1 5 Fe b 20 08 Quotients of unital A ∞ - categories

Assuming that B is a full A∞-subcategory of a unital A∞-category C we construct the quotient unital A∞-category D =‘C/B’. It represents the A u ∞-2-functor A 7→ A∞(C,A)modB, which associates with a given unital A∞-category A the A∞-category of unital A∞-functors C → A, whose restriction to B is contractible. Namely, there is a unital A∞-functor e : C → D such that the composition B →֒ C e −→ D is contractible, and for an arbitrary unital A∞-category A the restriction A∞-functor (e⊠ 1)M : A u ∞(D,A) → A u ∞(C,A)modB is an equivalence. Let Ck be the differential graded category of differential graded k-modules. We prove that the Yoneda A∞-functor Y : A → A u ∞(A op,C k ) is a full embedding for an arbitrary unital A∞-category A. In particular, such A is A∞-equivalent to a differential graded category with the same set of objects. Let A be an Abelian category. The question: what is the quotient {category of complexes in A}/{category of acyclic complexes}? admits several answers. The first answer – the derived category of A – was given by Grothendieck and Verdier [Ver77]. The second answer – a differential graded category D – is given by Drinfeld [Dri04]. His article is based on the work of Bondal and Kapranov [BK90] and of Keller [Kel99]. The derived category D(A) can be obtained as H(D). The third answer – an A∞-category of bar-construction type – is given by Lyubashenko and Ovsienko [LO06]. This A∞-category is especially useful when the basic ring k is a field. It is an A∞-version of one of the constructions of Drinfeld [Dri04]. The fourth answer – an A∞-category freely generated over the category of complexes in A – is given in this article. It is A∞-equivalent to the third answer and enjoys certain universal property of the quotient. Thus, it passes this universal property also to the third answer. Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine; [email protected] Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany; [email protected]