Remarks on exactness notions pertaining to pushouts

TL;DR

This paper explores the properties of diexact categories, establishing their equivalence to pretoposes and Barr-exact categories with specific pushout conditions, and characterizing their embeddings into Grothendieck toposes.

Contribution

It provides new characterizations of diexact categories, linking them to well-known categorical structures and embedding properties.

Findings

A category is a pretopos iff it is diexact with a strict initial object.

A category is diexact iff it is Barr-exact with stable pushouts of monomorphisms.

Small diexact categories with certain pushouts embed into Grothendieck toposes.

Abstract

We call a finitely complete category diexact if every Mal'cev relation admits a pushout which is stable under pullback and itself a pullback. We prove three results relating to diexact categories: firstly, that a category is a pretopos if and only if it is diexact with a strict initial object; secondly, that a category is diexact if and only if it is Barr-exact, and every pair of monomorphisms admits a pushout which is stable and a pullback; and thirdly, that a small category with finite limits and pushouts of Mal'cev spans is diexact if and only if it admits a full structure-preserving embedding into a Grothendieck topos.