Intersections, sums, and the Jordan-H\"older property for exact categories

TL;DR

This paper explores how intersection and sum concepts extend to exact categories, providing new characterizations of quasi-abelian and abelian categories, and introduces classes of exact categories with the Jordan-H"older property.

Contribution

It introduces generalized notions of intersection and sum in exact categories, characterizes Artin-Wedderburn exact structures, and links these to the Jordan-H"older property.

Findings

Characterization of quasi-abelian categories via admissible intersections

Identification of Artin-Wedderburn exact structures with the Jordan-H"older property

Explicit description of exact structures on Nakayama algebra representations

Abstract

We investigate how the concepts of intersection and sums of subobjects carry to exact categories. We obtain a new characterisation of quasi-abelian categories in terms of admitting admissible intersections in the sense of Hassoun and Roy. There are also many alternative characterisations of abelian categories as those that additionally admit admissible sums and in terms of properties of admissible morphisms. We then define a generalised notion of intersection and sum which every exact category admits. Using these new notions, we define and study classes of exact categories that satisfy the Jordan-H\"older property for exact categories, namely the Diamond exact categories and Artin-Wedderburn exact categories. By explicitly describing all exact structures on $\mathcal{A}= \mbox{rep}\, \Lambda$ for a Nakayama algebra $\Lambda$ we characterise all Artin-Wedderburn exact structures on $\mathcal{A}$ and show that these are precisely the exact structures with the Jordan-H\"{o}lder property.