Split epimorphisms and regular ones.

Question

It is well known that any split epimorphism is in particular a regular one, but the converse is not true in arbitrary categories. Is there any meaningful context where we have that the converse is always true ? At first, I thought about regular or abelian categories but it is not the case.

Answer 1

Some of those categories are called semisimple, which is quite a strong property! Which additionally demands every object to be a sum of its simples. For example finite dimensional vector spaces and finite sets have that property.

Also algebraic objects are usually called semisimple if the category of finitely generated modules over them is semisimple.