Comprehensive factorisation systems

TL;DR

This paper explores the relationship between comprehension schemes and orthogonal factorisation systems, applying these concepts to various categories to define universal coverings and a Galois-type fundamental group.

Contribution

It establishes a correspondence between comprehension schemes and complete orthogonal factorisation systems, extending the concept of comprehensive factorisation to multiple categories.

Findings

Unified framework for factorisation systems across categories

Definition of universal coverings in different categorical contexts

Introduction of a Galois-type fundamental group for based objects

Abstract

We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems exist for the categories of topological spaces, simplicial sets, small multicategories and Feynman categories. In each case comprehensive factorisation induces a natural notion of universal covering, leading to a Galois-type definition of fundamental group for based objects of the category.