Unifying exact completions

TL;DR

This paper introduces a generalized notion of exact completion within existential elementary doctrines, unifying existing concepts for regular and cartesian categories with weak pullbacks.

Contribution

It defines a new framework for exact completion relative to existential elementary doctrines and shows how it encompasses known exact completion constructions.

Findings

The forgetful functor from exact categories to doctrines has a left biadjoint.

The new notion unifies exact completions of regular and cartesian categories with weak pullbacks.

The framework provides a broader understanding of exact completions in category theory.

Abstract

We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2-category exact categories to existential elementary doctrines has a left biadjoint that can be obtained as a composite of two others. Finally, we conclude how this notion encompasses both that of the exact completion of a regular category as well as that of the exact completion of a cartesian category with weak pullbacks.