Abelian categories are additive (see 1.6.4), which excludes many interesting situations in which, nevertheless, several “exactness properties” of abelian categories are still valid, like the existence of images (see 1.5.5). The notion of a regular category recaptures many “exactness properties” of abelian categories, but avoids requiring additivity. For example, the category of sets and most “algebraic-like” categories are regular.

Convention: as a matter of convention, in the present chapter, the symbol ↠ will denote a regular epimorphism; no special notation will be used for ordinary epimorphisms.

Exactness properties of regular categories

We recall that an epimorphism is regular when it can be written as the coequalizer of some pair of morphisms (see 4.3.1, volume 1). Regular epimorphisms and kernel pairs are closely related via the properties:

• if a regular epimorphism has a kernel pair, it is the coequalizer of that kernel pair (see 2.5.7, volume 1);

• if a kernel pair has a coequalizer, it is the kernel pair of that regular epimorphism (see 2.5.8, volume 1).

The reader will also remember from 4.3.6 of volume 1 that

• every regular epimorphism is strong.

The properties of strong epimorphisms have been studied in section 4.3, volume 1.

There exist in the literature many different definitions of regular categories, which are all equivalent under the assumptions that finite limits and coequalizers exist. We choose here a somehow “weakest” possible definition.