On the derived DG functors

Abstract

Assume that abelian categories A, B over a field admit countable direct limits and that these limits are exact. Let F : D dg(A)→ D + dg(B) be a DG quasi-functor such that the functor Ho(F) : D+(A)→ D+(B) carries D≥0(A) to D≥0(B) and such that, for every i > 0, the functor HiF : A → B is effaceable. We prove that F is canonically isomorphic to the right derived DG functor RH0(F). We also prove a similar result for bounded derived DG categories and a formula that expresses Hochschild cohomology of the categories Db dg(A), D + dg(A) as the Ext groups in the abelian category of left exact functors A → IndA . The proofs are based on a description of Drinfeld’s category of quasi-functors as the derived category of a category of sheaves.