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.