Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.

1.1 A∞-algebras as spaces........................ 2

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 Au ∞-2-functor A ↦ → Au ∞(C, A)mod B, which associates with a given unital A∞-category A the A∞-category

Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A∞-functor Y: A op → A∞(A,C) is a full embedding for an arbitrary unital A∞-category A. 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∞-categories A and B, one can construct a third A∞-category A∞(A, B), whose objects are A∞-functors f: A → B, and morphisms are A∞-transformations (Fukaya [Fuk02], Kontsevich and Soibelman [KS02, KS], Lefèvre-Hasegawa [LH03], as well as [Lyu03]). For an A∞-category there is a notion of units up to a homotopy (homotopy identity morphisms) [Lyu03]. This allows to define a 2-category, whose objects are unital A∞-categories, 1-morphisms are unital A∞-functors and 2-morphisms are equivalence classes of natural A∞-transformations [Lyu03]. We continue to study this 2-category. The notations and conventions are explained in the first section.

For a differential graded k-quiver Q we define the free A∞-category FQ generated by Q. The main result is that the restriction A∞-functor A∞(FQ,A) → A1(Q,A) is an equivalence, where objects of the last A∞-category are morphisms of differential graded k-quivers Q → A. A∞-categories defined by Fukaya [Fuk93] and Kontsevich [Kon95] are generalizations of differential graded categories for which the binary composition is associative only up to a homotopy. They also generalize A∞-algebras introduced by Stasheff [Sta63, II]. A∞-functors are the corresponding generalizations of usual functors, see e.g. [Fuk93, Kel01]. Homomorphisms of A∞-algebras (e.g. [Kad82]) are particular cases of A∞-functors. A∞-transformations are certain coderivations. Examples of such structures are encountered in studies of mirror symmetry (e.g. [Kon95, Fuk02]) and in homological algebra. For an A∞-category there is a notion of units up to a homotopy (homotopy identity morphisms) [Lyu03]. Given two A∞-categories A and B, one can construct a third A∞-category A∞(A, B), whose objects are A∞-functors f: A → B, and morphisms are A∞-transformations between such functors (Fukaya [Fuk02], Kontsevich and Soibelman [KS02, KS], Lefèvre-Hasegawa [LH03], as well as [Lyu03]). This allows to define a 2-category, whose objects are unital A∞-categories, 1-morphisms are unital A∞-functors and 2-morphisms are equivalence classes of natural A∞-transformations [Lyu03]. We continue to study this 2-category. The notations and conventions are explained in the first section. We also describe AN-categories, AN-functors and AN-transformations – truncated at N < ∞ versions of A∞-categories. For instance, A1-categories and A1-functors are differential graded k-quivers and their morphisms. However, A1-transformations bring new 2-categorical features to the theory. In particular, for any differential graded k-quiver Q and any A∞-category

Abstract. We prove that three definitions of unitality for A∞-categories suggested by the first author, by Kontsevich and Soibelman, and by Fukaya are equivalent.