Very-Well-Behaved Epireflections for Categories of Models of Sketches

Firstly, precise conditions on how to obtain very-well-behaved epireflections are explored and improved from the author's previous papers; meaning that, beginning with a monad and a prefactorization system on a category, is produced a reflection with stable units (stronger than semi-left-exactness, also called admissibility in categorical Galois Theory) and an associated monotone-light factorization. Then, we were able to show that, for a pseudo-filtered category J in which every arrow is a monomorphism, the colimit functor on Set^J produces a very-well-behaved epireflection; if J = 2 the monotone-light factorization is non-trivial, as showed as an example. Then, new results are presented that grant very-well-behaved subreflections from the very-well-behaved reflections induced by an adjunction given by right Kan extensions for presheaves. These subreflections are obtained by restricting to the models of a sketch; it is showed finally that the known very-well-behaved reflection of n-categories into n-preorders is an example of this process (being n any positive integer).