{0}}→ sC(h^{X}X, h^{W} X)

σ^{−1} → C(h^{X}X, h^{W} X) ^{C(_{X}i_{0}^{𝒜} ,1)} → C(𝕜, h^{W} X) = h^{W} X].

It is a chain map. Indeed, pr_{0} is a chain map, and so is σ^{−1}C(_{X}i_{0}^{𝒜} , 1), since

σ^{−1}C(_{X}i_{0}^{𝒜} , 1)b_{1} = σ^{−1}C(_{X}i_{0}^{𝒜} , 1)C(1, b_{1})

= −σ^{−1}[C(1, b_{1}) − C(b_{1}, 1)]C(_{X}i_{0}^{𝒜} , 1) = b_{1}^{C}σ^{−1}C(_{X}i_{0}^{𝒜} , 1).

Let us prove that α is homotopy inverse to Y_{1}. First of all,

Y_{10} = −[h^{W} X ^{coev} → C(h^{X}Z, h^{X}Z ⊗ h^{W} X) ^{C(1,cb_{2})} → C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)].

Using this expression for Z = X we compute

Y_{1}α = −[h^{W} X ^{coev} → C(h^{X}X, h^{X}X ⊗ h^{W} X) ^{C(1,cb_{2})} → C(h^{X}X, h^{W} X)
C(_{X}i_{0}^{𝒜} ,1) → C(𝕜, h^{W} X) = h^{W} X]
= [h^{W} X ^{coev} → C(h^{X}X, h^{X}X ⊗ h^{W} X) ^{C(_{X}i_{0}^{𝒜} ,1)} → C(𝕜, h^{X}X ⊗ h^{W} X)
C(1,cb_{2}) → C(𝕜, h^{W} X) = h^{W} X]
= [h^{W} X ^{coev} → C(𝕜, 𝕜 ⊗ h^{W} X) ^{C(1,_{X}i_{0}^{𝒜} ⊗1)} → C(𝕜, h^{X}X ⊗ h^{W} X)
C(1,cb_{2}) → C(𝕜, h^{W} X) = h^{W} X]
= [h^{W} X = 𝕜 ⊗ h^{W} X ^{_{X}i_{0}^{𝒜} ⊗1} → h^{X}X ⊗ h^{W} X ^{cb_{2}}→ h^{W} X]
= [h^{W} X = h^{W} X ⊗ 𝕜 ^{1⊗_{X}i_{0}^{𝒜}} → h^{W} X ⊗ h^{X}X ^{b_{2}}→ h^{W} X].

Therefore, the map Y_{1}α is homotopic to identity map id : h^{W} X → h^{W} X by [Lyu03, Lemma 7.4]. Now we are going to prove that αY_{1} is homotopy invertible.

The graded 𝕜-module sA_{∞}(𝒜, C)(h^{X}, h^{W}) is V = ∏_{n=0}^{∞} V_{n}, where

V_{n} = ∏_{Z_{0}, ..., Z_{n}∈Ob 𝒜} C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{n−1}}Z_{n}, sC(h^{X}Z_{0}, h^{W} Z_{n}))

and all products are taken in the category of graded 𝕜-modules. In other terms, for d ∈ ℤ

V^{d} = ∏_{n=0}^{∞} V_{n}^{d},  V_{n}^{d} = ∏_{Z_{0}, ..., Z_{n}∈Ob 𝒜} C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{n−1}}Z_{n}, sC(h^{X}Z_{0}, h^{W} Z_{n}))^{d}.

We consider V_{n}^{d} as Abelian groups with discrete topology. The Abelian group V^{d} is equipped with the topology of the product. Thus, its basis of neighborhoods of 0 is given by 𝕜-submodules Φ_{m}^{d} = 0^{m−1} × ∏_{n=m}^{∞} V_{n}^{d}. They form a filtration V^{d} = Φ_{0}^{d} ⊃ Φ_{1}^{d} ⊃ Φ_{2}^{d} ⊃ . . .. We call a 𝕜-linear map a : V → V of degree p continuous if the induced maps a^{d,d+p} = a|_{V^{d}} : V^{d} → V^{d+p} are continuous for all d ∈ ℤ. This holds if and only if for any

d ∈ ℤ and m ∈ ℕ =^{def} ℤ_{⩾0} there exists an integer κ = κ(d, m) ∈ ℕ such that (Φ_{κ}^{d})a ⊂ Φ_{m}^{d+p}. We may assume that

m′ < m′′ implies κ(d, m′) ⩽ κ(d, m′′).  (A.9.1)

Indeed, a given function m ↦ κ(d, m) can be replaced with the function m ↦ κ′(d, m) = min_{n⩾m} κ(d, n) and κ′ satisfies condition (A.9.1). Continuous linear maps a : V → V of degree p are in bijection with families of ℕ × ℕ-matrices (A^{d,d+p})_{d∈ℤ} of linear maps A_{nm}^{d,d+p} : V_{n}^{d} → V_{m}^{d+p} with finite number of non-vanishing elements in each column of A^{d,d+p}. Indeed, to each continuous map a^{d,d+p} : V^{d} → V^{d+p} corresponds the inductive limit over m of κ(d, m) × m-matrices of maps V^{d}/Φ_{κ(d,m)}^{d} → V^{d+p}/Φ_{m}^{d+p}. On the other hand, to each family (A^{d,d+p})_{d∈ℤ} of ℕ×ℕ-matrices with finite number of non-vanishing elements in each column correspond obvious maps a^{d,d+p} : V^{d} → V^{d+p}, and they are continuous. Thus, a = (a^{d,d+p})_{d∈ℤ} is continuous. A continuous map a : V → V can be completely recovered from one ℕ × ℕ-matrix (a_{nm})_{n,m∈ℕ} of maps a_{nm} = (A_{nm}^{d,d+p})_{d∈ℤ} : V_{n} → V_{m} of degree p. Naturally, not any such matrix determines a continuous map, however, if the number of non-vanishing elements in each column of (a_{nm}) is finite, then this matrix does determine a continuous map.

The differential D =^{def} B_{1} : V → V, r ↦ (r)B_{1} = rb − (−)^{r}br is continuous and the function κ for it is simply κ(d, m) = m. Its matrix is given by

D = B_{1} = [\begin{matrix}D_{0,0} & D_{0,1} & D_{0,2} & … \\ 0 & D_{1,1} & D_{1,2} & … \\ 0 & 0 & D_{2,2} & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix}] ,

where

D_{k,k} = C(1, b_{1}^{C}) − C( ∑_{α+1+γ=k} 1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗γ}, 1) : V_{k} → V_{k},

r_{k}D_{k,k} = r_{k}b_{1}^{C} − (−)^{r} ∑_{α+1+γ=k}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗γ})r_{k},

(one easily recognizes the differential in the complex V_{k}),

r_{k}D_{k,k+1} = (r_{k} ⊗ h_{1}^{W})b_{2}^{C} + (h_{1}^{X} ⊗ r_{k})b_{2}^{C} − (−)^{r} ∑_{α+γ=k−1}(1^{⊗α} ⊗ b_{2} ⊗ 1^{⊗γ})r_{k}.

Further we will see that we do not need to compute other components.

Composition of αY_{1} with pr_{k} equals

αY_{1k} = −[sA_{∞}(𝒜, C)(h^{X}, h^{W}) ^{pr_{0}}→ sC(h^{X}X, h^{W} X) ^{σ^{−1}} → C(h^{X}X, h^{W} X)

C(_{X}i_{0}^{𝒜} ,1) → C(𝕜, h^{W} X) = h^{W} X ^{coev} →
C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} ⊗ h^{W} X)
C(1,τ_{c}^{k,1}b_{k+2}) → C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, h^{W} Z_{k})
∼→ C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}Z_{0}, h^{W} Z_{k}))

C(1,σ) → C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X}Z_{0}, h^{W} Z_{k}))].

Clearly, αY_{1} is continuous (take κ(d, m) = 1). Its ℕ × ℕ-matrix has the form

αY_{1} = [\begin{matrix}∗ & ∗ & ∗ & … \\ 0 & 0 & 0 & … \\ 0 & 0 & 0 & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix}] .

The reader might skip the proof of the following lemma at first reading.

A.10 Lemma. The map αY_{1} : V → V is homotopic to a continuous map V → V, whose ℕ × ℕ-matrix is upper-triangular with the identity maps id : V_{k} → V_{k} on the diagonal.

Proof. Define a continuous 𝕜-linear map H : sA_{∞}(𝒜, C)(h^{X}, h^{W}) → sA_{∞}(𝒜, C)(h^{X}, h^{W}) of degree −1 by its matrix

H = [\begin{matrix}0 & 0 & 0 & … \\ H_{1,0} & 0 & 0 & … \\ 0 & H_{2,1} & 0 & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix}] ,

so κ(d, m) = m+1, where H_{k+1,k} maps the factor indexed by (X, Z_{0}, …, Z_{k}) to the factor indexed by (Z_{0}, …, Z_{k}) as follows:

H_{k+1,k} = [C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X}X, h^{W} Z_{k}))
→^{C(1,σ^{−1})} C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}X, h^{W} Z_{k}))
→^{C(1,C(_{X}i_{0}^{𝒜} ,1))} C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(𝕜, h^{W} Z_{k}))
→^{∼} C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}Z_{0}, h^{W} Z_{k}))
→^{C(1,σ)} C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, sC(h^{X}Z_{0}, h^{W} Z_{k}))].

Other factors are ignored. Here the adjunction isomorphism is that of (A.1.1).

The composition of continuous maps V → V is continuous as well. In particular, one finds the matrices of B_{1}H and HB_{1}:

B_{1}H = [\begin{matrix}D_{0,1}H_{1,0} & D_{0,2}H_{2,1} & D_{0,3}H_{3,2} & … \\ D_{1,1}H_{1,0} & D_{1,2}H_{2,1} & D_{1,3}H_{3,2} & … \\ 0 & D_{2,2}H_{2,1} & D_{2,3}H_{3,2} & … \\ 0 & 0 & D_{3,3}H_{3,2} & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix}] ,

HB_{1} = [\begin{matrix}0 & 0 & 0 & … \\ H_{1,0}D_{0,0} & H_{1,0}D_{0,1} & H_{1,0}D_{0,2} & … \\ 0 & H_{2,1}D_{1,1} & H_{2,1}D_{1,2} & … \\ 0 & 0 & H_{3,2}D_{2,2} & … \\ ⋮ & ⋮ & ⋮ & ... \end{matrix}] .

We have D_{k+1,k+1}H_{k+1,k} +H_{k+1,k}D_{k,k} = 0 for all k ⩾ 0. Indeed, conjugating the expanded left hand side with C(1, σ) we come to the following identity

[C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}X, h^{W} Z_{k})) ^{−C(1,m_{1}^{C})+C(∑_{α+γ=k} 1^{⊗α}⊗b_{1}⊗1^{⊗γ},1)} →

C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}X, h^{W} Z_{k})) ^{C(1,C(_{X}i_{0}^{𝒜} ,1))} →

C(h^{X}Z_{0}⊗h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{k−1}}Z_{k}, C(𝕜, h^{W} Z_{k})) →^{∼} C(h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{k−1}}Z_{k}, C(h^{X}Z_{0}, h^{W} Z_{k}))]

+ [C(h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}X, h^{W} Z_{k})) ^{C(1,C(_{X}i_{0}^{𝒜} ,1))} →

C(h^{X}Z_{0} ⊗h^{Z_{0}}Z_{1} ⊗⋯⊗h^{Z_{k−1}}Z_{k}, C(𝕜, h^{W} Z_{k})) →^{∼} C(h^{Z_{0}}Z_{1} ⊗⋯⊗h^{Z_{k−1}}Z_{k}, C(h^{X}Z_{0}, h^{W} Z_{k}))

−C(1,m_{1}^{C})+C(∑_{α+γ=k−1} 1^{⊗α}⊗b_{1}⊗1^{⊗γ},1) → C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, C(h^{X}Z_{0}, h^{W} Z_{k}))] = 0.

After reducing all terms to the common form, beginning with C(1, C(_{X}i_{0}^{𝒜} , 1)), all terms cancel each other, so the identity is proven.

Therefore, the chain map a = αY_{1} + B_{1}H + HB_{1} is also represented by an uppertriangular matrix. Its diagonal elements are chain maps a_{kk} : V_{k} → V_{k}. We are going to show that they are homotopic to identity maps.

First of all, let us compute the matrix element a_{00} : V_{0} → V_{0} = ∏_{Z∈Ob 𝒜} sC(h^{X}Z, h^{W} Z). We have

a_{00} pr_{Z} = (αY_{10} + B_{1}H_{1,0}) pr_{Z} : V_{0} → V_{0} ^{pr_{Z}}→ sC(h^{X}Z, h^{W} Z).

In the expanded form these terms are as follows:

αY_{10} pr_{Z} = −[V_{0} ^{pr_{X}}→ sC(h^{X}X, h^{W} X) ^{σ^{−1}} → C(h^{X}X, h^{W} X) ^{C(_{X}i_{0}^{𝒜} ,1)} → C(𝕜, h^{W} X) = h^{W} X

coev → C(h^{X}Z, h^{X}Z ⊗ h^{W} X) ^{C(1,cb_{2})} → C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)],

B_{1}H_{1,0} pr_{Z} = [(1 ⊗ h_{1}^{W})b_{2} + (h_{1}^{X} ⊗ 1)b_{2}]H_{1,0} pr_{Z},

(1 ⊗ h_{1}^{W})b_{2} = [V_{0} ^{pr_{X}}→ sC(h^{X}X, h^{W} X) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ sC(h^{X}X, h^{W} X))

C(1,c(1⊗coev)) → C(h^{X}Z, sC(h^{X}X, h^{W} X) ⊗ C(h^{W} X, h^{W} X ⊗ h^{X}Z))
C(1,1⊗C(1,b_{2})) → C(h^{X}Z, sC(h^{X}X, h^{W} X) ⊗ C(h^{W} X, h^{W} Z)) ^{C(1,1⊗σ)} →

C(h^{X}Z, sC(h^{X}X, h^{W} X) ⊗ sC(h^{W} X, h^{W} Z)) ^{C(1,b_{2}^{C})} → C(h^{X}Z, sC(h^{X}X, h^{W} Z))],

(h_{1}^{X} ⊗ 1)b_{2} = [V_{0} ^{pr_{Z}}→ sC(h^{X}Z, h^{W} Z) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ sC(h^{X}Z, h^{W} Z))

C(1,coev ⊗1) → C(h^{X}Z, C(h^{X}X, h^{X}X ⊗ h^{X}Z) ⊗ sC(h^{X}Z, h^{W} Z))
C(1,C(1,b_{2})⊗1) → C(h^{X}Z, C(h^{X}X, h^{X}Z) ⊗ sC(h^{X}Z, h^{W} Z)) ^{C(1,σ⊗1)} →

C(h^{X}Z, sC(h^{X}X, h^{X}Z) ⊗ sC(h^{X}Z, h^{W} Z)) ^{C(1,b_{2}^{C})} → C(h^{X}Z, sC(h^{X}X, h^{W} Z))],

H_{1,0} pr_{Z} = [V_{1} ^{pr_{X,Z}} → C(h^{X}Z, sC(h^{X}X, h^{W} Z)) ^{C(1,σ^{−1})} → C(h^{X}Z, C(h^{X}X, h^{W} Z))

C(1,C(_{X}i_{0}^{𝒜} ,1)) → C(h^{X}Z, C(𝕜, h^{W} Z)) = C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)].

We claim that in the sum

a_{00} pr_{Z} = αY_{10} pr_{Z} +(1 ⊗ h_{1}^{W})b_{2}H_{1,0} pr_{Z} +(h_{1}^{X} ⊗ 1)b_{2}H_{1,0} pr_{Z}

the first two summands cancel each other, while the last, (h_{1}^{X} ⊗ 1)b_{2}H_{1,0} is homotopic to identity. Indeed, αY_{10} pr_{Z} +(1 ⊗ h_{1}^{W})b_{2}H_{1,0} pr_{Z} factors through

− [sC(h^{X}X, h^{W} X) ^{σ^{−1}} → C(h^{X}X, h^{W} X) ^{C(_{X}i_{0}^{𝒜} ,1)} → C(𝕜, h^{W} X) = h^{W} X

coev → C(h^{X}Z, h^{X}Z ⊗ h^{W} X) ^{C(1,cb_{2})} → C(h^{X}Z, h^{W} Z)]

+ [sC(h^{X}X, h^{W} X) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ sC(h^{X}X, h^{W} X))

C(1,c(1⊗coev)) → C(h^{X}Z, sC(h^{X}X, h^{W} X) ⊗ C(h^{W} X, h^{W} X ⊗ h^{X}Z))
C(1,1⊗C(1,b_{2})) → C(h^{X}Z, sC(h^{X}X, h^{W} X) ⊗ C(h^{W} X, h^{W} Z))

C(1,1⊗σ) → C(h^{X}Z, sC(h^{X}X, h^{W} X) ⊗ sC(h^{W} X, h^{W} Z)) ^{C(1,b_{2}^{C})} → C(h^{X}Z, sC(h^{X}X, h^{W} Z))

C(1,σ^{−1}) → C(h^{X}Z, C(h^{X}X, h^{W} Z)) ^{C(1,C(_{X}i_{0}^{𝒜} ,1))} → C(h^{X}Z, C(𝕜, h^{W} Z)) = C(h^{X}Z, h^{W} Z)].

