Internal monotone-light factorization for categories via preorders

Abstract

It is shown that, for a finitely-complete category C with coequalizers of kernel pairs, if every product-regular epi is also stably-regular then there exist the reflections (R)Grphs(C) → (R)Rel(C), from (reflexive) graphs into (reflexive) relations in C, and Cat(C) → Preord(C), from categories into preorders in C. Furthermore, such a sufficient condition ensures as well that these reflections do have stable units. This last property is equivalent to the existence of a monotone-light factorization system, provided there are sufficiently many effective descent morphisms with domain in the respective full subcategory. In this way, we have internalized the monotone-light factorization for small categories via preordered sets, associated with the reflection Cat → Preord, which is now just the special case C = Set.