1 7 Fe b 20 08 Free A ∞ - categories

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 Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine; [email protected] Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany; [email protected]