categorical property: every morphism can be decomposed into an epimorphism and monomorphism

Question

Certain categories have the property that for every morphism $f:A\to B$ can be decomposed into an epimorphism and monomorphism. with some intermediate object $C$. e.g. Grp has this property.

Is there a name for this so I can look it up?

Answer 1

Most commonly such factorizations exist where the epimorphism is even regular; in most interesting categories not every epi is regular. A category with nice (regular epi,mono) factorizations is, fittingly, called a regular category.

The key property making these factorization systems interesting is that every regular epi is “left orthogonal” to every mono. Orthogonal factorization systems are very interesting in their own right, as are their weakenings which play a central role in homotopical algebra. In exceptionally special categories like groups and sets, arbitrary epimorphisms manage to be left orthogonal to monos, and that is the situation described at Henno’s link.