2 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, 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 allow to construct a 2-category of unital A∞-categories. The latter notion is our generalization of strictly unital A∞-categories (cf. Keller [Kel01]). We discuss also units in unital A∞-categories, which are certain natural A∞-transformations, and unital A∞-functors. 1. Conventions We assume that all classes and sets are small sets with respect to some universe. k denotes a unital associative commutative ring. By abuse of notation it denotes also a chain complex, whose 0-th component is k, and the other components vanish. It is easy to understand the line A(X0, X1)⊗k A(X1, X2)⊗k · · · ⊗k A(Xn−1, Xn), and it is much harder to understand the order in A(Xn−1, Xn)⊗k · · · ⊗k A(X1, X2)⊗k A(X0, X1). That is why we use the right operators: the composition of two maps (or morphisms) f : X → Y and g : Y → Z is denoted fg : X → Z; a map is written on elements as f : x 7→ xf = (x)f . However, these conventions are not used systematically, and f(x) might be used instead. 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