Gluing derived equivalences together (Q1941146)

The Grothendieck construction of a diagram \(X\) of categories can be seen as a process to construct a single category \(\text{Gr}(X)\) by gluing categories in the diagram together. In this paper, the author formulates diagrams of categories as colax functors from a small category \(I\) to the 2-category \(\Bbbk\text{-Cat}\) of small \(\Bbbk\)-categories for a fixed commutative ring \(\Bbbk\). Roughly speaking two colax functors \(X, X' : I \to \Bbbk\text{-Cat}\) are derived equivalent if there is a derived equivalence from \(X(i)\) to \(X'(i)\) for all objects \(i\) in \(I\) satisfying some ``\(I\)-equivariance'' conditions. In this paper, the author glues the derived equivalences between \(X(i)\) and \(X'(i)\) together to obtain a derived equivalence between Grothendieck constructions \(\text{Gr}(X)\) and \(\text{Gr}(X')\), 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 \(\Bbbk\)-categories with a \(G\)-action for a group \(G\) are ``\(G\)-equivariantly'' derived equivalent, then their orbit categories are derived equivalent. As an easy application one can see by a unified proof that, if two \(\Bbbk\)-algebras \(A\) and \(A'\) are derived equivalent, then so are the path categories \(AQ\) and \(A'Q\) for any quiver \(Q\); so are the incidence categories \(AS\) and \(A'S\) for any poset \(S\); and so are the monoid algebras \(AG\) and \(A'G\) for any monoid \(G\). Also examples of gluing of many smaller derived equivalences together to have a larger derived equivalence are given.