Stacks associated to abelian tensor categories

TL;DR

This paper constructs stacks from abelian tensor categories and demonstrates their use in recovering schemes, dual stacks, and dual pairs of G-gerbes through categorical and glueing techniques.

Contribution

It introduces a method to associate stacks to abelian tensor categories and applies this to recover schemes and analyze dual stacks and G-gerbes.

Findings

Reconstructed schemes from categories of quasi-coherent sheaves.

Derived dual stacks of classifying stacks of finite groups.

Constructed dual pairs of G-gerbes via glueing local dual stacks.

Abstract

For an abelian tensor category a stack is constructed. As an application we show that our construction can be used to recover a quasi-compact separated scheme from the category of its quasi-coherent sheaves. In another application, we show how the "dual stack" of the classifying stack $BG$ of a finite group $G$ can be obtained by altering the tensor product on the category $\rep{G}$ of $G$-representations. Using glueing techniques we show that the dual pair of a $G$-gerbe, in the sense of [TT10], can be constructed by glueing local dual stacks.