Composing this sum with σ we get

− [C(h^{X}X, h^{W} X) ^{C(_{X}i_{0}^{𝒜} ,1)} → C(𝕜, h^{W} X) = h^{W} X

coev → C(h^{X}Z, h^{X}Z ⊗ h^{W} X) ^{C(1,cb_{2})} → C(h^{X}Z, h^{W} Z)]

− [C(h^{X}X, h^{W} X) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ C(h^{X}X, h^{W} X))

C(1,c(1⊗coev)) → C(h^{X}Z, C(h^{X}X, h^{W} X) ⊗ C(h^{W} X, h^{W} X ⊗ h^{X}Z))

C(1,1⊗C(1,b_{2})) → C(h^{X}Z, C(h^{X}X, h^{W} X) ⊗ C(h^{W} X, h^{W} Z)) ^{C(1,m_{2}^{C})} → C(h^{X}Z, C(h^{X}X, h^{W} Z))

C(1,C(_{X}i_{0}^{𝒜} ,1)) → C(h^{X}Z, C(𝕜, h^{W} Z)) = C(h^{X}Z, h^{W} Z)].

This expression vanishes by the properties of closed monoidal category C. Therefore, αY_{10} pr_{Z} +(1 ⊗ h_{1}^{W})b_{2}H_{1,0} pr_{Z} also vanishes.

Now we prove that (h_{1}^{X} ⊗ 1)b_{2}H_{1,0} is homotopic to identity. It maps each factor sC(h^{X}Z, h^{W} Z) into itself via the following map:

[sC(h^{X}Z, h^{W} Z) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ sC(h^{X}Z, h^{W} Z))

C(1,coev ⊗1) → C(h^{X}Z, C(h^{X}X, h^{X}X ⊗ h^{X}Z) ⊗ sC(h^{X}Z, h^{W} Z))
C(1,C(1,b_{2})⊗1) → C(h^{X}Z, C(h^{X}X, h^{X}Z) ⊗ sC(h^{X}Z, h^{W} Z))
C(1,σ⊗1) → C(h^{X}Z, sC(h^{X}X, h^{X}Z) ⊗ sC(h^{X}Z, h^{W} Z))
C(1,b_{2}^{C}) → C(h^{X}Z, sC(h^{X}X, h^{W} Z)) ^{C(1,σ^{−1})} → C(h^{X}Z, C(h^{X}X, h^{W} Z))

C(1,C(_{X}i_{0}^{𝒜} ,1)) → C(h^{X}Z, C(𝕜, h^{W} Z)) = C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)] = [sC(h^{X}Z, h^{W} Z) ^{σ^{−1}} → C(h^{X}Z, h^{W} Z) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ C(h^{X}Z, h^{W} Z))

C(1,coev ⊗1) → C(h^{X}Z, C(h^{X}X, h^{X}X ⊗ h^{X}Z) ⊗ C(h^{X}Z, h^{W} Z))

C(1,C(1,b_{2})⊗1) → C(h^{X}Z, C(h^{X}X, h^{X}Z) ⊗ C(h^{X}Z, h^{W} Z)) ^{C(1,m_{2}^{C})} → C(h^{X}Z, C(h^{X}X, h^{W} Z))

C(1,C(_{X}i_{0}^{𝒜} ,1)) → C(h^{X}Z, C(𝕜, h^{W} Z)) = C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)].

= −[sC(h^{X}Z, h^{W} Z) ^{σ^{−1}} → C(h^{X}Z, h^{W} Z) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ C(h^{X}Z, h^{W} Z))

C(1,(_{X}i_{0}^{𝒜} ⊗1)b_{2}⊗1) → C(h^{X}Z, h^{X}Z⊗C(h^{X}Z, h^{W} Z)) ^{C(1,ev)} → C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)].

Due to [Lyu03, Lemma 7.4] there exists a homotopy h′′ : h^{X}Z → h^{X}Z (a map of degree −1), such that (_{X}i_{0}^{𝒜} ⊗ 1)b_{2} = −1 + h′′b_{1} + b_{1}h′′. Therefore, the map considered above equals to

id_{sC(h^{X}Z,h^{W} Z)} −[sC(h^{X}Z, h^{W} Z) ^{σ^{−1}} → C(h^{X}Z, h^{W} Z) ^{coev} → C(h^{X}Z, h^{X}Z ⊗ C(h^{X}Z, h^{W} Z))

C(1,(h′′b_{1}+b_{1}h′′)⊗1) → C(h^{X}Z, h^{X}Z ⊗C(h^{X}Z, h^{W} Z)) ^{C(1,ev)} → C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)]

= (id_{sC(h^{X}Z,h^{W} Z)} +b_{1}^{C}H′_{00} + H′_{00}b_{1}^{C}),

where

H′_{00} = [sC(h^{X}Z, h^{W} Z) ^{σ^{−1}} → C(h^{X}Z, h^{W} Z) ^{C(h′′,1)} → C(h^{X}Z, h^{W} Z) ^{σ}→ sC(h^{X}Z, h^{W} Z)].

Therefore, (h_{1}^{X} ⊗ 1)b_{2}H_{1,0} and a_{00} are homotopic to identity.

Now we are proving that diagonal elements a_{kk} : V_{k} → V_{k} are homotopic to identity maps for k > 0. Using the explicit formula for adjunction isomorphism we can re-write the formula for H as follows. An element r_{k+1} ∈ V_{k+1} is mapped to direct product over Z_{0}, …, Z_{k} ∈ Ob 𝒜 of

r_{k+1}H_{k+1,k} = coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, r_{k+1}^{X,Z_{0},...,Z_{k}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ : h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}).

Here r_{k+1}^{X,Z_{0},...,Z_{k}} : h^{X}Z_{0}⊗h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{k−1}}Z_{k} → sC(h^{X}X, h^{W} Z_{k}) is one of the coordinates of r_{k+1}.

Thus, r_{k}D_{k,k+1}H_{k+1,k} is the sum of three terms (A.10.1a)–(A.10.1c):

coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, (r_{k}^{X,Z_{0},...,Z_{k−1}} ⊗ h_{1}^{W})b_{2}^{C}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ :

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}), (A.10.1a)

where r_{k}^{X,Z_{0},...,Z_{k−1}} : h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−2}}Z_{k−1} → sC(h^{X}X, h^{W} Z_{k−1}). Since b_{2}^{C} = (σ ⊗ σ)^{−1}m_{2}σ = −(σ^{−1} ⊗ σ^{−1})m_{2}σ, we have

(r_{k} ⊗ h_{1}^{W})b_{2}^{C}σ^{−1} = −(r_{k}σ^{−1} ⊗ h_{1}^{W} σ^{−1})m_{2}^{C}.

Further, for any a : X → A the following equation holds:

(C(A, B) ⊗ C(B, C) ^{m_{2}}→ C(A, C) ^{C(a,C)} → C(X, C))

= (C(A, B) ⊗ C(B, C) ^{C(a,B)⊗1} → C(X, B) ⊗ C(B, C) ^{m_{2}}→ C(X, C)).

Thus,

(r_{k} ⊗ h_{1}^{W})b_{2}^{C}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) = −(r_{k}σ^{−1} ⊗ h_{1}^{W} σ^{−1})m_{2}^{C}C(_{X}i_{0}^{𝒜} , 1)

= −(r_{k}σ^{−1} ⊗ h_{1}^{W} σ^{−1})(C(_{X}i_{0}^{𝒜} , 1) ⊗ 1)m_{2}^{C} = (r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ h_{1}^{W} σ^{−1})m_{2}^{C}

(we have used the fact that C(_{X}i_{0}^{𝒜} , 1) has degree −1 and h_{1}^{W} σ^{−1} has degree 1).

coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, (h_{1}^{X} ⊗ r_{k}^{Z_{0},...,Z_{k}})b_{2}^{C}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ :

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}), (A.10.1b)

where r_{k}^{Z_{0},...,Z_{k}} : h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}). Similarly to above

(h_{1}^{X} ⊗ r_{k})b_{2}^{C}σ^{−1} = (−)^{r+1}(h_{1}^{X}σ^{−1} ⊗ r_{k}σ^{−1})m_{2}^{C},

so that

(h_{1}^{X} ⊗ r_{k})b_{2}^{C}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) = (−)^{r+1}(h_{1}^{X}σ^{−1} ⊗ r_{k}σ^{−1})(C(_{X}i_{0}^{𝒜} , 1) ⊗ 1)m_{2}^{C}

= (h_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ r_{k}σ^{−1})m_{2}^{C}

(r_{k}σ^{−1} has degree deg r + 1 and C(_{X}i_{0}^{𝒜} , 1) has degree −1).

For each α, γ, such that α + γ = k − 1

coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, (1^{⊗α} ⊗ b_{2} ⊗ 1^{⊗γ})r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ :

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}), (A.10.1c)

where r_{k} means r_{k}^{X,Z_{0},...,Z_{α−1},Z_{α+1},...,Z_{k}} : h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{α−1}}Z_{α+1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}X, h^{W} Z_{k}), and Z_{−1} = X.

Thus, r_{k}D_{k,k+1}H_{k+1,k} = coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, Σ_{1})σ, where

Σ_{1} = (r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ h_{1}^{W} σ^{−1})m_{2}^{C} + (h_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ r_{k}^{Z_{0},...,Z_{k}}σ^{−1})m_{2}^{C}

− (−)^{r} ∑_{α+γ=k−1}(1^{⊗α} ⊗ b_{2} ⊗ 1^{⊗γ})r_{k}^{X,Z_{0},...,Z_{α−1},Z_{α+1},...,Z_{k}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) :

h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → h^{W} Z_{k}.

Similarly, r_{k}H_{k,k−1}D_{k−1,k} is the sum of three terms (A.10.2a)–(A.10.2c).

(coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ ⊗ coev_{h^{W} Z_{k−1},h^{Z_{k−1}}Z_{k}} C(h^{W} Z_{k−1}, b_{2})σ)b_{2}^{C}

= −(coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1)) ⊗ coev_{h^{W} Z_{k−1},h^{Z_{k−1}}Z_{k}} C(h^{W} Z_{k−1}, b_{2}))m_{2}σ

= − coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, (r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ 1)b_{2})σ :

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}), (A.10.2a) where r_{k} means r_{k}^{X,Z_{0},...,Z_{k−1}} : h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−2}}Z_{k−1} → sC(h^{X}X, h^{W} Z_{k−1}). Here we use (A.1.2) (compare with (A.2.2)).

(coev_{h^{X}Z_{0},h^{Z_{0}}Z_{1}} C(h^{X}Z_{0}, b_{2})σ ⊗ coev_{h^{X}Z_{1},∗} C(h^{X}Z_{1}, r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ)b_{2}^{C}

= (−)^{r}(coev_{h^{X}Z_{0},h^{Z_{0}}Z_{1}} C(h^{X}Z_{0}, b_{2}) ⊗ coev_{h^{X}Z_{1},∗} C(h^{X}Z_{1}, r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1)))m_{2}^{C}σ
= (−)^{r} coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, (b_{2} ⊗ 1^{⊗k−1})r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ :

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}), (A.10.2b)

where r_{k} means the coordinate r_{k}^{X,Z_{1},...,Z_{k}} : h^{X}Z_{1}⊗h^{Z_{1}}Z_{2}⊗⋯⊗h^{Z_{k−1}}Z_{k} → sC(h^{X}X, h^{W} Z_{k}). We have used that r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) has degree deg r and formula (A.1.2).

(1^{⊗α} ⊗ b_{2} ⊗ 1^{⊗γ}) coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ

= coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, 1^{⊗α+1} ⊗ b_{2} ⊗ 1^{⊗γ})C(h^{X}Z_{0}, r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ
= coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, (1^{⊗α+1} ⊗ b_{2} ⊗ 1^{⊗γ})r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1))σ :

h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}Z_{0}, h^{W} Z_{k}), (A.10.2c)

where r_{k} means the coordinate r_{k}^{X,Z_{0},...,Z_{α},Z_{α+2},...,Z_{k}} : h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{α}}Z_{α+2} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → sC(h^{X}X, h^{W} Z_{k}). Here we use functoriality of coev (compare with (A.2.3)).

Thus, r_{k}H_{k,k−1}D_{k−1,k} = coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, Σ_{2})σ, where

Σ_{2} = −(r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ 1)b_{2} + (−)^{r}(b_{2} ⊗ 1^{⊗k−1})r_{k}^{X,Z_{1},...,Z_{k}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1)

− (−)^{r−1} ∑_{α+γ=k−2}(1^{⊗α+1} ⊗ b_{2} ⊗ 1^{⊗γ})r_{k}^{X,Z_{0},...,Z_{α},Z_{α+2},...,Z_{k}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) :

h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → h^{W} Z_{k}.

The element rH has degree deg r − 1, so the sign (−)^{r−1} arises. Combining this with the expression for r_{k}D_{k,k+1}H_{k+1,k} we obtain

r_{k}D_{k,k+1}H_{k+1,k} + r_{k}H_{k,k−1}D_{k−1,k} = coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, Σ)σ,

where Σ = Σ_{1} + Σ_{2}. We claim that Σ = (h_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ r_{k}^{Z_{0},...,Z_{k}}σ^{−1})m_{2}. Indeed, first of all,

(b_{2} ⊗ 1^{⊗k−1})r_{k}^{X,Z_{1},...,Z_{k}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1)

+ ∑_{α+γ=k−2}(1^{⊗α+1} ⊗ b_{2} ⊗ 1^{⊗γ})r_{k}^{X,Z_{0},...,Z_{α},Z_{α+2},...,Z_{k}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1)

= ∑_{α+γ=k−1}(1^{⊗α} ⊗ b_{2} ⊗ 1^{⊗γ})r_{k}^{X,Z_{0},...,Z_{α−1},Z_{α+1},...,Z_{k}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1),

so that

Σ = (r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ h_{1}^{W} σ^{−1})m_{2}^{C} + (h_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ r_{k}^{Z_{0},...,Z_{k}}σ^{−1})m_{2}^{C}

− (r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ 1)b_{2} : h^{X}Z_{0} ⊗ h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k} → h^{W} Z_{k}.

Further, for all y ∈ h^{X}Z_{0}, z_{i} ∈ h^{Z_{i−1}}Z_{i} we have the action of the third summand

(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1} ⊗ z_{k})(r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ 1)b_{2}

= (−)^{rz_{k}}((y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1})r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ z_{k})b_{2} = (−)^{rz_{k}+(r+1)+(y+z_{1}+···+z_{k−1})}((_{X}i_{0}^{𝒜})(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1})r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1} ⊗ z_{k})b_{2}.

Notice that r_{k}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) has degree deg r, and (y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1})r_{k}σ^{−1} has degree (r + 1) + (y + z_{1} + ⋯ + z_{k−1}). Similarly, the action of the first summand is

(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1} ⊗ z_{k})(r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ h_{1}^{W} σ^{−1})m_{2}^{C}

= (−)^{rz_{k}+(r+1)+(y+z_{1}+···+z_{k})}((_{X}i_{0}^{𝒜})(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1})r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1} ⊗ z_{k}h_{1}^{W} σ^{−1})m_{2}^{C}, and since z_{k}h_{1}^{W} σ^{−1} = z_{k} coev_{h^{W} Z_{k−1},h^{Z_{k−1}}Z_{k}} C(h^{W} Z_{k−1}, b_{2}) = (t ↦ (t ⊗ z_{k})b_{2}), we obtain

(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1} ⊗ z_{k})(r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ h_{1}^{W} σ^{−1})m_{2}^{C}

= (−)^{rz_{k}+(r+1)+(y+z_{1}+···+z_{k})}((_{X}i_{0}^{𝒜})(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k−1})r_{k}^{X,Z_{0},...,Z_{k−1}}σ^{−1} ⊗ z_{k})b_{2}.

We see that the first and the third terms in Σ cancel each other. Hence, only the second summand remains in Σ = (h_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ r_{k}^{Z_{0},...,Z_{k}}σ^{−1})m_{2}^{C},

(y ⊗ z_{1} ⊗ ⋯ ⊗ z_{k})(h_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ r_{k}^{Z_{0},...,Z_{k}}σ^{−1})m_{2}^{C}

= (yh_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) ⊗ (z_{1} ⊗ ⋯ ⊗ z_{k})r_{k}^{Z_{0},...,Z_{k}}σ^{−1})m_{2}^{C}.

We have

yh_{1}^{X}σ^{−1} = (y) coev_{h^{X}X,h^{X}Z_{0}} C(h^{X}X, b_{2}) = (t ↦ (t ⊗ y)b_{2})

so that yh_{1}^{X}σ^{−1}C(_{X}i_{0}^{𝒜} , 1) = (−)^{y+1}(_{X}i_{0}^{𝒜} ⊗ y)b_{2} = −(y)(_{X}i_{0}^{𝒜} ⊗ 1)b_{2} = y − yh′′b_{1} − yb_{1}h′′. Then (z_{1}⊗⋯⊗z_{k}) coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, Σ) = (y ↦ (y−yh′′b_{1}−yb_{1}h′′)(z_{1}⊗⋯⊗z_{k})(r_{k}^{Z_{0},...,Z_{k}}σ^{−1})) and (z_{1} ⊗ ⋯ ⊗ z_{k}) coev_{h^{X}Z_{0},∗} C(h^{X}Z_{0}, Σ)σ = (y ↦ (y − yh′′b_{1} − yb_{1}h′′)(z_{1} ⊗ ⋯ ⊗ z_{k})r_{k}^{Z_{0},...,Z_{k}}).

We see that for k > 0 a_{kk} = D_{k,k+1}H_{k+1,k} + H_{k,k−1}D_{k−1,k} = 1 + f :

C(h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{k−1}}Z_{k}, sC(h^{X}Z_{0}, h^{W} Z_{k})) → C(h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{k−1}}Z_{k}, sC(h^{X}Z_{0}, h^{W} Z_{k})), where f = −C(h^{Z_{0}}Z_{1}⊗⋯⊗h^{Z_{k−1}}Z_{k}, σ^{−1}C(h′′b_{1}+b_{1}h′′, 1)σ). More precisely, D_{k,k+1}H_{k+1,k}+ H_{k,k−1}D_{k−1,k} is a diagonal map, whose components are 1 + f. We claim that f = m_{1}^{C}H′_{kk} + H′_{kk}m_{1}^{C}, where H′_{kk} = C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, σ^{−1}C(h′′, 1)σ). Indeed, m_{1}^{C} = C(1, b_{1}^{C}) − C(∑_{α+γ=k−1}(1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗γ}), 1), so that

m_{1}^{C}H′_{kk} = (C(1, b_{1}^{C}) − C( ∑_{α+γ=k−1} 1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗γ}, 1))C(1, σ^{−1}C(h′′, 1)σ)

= C(1, b_{1}^{C}σ^{−1}C(h′′, 1)σ) + C(1, σ^{−1}C(h′′, 1)σ)C( ∑_{α+γ=k−1} 1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗γ}, 1),

H′_{kk}m_{1}^{C} = C(1, σ^{−1}C(h′′, 1)σ)(C(1, b_{1}^{C}) − C( ∑_{α+γ=k−1} 1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗γ}, 1))

= C(1, σ^{−1}C(h′′, 1)σb_{1}^{C}) − C(1, σ^{−1}C(h′′, 1)σ)C( ∑_{α+γ=k−1} 1^{⊗α} ⊗ b_{1} ⊗ 1^{⊗γ}, 1).

Therefore,

H′_{kk}m_{1}^{C} + m_{1}^{C}H′_{kk} = C(1, b_{1}^{C}σ^{−1}C(h′′, 1)σ + σ^{−1}C(h′′, 1)σb_{1}^{C})
= −C(1, σ^{−1}(m_{1}^{C}C(h′′, 1) + C(h′′, 1)m_{1}^{C})σ).

Since m_{1}^{C} = C(1, b_{1}) − C(b_{1}, 1), we have

m_{1}^{C}C(h′′, 1) + C(h′′, 1)m_{1}^{C}

= C(1, b_{1})C(h′′, 1) − C(b_{1}, 1)C(h′′, 1) + C(h′′, 1)C(1, b_{1}) − C(h′′, 1)C(b_{1}, 1)

= C(h′′b_{1} + b_{1}h′′, 1),

so that H′_{kk}m_{1}^{C} + m_{1}^{C}H′_{kk} = −C(1, σ^{−1}C(h′′b_{1} + b_{1}h′′, 1)σ) = f.

Summing up, we have proved that

a = αY_{1} + B_{1}H + HB_{1} = 1 + B_{1}H′ + H′B_{1} + N,

where H′ : sA_{∞}(𝒜, C)(h^{X}, h^{W}) → sA_{∞}(𝒜, C)(h^{X}, h^{W}) is a continuous 𝕜-linear map of degree −1 determined by a diagonal matrix with the matrix elements

H′_{kk} = C(h^{Z_{0}}Z_{1} ⊗ ⋯ ⊗ h^{Z_{k−1}}Z_{k}, σ^{−1}C(h′′, 1)σ) : V_{k} → V_{k},

H′_{kl} = 0 for k ≠ l, and the matrix of the remainder N is strictly upper-triangular: N_{kl} = 0 for all k ⩾ l.

The continuous map of degree 0

αY_{1} + B_{1}(H − H′) + (H − H′)B_{1} = 1 + N : V → V,

obtained in Lemma A.10, is invertible (its inverse is determined by the upper-triangular matrix ∑_{i=0}^{∞}(−N)^{i}, which is well-defined). Therefore, αY_{1} is homotopy invertible. We have proved earlier that Y_{1}α is homotopic to identity. Viewing α, Y_{1} as morphisms of homotopy category H^{0}(C), we see that both of them are homotopy invertible. Hence, they are homotopy inverse to each other.

Let us define a full subcategory RepA_{∞}^{u}(𝒜, C) of the 𝒰′-small differential graded 𝒰 -category A_{∞}^{u}(𝒜, C) as follows. Its objects are all A_{∞}-functors h^{X} : 𝒜 → C for X ∈ Ob 𝒜. They are unital by Lemma A.3. The differential graded category RepA_{∞}^{u}(𝒜, C) is 𝒰 -small. Thus, the Yoneda A_{∞}-functor Y : 𝒜^{op} → A_{∞}(𝒜, C) takes values in the 𝒰 -small subcategory RepA_{∞}^{u}(𝒜, C).

A.11 Theorem. Let 𝒜 be a unital A_{∞}-category. Then the restricted Yoneda A_{∞}-functor Y : 𝒜^{op} → RepA_{∞}^{u}(𝒜, C) is an equivalence.

Proof. Let us verify the assumptions of Corollary 1.9. Proposition A.9 shows that the maps Y_{1} are homotopy invertible. The map Ob Y : Ob 𝒜 → Ob RepA_{∞}^{u}(𝒜, C), X ↦ h^{X} is surjective (in our conventions it may happen that X ≠ Y, but h^{X} = h^{Y}). Thus, assumptions of Corollary 1.9 are satisfied. We deduce by it that Y is an equivalence.

A.12 Corollary. Each 𝒰 -small unital A_{∞}-category 𝒜 is A_{∞}-equivalent to a 𝒰 -small differential graded category RepA_{∞}^{u}(𝒜, C)^{op}.

A.13 Remark. We may use the surjective map Ob Y : Ob 𝒜 → Ob RepA_{∞}^{u}(𝒜, C) to transfer the differential graded category structure of RepA_{∞}^{u}(𝒜, C) to Ob 𝒜. This new 𝒰 -small differential graded category is denoted R̃epA_{∞}^{u}(𝒜, C). Thus, its set of objects is Ob 𝒜, the sets of morphisms are

R̃epA_{∞}^{u}(𝒜, C)(X, Y) = A_{∞}(𝒜, C)(h^{X}, h^{Y}),

and the operations are those of A_{∞}(𝒜, C). It is equivalent to RepA_{∞}^{u}(𝒜, C) by surjectivity of Ob Y. The Yoneda A_{∞}-functor can be presented as an A_{∞}-equivalence Ỹ : 𝒜^{op} → R̃epA_{∞}^{u}(𝒜, C), identity on objects, whose components Ỹ_{n} = Y_{n} are given by (A.6.1). A quasi-inverse to Ỹ equivalence R̃epA_{∞}^{u}(𝒜, C) → 𝒜^{op} can be chosen so that it induces the identity map on objects as well by Corollary 1.9.

B. Strict A_{∞}-2-functor.

The goal of this section is to show that the problem of representing the A_{∞}-2-functor 𝒜 ↦ A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} for a pair (𝒞, ℬ) of a unital A_{∞}-category 𝒞 and its full subcategory ℬ reduces to the case of differential graded 𝒞.

B.1. An A_{∞}-functor. For arbitrary A_{∞}-categories 𝒳, 𝒴, 𝒵 the left hand side of the equation

[T sA_{∞}(𝒴, 𝒵) ⊠ T sA_{∞}(𝒳, 𝒴) ^{c}→ T sA_{∞}(𝒳, 𝒴) ⊠ T sA_{∞}(𝒴, 𝒵) ^{M}→ T sA_{∞}(𝒳, 𝒵)]

= [T sA_{∞}(𝒴, 𝒵) ⊠ T sA_{∞}(𝒳, 𝒴) ^{1⊠A_{∞}( ,𝒵)} → T sA_{∞}(𝒴, 𝒵) ⊠ T sA_{∞}(A_{∞}(𝒴, 𝒵), A_{∞}(𝒳, 𝒵))

α→ T sA_{∞}(𝒳, 𝒵)] (B.1.1)

is an A_{∞}-functor. Therefore, by Proposition 5.5 of [Lyu03] there exists a unique A_{∞}-func-tor

A_{∞}( , 𝒵) : A_{∞}(𝒳, 𝒴) → A_{∞}(A_{∞}(𝒴, 𝒵), A_{∞}(𝒳, 𝒵))

in the right hand side, which makes equation (B.1.1) hold true. The proof of Proposition 3.4 of [Lyu03] contains a recipe for finding the components of A_{∞}( , 𝒵). Namely, the equation

(p ⊠ 1)M = [p.A_{∞}( , 𝒵)]θ  (B.1.2)

has to hold for all p ∈ T sA_{∞}(𝒳, 𝒴). In particular,

f.A_{∞}( , 𝒵) = (f ⊠ 1)M = (f ⊠ 1_{𝒵})M : A_{∞}(𝒴, 𝒵) → A_{∞}(𝒳, 𝒵) for f ∈ Ob A_{∞}(𝒳, 𝒴),
r.A_{∞}( , 𝒵)_{1} = (r ⊠ 1)M = (r ⊠ 1_{𝒵})M : (f ⊠ 1)M → (g ⊠ 1)M for r ∈ sA_{∞}(𝒳, 𝒴)(f, g).

Other components of A_{∞}( , 𝒵) are obtained from the recurrent relation, which is equation (B.1.2) written for p = p^{1} ⊗ ⋯ ⊗ p^{n}:

(p^{1} ⊗ ⋯ ⊗ p^{n})A_{∞}( , 𝒵)_{n} = (p^{1} ⊗ ⋯ ⊗ p^{n} ⊠ 1)M

− ∑_{i_{1}+···+i_{l}=n}^{l>1}[(p^{1} ⊗ ⋯ ⊗ p^{n}).(A_{∞}( , 𝒵)_{i_{1}} ⊗ A_{∞}( , 𝒵)_{i_{2}} ⊗ ⋯ ⊗ A_{∞}( , 𝒵)_{i_{l}})]θ. (B.1.3)

In particular, for r ⊗ t ∈ T^{2}sA_{∞}(𝒳, 𝒴) we get

(r ⊗ t)A_{∞}( , 𝒵)_{2} = (r ⊗ t ⊠ 1)M − [(r ⊠ 1)M ⊗ (t ⊠ 1)M]θ.

Given g^{0} ^{p^{1}}→ g^{1} ^{p^{2}}→ … g^{n−1} ^{p^{n}}→ g^{n} we find from (B.1.3) the components of the A_{∞}-transformation

(p^{1} ⊗ ⋯ ⊗ p^{n})A_{∞}( , 𝒵)_{n} ∈ sA_{∞}(A_{∞}(𝒴, 𝒵), A_{∞}(𝒳, 𝒵))((g^{0} ⊠ 1)M, (g^{n} ⊠ 1)M)

in the form

[(p^{1} ⊗ ⋯ ⊗ p^{n})A_{∞}( , 𝒵)_{n}]_{m} = (p^{1} ⊗ ⋯ ⊗ p^{n} ⊠ 1)M_{nm}.

So they vanish for m > 1.

If 𝒵 is unital, then the A_{∞}-functor A_{∞}( , 𝒵) takes values in the subcategory

A_{∞}^{u}(A_{∞}(𝒴, 𝒵), A_{∞}(𝒳, 𝒵)),

because the A_{∞}-functors (f ⊠ 1_{𝒵})M commute with (1 ⊠ i^{𝒵})M, so they are unital.

B.2 Proposition. For arbitrary A_{∞}-categories 𝒳, 𝒴 and unital A_{∞}-categories 𝒞, 𝒟 we have

[T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒳, 𝒴) ^{c}→ T sA_{∞}(𝒳, 𝒴) ⊠ T sA_{∞}^{u}(𝒞, 𝒟) ^{A_{∞}( ,𝒞)⊠A_{∞}(𝒳, )}→ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒞)) ⊠ T sA_{∞}^{u}(A_{∞}(𝒳, 𝒞), A_{∞}(𝒳, 𝒟))

M→ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒟))]

= [T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒳, 𝒴) ^{A_{∞}(𝒴, )⊠A_{∞}( ,𝒟)}→

T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒴, 𝒟)) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒟), A_{∞}(𝒳, 𝒟))

M→ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒟))]. (B.2.1)

The same statement holds true if one removes the unitality superscript u, and do not assume 𝒞, 𝒟 unital. The same equation holds true if all four A_{∞}-categories 𝒳, 𝒴, 𝒞, 𝒟 are unital and all A_{∞}-categories A_{∞}(,) are replaced with their subcategories A_{∞}^{u}(,).

Proof. Due to Proposition 3.4 of [Lyu03] equation (B.2.1) is equivalent to the following one:

[T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(𝒞, 𝒟)⊠T sA_{∞}(𝒳, 𝒴) ^{1⊠c} → T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}(𝒳, 𝒴)⊠T sA_{∞}^{u}(𝒞, 𝒟)

1⊠A_{∞}( ,𝒞)⊠A_{∞}(𝒳, ) → T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒞))⊠T sA_{∞}^{u}(A_{∞}(𝒳, 𝒞), A_{∞}(𝒳, 𝒟))

1⊠M → T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒟)) ^{α}→ T sA_{∞}(𝒳, 𝒟)]

= [T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒳, 𝒴) ^{1⊠A_{∞}(𝒴, )⊠A_{∞}( ,𝒟)}→ T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒴, 𝒟)) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒟), A_{∞}(𝒳, 𝒟))

1⊠M → T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒟)) ^{α}→ T sA_{∞}(𝒳, 𝒟)].

Using the definition of M [Lyu03, diagram (4.0.1)] we transform this equation into

[T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(𝒞, 𝒟)⊠T sA_{∞}(𝒳, 𝒴) ^{1⊠c} → T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}(𝒳, 𝒴)⊠T sA_{∞}^{u}(𝒞, 𝒟)

1⊠A_{∞}( ,𝒞)⊠A_{∞}(𝒳, ) → T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒞))⊠T sA_{∞}^{u}(A_{∞}(𝒳, 𝒞), A_{∞}(𝒳, 𝒟))

α⊠1 → T sA_{∞}(𝒳, 𝒞) ⊠ T sA_{∞}^{u}(A_{∞}(𝒳, 𝒞), A_{∞}(𝒳, 𝒟)) ^{α}→ T sA_{∞}(𝒳, 𝒟)]

= [T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒳, 𝒴) ^{1⊠A_{∞}(𝒴, )⊠A_{∞}( ,𝒟)}→ T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒴, 𝒟)) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒟), A_{∞}(𝒳, 𝒟))

α⊠1 → T sA_{∞}(𝒴, 𝒟) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒟), A_{∞}(𝒳, 𝒟)) ^{α}→ T sA_{∞}(𝒳, 𝒟)].

Using the definitions of A_{∞}( , 𝒞) and A_{∞}(𝒴, ) [Lyu03, (6.1.2)] we rewrite this equation as follows:

[T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(𝒞, 𝒟)⊠T sA_{∞}(𝒳, 𝒴) ^{c_{(123)}} → T sA_{∞}(𝒳, 𝒴)⊠T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(𝒞, 𝒟)

M⊠A_{∞}(𝒳, ) → T sA_{∞}(𝒳, 𝒞) ⊠ T sA_{∞}^{u}(A_{∞}(𝒳, 𝒞), A_{∞}(𝒳, 𝒟)) ^{α}→ T sA_{∞}(𝒳, 𝒟)]
= [T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒳, 𝒴) ^{M⊠A_{∞}( ,𝒟)} →

T sA_{∞}(𝒴, 𝒟) ⊠ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒟), A_{∞}(𝒳, 𝒟)) ^{α}→ T sA_{∞}(𝒳, 𝒟)].

Now we use definitions of A_{∞}(𝒳, ) and A_{∞}( , 𝒟) to get an equivalent form of the required equation:

[T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(𝒞, 𝒟)⊠T sA_{∞}(𝒳, 𝒴) ^{c_{(123)}} → T sA_{∞}(𝒳, 𝒴)⊠T sA_{∞}(𝒴, 𝒞)⊠T sA_{∞}^{u}(𝒞, 𝒟)

M⊠1 → T sA_{∞}(𝒳, 𝒞) ⊠ T sA_{∞}^{u}(𝒞, 𝒟) ^{M}→ T sA_{∞}(𝒳, 𝒟)]

= [T sA_{∞}(𝒴, 𝒞) ⊠ T sA_{∞}^{u}(𝒞, 𝒟) ⊠ T sA_{∞}(𝒳, 𝒴) ^{M⊠1} → T sA_{∞}(𝒴, 𝒟) ⊠ T sA_{∞}(𝒳, 𝒴)

c→ T sA_{∞}(𝒳, 𝒴) ⊠ T sA_{∞}(𝒴, 𝒟) ^{M}→ T sA_{∞}(𝒳, 𝒟)].

This equation holds true due to associativity of M, since M has degree 0.

Other statements are similar or follow from the already proven one.

B.3. An A_{∞}-2-functor. Let A_{∞}-category 𝒜 be unital. The A_{∞}-category ℰ is pseudounital with distinguished elements equal to the unit elements of 𝒞.

