The monotone-light factorization for 2-categories via 2-prorders

Abstract

It is shown that the reflection 2Cat --> 2Preord of the category of all 2-categories into the category of 2-preorders determines a monotone-light factorization system on 2Cat and that the light morphisms are precisely the 2-functors faithful on 2-cells with respect to the vertical structure. In order to achieve such result it was also proved that the reflection 2Cat --> 2Preord has stable units, a stronger condition than admissibility in categorical Galois theory, and that 2-functors surjective both on horizontally composable triples of vertically composable pairs and on vertically composable triples of horizontally composable pairs of 2-cells are effective descent morphisms in 2Cat.