Monotone-light factorizations, very-well-behaved epireflections and categories of models of sketches

The genesis of this work is in [1]. Its main results show how to obtain the monotone-light factorization from n-categories via n-preorders ($n\ge 1$), using sketches of presheaves, without resorting to some more complicated tools or $𝒱$-categories (as in [5] and [6]). There are also some other new results here (cf. [7]), some as displayed below. The provided theoretical framework will hopefully be applied to new settings, as the algebraic one for instance.

Firstly, precise conditions on how to obtain very-well-behaved epireflections are explored and improved from the author’s previous papers [3] and [4]; 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.

Secondly, as a first application of the conditions above, deriving from adjunctions given by left Kan extensions for presheaves, we will show that, for a pseudo-filtered category $𝕁$ in which every arrow is a monomorphism, the colimit functor on $Se{t}^{𝕁}$ produces a very-well-behaved epireflection; astonishingly, in the very simple case with $𝕁=\mathrm{𝟐}$, the monotone-light factorization is non-trivial.

Thirdly and more importantly, 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).