Strict A_{∞}-2-functors are defined in [LM03, Definition 3.1]. There is a strict A_{∞}-2-functor F, given by the following data:

1. the mapping of objects F : Ob A_{∞}^{u} → Ob A_{∞}^{u}, 𝒜 ↦ F 𝒜 = A_{∞}(𝒞, 𝒜)_{mod ℬ} (F 𝒜 is a full subcategory of the unital A_{∞}-category A_{∞}(𝒞, 𝒜), hence it is unital as well);

2. the strict unital A_{∞}-functor F = F_{𝒜_{1},𝒜_{2}} : A_{∞}^{u}(𝒜_{1}, 𝒜_{2}) → A_{∞}^{u}(F 𝒜_{1}, F 𝒜_{2}) for each pair of unital A_{∞}-categories 𝒜_{1}, 𝒜_{2} given as follows:

Ob F : g ↦ gF = (1 ⊠ g)M|_{F 𝒜_{1}} ,

where (1 ⊠ g)M : A_{∞}(𝒞, 𝒜_{1}) → A_{∞}(𝒞, 𝒜_{2}). Indeed, if ℬ^{⊂} → 𝒞 ^{f}→ 𝒜_{1} is a contractible A_{∞}-functor, then so is ℬ^{⊂} → 𝒞 ^{fg}→ 𝒜_{2}. Actually, if _{X}i_{0}f_{1} = w_{X}b_{1} for some w_{X} ∈ (s𝒜_{1})^{−2}(Xf, Xf), X ∈ Ob ℬ, then _{X}i_{0}f_{1}g_{1} = w_{X}b_{1}g_{1} = (w_{X}g_{1})b_{1}, where w_{X}g_{1} ∈ (s𝒜_{2})^{−2}(Xfg, Xfg). Furthermore,

F_{1} : A_{∞}^{u}(𝒜_{1}, 𝒜_{2})(g, h) → A_{∞}^{u}(F 𝒜_{1}, F 𝒜_{2})(gF, hF),
(r : g → h : 𝒜_{1} → 𝒜_{2}) ↦ rF_{1} = (1 ⊠ r)M|_{F 𝒜_{1}} ,

or more precisely, [sF 𝒜_{1}(f_{0}, f_{1}) ⊗ ⋯ ⊗ sF 𝒜_{1}(f_{n−1}, f_{n}) ^{[rF_{1}]_{n}} → sF 𝒜_{2}(f_{0}g, f_{n}h)]

= [sA_{∞}(𝒞, 𝒜_{1})(f_{0}, f_{1}) ⊗ ⋯ ⊗ sA_{∞}(𝒞, 𝒜_{1})(f_{n−1}, f_{n}) ^{(1⊠r)M_{n1}} → sA_{∞}(𝒞, 𝒜_{2})(f_{0}g, f_{n}h)] .

The necessary equations for F_{1} are consequences of those for A_{∞}(𝒞, ) [Lyu03, Proposition 6.2]. Clearly, F is a unital A_{∞}-functor. Let us check that the following diagram commutes:

T sA_{∞}^{u}(𝒜_{0}, 𝒜_{1}) ⊠ T sA_{∞}^{u}(𝒜_{1}, 𝒜_{2})  \overline{^{M} →} T sA_{∞}^{u}(𝒜_{0}, 𝒜_{2})

=

T sA_{∞}^{u}(F 𝒜_{0}, F 𝒜_{1}) ⊠ T sA_{∞}^{u}(F 𝒜_{1}, F 𝒜_{2}) ↓^{F_{𝒜_{0},𝒜_{1}} ⊗F_{𝒜_{1},𝒜_{2}}}  →^{M} T sA_{∞}^{u}(F 𝒜_{0}, F 𝒜_{2}). ↓^{F_{𝒜_{0},𝒜_{2}}}

It follows from a similar diagram for A_{∞}(𝒞, ) in place of F (equation (3.3.1) of [LM03]). The commutativity is clear on objects; since both sides of the required identity are cocategory homomorphisms it suffices to show that

(F_{𝒜_{0},𝒜_{1}} ⊠ F_{𝒜_{1},𝒜_{2}} )M pr_{1} = MF_{𝒜_{0},𝒜_{1}} pr_{1} :

T sA_{∞}^{u}(𝒜_{0}, 𝒜_{1}) ⊠ T sA_{∞}^{u}(𝒜_{1}, 𝒜_{2}) → sA_{∞}^{u}(F 𝒜_{0}, F 𝒜_{2}).

Since F_{−,−} are strict A_{∞}-functors, we must show that for any non-negative integers n, m

((F_{𝒜_{0},𝒜_{1}} )_{1}^{⊗n} ⊠ (F_{𝒜_{1},𝒜_{2}} )_{1}^{⊗m})M_{nm} = M_{nm}(F_{𝒜_{0},𝒜_{2}} )_{1} : T^{n}sA_{∞}^{u}(𝒜_{0}, 𝒜_{1}) ⊠ T^{m}sA_{∞}^{u}(𝒜_{1}, 𝒜_{2}) → sA_{∞}^{u}(F 𝒜_{0}, F 𝒜_{2}).

Since M_{nm} vanishes whenever m > 1, we restrict our attention to m = 0 and m = 1. These cases are similar and we will give verification in the case m = 1. Given diagrams of A_{∞}-functors and A_{∞}-transformations

f^{0} ^{r^{1}}→ f^{1} → … ^{r^{n}}→ f^{n} : 𝒜_{0} → 𝒜_{1}, g^{0} ^{t^{1}}→ g^{1} : 𝒜_{1} → 𝒜_{2},

we must show that (1 ⊠ (r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ t^{1})M)M|_{F 𝒜_{0}} = ((1 ⊠ r^{1})M|_{F 𝒜_{0}} ⊗ ⋯ ⊗ (1 ⊠ r^{n})M|_{F 𝒜_{0}} ⊠(1⊠t^{1})M|_{F 𝒜_{1}} )M. This is a particular case of equation (3.3.1) of [LM03] since it coincides with a similar equation for A_{∞}(𝒞, ) in place of F.

Let 𝒟 be the A_{∞}-category defined in Section 5.2. We claim that restr : A_{∞}(𝒟, 𝒜) → F 𝒜 is an A_{∞}-2-transformation as defined in [LM03, Definition 3.2]. The strict A_{∞}-functor restr is induced by the inclusion ι : 𝒞^{⊂} → 𝒟:

f ↦ (𝒞^{⊂} → 𝒟 ^{f}→ 𝒜) = f|_{𝒞},
restr_{1} : A_{∞}(𝒟, 𝒜)(f, g) → F 𝒜(f|_{𝒞}, g|_{𝒞}) = A_{∞}(𝒞, 𝒜)(f|_{𝒞}, g|_{𝒞}).

The restriction f|_{ℬ} is a contractible A_{∞}-functor, for_{X}i_{0}f_{1} = ε_{X}b_{1}f_{1} = (ε_{X}f_{1})b_{1} for X ∈ ℬ. Let us check that the following diagram of A_{∞}-functors commutes:

A_{∞}^{u}(𝒜_{1}, 𝒜_{2}) \overline{^{A_{∞}(𝒟,−)} →} A_{∞}^{u}(A_{∞}(𝒟, 𝒜_{1}), A_{∞}(𝒟, 𝒜_{2}))

=

A_{∞}^{u}(F 𝒜_{1}, F 𝒜_{2}) ↓^{F}  \overline{^{(restr_{𝒜_{1}} ⊠1)M} →} A_{∞}^{u}(A_{∞}(𝒟, 𝒜_{1}), F 𝒜_{2}) ↓^{(1⊠restr_{𝒜_{2}} )M}

All functors in the diagram above are strict (the proof is given in [LM03, Section 3.4]). We must verify the equation

A_{∞}(𝒟, −)(1 ⊠ restr_{𝒜_{2}} )M = F (restr_{𝒜_{1}} ⊠1)M.

On objects: given a unital A_{∞}-functor g : 𝒜_{1} → 𝒜_{2}, we are going to check that

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

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

f^{0} ^{r^{1}}→ f^{1} → … ^{r^{n}}→ f^{n} : 𝒟 → 𝒜_{1}

we have

{(r^{1} ⊗ ⋯ ⊗ r^{n})[(1 ⊠ g)M]_{n}}_{k}|_{𝒞} = [(r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ g)M_{n0}]_{k}|_{𝒞}

= {∑_{l}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}g_{l}}|_{𝒞}

= ∑_{l}(r^{1}|_{𝒞} ⊗ ⋯ ⊗ r^{n}|_{𝒞})θ_{kl}g_{l}|_{𝒞}

= {(r^{1}|_{𝒞} ⊗ ⋯ ⊗ r^{n}|_{𝒞})[(1 ⊠ g|_{𝒞})M]_{n}}_{k}.

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}}_{k}|_{𝒞} = [(r^{1} ⊗ ⋯ ⊗ r^{n} ⊠ t)M_{n1}]_{k}|_{𝒞}

= {∑_{l}(r^{1} ⊗ ⋯ ⊗ r^{n})θ_{kl}t_{l}}|_{𝒞}

= ∑_{l}(r^{1}|_{𝒞} ⊗ ⋯ ⊗ r^{n}|_{𝒞})θ_{kl}t_{l}|_{𝒞}

= {(r^{1}|_{𝒞} ⊗ ⋯ ⊗ r^{n}|_{𝒞})[(1 ⊠ t|_{𝒞})M]_{n}}_{k}.

We are more interested in the following A_{∞}-2-subfunctor of F, denoted G : Ob A_{∞}^{u} → Ob A_{∞}^{u}, 𝒜 ↦ G𝒜 = A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} ⊂ F 𝒜. All the structure data of G are restrictions of those of F. Hence, the conclusion that restr : A_{∞}^{ψu}(𝒟, 𝒜) → G𝒜 is an A_{∞}-2-natural transformation remains valid. We prove in Theorem 5.13 that this A_{∞}-functor is an equivalence. Therefore, this restriction A_{∞}-functor is a 2-natural A_{∞}-equivalence. If 𝒞 is strictly unital, then 𝒟 = Q(𝒞|ℬ) is unital by Theorem 6.5. Whenever 𝒟 is unital, we say that 𝒟 unitally represents G.

We are going to discuss how an A_{∞}-2-functor represented by an A_{∞}-category 𝒳 depends on 𝒳.

B.4 Proposition. Let f : 𝒳 → 𝒴 be an A_{∞}-functor. Then

λ_{𝒜} = (f ⊠ 1)M : A_{∞}(𝒴, 𝒜) → A_{∞}(𝒳, 𝒜), g ↦ fg

is a strict A_{∞}-2-transformation between two A_{∞}-2-functors of 𝒜 ∈ Ob A_{∞}^{u}. If f : 𝒳 → 𝒴 is a unital A_{∞}-functor, then

(f ⊠ 1)M : A_{∞}^{u}(𝒴, 𝒜) → A_{∞}^{u}(𝒳, 𝒜)

is also a strict A_{∞}-2-transformation.

Proof. The A_{∞}-functor (f ⊠ 1)M strictly commutes with the unit transformations (1 ⊠ i^{𝒜} )M in A_{∞}(𝒴, 𝒜) and A_{∞}(𝒳, 𝒜):

(f ⊠ 1_{𝒜} )M · (1_{𝒳} ⊠ i^{𝒜} )M = (1_{𝒴} ⊠ i^{𝒜} )M · (f ⊠ 1_{𝒜} )M

due to associativity of M. Therefore, (f ⊠ 1)M is unital.

We have to prove that the diagram of A_{∞}-functors

\begin{matrix}A_{∞}^{u}(𝒞, 𝒟) & \overline{→^{A_{∞}(𝒴, )}} & A_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒴, 𝒟)) \\ A_{∞}^{u}(A_{∞}(𝒳, 𝒞), A_{∞}(𝒳, 𝒟)) & ↓^{A_{∞}(𝒳, )} & \overline{→^{(λ_{𝒞} ⊠1)M}} & A_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒟)) & ↓^{(1⊠λ_{𝒟} )M} \end{matrix}  (B.4.1)

commutes. All four A_{∞}-functors in this diagram are strict. So it suffices to check commutativity on objects and for the first components.

If h : 𝒞 → 𝒟 is a unital A_{∞}-functor, then

h.A_{∞}(𝒴, )(1 ⊠ λ_{𝒟} )M = (1 ⊠ h)M · (f ⊠ 1)M = (f ⊠ 1 ⊠ h)(1 ⊠ M)M

equals to

h.A_{∞}(𝒳, )(λ_{𝒞} ⊠ 1)M = (f ⊠ 1)M · (1 ⊠ h)M = (f ⊠ 1 ⊠ h)(M ⊠ 1)M

due to associativity of M.

If t ∈ sA_{∞}^{u}(𝒞, 𝒟)(g, h), then it is mapped by the first components of A_{∞}-functors in diagram (B.4.1) to

t.A_{∞}(𝒴, )_{1}(1 ⊠ λ_{𝒟} )M_{10} = (1 ⊠ t)M · (f ⊠ 1)M = (f ⊠ 1 ⊠ t)(1 ⊠ M)M

and

t.A_{∞}(𝒳, )_{1}(λ_{𝒞} ⊠ 1)M_{01} = (f ⊠ 1)M · (1 ⊠ t)M = (f ⊠ 1 ⊠ t)(M ⊠ 1)M.

These expressions are equal due to associativity of M.

The case of unital f and A_{∞}-2-subfunctors A_{∞}^{u}(𝒴, 𝒜), A_{∞}^{u}(𝒳, 𝒜) follows from the general case.

With two strict A_{∞}-2-transformations λ, μ in F ^{λ}→ G ^{μ}→ H : A_{∞}^{u} → A_{∞}^{u} is associated the third A_{∞}-2-transformation λμ : F → H : A_{∞}^{u} → A_{∞}^{u} – their composition, specified by the family of unital A_{∞}-functors

(λμ)_{𝒞} = λ_{𝒞}μ_{𝒞} : F 𝒞 → H𝒞, 𝒞 ∈ Ob A_{∞}^{u}.

In order to verify that λμ is indeed a strict A_{∞}-2-transformation, we have to check equation (3.2.1) of [LM03]:

F · (1 ⊠ (λμ)_{𝒟} )M = H · ((λμ)_{𝒞} ⊠ 1)M : A_{∞}^{u}(𝒞, 𝒟) → A_{∞}^{u}(F 𝒞, H𝒟).

We do it as follows:

[A_{∞}^{u}(𝒞, 𝒟) ^{F}→ A_{∞}^{u}(F 𝒞, F 𝒟) \begin{matrix}(1⊠λ_{𝒟} μ_{𝒟} )M \\ (1⊠λ_{𝒟} )M·(1⊠μ_{𝒟} )M \end{matrix} → A_{∞}^{u}(F 𝒞, H𝒟)]
= [A_{∞}^{u}(𝒞, 𝒟) ^{G}→ A_{∞}^{u}(G𝒞, G𝒟) \begin{matrix}(λ_{𝒞} ⊠1)M·(1⊠μ_{𝒟} )M \\ (1⊠μ_{𝒟} )M·(λ_{𝒞} ⊠1)M \end{matrix} → A_{∞}^{u}(F 𝒞, H𝒟)]
= [A_{∞}^{u}(𝒞, 𝒟) ^{H}→ A_{∞}^{u}(H𝒞, H𝒟) \begin{matrix}(μ_{𝒞} ⊠1)M·(λ_{𝒞} ⊠1)M \\ (λ_{𝒞} μ_{𝒞} ⊠1)M \end{matrix} → A_{∞}^{u}(F 𝒞, H𝒟)].

B.5 Definition. A strict modification m : λ → μ : F → G : A_{∞}^{u} → A_{∞}^{u} of strict A_{∞}-2-transformations λ, μ is

1. a family of A_{∞}-transformations m_{𝒞} : λ_{𝒞} → μ_{𝒞} : F 𝒞 → G𝒞 for 𝒞 ∈ Ob A_{∞}^{u} such that

2. for any pair of unital A_{∞}-categories 𝒞, 𝒟 the A_{∞}-transformations

(F_{𝒞,𝒟} ⊠λ_{𝒟} )·M = (λ_{𝒞} ⊠G_{𝒞,𝒟} )·M ^{(m_{𝒞} ⊠G_{𝒞,𝒟} )·M} → (μ_{𝒞} ⊠G_{𝒞,𝒟} )·M : A_{∞}^{u}(𝒞, 𝒟) → A_{∞}^{u}(F 𝒞, G𝒟),

(F_{𝒞,𝒟} ⊠λ_{𝒟} )·M ^{(F_{𝒞,𝒟} ⊠m_{𝒟} )·M} → (F_{𝒞,𝒟} ⊠μ_{𝒟} )·M = (μ_{𝒞} ⊠G_{𝒞,𝒟} )·M : A_{∞}^{u}(𝒞, 𝒟) → A_{∞}^{u}(F 𝒞, G𝒟)

are equal, in short,

G_{𝒞,𝒟} (m_{𝒞} ⊠ 1)M = F_{𝒞,𝒟} (1 ⊠ m_{𝒟} )M : F_{𝒞,𝒟} (1 ⊠ λ_{𝒟} )M → G_{𝒞,𝒟} (μ_{𝒞} ⊠ 1)M.

A modification m is natural if all A_{∞}-transformations m_{𝒞} are natural.

B.6. Example of strict modification. We claim that for an arbitrary A_{∞}-transfor-mation r : f → g : 𝒳 → 𝒴 the family of A_{∞}-transformations

m_{𝒞} = (r ⊠ 1)M : λ_{𝒞} = (f ⊠ 1)M → μ_{𝒞} = (g ⊠ 1)M : F 𝒞 = A_{∞}(𝒴, 𝒞) → G𝒞 = A_{∞}(𝒳, 𝒞)

is a strict modification. In order to prove it, we notice first of all that (r ⊠ 1)M is a ((f ⊠ 1)M, (g ⊠ 1)M)-coderivation for an arbitrary 𝒞, since M is a cocategory homomorphism.

When we want to indicate the category 𝒞, we write this coderivation as (r ⊠ 1_{𝒞})M ∈ sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒞))((f ⊠ 1_{𝒞})M, (g ⊠ 1_{𝒞})M). The equation to verify is

[T sA_{∞}^{u}(𝒞, 𝒟) ^{(r⊠1_{𝒞} )M⊠A_{∞}(𝒳, )} → T^{1}sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒞))⊠T sA_{∞}^{u}(A_{∞}(𝒳, 𝒞), A_{∞}(𝒳, 𝒟))

M→ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒟))]

= [T sA_{∞}^{u}(𝒞, 𝒟) ^{A_{∞}(𝒴, )⊠(r⊠1_{𝒟} )M} → T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒴, 𝒟))⊠T^{1}sA_{∞}^{u}(A_{∞}(𝒴, 𝒟), A_{∞}(𝒳, 𝒟))

M→ T sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒟))].

