Category of a ∞ -categories

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. 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