Gluing derived equivalences together

TL;DR

This paper demonstrates how to combine derived equivalences of categories in a diagram to establish derived equivalences of their Grothendieck constructions, generalizing known results and providing new methods for constructing such equivalences.

Contribution

It formulates a method to glue derived equivalences of categories in a diagram to obtain derived equivalences of their Grothendieck constructions, extending previous results.

Findings

Derived equivalences of categories can be glued to produce equivalences of their Grothendieck constructions.

If two algebras are derived equivalent, their path, incidence, and monoid categories are also derived equivalent.

Provides examples of constructing larger derived equivalences from smaller ones.

Abstract

The Grothendieck construction of a diagram $X$ of categories can be seen as a process to construct a single category $\Gr(X)$ by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category $I$ to the 2-category $\kCat$ of small $\k$-categories for a fixed commutative ring $\k$. In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors $X, X' \colon I \to \kCat$ 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 we glue the derived equivalences between $X(i)$ and $X'(i)$ together to obtain a derived equivalence between Grothendieck constructions $\Gr(X)$ and $\Gr(X')$, 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 $\k$-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 we 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 we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence.