Regular category

Remarks

A regular epimorphism is an epimorphism. In a regular category every morphism $f$ factors as $f = m \circ e$ with $e$ a regular epimorphism and $m$ a monomorphism. Here $e$ is the coequalizer of the kernel pair of $f$, and $m$ is called the image of $f$.

Example: the category of sets is regular. It has all finite limits, and the kernel pair of $f \colon A \to B$ is the relation $\set{(a, a') : f(a) = f(a')}$, whose coequalizer is the quotient of $A$ by it. Regular epimorphisms are the surjections, and a pullback of a surjection is a surjection.

Non-example: the category of topological spaces satisfies clauses 1 and 2 but not clause 3. Its regular epimorphisms are the quotient maps, and a pullback of a quotient map need not be a quotient map.