A note on the "Smith is Huq" condition

TL;DR

This paper demonstrates the equivalence of two conditions in category theory related to commutator theory and internal crossed modules, answering a question posed by George Janelidze.

Contribution

It establishes that the star-multiplicative graph condition is equivalent to the commutation of effective equivalence relations when their normalizations commute, linking two known conditions.

Findings

Star-multiplicative graphs are multiplicative under certain conditions

Effective equivalence relations commute when their normalizations do

Answers a previously open question by George Janelidze

Abstract

We show that two known conditions which arose naturally in commutator theory and in the theory of internal crossed modules coincide: every star-multiplicative graph is multiplicative if and only if every two effective equivalence relations commute as soon as their normalisations do. This answers a question asked by George Janelidze.