It is an immediate consequence of equation (B.2.1) restricted to the element r ∈ T^{1}sA_{∞}(𝒳, 𝒴).

B.7 Lemma. If m : λ → μ : F → G : A_{∞}^{u} → A_{∞}^{u} is a strict modification, then so is mB_{1} : λ → μ : F → G : A_{∞}^{u} → A_{∞}^{u}, where (mB_{1})_{𝒞} = m_{𝒞}B_{1} : λ_{𝒞} → μ_{𝒞} : F 𝒞 → G𝒞 for all 𝒞 ∈ Ob A_{∞}^{u}. The 𝕜-linear map

sA_{∞}(𝒳, 𝒴)(f, g) ∋ r ↦ (r ⊠ 1)M ∈ sA_{∞}^{u}(A_{∞}(𝒴, 𝒞), A_{∞}(𝒳, 𝒞))((f ⊠ 1)M, (g ⊠ 1)M)

is a chain map.

Proof. Let us prove that the family m_{𝒞}B_{1} constitutes a strict modification. The identity (1 ⊠ B + B ⊠ 1)M = MB implies that

(F_{𝒞,𝒟} ⊠ m_{𝒟} .B_{1})M = (F_{𝒞,𝒟} ⊠ m_{𝒟} )MB − (−)^{m}B(F_{𝒞,𝒟} ⊠ m_{𝒟} )M = [(F_{𝒞,𝒟} ⊠ m_{𝒟} )M].B_{1} = [(m_{𝒞} ⊠G_{𝒞,𝒟} )M].B_{1} = (m_{𝒞} ⊠G_{𝒞,𝒟} )MB −(−)^{m}B(m_{𝒞} ⊠G_{𝒞,𝒟} )M = (m_{𝒞}.B_{1} ⊠G_{𝒞,𝒟} )M.

Here we use the fact that m_{𝒟} .B = m_{𝒟} .B_{1} due to m_{𝒟} ∈ T^{1}sA_{∞}(F 𝒟, G𝒟)(λ_{𝒟} , μ_{𝒟} ), and, similarly, m_{𝒞}.B = m_{𝒞}.B_{1}. The equation

[(r ⊠ 1)M].B_{1} = (r ⊠ 1)MB − (−)^{r}B(r ⊠ 1)M = (r.B ⊠ 1)M = (r.B_{1} ⊠ 1)M (B.7.1)

proves that r ↦ (r ⊠ 1)M is a chain map.

B.8 Lemma. Let m, n in λ ^{m}→ μ ^{n}→ ν : F → G : A_{∞}^{u} → A_{∞}^{u} be strict modifications. Then

F_{𝒞,𝒟} [1 ⊠ (m_{𝒟} ⊗ n_{𝒟} )B_{2}]M = G_{𝒞,𝒟} [(m_{𝒞} ⊗ n_{𝒞})B_{2} ⊠ 1]M + G_{𝒞,𝒟} [(m_{𝒞} ⊗ n_{𝒞})(1 ⊗ B_{1} + B_{1} ⊗ 1)A_{∞}( , G𝒟)_{2}] − G_{𝒞,𝒟} [(m_{𝒞} ⊗ n_{𝒞})A_{∞}( , G𝒟)_{2}B_{1}]

Proof. Since A_{∞}( , G𝒟) is an A_{∞}-functor we have

[(m_{𝒞} ⊠ 1)M ⊗ (n_{𝒞} ⊠ 1)M]B_{2} + (m_{𝒞} ⊗ n_{𝒞}).A_{∞}( , G𝒟)_{2}B_{1}
= [m_{𝒞}.A_{∞}( , G𝒟)_{1} ⊗ n_{𝒞}.A_{∞}( , G𝒟)_{1}]B_{2} + (m_{𝒞} ⊗ n_{𝒞}).A_{∞}( , G𝒟)_{2}B_{1}
= (m_{𝒞} ⊗ n_{𝒞})B_{2}.A_{∞}( , G𝒟)_{1} + (m_{𝒞} ⊗ n_{𝒞})(1 ⊗ B_{1} + B_{1} ⊗ 1)A_{∞}( , G𝒟)_{2}
= [(m_{𝒞} ⊗ n_{𝒞})B_{2} ⊠ 1]M + (m_{𝒞} ⊗ n_{𝒞})(1 ⊗ B_{1} + B_{1} ⊗ 1)A_{∞}( , G𝒟)_{2}.  (B.8.1)

Using this identity we find

F_{𝒞,𝒟} [1 ⊠ (m_{𝒟} ⊗ n_{𝒟} )B_{2}]M = F_{𝒞,𝒟} [(1 ⊠ m_{𝒟} )M ⊗ (1 ⊠ n_{𝒟} )M]B_{2} = [F_{𝒞,𝒟} (1 ⊠ m_{𝒟} )M ⊗ F_{𝒞,𝒟} (1 ⊠ n_{𝒟} )M]B_{2} = [G_{𝒞,𝒟} (m_{𝒞} ⊠ 1)M ⊗ G_{𝒞,𝒟} (n_{𝒞} ⊠ 1)M]B_{2} = G_{𝒞,𝒟} [(m_{𝒞} ⊠ 1)M ⊗ (n_{𝒞} ⊠ 1)M]B_{2} = G_{𝒞,𝒟} [(m_{𝒞} ⊗ n_{𝒞})B_{2} ⊠ 1]M + G_{𝒞,𝒟} [(m_{𝒞} ⊗ n_{𝒞})(1 ⊗ B_{1} + B_{1} ⊗ 1)A_{∞}( , G𝒟)_{2}] − G_{𝒞,𝒟} [(m_{𝒞} ⊗ n_{𝒞})A_{∞}( , G𝒟)_{2}B_{1}],

so the lemma is proven.

B.9 Proposition. If unital A_{∞}-categories 𝒳, 𝒴 are equivalent, then A_{∞}-2-functors 𝒜 ↦ A_{∞}^{u}(𝒳, 𝒜) and 𝒜 ↦ A_{∞}^{u}(𝒴, 𝒜) are naturally A_{∞}-2-equivalent.

Proof. Let ϕ : 𝒳 → 𝒴, ψ : 𝒴 → 𝒳 be A_{∞}-equivalences, quasi-inverse to each other. Then there are natural A_{∞}-transformations

r : ϕψ → id_{𝒳} : 𝒳 → 𝒳, p : id_{𝒳} → ϕψ : 𝒳 → 𝒳,

inverse to each other, that is,

(r ⊗ p)B_{2} ≡ ϕψi^{𝒳} , (p ⊗ r)B_{2} ≡ i^{𝒳} .

The A_{∞}-transformations

(r ⊠ 1)M : (ψ ⊠ 1)M(ϕ ⊠ 1)M → id : A_{∞}^{u}(𝒳, 𝒜) → A_{∞}^{u}(𝒳, 𝒜),  (B.9.1)
(p ⊠ 1)M : id → (ψ ⊠ 1)M(ϕ ⊠ 1)M : A_{∞}^{u}(𝒳, 𝒜) → A_{∞}^{u}(𝒳, 𝒜)  (B.9.2)

are also natural by (B.7.1). We are going to prove that these A_{∞}-transformations are inverse to each other.

The natural A_{∞}-transformation

(1 ⊠ i^{𝒜} )M : id → id : A_{∞}^{u}(𝒳, 𝒜) → A_{∞}^{u}(𝒳, 𝒜)

is a unit transformation of A_{∞}-category A_{∞}^{u}(𝒳, 𝒜) by Proposition 7.7 of [Lyu03]. Another unit transformation is given by

(i^{𝒳} ⊠ 1)M : id → id : A_{∞}^{u}(𝒳, 𝒜) → A_{∞}^{u}(𝒳, 𝒜).  (B.9.3)

Indeed, values of 0-th components of the both natural transformations on an object f of A_{∞}^{u}(𝒳, 𝒜) differ by a boundary since

f[(i^{𝒳} ⊠ 1)M]_{0} = (i^{𝒳} ⊠ f)M_{10} = i^{𝒳} f ≡ fi^{𝒜} = (f ⊠ i^{𝒜} )M_{01} = _{f}[(1 ⊠ i^{𝒜} )M]_{0}.

Therefore, [1 ⊗ (i^{𝒳} ⊠ 1)M_{10}]B_{2} and [(i^{𝒳} ⊠ 1)M_{10} ⊗ 1]B_{2} are homotopy invertible. Furthermore, by (B.8.1)

[(i^{𝒳} ⊠ 1)M ⊗ (i^{𝒳} ⊠ 1)M]B_{2} ≡ [(i^{𝒳} ⊗ i^{𝒳})B_{2} ⊠ 1]M ≡ (i^{𝒳} ⊠ 1)M

due to Lemma B.7. Therefore, homotopy idempotent (B.9.3) is a unit transformation of A_{∞}^{u}(𝒳, 𝒜) by [Lyu03, Definition 7.6]. Since unit transformation is unique up to equivalence by [Lyu03, Corollary 7.10] we have

(i^{𝒳} ⊠ 1)M ≡ (1 ⊠ i^{𝒜} )M : id → id : A_{∞}^{u}(𝒳, 𝒜) → A_{∞}^{u}(𝒳, 𝒜).

Composing natural A_{∞}-transformations (r ⊠ 1)M and (p ⊠ 1)M given by (B.9.1) and (B.9.2) we get

[(r ⊠ 1)M ⊗ (p ⊠ 1)M]B_{2} ≡ [(r ⊗ p)B_{2} ⊠ 1]M ≡ (ϕψi^{𝒳} ⊠ 1)M

≡ (i^{𝒳} ϕψ ⊠ 1)M = (ψ ⊠ 1)M(ϕ ⊠ 1)M(i^{𝒳} ⊠ 1)M,

[(p ⊠ 1)M ⊗ (r ⊠ 1)M]B_{2} ≡ [(p ⊗ r)B_{2} ⊠ 1]M ≡ (i^{𝒳} ⊠ 1)M

by (B.8.1) and Lemma B.7. Since (i^{𝒳} ⊠ 1)M is a unit transformation, the A_{∞}-transfor-mations (r ⊠ 1)M and (p ⊠ 1)M are inverse to each other.

The obtained statement together with one more statement in which ϕ and ψ exchange their places implies that A_{∞}-functors

A_{∞}^{u}(ϕ, 𝒜) = (ϕ ⊠ 1)M : A_{∞}^{u}(𝒴, 𝒜) → A_{∞}^{u}(𝒳, 𝒜),
A_{∞}^{u}(ψ, 𝒜) = (ψ ⊠ 1)M : A_{∞}^{u}(𝒳, 𝒜) → A_{∞}^{u}(𝒴, 𝒜)

are quasi-inverse to each other.

They form strict A_{∞}-2-transformations by Proposition B.4. Therefore, A_{∞}^{u}(ϕ, 𝒜) and A_{∞}^{u}(ψ, 𝒜) are natural A_{∞}-2-equivalences.

Given a pair (𝒞, ℬ) consisting of a unital A_{∞}-category 𝒞 and its full subcategory ℬ, we shall construct another pair (𝒞̃, ℬ̃) consisting of a differential graded category 𝒞̃ and its full subcategory ℬ̃ as follows. Set 𝒞̃ to be the differential graded category R̃epA_{∞}^{u}(𝒞^{op}, C) of A_{∞}-functors, represented by objects of 𝒞, see Remark A.13. Thus, Ob 𝒞̃ = Ob 𝒞. The category 𝒞̃ is equivalent to 𝒞, the Yoneda A_{∞}-equivalence Ỹ : 𝒞 → 𝒞̃ and its quasi-inverse Ψ : 𝒞̃ → 𝒞 are identity on objects by Remark A.13. We take ℬ̃ to be the full subcategory of 𝒞̃ with the set of objects Ob ℬ̃ = Ob ℬ. Therefore, the A_{∞}-functors Ỹ and Ψ can be restricted to quasi-inverse to each other A_{∞}-equivalences Y′ = Ỹ|_{ℬ} : ℬ → ℬ̃ and Ψ′ = Ψ|ℬ̃ : ℬ̃ → ℬ.

B.10 Corollary. Let 𝒜 be a unital A_{∞}-category. The A_{∞}-functors

(Ψ ⊠ 1)M : A_{∞}^{u}(𝒞, 𝒜) → A_{∞}^{u}(𝒞̃, 𝒜),  f ↦ Ψf,
(Ỹ ⊠ 1)M : A_{∞}^{u}(𝒞̃, 𝒜) → A_{∞}^{u}(𝒞, 𝒜),  g ↦ Ỹg,

are quasi-inverse to each other A_{∞}-equivalences. The first maps objects of A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} to objects of A_{∞}^{u}(𝒞̃, 𝒜)_{mod ℬ̃}, the second does vice versa. Therefore, their restrictions determine quasi-inverse to each other A_{∞}-equivalences

(Ψ ⊠ 1)M : A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} → A_{∞}^{u}(𝒞̃, 𝒜)_{mod ℬ̃},
(Ỹ ⊠ 1)M : A_{∞}^{u}(𝒞̃, 𝒜)_{mod ℬ̃} → A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} .

Proof. Let f : 𝒞 → 𝒜 be a unital A_{∞}-functor such that f|_{ℬ} is contractible. Then for each object X of ℬ the complex (s𝒜(Xf, Xf), b_{1}) is contractible [LO02, Proposition 6.1]. Equivalently we may say that for each object Z of ℬ̃ the complex (s𝒜(ZΨf, ZΨf), b_{1}) is contractible, as the following commutative diagram shows:

\begin{aligned}ℬ̃ & ^{⊂}  → & 𝒞̃ & \overline{^{Ψf} →} & 𝒜 \\ ℬ & ↓^{Ψ′} & ^{⊂}  → & 𝒞 & ↓^{Ψ} & \overline{^{f} →} & 𝒜 & ‖ \end{aligned}

This implies contractibility of the A_{∞}-functor Ψf|ℬ̃. Similarly, if g ∈ A_{∞}^{u}(𝒞̃, 𝒜)_{mod ℬ̃}, then Ỹg ∈ A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} . B.11 Corollary. The A_{∞}-2-functors 𝒜 ↦ A_{∞}^{u}(𝒞, 𝒜)_{mod ℬ} and 𝒜 ↦ A_{∞}^{u}(𝒞̃, 𝒜)_{mod ℬ̃} are naturally A_{∞}-2-equivalent. Therefore, if one of them is representable, then so is the other.

C. The Yoneda Lemma for 2-categories

C.1. 2-categories. We recall the definitions of strict 2-categories and associated weak notions originating in [Bén67]. We use the form and the notation of [Lyu99]. C.2 Definition. A (strict) 2-category 𝔄 consists of

1. a set of objects Ob 𝔄;

2. for any pair of objects X, Y ∈ Ob 𝔄 a category 𝔄(X, Y);

3. (a) for any object X ∈ Ob 𝔄 an object 1_{X} of 𝔄(X, X);

(b) for any triple of objects X, Y, Z ∈ Ob 𝔄 a functor • : 𝔄(X, Y) × 𝔄(Y, Z) → 𝔄(X, Z), (F, G) ↦ F G = F •G = G ∘ F;

such that the following functors are equal

4. F •1 = F = 1•F, F (GH) = (F G)H.

The 2-category of (𝒰 -small) categories is denoted 𝒞at. C.3 Definition. A weak 2-functor (a homomorphism in [Bén67]) between 2-categories 𝔄 and ℭ consists of

1. a function F : Ob 𝔄 → Ob ℭ;

2. a functor F = F_{X,Y} : 𝔄(X, Y) → ℭ(F X, F Y) for each pair of objects X, Y ∈ Ob 𝔄; 3. (a) an isomorphism ϕ_{0} : 1_{F X} → F 1_{X};

(b) an invertible (natural) transformation

\begin{matrix}𝔄(X, Y) × 𝔄(Y, Z) & \overline{^{•} →} & 𝔄(X, Z) \\ ℭ(F X, F Y) × ℭ(F Y, F Z) & ↓^{F_{X,Y} ×F_{Y,Z}} & →^{•} & \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}^{ϕ_{2}} & ℭ(F X, F Z) & ↓^{F_{X,Z}} \end{matrix}

for each triple X, Y, Z ∈ Ob 𝔄; such that

4. (a) for any object M ∈ 𝔄(X, Y) the composites

F M = F M •1_{F Y} ^{F M •ϕ_{0}} → F M •F 1_{Y} ^{ϕ_{2}}→ F (M •1_{Y}) = F M  (C.3.1)
F M = 1_{F X} •F M ^{ϕ_{0}•F M} → F 1_{X} •F M ^{ϕ_{2}}→ F (1_{X} •M) = F M  (C.3.2)

are identity morphisms in ℭ(F M, F M);

(b) For any objects W, X, Y, Z ∈ Ob 𝔄 and any object

(K, L, M) ∈ 𝔄(W, X) × 𝔄(X, Y) × 𝔄(Y, Z)

there is an equation

(F K •(F L•F M) ^{F K•ϕ_{2}} → F K •F (L•M) ^{ϕ_{2}}→ F (K •(L•M)))

= ((F K •F L)•F M ^{ϕ_{2}•F M} → F (K •L)•F M ^{ϕ_{2}}→ F ((K •L)•M)).

