A note on the "Smith is Huq" condition

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.

11 pages

References in corpus (1)

• On the second cohomology group in semi-abelian categories

Cited by in corpus (14)

• The ternary commutator obstruction for internal crossed modules
• On the normality of Higgins commutators
• A characterisation of Lie algebras amongst anti-commutative algebras
• Higher central extensions and cohomology
• Algebras with representable representations
• Further remarks on the "Smith is Huq" condition
• Peri-abelian categories and the universal central extension condition
• A decomposition formula for the weighted commutator
• Universal central extensions of internal crossed modules via the non-abelian tensor product
• A comonadic interpretation of Baues-Ellis homology of crossed modules
• On the "three subobjects lemma" and its higher-order generalisations
• What is an ideal a zero-class of?
• A universal Kaluzhnin--Krasner embedding theorem
• Categorical-algebraic aspects of Heyting semilattices