Comparison of the definitions 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 in [4] and [2]. One is the relatively exact 2-category defined in [4], and the other(s) is the (2-)abelian Gpd-category defined by Dupont [2]. We compare these notions, using the arguments in [4] and [2]. 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.