If ϕ_{2} and ϕ_{0} are identity isomorphisms, F is called a strict 2-functor. C.4 Definition. A weak 2-transformation (pseudo-natural transformation [Gra74]) λ : (F, ϕ_{2}, ϕ_{0}) → (G, ψ_{2}, ψ_{0}) : 𝔄 → ℭ is

1. a family of 1-morphisms λ_{X} : F X → GX, X ∈ Ob 𝔄;

2. for any 1-morphism f : X → Y in 𝔄 a 2-isomorphism in ℭ

λ_{f} : F f •λ_{Y} ^{∼}→ λ_{X} •Gf : F X → GY,

which is an isomorphism of functors λ_{−} : F − •λ_{Y} → λ_{X} •G− : 𝔄(X, Y) → ℭ(F X, GY), that is, for any 2-morphism ξ : f → g : X → Y

F X \overline{^{F f} →} F Y ; ⟸^{λ_{f}} ; GX ↓^{λ_{X}}  \overline{^{Gf} →} ; ⇓Gξ ; \overline{_{_{Gg} →}} GY ↓^{λ_{Y}} = \begin{matrix}F X & \overline{^{F f} →} ; ⇓F ξ ; \overline{_{_{F g} →}} & F Y \\ GX & ↓^{λ_{X}} & \overline{_{_{Gg} →}} & ⟸^{λ_{g}} & GY & ↓^{λ_{Y}} \end{matrix} ,

such that

3. (a) for any object X ∈ Ob 𝔄

\begin{matrix}F X & \overline{^{1_{F X}} →} ; ⇓ ϕ_{0} ; \overline{_{_{F 1_{X}} →}} & F X \\ GX & ↓^{λ_{X}} & \overline{_{_{G1_{X}} →}} & ⟸^{λ_{1_{X}}} & GX & ↓^{λ_{X}} \end{matrix} = \begin{aligned}F X & \overline{^{1_{F X}} →} & F X \\ & = \\ GX & ↓^{λ_{X}} & \overline{^{1_{GX}} →} ; ⇓ ψ_{0} ; \overline{_{_{G1_{X}} →}} & GX & ↓^{λ_{X}} \end{aligned} ;

(b) for any pair of composable 1-morphisms f, g ∈ 𝔄_{1}

\begin{aligned}& F Y \\ F X & \overline{_{_{F (fg)}  →}} & ^{F f}  \overline{→} & ^{^{ϕ_{2}} ‖} & F Z & ^{^{F g} \overline{→}} \\ GX & ↓^{λ_{X}} & \overline{_{_{G(fg)}  →}} & ⟸^{λ_{fg}} & GZ & ↓^{λ_{Z}} \end{aligned} = \begin{aligned}& F Y \\ F X & ^{F f}  \overline{→} & GY & ↓^{λ_{Y}} & ⟸^{λ_{g}} & F Z & ^{^{F g} \overline{→}} \\ GX & ↓^{λ_{X}} & \overline{_{_{G(fg)}  →}} & ⟸^{λ_{f}} & ^{Gf}  \overline{→} & ^{^{ψ_{2}} ‖} & GZ & ↓^{λ_{Z}} & ^{^{Gg}  \overline{→}} \end{aligned} .

If λ_{f} are identity isomorphisms, λ is called a strict 2-transformation. A weak 2-transfor-mation λ = (λ_{X}) for which λ_{X} are equivalences is called a 2-natural equivalence. C.5 Definition. A modification m : λ → μ : F → G : 𝔄 → ℭ is

1. a family of 2-morphisms m_{X} : λ_{X} → μ_{X}, X ∈ Ob 𝔄 such that

2. for any 1-morphism f : X → Y in 𝔄

\begin{matrix}F X & \overline{^{F f} →} & F Y \\ GX & ↓^{μ_{X}}^{⇐^{m_{X}} ↓^{λ_{X}}} & \overline{^{Gf} →} & ⟸^{λ_{f}} & GY & ↓^{λ_{Y}} \end{matrix}  =  \begin{matrix}F X & \overline{^{F f} →} & F Y \\ GX & ↓^{μ_{X}} & \overline{^{Gf} →} & ⟸^{μ_{f}} & GY & ↓^{μ_{Y}}^{⇐^{m_{Y}} ↓^{λ_{Y}}} \end{matrix}

C.6 Proposition (Invertibility of 2-natural equivalences). Let λ : F → G : 𝔄 → ℭ be a 2-natural equivalence. Then there exist a weak 2-transformation μ : G → F : 𝔄 → ℭ and invertible modifications ε : λμ → 1_{F} : F → F : 𝔄 → ℭ and η : 1_{G} → μλ : G → G : 𝔄 → ℭ. Thus, μ is quasi-inverse to λ.

Proof. Since λ_{X} : F X → GX is an equivalence for every X ∈ Ob 𝔄, we obtain: for every X ∈ Ob 𝔄 there exist a 1-morphism μ_{X} : GX → F X and invertible 2-morphisms ε_{X} : λ_{X}μ_{X} → 1_{F X} : F X → F X, β_{X} : 1_{GX} → μ_{X}λ_{X} : GX → GX.

C.7 Lemma. There exist such invertible 2-morphisms η_{X} : 1_{GX} → μ_{X}λ_{X} : GX → GX that the following equations hold true:

(μ_{X} ^{η_{X} •μ_{X}} → μ_{X}λ_{X}μ_{X} ^{μ_{X} •ε_{X}} → μ_{X}) = 1_{μ_{X}},  (C.7.1)
(λ_{X} ^{λ_{X} •η_{X}} → λ_{X}μ_{X}λ_{X} ^{ε_{X} •λ_{X}} → λ_{X}) = 1_{λ_{X}}.  (C.7.2)

Proof. Consider the following functors:

ℭ(GX, GX) ⟶ ℭ(GX, F X),  ℭ(GX, GX) ⟶ ℭ(F X, GX),  (C.7.3)
f ⟼ fμ_{X},  f ⟼ λ_{X}f,
ϕ : f → g ⟼ ϕ•μ_{X} : fμ_{X} → gμ_{X},  ϕ : f → g ⟼ λ_{X} •ϕ : λ_{X}f → λ_{X}g, (C.7.4)
ℭ(GX, F X) ⟶ ℭ(GX, GX),  ℭ(F X, GX) ⟶ ℭ(GX, GX),
h ⟼ hλ_{X},  h ⟼ μ_{X}h,
χ : h → k ⟼ χ•λ_{X} : hλ_{X} → kλ_{X},  χ : h → k ⟼ μ_{X} •χ : μ_{X}h → μ_{X}k.

These functors are faithful. Let us prove it for the first one. Indeed, if ϕ•μ_{X} = ψ•μ_{X} : fμ_{X} → gμ_{X}, then ϕ•μ_{X} •λ_{X} = ψ•μ_{X} •λ_{X} : fμ_{X}λ_{X} → gμ_{X}λ_{X} and (f •β_{X})(ϕ•μ_{X} •λ_{X}) = (f •β_{X})(ψ•μ_{X} •λ_{X}) : f → gμ_{X}λ_{X}, i.e., ϕ•(gβ_{X}) = ψ•(gβ_{X}) : f → gμ_{X}λ_{X}, hence ϕ = ψ, because β_{X} is invertible. Similarly or by symmetry the other 3 functors are also faithful.

Functors (C.7.3) are full. Let us prove it for the first one. Given ψ : fμ_{X} → gμ_{X}, we set ϕ = (f •β)(ψ•λ_{X})(g•β_{X}^{−1}) : f → g. Then (f •β_{X})(ϕ•μ_{X} •λ_{X})(g•β_{X}^{−1}) = (f •β_{X})(f •β_{X}^{−1})ϕ = ϕ = (f •β_{X})(ψ•λ_{X})(g•β_{X}^{−1}). This yields ϕ•μ_{X} •λ_{X} = ψ•λ_{X}. Since multiplication with λ_{X} is a faithful functor, we obtain ϕ•μ_{X} = ψ. Notice that if ψ is a 2-isomorphism, then the composition of 2-isomorphisms ϕ is a 2-isomorphism as well. Similarly we prove that the second functor of (C.7.3) is full.

Now let us take f = 1_{GX}, g = μ_{X}λ_{X}. By the first bijection of (C.7.4) we find a 2-iso-morphism η_{X} : 1_{GX} → μ_{X}λ_{X} corresponding to the 2-isomorphism ψ = μ_{X} •ε_{X}^{−1} : μ_{X} → μ_{X}λ_{X}μ_{X}. Then η_{X} •μ_{X} = μ_{X} •ε_{X}^{−1}, that is, (η_{X} •μ_{X})(μ_{X} •ε_{X}) = 1_{μ_{X}}. By the second bijection of (C.7.4) we find a 2-isomorphism γ_{X} : 1_{GX} → μ_{X}λ_{X} corresponding to the 2-isomorphism ξ = ε_{X}^{−1}•λ_{X} : λ_{X} → λ_{X}μ_{X}λ_{X}. Then λ_{X} •γ_{X} = ε_{X}^{−1}•λ_{X}, that is, (λ_{X} •γ_{X})(ε_{X} •λ_{X}) = 1_{λ_{X}}. We have

γ = (1_{GX} ^{γ}→ μλ ^{1_{μ}λ} → μλ) = (1_{GX} ^{γ}→ μλ ^{ημλ} → μλμλ ^{μελ} → μλ)

= (1_{GX} ^{η}→ μλ ^{μλγ} → μλμλ ^{μελ} → μλ) = (1_{GX} ^{η}→ μλ ^{μ1_{λ}}→ μλ) = η.

Therefore, the 2-isomorphism η fulfills both equations (C.7.1) and (C.7.2).

For any 1-morphism f : X → Y in 𝔄 we define a 2-isomorphism μ_{f} as the pasting of

GX \overline{^{μ_{X}} →} F X \overline{^{F f} →} F Y ; \begin{matrix}= \\ = \\ = \\ ⇒ \end{matrix}^{η_{X}} ; GX ↓^{λ_{X}}  \overline{_{_{Gf} →}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}^{λ_{f}^{−1}}^{^{1_{GX}}  \overline{→}}  GY ↓^{λ_{Y}}  \overline{_{_{μ_{Y}} →}} \begin{matrix}= \\ = \\ ⇒ \end{matrix}^{ε_{Y}} F Y^{^{1_{F Y}}  \overline{→}} .

Since λ_{f} determine an isomorphism of functors

λ_{−} : F − •λ_{Y} → λ_{X} •G− : 𝔄(X, Y) → ℭ(F X, GY),

for any 2-morphism ξ : f → g : X → Y equation 2 of Definition C.4 holds. It implies

GX \overline{^{μ_{X}} →} F X \overline{^{F g} →} ; ⇑F ξ ; \overline{_{_{F f} →}} F Y ; \begin{matrix}= \\ = \\ = \\ ⇒ \end{matrix}^{η_{X}} ; GX ↓^{λ_{X}}  \overline{_{_{Gf} →}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}^{λ_{f}^{−1}}^{^{1_{GX}}  \overline{→}}  GY ↓^{λ_{Y}}  \overline{_{_{μ_{Y}} →}} \begin{matrix}= \\ = \\ ⇒ \end{matrix}^{ε_{Y}} F Y^{^{1_{F X}} \overline{→}} = GX \overline{^{μ_{X}} →} F X \overline{^{F g} →} F Y ; \begin{matrix}= \\ = \\ = \\ ⇒ \end{matrix}^{η_{X}} ; GX ↓^{λ_{X}} \overline{^{Gg} →} ; ⇑Gξ ; \overline{_{_{Gf} →}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}^{λ_{g}^{−1}}^{^{1_{GX}}  \overline{→}}  GY ↓^{λ_{Y}}  \overline{_{_{μ_{Y}} →}} \begin{matrix}= \\ = \\ ⇒ \end{matrix}^{ε_{Y}} F Y^{^{1_{F X}} \overline{→}} .

This shows that the collection of 2-isomorphisms μ_{f} determines an isomorphism of functors

μ_{−} : G − •μ_{Y} → μ_{X} •F − : 𝔄(X, Y) → ℭ(GX, F Y).

Let us check conditions 3(a),(b) of Definition C.4 for μ. For 3(a) we have

GX \overline{^{μ_{X}} →} F X \overline{^{F 1_{X}}→} F X ; \begin{matrix}= \\ = \\ = \\ ⇒ \end{matrix}^{η_{X}} ; GX ↓^{λ_{X}} \overline{^{G1_{X}}→} ; ⇑ψ_{0} ; \overline{_{_{1_{GX}}→}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}^{λ_{1_{X}}^{−1}}^{^{1_{GX}}  \overline{→}}  GX ↓^{λ_{X}}  \overline{_{_{μ_{X}} →}} \begin{matrix}= \\ = \\ ⇒ \end{matrix}^{ε_{X}} F X^{^{1_{F X}} \overline{→}} = \begin{aligned}GX & \overline{^{μ_{X}} →} & F X & \overline{^{F 1_{X}}→} ; ⇑ϕ_{0} ; \overline{_{_{1_{F X}}→}} & F X \\ & \begin{matrix}= \\ = \\ = \\ ⇒ \end{matrix}^{η_{X}} & = \\ GX & ↓^{λ_{X}} & \overline{_{_{1_{GX}}→}} & ^{^{1_{GX}}  \overline{→}} & GX & ↓^{λ_{X}} & \overline{_{_{μ_{X}} →}} & \begin{matrix}= \\ = \\ ⇒ \end{matrix}^{ε_{X}} & F X & ^{^{1_{F X}} \overline{→}} \end{aligned}

and the required equation follows from (C.7.1).

Equation (C.7.2) and the corresponding property for λ imply

GY

GX  \overline{_{_{G(fg)}  →}}^{Gf}  \overline{→}^{^{ψ_{2}} ‖}  GZ \overline{^{μ_{Z}} →}^{^{Gg} \overline{→}}  F Z
ε_{Z} \begin{matrix}= \\ = \\ = \\ ⇒ \end{matrix}
GX \overline{_{_{μ_{X}} →}}^{1_{GX}}  \overline{→} F X^{↑^{λ_{X}}}  \overline{_{_{F (fg)}  →}}^{^{η_{X}} \begin{matrix}= \\ = \\ ⇒ \end{matrix}}  F Z^{^{λ_{Z}} ↑}  ^{1_{F Z}}  \overline{→}^{^{λ_{fg}^{−1}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}}

= \begin{matrix}GY & \overline{^{1_{GY}}  →} & GY \\ GX & =^{λ_{f}^{−1}} ⇒ & ^{Gf}  \overline{→} & F Y & ^{^{λ_{Y}} ↑} & \overline{_{_{1_{F Y}}  →}} & ^{^{ε_{Y}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}} & F Y & ^{↑^{λ_{Y}}} & ^{^{η_{Y}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}} & ^{^{μ_{Y}}  \overline{→}} & GZ & \overline{^{μ_{Z}} →} & ^{^{Gg} \overline{→}} & F Z \\ ε_{Z} & \begin{matrix}= \\ = \\ = \\ ⇒ \end{matrix} \\ GX & \overline{_{_{μ_{X}} →}} & ^{1_{GX}}  \overline{→} & F X & ^{↑^{λ_{X}}} & \overline{_{_{F (fg)}  →}} & ^{F f}  \overline{→} & ^{^{η_{X}} \begin{matrix}= \\ = \\ ⇒ \end{matrix}} & ^{^{ϕ_{2}} ‖} & F Z & ^{↑^{λ_{Z}}} & ^{^{λ_{g}^{−1}} \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ = \\ ⇒ \end{matrix}} & ^{1_{F Z}}  \overline{→} & ^{^{F g}  \overline{→}} \end{matrix}

and the assertion 3(b) for μ follows.

Let us verify that collections ε = (ε_{X}) and η = (η_{X}) determine modifications. Equation (C.7.2) implies that

\begin{matrix}F X & \overline{^{F f}  →} & F Y \\ = \\ ⟸^{ε_{X}} & GX & \overline{^{1_{GX}} →} & ^{^{λ_{X}}  \overline{→}} & GX & ^{^{λ_{f}} ‖} & \overline{^{Gf} →} & ^{^{λ_{X}}  \overline{→}} & GY & ^{^{λ_{Y}}  \overline{→}} \\ F X & ↓^{1_{F X}} & \overline{_{_{F f}  →}} & ^{λ_{X}}  \overline{→} & ⟸^{η_{X}} & \underline{←}^{μ_{X}} & F Y & ^{^{λ_{f}^{−1}} ‖} & \overline{_{_{1_{F Y}} →}} & ^{λ_{Y}}  \overline{→} & ⟸^{ε_{Y}} & F Y & ↓^{μ_{Y}} \end{matrix} =  \begin{matrix}F X & \overline{→^{F f}} & F Y \\ = & ⟸^{ε_{Y}} & GY & ^{^{λ_{Y}}  \overline{→}} \\ F X & ↓^{1_{F X}} & \overline{_{_{F f}→}} & F Y & ↓^{1_{F Y}} & \underline{←}^{μ_{Y}} \end{matrix} .

This means that ε : λμ → 1_{F} : F → F : 𝔄 → ℭ is a modification.

