An observation on n-permutability
Nelson Martins-Ferreira, Diana Rodelo, Tim Van der Linden

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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
