3 J an 2 00 3 Category of A ∞ - categories Volodymyr

We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta63, II] Stasheff defines also A∞-algebras and their homotopy-bar constructions. These algebras and their applications to topology were actively studied, for instance, by Smirnov [Smi80] and Kadeishvili [Kad80, Kad82]. We adopt some notations of Getzler and Jones [GJ90], which reduce the number of signs in formulas. The notion of an A∞-category is a natural generalization of A∞-algebras. It arose in connection with Floer homology in Fukaya’s work [Fuk93, Fuk] and was related by Kontsevich to mirror symmetry [Kon95]. See Keller [Kel01] for a survey on A∞-algebras and categories. In the present article we show that 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 natural A∞-transformations between such functors. This result was also obtained by Fukaya [Fuk] and by Kontsevich and Soibelman [KS], independently and, apparently, earlier. We describe compositions between such categories of A∞-functors, which would allow us to construct a 2-category of unital A∞-categories. The latter notion is our generalization of strictly unital A∞-categories (cf. Keller [Kel01]). We also discuss unit elements in unital A∞-categories, unital natural A∞-transformations, and unital A∞-functors. Plan of the article with comments and explanations. The first section describes some notation, sign conventions, composition convention, etc. used in the article. The Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine The research was supported in part by grant 01.07/132 of State Fund for Fundamental Research of Ukraine