O ct 2 00 6 A construction of quotient A ∞ - categories

We construct an A ∞ -category D(C|B) from a given A ∞ -category C and its full subcategory B. The construction is similar to a particular case of Drinfeld’s construction of the quotient of differential graded categories [Dri04]. We use D(C|B) to construct an A ∞ -functor of K-injective resolutions of a complex, when the ground ring is a field. The conventional derived category is obtained as the 0-th cohomology of the quotient of the differential graded category of complexes over acyclic complexes. This result follows also from Drinfeld’s theory of quotients of differential graded categories [Dri04]. In [Dri04] Drinfeld reviews and develops Keller’s construction of the quotient of differential graded categories [Kel99] and gives a new construction of the quotient. This construction consists of two parts. The first part replaces given pair B ⊂ C of a differential graded category C and its full subcategory B with another such pair B̃ ⊂ C̃, where C̃ is homotopically flat over the ground ring k (K-flat) [Dri04, Section 3.3], and there is a quasi-equivalence C̃ → C [Dri04, Section 2.3]. The first step is not needed, when C is already homotopically flat, for instance, when k is a field. In the second part a new differential graded category C/B is produced from a given pair B ⊂ C, by adding to C new morphisms εU : U → U of degree −1 for every object U of B, such that d(εU) = idU . In the present article we study an A∞-analogue of the second part of Drinfeld’s construction. Namely, to a given pair B ⊂ C of an A∞-category C and its full subcategory B we associate another A∞-category D(C|B) via a construction related to the bar resolution of C. The A∞-category D(C|B) plays the role of the quotient of C over B in some cases, for instance, when k is a field. When C is a differential graded category, D(C|B) is precisely the category C/B constructed by Drinfeld [Dri04, Section 3.1]. There exists another construction of quotient A∞-category q(C|B), see [LM04]. It enjoys some universal property, and is significantly bigger in size than D(C|B). However, 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]