Comparison of the definitions of Abelian 2-categories

TL;DR

This paper compares different definitions of abelian 2-categories, aiming to unify and clarify their relationships within higher category theory.

Contribution

It systematically compares the notions of relatively exact 2-categories and (2-)abelian Gpd-categories, clarifying their connections.

Findings

Identifies relationships between different abelian 2-category definitions

Unifies various approaches in higher category theory

Clarifies the structure of abelian 2-categories

Abstract

In the efforts to define a 2-categorical analog of an abelian category, two (or three) notions of "abelian 2-categories" are defined. One is the relatively exact 2-category, and the other(s) is the (2-)abelian Gpd-category. We compare these notions. Since they proceed independently in their own way, in different settings and terminologies, it will be worth while to collect and unify them. In this paper, by comparing their definitions and arguments, we show the relationship among these classes of 2-categories.