Relative Commutator Theory in Semi-Abelian Categories

Abstract

Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined by the abelian objects in A, it coincides with Huq's commutator; and when the category A is a variety of omega-groups, it coincides with the relative commutator introduced by the first author.

22 pages

References in corpus (6)

• Higher Hopf formulae for homology via Galois Theory
• On the second cohomology group in semi-abelian categories
• Higher central extensions and Hopf formulae
• Protoadditive functors, derived torsion theories and homology
• Relative commutator theory in varieties of omega-groups
• Galois theory and commutators

Cited by in corpus (12)

• The ternary commutator obstruction for internal crossed modules
• Resolutions, higher extensions and the relative Mal'tsev axiom
• Higher central extensions and cohomology
• Further remarks on the "Smith is Huq" condition
• Weighted commutators in semi-abelian categories
• Universal central extensions of internal crossed modules via the non-abelian tensor product
• Higher coverings of racks and quandles -- Part I
• On Crossed Modules in Modified Categories of Interest
• On some properties of $\mathsf{Lie}$-centroids of Leibniz algebras
• Galois theory and the categorical Peiffer commutator
• Variedades de álgebras topologicas
• Categorical-algebraic aspects of Heyting semilattices