On closure operators and reflections in Goursat categories

TL;DR

This paper explores the relationship between closure operators on effective equivalence relations and reflective subcategories in Goursat categories, providing explicit descriptions and characterizations within this framework.

Contribution

It establishes a bijective correspondence between closure operators and reflective subcategories in Goursat categories, extending to Birkhoff closure operators and subcategories.

Findings

Bijection between closure operators and reflective subcategories

Explicit description of closure in exact Goursat categories

Characterization of congruence distributive Goursat categories

Abstract

By defining a closure operator on effective equivalence relations in a regular category $C$, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories $L$ of $C$. When $C$ is an exact Goursat category this correspondence restricts to a bijection between the Birkhoff closure operators on effective equivalence relations and the Birkhoff subcategories of $C$. In this case it is possible to provide an explicit description of the closure, and to characterise the congruence distributive Goursat categories.