Regular category
In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.
Definition
A category C is called regular if it satisfies the following three properties:
C is finitely complete.
If f:X→Y is a morphism in C, and
If f:X→Y is a morphism in C, and
Examples
Examples of regular categories include:
Set, the category of sets and functions between the sets
More generally, every elementary topos
Grp, the category of groups and group homomorphisms
The category of rings and ring homomorphisms
More generally, the category of models of any variety
Every bounded meet-semilattice, with morphisms given by the order relation
