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

TL;DR

This paper establishes a monotone-light factorization system for 2-categories via 2-preorders, characterizing light morphisms as faithful 2-functors and exploring properties like stable units and effective descent morphisms.

Contribution

It introduces a monotone-light factorization system for 2-categories through the reflection into 2-preorders, with new insights into the nature of light morphisms and descent morphisms.

Findings

Light morphisms are faithful 2-functors on 2-cells.

The reflection 2Cat --> 2Preord has stable units.

Certain surjective 2-functors are effective descent morphisms.

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.