2 00 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, (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 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 unit elements in unital A∞-categories, unit 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 ground commutative ring k is not assumed to be a field. This is suggested by development of homological algebra [Dri02]. Working over a ring k instead of a field has strong consequences. For instance, one may not hope for Kadeishvili’s theorem [Kad82] to hold for all A∞-algebras over k. In the second section we recall or give definitions of the main objects. A k-quiver is such a graph that the set of arrows (morphisms) between two vertices (objects) is a k-module (Definition 2.1). We view quivers as categories without multiplication and units. Cocategories are k-quivers and k-coalgebras with a matrix type decomposition into k-submodules, indexed by pairs of objects (Definition 2.2). A∞-categories are defined as a special kind of differential graded cocategories – the ones of the form of tensor cocategory TA of a k-quiver A (Definition 2.3). A∞-functors are homomorphisms of cocategories that commute with the differential (Definition 2.4). A∞-transformations between 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