Also equation (C.7.2) implies that

\begin{aligned}& GX & \overline{→^{Gf}} & GY \\ F X & ⟸^{η_{X}} & \underline{←}^{μ_{X}} & = \\ GX & ↓^{1_{GX}} & \overline{_{_{Gf}→}} & ^{^{λ_{X}}  \overline{→}} & GY & ↓^{1_{GY}} \end{aligned}  = \begin{aligned}GX & \overline{^{1_{GX}} →} & GX & \overline{^{Gf}  →} & GY \\ F X & ↓^{μ_{X}} & \overline{_{_{F f} →}} & ^{λ_{X}}  \overline{→} & ⟸^{η_{X}} & F Y & ^{^{λ_{f}^{−1}} ‖} & \overline{_{_{1_{F Y}} →}} & ^{λ_{Y}}  \overline{→} & ⟸^{ε_{Y}} & F Y & ⟸^{η_{Y}} & \underline{←}^{μ_{Y}} \\ & = \\ GX & ^{^{λ_{f}} ‖} & \overline{_{_{Gf}  →}} & ^{^{λ_{X}}  \overline{→}} & GY & ↓^{1_{GX}} & ^{^{λ_{Y}} \overline{→}} & ^{^{λ_{Y}}  \overline{→}} \end{aligned}

This means that η : 1_{G} → μλ : G → G : 𝔄 → ℭ is a modification.

Therefore, μ is a weak 2-transformation quasi-inverse to λ.

C.8 Remark. Clearly, if λ : F → G : 𝔄 → ℭ and μ : G → F : 𝔄 → ℭ are weak 2-transformations, quasi-inverse to each other, then both are 2-natural equivalences.

C.9. The 2-functor 𝔄(A, ). We fix a strict 2-category 𝔄 till the end of this section. An arbitrary object A ∈ Ob 𝔄 gives rise to a strict 2-functor 𝔄(A, ) : 𝔄 → 𝒞at. It is specified by the following data:

1. the function Ob 𝔄 → Ob 𝒞at, X ↦ 𝔄(A, X); 2. the functor 𝔄(A, )_{XY} : 𝔄(X, Y) → 𝒞at(𝔄(A, X), 𝔄(A, Y)) for each pair of objects

X, Y ∈ Ob 𝔄.

The functor 𝔄(A, )_{XY} is given as follows. For any 1-morphism f : X → Y the functor 𝔄(A, f) : 𝔄(A, X) → 𝔄(A, Y) is given by the following formulas:

(ϕ : A → X) ↦ (ϕf : A → Y),
(π : ϕ → ψ : A → X) ↦ (π•f : ϕf → ψf : A → Y).

For any 2-morphism α : f → g : X → Y the natural transformation 𝔄(A, α) : 𝔄(A, f) → 𝔄(A, g) : 𝔄(A, X) → 𝔄(A, Y) is given explicitly by its components:

(ϕ)𝔄(A, α) = ϕ•α : (ϕ)𝔄(A, f) = ϕf → ϕg = (ϕ)𝔄(A, g), ϕ ∈ 𝔄(A, X).

We want to generalize the classical Yoneda Lemma to 2-categories. If we were using strict 2-functors and strict 2-transformations, we would view 2-categories as cat-categories, where cat is the category of categories. The Yoneda Lemma for enriched categories [EK66, Theorem I.8.6] would be sufficient. However, we need weak 2-transformations, so we prove a direct generalization in the weak setup.

Let [𝔄, 𝒞at] denote the strict 2-category of weak 2-functors 𝔄 → 𝒞at, their weak 2-transformations and their modifications, see e.g. [Lyu99, Appendix A.1.5].

A weak 2-transformation λ : 𝔄(A, ) → G : 𝔄 → 𝒞at involves, in particular, a functor λ_{A} : 𝔄(A, A) → GA. Evaluating it on the object 1_{A} ∈ Ob 𝔄(A, A) we get an object (1_{A})λ_{A} ∈ Ob GA. A modification m : λ → μ : 𝔄(A, ) → G : 𝔄 → 𝒞at involves, in particular, a natural transformation m_{A} : λ_{A} → μ_{A} : 𝔄(A, A) → GA. Evaluating it on the object 1_{A} ∈ Ob 𝔄(A, A) we get a morphism (1_{A})m_{A} : (1_{A})λ_{A} → (1_{A})μ_{A} of GA.

C.10 Proposition. Let G : 𝔄 → 𝒞at be a weak 2-functor. Let A be an object of 𝔄. Then the functor

ev_{1_{A}} : [𝔄, 𝒞at](𝔄(A, ), G) ⟶ GA,
λ ⟼ (1_{A})λ_{A},
m : λ → μ ⟼ (1_{A})m_{A} : (1_{A})λ_{A} → (1_{A})μ_{A},

is full and faithful.

Proof. Clearly, the assignment ev_{1_{A}} gives a functor. Let us show that it is faithful. Let two modifications m, n : λ → μ : 𝔄(A, ) → G : 𝔄 → 𝒞at be given such that (1_{A})m_{A} = (1_{A})n_{A}. The modification m is a family of natural transformations (m_{C})_{C∈Ob 𝔄} satisfying the equation

\begin{matrix}𝔄(A, B) & \overline{→^{𝔄(A,f)}} & 𝔄(A, C) \\ GB & ↓^{μ_{B}}^{⇐^{m_{B}} ↓^{λ_{B}}} & \overline{^{Gf} →} & ⟸^{λ_{f}} & GC & ↓^{λ_{C}} \end{matrix} = \begin{matrix}𝔄(A, B) & \overline{→^{𝔄(A,f)}} & 𝔄(A, C) \\ GB & ↓^{μ_{B}} & \overline{^{Gf} →} & ⟸^{μ_{f}} & GC & ↓^{μ_{C}}^{⇐^{m_{C}} ↓^{λ_{C}}} \end{matrix}  (C.10.1)

for an arbitrary 1-morphism f : B → C of 𝔄. In particular, it holds for B = A. Restrict this equation to the object 1_{A} of 𝔄(A, A). Then it gives for an arbitrary 1-morphism f : A → C the equation

[(f)λ_{C} ^{(1_{A})λ_{f}} → ((1_{A})λ_{A})(Gf) ^{((1_{A})m_{A})(Gf)} → ((1_{A})μ_{A})(Gf)]

= [(f)λ_{C} ^{(f)m_{C}} → (f)μ_{C} ^{(1_{A})μ_{f}} → ((1_{A})μ_{A})(Gf)].

Therefore, the value of m_{C} on an arbitrary object f of 𝔄(A, C) is completely determined by the morphism (1_{A})m_{A}:

(f)m_{C} = [(f)λ_{C} ^{(1_{A})λ_{f}} → ((1_{A})λ_{A})(Gf) ^{((1_{A})m_{A})(Gf)} → ((1_{A})μ_{A})(Gf) ^{(1_{A})μ_{f}^{−1}} → (f)μ_{C}].

Thus, m = n and ev_{1_{A}} is faithful.

Let us prove that ev_{1_{A}} is full. Let λ, μ : 𝔄(A, ) → G : 𝔄 → 𝒞at be weak 2-trans-formations. Let ϕ : (1_{A})λ_{A} → (1_{A})μ_{A} be a morphism of GA. We claim that there is a modification m : λ → μ such that (1_{A})m_{A} = ϕ. The value of m_{C} on an arbitrary 1-morphism f : A → C can be only

(f)m_{C} = [(f)λ_{C} ^{(1_{A})λ_{f}} → ((1_{A})λ_{A})(Gf) ^{(ϕ)(Gf)} → ((1_{A})μ_{A})(Gf) ^{(1_{A})μ_{f}^{−1}} → (f)μ_{C}], (C.10.2)

as we have seen. Let us verify that, indeed, this formula determines a modification.

First of all, each m_{C} is a natural transformation. Indeed, for each 2-morphism ξ : f → g : A → C of 𝔄 the following diagram commutes:

\begin{matrix}(f)λ_{C} & \overline{→^{(1_{A})λ_{f}}} & ((1_{A})λ_{A})(Gf) & →^{(ϕ)(Gf)} & ((1_{A})μ_{A})(Gf) & \overline{→^{(1_{A})μ_{f}^{−1}}} & (f)μ_{C} \\ (g)λ_{C} & ↓^{(ξ)λ_{C}} & \overline{→^{(1_{A})λ_{g}}} & ((1_{A})λ_{A})(Gg) & ↓^{((1_{A})λ_{A})(Gξ)} & \overline{→^{(ϕ)(Gg)}} & ((1_{A})μ_{A})(Gg) & ↓^{((1_{A})μ_{A})(Gξ)} & \overline{→^{(1_{A})μ_{g}^{−1}}} & (g)μ_{C} & ↓^{(ξ)μ_{C}} \end{matrix} (C.10.3)

The central square commutes because Gξ : Gf → Gg is a natural transformation. Condition 2 for λ from Definition C.4 implies, in particular, equation

𝔄(A, A) \overline{→^{𝔄(A,f)}} 𝔄(A, C) ; ⟸^{λ_{f}} ; GA ↓^{λ_{A}}  \overline{^{Gf} →} ; ⇓Gξ ; \overline{_{_{Gg} →}} GC ↓^{λ_{C}} = \begin{matrix}𝔄(A, A) & \overline{\frac{𝔄(A,f)}{→}} ; ⇓𝔄(A,ξ) ; \overline{_{_{𝔄(A,g)}→}} & 𝔄(A, C) \\ GA & ↓^{λ_{A}} & \overline{_{_{Gg} →}} & ⟸^{λ_{g}} & GC & ↓^{λ_{C}} \end{matrix}.

Restricting this equation to the object 1_{A} of 𝔄(A, A) we get an equation, which expresses precisely commutativity of the left square of diagram (C.10.3). The right square of (C.10.3) commutes by the same reasoning applied to μ instead of λ. Thus, m_{C} is a natural transformation.

Secondly, we have to prove equation (C.10.1) for the family (m_{C}) and for an arbitrary 1-morphism f : B → C of 𝔄. On an arbitrary object g : A → B of 𝔄(A, B) this equation reads:

[(gf)λ_{C} ^{(g)λ_{f}} → ((g)λ_{B})(Gf) ^{((g)m_{B})(Gf)} → ((g)μ_{B})(Gf)]

= [(gf)λ_{C} ^{(gf)m_{C}} → (gf)μ_{C} ^{(g)μ_{f}} → ((g)μ_{B})(Gf)].

Substituting definition (C.10.2) of m_{C} we get an equation, which expresses commutativity of the exterior of the following diagram:

(gf)λ_{C}^{^{(g)λ_{f}} ↑}  \overline{^{(1_{A})λ_{gf}} →} ((1_{A})λ_{A})(G(gf)) ↓^{((1_{A})λ_{A})(g,f)ψ_{2}}  \overline{→^{(ϕ)(G(gf))}} ((1_{A})μ_{A})(G(gf)) ↓^{((1_{A})μ_{A})(g,f)ψ_{2}} \overline{^{(1_{A})μ_{gf}^{−1}} →} (gf)μ_{C}^{↑^{(g)μ_{f}}}^{((g)λ_{B})(Gf) \overline{→^{((1_{A})λ_{g})(Gf)}} ((1_{A})λ_{A})(Gg)(Gf) \overline{→^{(ϕ)(Gg)(Gf)}} ((1_{A})μ_{A})(Gg)(Gf) \overline{→^{((1_{A})μ_{g}^{−1})(Gf)}} ((g)μ_{B})(Gf)}

(C.10.4)

The middle square commutes, because (g, f)ψ_{2} : (Gg)(Gf) → G(gf) is a morphism of functors. Property 3(b) of Definition C.4 for λ implies, in particular, the equation

\begin{matrix}𝔄(A, B) \\ 𝔄(A, A) & \overline{_{_{𝔄(A,gf)} →}} & ^{𝔄(A,g)}  \overline{→} & ^{⇃⇂^{=} ‖} & 𝔄(A, C) & ^{^{𝔄(A,f)} \overline{→}} \\ GA & ↓^{λ_{A}} & \overline{_{_{G(gf)}  →}} & ⟸^{λ_{gf}} & GC & ↓^{λ_{C}} \end{matrix} = \begin{matrix}𝔄(A, B) \\ 𝔄(A, A) & ^{𝔄(A,g)}  \overline{→} & GB & ↓^{λ_{B}} & ⟸^{λ_{f}} & 𝔄(A, C) & ^{^{𝔄(A,f)} \overline{→}} \\ GA & ↓^{λ_{A}} & \overline{_{_{G(gf)}  →}} & ⟸^{λ_{g}} & ^{Gg}  \overline{→} & ^{^{ψ_{2}} ‖} & GC & ↓^{λ_{C}} & ^{^{Gf}  \overline{→}} \end{matrix} .

Restricting this equation to the object 1_{A} of 𝔄(A, A), we will get precisely the left square of diagram (C.10.4), therefore, it commutes. The right square of (C.10.4) commutes by the same reasoning applied to μ instead of λ. Therefore, m is a modification, and ev_{1_{A}} is full.

C.11 Proposition. For each object x of GA there is a weak 2-transformation

λ^{x} = λ^{A,x} = ^{G}λ^{A,x} : 𝔄(A, ) → (G, ψ_{2}, ψ_{0}) : 𝔄 → 𝒞at,

specified by the family of functors λ_{C}^{x}, C ∈ Ob 𝔄:

λ_{C}^{x} : 𝔄(A, C) ⟶ GC,
f : A → C ⟼ (x)(Gf),
ξ : f → g : A → C ⟼ (x)(Gξ) : (x)(Gf) → (x)(Gg),

and by the family of invertible natural transformations

λ_{f}^{x} : 𝔄(A, f)•λ_{C}^{x} → λ_{B}^{x} •Gf : 𝔄(A, B) → GC,

f ∈ Ob 𝔄(B, C), which map an object g ∈ Ob 𝔄(A, B) to the isomorphism of GC:

(g)λ_{f}^{x} =^{def} (x)(g, f)ψ_{2}^{−1} : (gf)λ_{C}^{x} = (x)(G(gf)) → (x)(Gg)(Gf) = (g)λ_{B}^{x}(Gf). (C.11.1)

For each morphism u : x → y of GA there is a modification

λ^{u} = λ^{A,u} = ^{G}λ^{A,u} : λ^{x} → λ^{y} : 𝔄(A, ) → (G, ψ_{2}, ψ_{0}) : 𝔄 → 𝒞at,

specified by the family of natural transformations λ_{C}^{u}, C ∈ Ob 𝔄:

λ_{C}^{u} : λ_{C}^{x} → λ_{C}^{y} : 𝔄(A, C) → GC,

λ_{C}^{u} : (f : A → C) ⟼ ((f)λ_{C}^{u} = (u)(Gf) : (f)λ_{C}^{x} = (x)(Gf) → (y)(Gf) = (f)λ_{C}^{y}). (C.11.2)

The correspondence

Λ : GA ⟶ [𝔄, 𝒞at](𝔄(A, ), G),
x ⟼ λ^{x},
u : x → y ⟼ λ^{u} : λ^{x} → λ^{y},

is a functor.

Proof. As G : 𝔄(A, C) → 𝒞at(GA, GC) is a functor, G1_{f} = 1_{Gf} for the unit 2-morphism 1_{f} : f → f : A → C of 𝔄, and for each pair of composable 2-morphisms f ^{ξ}→ g ^{χ}→ h : A → C of 𝔄 we have

G(ξχ) = (Gf ^{Gξ}→ Gg ^{Gχ}→ Gh).

Evaluating these equations on x we get (1_{f})λ_{C}^{x} = (x)1_{Gf} = 1_{(x)(Gf)} and (ξχ)λ_{C}^{x} = (ξλ_{C}^{x})(χλ_{C}^{x}), thus, λ_{C}^{x} is a functor.

We claim that λ_{f}^{x} given by (C.11.1) is a natural transformation. Indeed, naturality of ψ_{2}, expressed by

G(gf) ←^{(g,f)ψ_{2}} (Gg)(Gf)

=

G(hf) ↓^{G(ξ•f)} ←^{(h,f)ψ_{2}} (Gh)(Gf) ↓^{Gξ•Gf}

implies commutativity of

(x)(G(gf)) \overline{→^{(x)(g,f)ψ_{2}^{−1}}} (x)(Gg)(Gf)

=

(x)(G(hf)) ↓^{(x)(G(ξ•f))}  \overline{→^{(x)(h,f)ψ_{2}^{−1}}} (x)(Gh)(Gf) ↓^{(x)(Gξ)(Gf)}

which is nothing else, but naturality of λ_{f}^{x}:

(g)𝔄(A, f)λ_{C}^{x} \overline{→^{(g)λ_{f}^{x}}} (g)λ_{B}^{x}(Gf)

=

(h)𝔄(A, f)λ_{C}^{x} ↓^{(ξ)𝔄(A,f)λ_{C}^{x}}  \overline{→^{(h)λ_{f}^{x}}} (h)λ_{B}^{x}(Gf) ↓^{(ξ)λ_{B}^{x}(Gf)}

We claim that

λ_{−}^{x} : 𝔄(A, )λ_{C}^{x} → λ_{B}^{x} •G− : 𝔄(A, B) → GC

is a morphism of functors. That is, for each 2-morphism ξ : f → g : B → C of 𝔄 the following equation holds:

𝔄(A, B) \overline{→^{𝔄(A,f)}} 𝔄(A, C) ; ⟸^{λ_{f}^{x}} ; GB ↓^{λ_{B}^{x}}  \overline{^{Gf} →} ; ⇓Gξ ; \overline{_{_{Gg} →}} GC ↓^{λ_{C}^{x}} = \begin{matrix}𝔄(A, B) & \overline{\frac{𝔄(A,f)}{→}} ; ⇓𝔄(A,ξ) ; \overline{_{_{𝔄(A,g)}→}} & 𝔄(A, C) \\ GB & ↓^{λ_{B}^{x}} & \overline{_{_{Gg} →}} & ⟸^{λ_{g}^{x}} & GC & ↓^{λ_{C}^{x}} \end{matrix}.  (C.11.3)

Indeed, for each 1-morphism h : A → B of 𝔄 we have

(x)(G(hf)) \overline{→^{(x)(h,f)ψ_{2}^{−1}}} (x)(Gh)(Gf)

=

(x)(G(hg)) ↓^{(x)(G(h•ξ))}  \overline{→^{(x)(h,g)ψ_{2}^{−1}}} (x)(Gh)(Gg) ↓^{(x)(Gh)(Gξ)}

by naturality of ψ_{2}. Rewriting this equation in the form

(hf)λ_{C}^{x} \overline{→^{(h)λ_{f}^{x}}} (h)λ_{B}^{x}(Gf)

=

(hg)λ_{C}^{x} ↓^{(h•ξ)λ_{C}^{x}}  \overline{→^{(h)λ_{g}^{x}}} (h)λ_{B}^{x}(Gg) ↓^{((h)λ_{B}^{x})(Gξ)}

we deduce that (C.11.3) holds on h. Therefore, condition 2 of Definition C.4 is satisfied. Let us verify condition 3(a) of Definition C.4, that is, equation

\begin{matrix}𝔄(A, B) & \overline{→^{\begin{matrix}1_{𝔄(A,B)} \\ 𝔄(A,1_{B}) \end{matrix}}} & 𝔄(A, B) \\ GB & ↓^{λ_{B}^{x}} & \overline{_{_{G1_{B}} →}} & ⟸^{λ_{1_{B}}^{x}} & GB & ↓^{λ_{B}^{x}} \end{matrix} = \begin{aligned}𝔄(A, B) & \overline{→^{1_{𝔄(A,B)}}} & 𝔄(A, B) \\ & = \\ GB & ↓^{λ_{B}^{x}} & \overline{^{1_{GB}} →} ; ⇓ ψ_{0} ; \overline{_{_{G1_{B}} →}} & GB & ↓^{λ_{B}^{x}} \end{aligned}.

