The notion of category over an algebraic stack

TL;DR

This paper clarifies the concept of abelian categories over algebraic stacks, aiming to formalize a well-understood idea and setting the stage for future work on triangulated categories in similar contexts.

Contribution

It explicitly defines the notion of abelian category over an algebraic stack, providing a foundation for further exploration of triangulated categories over stacks.

Findings

Clarification of abelian categories over algebraic stacks

Establishment of a formal framework for future research

Preparation for extending to triangulated categories

Abstract

The goal of this note is to spell out the (apparently well-known and intuitively clear) notion of abelian category over an algebraic stack. In the future we will discuss the (much less evident) notion, when instead of an abelian category one considers a triangulated one.