Quotients of span categories that are allegories and the representation of regular categories

TL;DR

This paper explores how quotients of span categories can form allegories and represents regular categories through these structures, providing criteria and constructions for such representations.

Contribution

It establishes a general criterion for quotients of span categories to be allegories and constructs a reflection of regular categories into allegories using these quotients.

Findings

Quotients of span categories can be allegories under certain conditions.

A specific quotient category is isomorphic to M-relations in C when M is monomorphisms.

Every finitely complete category with a stable factorization system can be reflected into the 2-category of regular categories.

Abstract

We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E, M)-factorization system, we establish a quotient category Span_E(C) that is isomorphic to the category Rel_M(C) of M-relations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can still find a least pullback-stable and composition-closed class E. containing E such that Span_E.(C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2-category of all regular categories.