On an object f : A → B of 𝔄(A, B) it reads:

(f)λ_{1_{B}}^{x} = ((f)λ_{B}^{x})ψ_{0} : (f)λ_{B}^{x} → (f)λ_{B}^{x}(G1_{B}).  (C.11.4)

It follows from condition (C.3.1) for G,

[Gf ^{Gf •ψ_{0}} → (Gf)(G1_{B}) ^{(f,1_{B})ψ_{2}} → Gf] = 1_{Gf} : Gf → Gf : GA → GB,

which, evaluated on x ∈ Ob GA, can be written as

(x)(f, 1_{B})ψ_{2}^{−1} = ((x)(Gf))ψ_{0} : (x)(Gf) → (x)(Gf)(G1_{B}).

This is precisely (C.11.4).

Let us verify condition 3(b) of Definition C.4, that is, equation

\begin{matrix}𝔄(A, C) \\ 𝔄(A, B) & \overline{_{_{𝔄(A,fg)} →}} & ^{𝔄(A,f)}  \overline{→} & ^{⇃⇂^{=} ‖} & 𝔄(A, D) & ^{^{𝔄(A,g)} \overline{→}} \\ GB & ↓^{λ_{B}^{x}} & \overline{_{_{G(fg)}  →}} & ⟸^{λ_{fg}^{x}} & GD & ↓^{λ_{D}} \end{matrix} = \begin{matrix}𝔄(A, C) \\ 𝔄(A, B) & ^{𝔄(A,f)}  \overline{→} & GC & ↓^{λ_{C}^{x}} & ⟸^{λ_{g}^{x}} & 𝔄(A, D) & ^{^{𝔄(A,g)} \overline{→}} \\ GB & ↓^{λ_{B}^{x}} & _{G(fg)}  → & ⟸^{λ_{f}^{x}} & ^{Gf}  \overline{→} & ^{^{ψ_{2}} ‖} & GD & ↓^{λ_{D}^{x}} & ^{^{Gg}  \overline{→}} \end{matrix}

(C.11.5)

for arbitrary pair of composable 1-morphisms B ^{f}→ C ^{g}→ D of 𝔄. We have to check this equation on an arbitrary 1-morphism h : A → B. Condition 4(b) of Definition C.3 for G is the equation

(Gh)(Gf)(Gg) \overline{→^{Gh•(f,g)ψ_{2}}} (Gh)•G(fg)

=

G(hf).Gg ↓^{(h,f)ψ_{2}•Gg}  \overline{^{(hf,g)ψ_{2}} →} G(hfg) ↓^{(h,fg)ψ_{2}}

Evaluating it on x we get the equation

(x)(h, fg)ψ_{2}^{−1} = [(x)(G(hfg)) ^{(x)(hf,g)ψ_{2}^{−1}} → (x)(G(hf))(Gg)

((x)(h,f)ψ_{2}^{−1})(Gg) → (x)(Gh)(Gf)(Gg) ^{((x)(Gh))(f,g)ψ_{2}} → (x)(Gh)(G(fg))],

which can be rewritten as

(h)λ_{fg}^{x} = [(hfg)λ_{D}^{x} ^{(hf)λ_{g}^{x}} → (hf)λ_{C}^{x}(Gg)

((h)λ_{f}^{x})(Gg) → (h)λ_{B}^{x}(Gf)(Gg) ^{((h)λ_{B}^{x})(f,g)ψ_{2}} → (h)λ_{B}^{x}(G(fg))].

And this is precisely (C.11.5), evaluated on h : A → B.

Therefore, all conditions of Definition C.4 are satisfied, and λ^{x} is a weak 2-transfor-mation.

Let us show that correspondence (C.11.2) defines a natural transformation. Indeed, for each 2-morphism ξ : f → g : A → C of 𝔄 the diagram

\begin{matrix}(f)λ_{C}^{x} = (x)(Gf) & \overline{→^{\begin{matrix}(f)λ_{C}^{u} \\ (u)(Gf) \end{matrix}}} & (y)(Gf) = (f)λ_{C}^{y} \\ (g)λ_{C}^{x} = (x)(Gg) & ↓^{(ξ)λ_{C}^{x}= (x)(Gξ)} & \overline{→^{\begin{matrix}(u)(Gg) \\ (g)λ_{C}^{u} \end{matrix}}} & (y)(Gg) = (g)λ_{C}^{y} & ↓^{(y)(Gξ) =(ξ)λ_{C}^{y}} \end{matrix}

commutes due to Gξ : Gf → Gg : GA → GC being a natural transformation.

We claim that property 2 of Definition C.5 holds for λ^{u}. For an arbitrary 1-morphism f : B → C of 𝔄 we have to prove the equation

\begin{matrix}𝔄(A, B) & \overline{→^{𝔄(A,f)}} & 𝔄(A, C) \\ GB & ↓^{λ_{B}^{y}}^{⇐^{λ_{B}^{u}}↓^{λ_{B}^{x}}} & \overline{^{Gf} →} & ⟸^{λ_{f}^{x}} & GC & ↓^{λ_{C}^{x}} \end{matrix} = \begin{matrix}𝔄(A, B) & \overline{→^{𝔄(A,f)}} & 𝔄(A, C) \\ GB & ↓^{λ_{B}^{y}} & \overline{^{Gf} →} & ⟸^{λ_{f}^{y}} & GC & ↓^{λ_{C}^{y}}^{⇐^{λ_{C}^{u}}↓^{λ_{C}^{x}}} \end{matrix} .

On the object g : A → B of 𝔄(A, B) this equation reads

(gf)λ_{C}^{x} = (x)(G(gf)) \overline{→^{\begin{matrix}(g)λ_{f}^{x} \\ (x)(g,f)ψ_{2}^{−1} \end{matrix}}} (x)(Gg)(Gf) = (g)λ_{B}^{x}(Gf)

=

(gf)λ_{C}^{y} = (y)(G(gf)) ↓^{(gf)λ_{C}^{u}= (u)(G(gf))} \overline{→^{\begin{matrix}(y)(g,f)ψ_{2}^{−1} \\ (g)λ_{f}^{y} \end{matrix}}} (y)(Gg)(Gf) = (g)λ_{B}^{y}(Gf) ↓^{(u)(Gg)(Gf) =(g)λ_{B}^{u}(Gf)}

It holds due to (g, f)ψ_{2}^{−1} : G(gf) → (Gg)(Gf) : GA → GC being a natural transformation. Therefore, λ^{u} is a modification.

The unit morphism 1_{x} : x → x of GA goes to the identity transformation

λ_{C}^{1_{x}} : (f : A → C) ⟼ ((1_{x})(Gf) = 1_{(x)(Gf)} : (x)(Gf) → (x)(Gf)),

because Gf is a functor. For a pair of composable morphisms x ^{u}→ y ^{v}→ z of GA we have λ_{C}^{uv} = (λ_{C}^{x} ^{λ_{C}^{u}}→ λ_{C}^{y} ^{λ_{C}^{v}}→ λ_{C}^{z}), since (uv)(Gf) = (u)(Gf) · (v)(Gf) due to Gf being a functor. Therefore, Λ is a functor.

C.12 Proposition. Functors

ev_{1_{A}} : [𝔄, 𝒞at](𝔄(A, ), G) → GA and Λ : GA → [𝔄, 𝒞at](𝔄(A, ), G)

are equivalences, quasi-inverse to each other.

Proof. We have

[GA ^{Λ}→ [𝔄, 𝒞at](𝔄(A, ), G) ^{ev_{1_{A}}} → GA] = G1_{A}.

Indeed, for any object x of GA

(x)Λ ev_{1_{A}} = (λ^{x}) ev_{1_{A}} = (1_{A})λ_{A}^{x} = (x)(G1_{A}),

for any morphism u : x → y of GA

(u)Λ ev_{1_{A}} = (λ^{u}) ev_{1_{A}} = (1_{A})λ_{A}^{u} = (u)(G1_{A}).

An isomorphism of functors ψ_{0} : 1_{GA} → G1_{A} implies that an arbitrary object x of GA is isomorphic to (x)(G1_{A}) = ((x)Λ) ev_{1_{A}}. Thus, ev_{1_{A}} is essentially surjective on objects. By Proposition C.10 ev_{1_{A}} is an equivalence. Therefore, Λ is isomorphic to a functor quasi-inverse to ev_{1_{A}}. Hence, Λ itself is an equivalence quasi-inverse to ev_{1_{A}}.

C.13. Example of strict 2-functor G = 𝔄(B, ). Applying Proposition C.11 to the strict 2-functor G = 𝔄(B, ) : 𝔄 → 𝒞at, we get the following. An arbitrary 1-morphism f : B → A gives rise to the strict 2-transformation f^{∗} = ^{𝔄(B,−)}λ^{A,f} : 𝔄(A, ) → 𝔄(B, ). It is specified by the family of functors f_{C}^{∗}, C ∈ Ob 𝔄:

f_{C}^{∗} = 𝔄(f, C) : 𝔄(A, C) ⟶ 𝔄(B, C)
(ϕ : A → C) ⟼ (f)𝔄(B, ϕ) = fϕ : B → C,
(π : ϕ → ψ : A → C) ⟼ (f)𝔄(B, π) = f •π : fϕ → fψ : B → C.

An arbitrary 2-morphism α : f → g : B → A gives rise to the modification α^{∗} = 𝔄(B,−)λ^{A,α} : f^{∗} → g^{∗} : 𝔄(A, ) → 𝔄(B, ) given by the family of natural transformations α_{C}^{∗} : f_{C}^{∗} → g_{C}^{∗} : 𝔄(A, C) → 𝔄(B, C), C ∈ Ob 𝔄. The transformation α_{C}^{∗} is specified by its components:

α_{C}^{∗} = 𝔄(α, C) : (ϕ : A → C) ⟼ (α)𝔄(B, ϕ) = α•ϕ : (ϕ)f_{C}^{∗} = fϕ → gϕ = (ϕ)g_{C}^{∗}.

By Proposition C.11 the correspondence f ↦ f^{∗}, α ↦ α^{∗} determines a functor Y_{AB} : 𝔄^{op}(A, B) = 𝔄(B, A) → [𝔄, 𝒞at](𝔄(A, ), 𝔄(B, )). One easily verifies that in fact we have a strict 2-functor Y : 𝔄^{op} → [𝔄, 𝒞at].

C.14 Corollary. Y is a local equivalence, i.e., for each pair of objects A, B ∈ Ob 𝔄^{op} the functor Y_{AB} is an equivalence.

C.15 Proposition. Let G : 𝔄 → 𝒞at be a weak 2-functor. Then the following conditions are equivalent:

1. there exists an object A of 𝔄 and a 2-natural equivalence λ : 𝔄(A, ) → G; 2. there exists an object A of 𝔄 and an object x of GA such that the weak 2-transfor-

mation λ^{x} : 𝔄(A, ) → G is a 2-natural equivalence.

Proof. Clearly, the second property implies the first one. Assume that condition 1) holds. By Proposition C.12 the weak 2-transformation λ is isomorphic to λ^{x} for some x ∈ Ob GA. By Proposition C.6 λ is a quasi-invertible 1-morphism of [𝔄, 𝒞at], hence, so is λ^{x}. By Remark C.8 condition 2) holds.

C.16 Definition. A weak 2-functor G : 𝔄 → 𝒞at is representable if it satisfies equivalent conditions of Proposition C.15. A pair (A, x) consisting of an object A of 𝔄 and an object x of GA is said to represent G, if λ^{x} = λ^{A,x} = ^{G}λ^{A,x} : 𝔄(A, ) → G is a 2-natural equivalence.

C.17. Uniqueness of the representing pair. Let two pairs (A, x) and (B, y) represent G. Then there is a quasi-inverse to λ^{B,y} : 𝔄(B, ) → G weak 2-transformation λ^{B,y−} : G → 𝔄(B, ). Define a 2-natural equivalence μ = λ^{A,x}•λ^{B,y−} : 𝔄(A, ) → 𝔄(B, ). It is isomorphic to the 2-transformation ^{𝔄(B,−)}λ^{A,f} for some f ∈ Ob 𝔄(B, A). There is an invertible modification m:

\begin{matrix}𝔄(A, ) & \overline{^{^{G}λ^{A,x}}  →} & G \\ 𝔄(B, ) & ^{^{m} ‖} & ^{^{G}λ^{B,y}} \overline{→} & ^{^{^{𝔄(B,−)}λ^{A,f}}  \overline{→}} \end{matrix}

Then (1_{A})m_{A} : (x)(G1_{A}) → (y)(Gf) is an isomorphism of GA. Therefore,

x = (x)(1_{GA}) ^{(x)ψ_{0}} → (x)(G1_{A}) ^{(1_{A})m_{A}} → (y)(Gf)

is an isomorphism of GA. By symmetry we get a 1-morphism g : A → B of 𝔄 and an isomorphism y ^{∼}→ (x)(Gg) of GB. By construction the strict 2-transformations

𝔄(B,−)λ^{A,f} : 𝔄(A, ) → 𝔄(B, ),
𝔄(A,−)λ^{B,g} : 𝔄(B, ) → 𝔄(A, )

are quasi-inverse to each other. In particular,

1_{A} ≃ ((1_{A})^{𝔄(B,−)}λ_{A}^{A,f})^{𝔄(A,−)}λ_{A}^{B,g} = ((f)𝔄(B, 1_{A}))^{𝔄(A,−)}λ_{A}^{B,g}

= (f)^{𝔄(A,−)}λ_{A}^{B,g} = (g)𝔄(A, f) = gf,

and by symmetry 1_{B} ≃ fg. Therefore, 1-morphisms f and g are quasi-inverse to each other.

Summing up, a pair (A ∈ Ob 𝔄, x ∈ Ob GA) representing a weak 2-functor G : 𝔄 → 𝒞at is unique up to equivalence f of the first objects, such that Gf preserves the second object up to an isomorphism.

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Contents

1 Introduction 2

1.1 Differential for A_{∞}-transformations compared with the Hochschild differ-

ential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Main result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Uniqueness of the representing A_{∞}-category. . . . . . . . . . . . . . . . . . 5 1.7 Basic properties of the main construction. . . . . . . . . . . . . . . . . . . 6 1.8 Description of various results. . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.10 Conventions and preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Quotients of free A_{∞}-categories 9 2.2 A_{∞}-functors from a quotient of a free A_{∞}-category. . . . . . . . . . . . . . 10 2.5 Transformations from a free A_{∞}-category. . . . . . . . . . . . . . . . . . . 11 2.8 Transformations from a quotient of a free A_{∞}-category. . . . . . . . . . . . 12

3 A simple example 14

4 A_{∞}-categories and quivers 16 4.2 A_{∞1}-functors and transformations. . . . . . . . . . . . . . . . . . . . . . . 17 4.4 A strict A_{∞}-2-functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.5 An A_{∞}-category freely generated over an A_{∞}-category. . . . . . . . . . . . 20

5 Relatively free A_{∞}-categories 28 5.2 The main construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.5 The first equivalence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Unitality of 𝒟 40 6.1 A multicategory operating in 𝒟. . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3 Examples of right derivations. . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 The example of complexes 48 A The Yoneda Lemma for unital A_{∞}-categories 49

A.1 Basic identities in symmetric closed monoidal category of complexes. … 49 A.2 A_{∞}-functor h^{X}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 A.4 The opposite A_{∞}-category. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 A.6 The Yoneda A_{∞}-functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A.8 The Yoneda embedding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

B Strict A_{∞}-2-functor. 70 B.1 An A_{∞}-functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 B.3 An A_{∞}-2-functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 B.6 Example of strict modification. . . . . . . . . . . . . . . . . . . . . . . . . 77

C The Yoneda Lemma for 2-categories 81 C.1 2-categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 C.9 The 2-functor 𝔄(A, ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 C.13 Example of strict 2-functor G = 𝔄(B, ). . . . . . . . . . . . . . . . . . . . 94 C.17 Uniqueness of the representing pair. . . . . . . . . . . . . . . . . . . . . . . 95