Internal monotone-light factorization for categories via preorders

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

Well-behaved Epireflections for Kan Extensions

Let K : B -> A be a functor such that the image of the objects in B is a cogenerating set of objects for A. Then, the right Kan extensions adjunction Set(K) (sic) Ran(K) induces necessarily an epireflection with stable units and a monotone-light factorization. This result follows from the one stating that an adjunction produces an epireflection in a canonical way, provided there exists a prefactorization system which factorizes all of its unit morphisms through epimorphisms. The stable units property, for the so obtained epireflections, is thereafter equivalently restated in such a manner it is easily recognizable in the examples. Furthermore, having stable units, there is a strong but quite simple sufficient condition for the existence of an associated monotone-light factorization, which has proved to be fruitful