Gluing of derived equivalences of dg categories

Abstract

A diagram consisting of differential graded (dg for short) categories and dg functors is formutated as a colax functor X from a small category I to the 2category k-dgCat of dg categories, dg functors and natural transformations for a fixed commutative ring k. The dg categories X(i) with i objects of I can be glued together to have a single dg category Gr(X), called the Grothendieck construction of X. In this paper, we consider colax functors X and X ′ from I to k-dgCat such that X(i) and X ′ (i) are derived equivalent for all objects i of I, and give a way to glue these derived equivalences together and a sufficient condition for this gluing to be a derived equivalence between their Grothendieck constructions Gr(X) and Gr(X ′ ). This generalizes the main result of [8] to the dg case. Finally, we give some examples to illustrate our main theorem.