Gluing derived equivalences together

Abstract: The Grothendieck construction of a diagram of categories can be seen as a process to construct a single category by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category to the 2-category of small -categories for a fixed commutative ring . In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors are derived equivalent if there is a derived equivalence from to for all objects in satisfying some “-equivariance” conditions. In this paper we glue the derived equivalences between and together to obtain a derived equivalence between Grothendieck constructions and , 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 -categories with a -action for a group are “-equivariantly” derived equivalent, then their orbit categories are derived equivalent. As an easy application we see by a unified proof that if two -algebras and are derived equivalent, then so are the path categories and for any quiver ; so are the incidence categories and for any poset ; and so are the monoid algebras and for any monoid . Also we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence. [Copyright &y& Elsevier]