The ternary commutator obstruction for internal crossed modules

Abstract

In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones. Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.

Revised version; 32 pages

References in corpus (6)

• Higher Hopf formulae for homology via Galois Theory
• A note on the "Smith is Huq" condition
• On the second cohomology group in semi-abelian categories
• Higher central extensions and Hopf formulae
• Relative Commutator Theory in Semi-Abelian Categories
• Internal object actions in homological categories

Cited by in corpus (15)

• On the normality of Higgins commutators
• Algebras with representable representations
• Further remarks on the "Smith is Huq" condition
• Peri-abelian categories and the universal central extension condition
• Central reflections and nilpotency in exact Mal'tsev categories
• A decomposition formula for the weighted commutator
• Universal central extensions of internal crossed modules via the non-abelian tensor product
• Associativity and the cosmash product in operadic varieties of algebras
• Hopf algebras in non-associative Lie theory
• On the "three subobjects lemma" and its higher-order generalisations
• What is an ideal a zero-class of?
• Categorical-algebraic aspects of Heyting semilattices
• Commutator theory for loops
• Twisted Commutators and Internal Crossed Modules
• Projective crossed modules in semi-abelian categories