On the cocartesian image of preorders and equivalence relations in regular categories

TL;DR

This paper explores the conditions under which cocartesian images of preorders exist in regular categories, linking them to supremum existence and applying to toposes and n-permutable categories.

Contribution

It characterizes the existence of cocartesian images of preorders in regular categories and connects this to the existence of certain suprema, extending understanding in categorical logic.

Findings

Existence of cocartesian images is equivalent to certain supremum conditions.

Applicable to toposes with chain suprema and n-permutable regular categories.

Provides conditions ensuring these images and suprema exist.

Abstract

In a regular category $\mathbb E$, the direct image along a regular epimorphism $f$ of a preorder is not a preorder in general. In $Set$, its best preorder approximation is then its cocartesian image above $f$. In a regular category, the existence of such a cocartesian image above $f$ of a preorder $S$ is actually equivalent to the existence of the supremum $R[f]\vee S$ among the preorders. We investigate here some conditions ensuring the existence of these cocartesian images or equivalently of these suprema. They applied to two very dissimilar contexts: any topos $\mathbb E$ with suprema of chains of subobjects or any $n$-permutable regular category.