Nice exact categories are coexact

TL;DR

The paper explores conditions under which categories are both exact and coexact, extending known results for abelian categories and elementary toposes to a broader class of categories.

Contribution

It introduces a weaker condition than extensivity, equivalent to additivity, that ensures coexactness and coprotomodularity in finitely cocomplete categories.

Findings

Finitely cocomplete categories with the new condition are coexact.

A finitely cocomplete pretopos is coexact.

The study generalizes known duality results for abelian categories and toposes.

Abstract

Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by Tierney's characterization of them as additive exact categories. The next type consists of elementary toposes which are well-known to be exact, but have also been shown to be coexact and coprotomodular by Bourn. In this paper we study a condition weaker than extensivity and equivalent to additivity for pointed categories. We show that for a finitely cocomplete category this condition together with exactness implies coexactness and coprotomodularity. As a special case we obtain that a finitely cocomplete pretopos is coexact.