Gluing derived equivalences together

The Grothendieck construction of a diagram XX of categories can be seen as a process to construct a single category Gr(X)Gr(X) by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category II to the 2-category k-Cat of small kk-categories for a fixed commutative ring kk. In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors X,X′:I→k-Cat are derived equivalent if there is a derived equivalence from X(i)X(i) to X′(i)X′(i) for all objects ii in II satisfying some “II-equivariance” conditions. In this paper we glue the derived equivalences between X(i)X(i) and X′(i)X′(i) together to obtain a derived equivalence between Grothendieck constructions Gr(X)Gr(X) and Gr(X′)Gr(X′), which shows that if colax functors are derived equivalent, then so are their Grothendieck constructions. This generalizes and well formulates the fact that if two kk-categories with a GG-action for a group GG are “GG-equivariantly” derived equivalent, then their orbit categories are derived equivalent. As an easy application we see by a unified proof that if two kk-algebras AA and A′A′ are derived equivalent, then so are the path categories AQAQ and A′QA′Q for any quiver QQ; so are the incidence categories ASAS and A′SA′S for any poset SS; and so are the monoid algebras AGAG and A′GA′G for any monoid GG. Also we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence.