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.
Keywords: (reflexive) graph, (reflexive) relation, category, preorder, factorization system, localization, stabilization, descent theory, Galois theory, monotone-light factorization
2000 MSC: 18A32, 12F10
Theory and Applications of Categories,
Vol. 13, 2004,
No. 15, pp 235-251.