An observation on n-permutability

TL;DR

This paper establishes a deep connection between n-permutability in regular categories and the symmetry of reflexive transitive relations, showing that certain conditions imply all internal categories are groupoids.

Contribution

It proves that in regular categories, n-permutability ensures all reflexive transitive relations are symmetric and characterizes when internal categories become groupoids.

Findings

Reflexive transitive relations are symmetric in n-permutable categories.

All internal categories are groupoids under n-permutability.

Conditions for symmetry relate to n-permutability in regular categories.

Abstract

We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these conditions hold when the category is n-permutable for some n.