---
tag: 2SM3JRJXA4
kind: concept
name: Regular category
msc: ["18E08"]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_category
same_as:
  wikidata: Q2844249
  wikipedia: Regular category
---

## Summary

A regular category is a finitely complete category in which every morphism has a well-behaved image.
Precisely, coequalizers of kernel pairs exist and regular epimorphisms are stable under pullback.
In a regular category every morphism factors as a regular epimorphism followed by a monomorphism, and relations can be composed.
Sets, groups, modules, every abelian category and every topos are regular, and regular categories are the semantics of regular logic.

## Explanations

### L1 human

In sets, every function factors through its image: first onto the image, then into the target.
A regular category is one where this factorization exists for all morphisms and behaves well when one restricts to a part of the target.
This is what is needed to talk about images, to compose relations, and to interpret statements built from "and", "true" and "there exists".
The category of groups is regular, while the category of topological spaces is not.