Remarks on partially abelian exact categories

TL;DR

This paper clarifies the relationships between various classes of exact categories, showing that certain conditions characterize abelian, quasi-abelian, and Laumon-type additive categories, with all morphisms having kernels and coimages.

Contribution

It establishes equivalences between classes of exact categories and well-known categories like abelian, quasi-abelian, and Laumon-type categories, clarifying their structural properties.

Findings

Partially abelian exact categories are precisely abelian categories.

AIC and AIC{ extdegree} axioms characterize quasi-abelian categories.

AIC axiom characterizes certain additive categories related to filtered ${ m D}$-modules.

Abstract

The purpose of this short and elementary note is to identify some classes of exact categories introduced in L. Previdi's thesis. Among other things we show: (1) An exact category is partially abelian exact if and only if it is abelian. (2) An exact category satisfies the axioms AIC and AIC{\deg} if and only if it is quasi-abelian in the sense of J.-P. Schneiders. (3) An exact category satisfies AIC if and only if it is an additive category of the type considered by G. Laumon in his work on derived categories of filtered ${\cal D}$-modules. In all of the above classes all morphisms have kernels and coimages and the exact structure must be given by all kernel-cokernel pairs.