A∞-categories and the Yoneda lemma

Let C be the differential graded category of differential graded k-modules. We prove that the Yoneda A ∞ -functor Y : A → A ∞ (A,C) is a full embedding for an arbitrary unital A ∞ -category A. For a differential graded k-quiver B we define the free A ∞ -category FB generated by B. The main result is that the restriction A ∞ -functor A ∞ (FB,A) → A1(B,A) is an equivalence, where objects of the last A ∞ -category are morphisms of differential graded k-quivers B → A. A∞-categories defined by 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. [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 [LH02], 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 Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine; [email protected] Department of Algebra, Faculty of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 64 Volodymyrska st., Kyiv, 01033, Ukraine; [email protected]