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. 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.
Homology, Homotopy and Applications, Vol. 8 (2006), No. 2, pp.157-203.