3 J an 2 00 3 A construction of an A ∞ - category

We construct an A ∞ -category D(C|B) from a given A ∞ -category C and its full subcategory B. The construction resembles a particular case of Drinfeld’s quotient of differential graded categories [Dri02]. We use D(C|B) to construct an A ∞ -functor of K-injective resolutions of a complex. The conventional derived category is obtained as the 0-th cohomology of the quotient of the differential graded category of complexes over acyclic complexes. We continue to study the 2-category of A∞-categories introduced in [Lyu02]. Our main subject is a quotient-like A∞-category D(C|B) obtained from a given A∞-category C and its full subcategory B. Originally it has been defined by Drinfeld for differential graded categories [Dri02]. Bondal and Kapranov proposed to produce triangulated categories out of differential graded categories [BK90]. Drinfeld’s construction deals with their quotients, in particular, it produces derived categories. The usefulness of A∞-approach is explained by our construction of an A∞-functor, which assigns to a complex its K-injective resolution, when the ground ring is a field. Plan of the article with comments. In the first section we describe conventions and notations used in the article. In particular, we recall some conventions and useful formulas from [Lyu02]. In the second section we describe a construction of an A∞-category D(C|B), departing from an A∞-category B fully embedded into an A∞-category C. When B ⊂ C are differential graded categories, D(C|B) is the first Drinfeld’s construction of a quotient category, and under additional assumptions it, indeed, gives a quotient C/B [Dri02]. The underlying quiver of D(C|B) is described in Definition 2.1. Its particular case D(C|C) is sTsC = ⊕n>0s T sC, where sC = C[1] stands for suspended quiver C. We introduce two A∞-category structures for s TsC. The first, C = (sTsC, b) uses the Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine, [email protected] The research of V. L. was supported in part by grant 01.07/132 of State Fund for Fundamental Research of Ukraine Department of Algebra, Faculty of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 64 Volodymyrska st., Kyiv, 01033, Ukraine, [email protected]