Two-dimensional regularity and exactness

TL;DR

This paper introduces three new notions of regularity and exactness for 2-categories, based on different factorizations of functors, and develops an abstract theory extending previous work in the area.

Contribution

It defines and justifies three new notions of regularity and exactness for 2-categories, extending the theory with a kernel--quotient factorisation framework.

Findings

Validated the correctness of the new notions using lex colimits.

Extended the theory of regularity and exactness to a kernel--quotient context.

Provided a unified approach based on canonical functor factorizations.

Abstract

We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel--quotient factorisation, extending earlier work of Street and others.