Abelian categories versus abelian 2-categories

TL;DR

This paper explores the relationship between abelian categories and abelian 2-categories, providing a partial answer to Dupont's question about their equivalence under certain conditions.

Contribution

It offers a simple solution to Dupont's problem for abelian categories with enough projective or injective objects.

Findings

Categories of discrete and codiscrete objects in an abelian 2-category are equivalent to abelian categories.

Provides a partial answer to Dupont's question about the characterization of abelian categories.

Highlights conditions under which abelian categories can be realized as categories of objects in abelian 2-categories.

Abstract

Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.