A Galois Theory with Stable Units for Simplicial Sets

We recall and reformulate certain known constructions, in order to make a convenient setting for obtaining generalized monotone-light factorizations in the sense of A. Carboni, G. Janelidze, G. M. Kelly and R. Paré. This setting is used to study the existence of monotone-light factorizations both in categories of simplicial objects and in categories of internal categories. It is shown that there is a non-trivial monotone-light factorization for simplicial sets, such that the monotone-light factorization for reflexive graphs via reflexive relations is a special case of it, obtained by truncation. More generally, we will show that there exists a monotone-light factorization associated with every full subcategory Mono(Fn), n ≥ 0, consisting of all simplicial sets whose unit morphisms are monic for the localization Fn : Set op → Set∆op n , which truncates each simplicial set after the object of n-simplices. The monotone-light factorization for categories via preorders is as well derived from the proposed setting. We also show that, for regular Mal’cev categories, the reflection of internal groupoids into internal equivalence relations necessarily produces monotone-light factorizations. It turns out that all these reflections do have stable units, in the sense of C. Cassidy, M. Hébert and G. M. Kelly, giving rise to Galois theories.