A note on the ``Smith is Huq condition

DOI10.1007/s10485-010-9231-2zbMath1255.18008arXiv1006.1768OpenAlexW3099969302MaRDI QIDQ453593

Tim Van der Linden, Nelson Martins-Ferreira

Publication date: 27 September 2012

Published in: Applied Categorical Structures (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1006.1768

zbMATH Keywords

commutatorinternal groupoidinternal reflexive graphsemiabelian categorystar-multiplication

Mathematics Subject Classification ID

Nonabelian homological algebra (category-theoretic aspects) (18G50) Category of groups (20J15)

Related Items (33)

Aspects of the category SKB of skew braces ⋮ Extension theory and the calculus of butterflies ⋮ Profunctors in Mal'tsev categories and fractions of functors ⋮ Commutators for near-rings: Huq \({\neq}\) Smith ⋮ Complete objects in categories ⋮ Algebraic exponentiation for Lie algebras ⋮ Butterflies in a semi-abelian context ⋮ Cancellative conjugation semigroups and monoids ⋮ Hall's criterion for nilpotence in semi-abelian categories ⋮ A natural extension of the universal enveloping algebra functor to crossed modules of Leibniz algebras ⋮ S-protomodularity of the category of cocommutative bialgebras ⋮ Higher central extensions and cohomology ⋮ What is an ideal a zero-class of? ⋮ On the normality of Higgins commutators ⋮ Unnamed Item ⋮ Crossed modules and Whitehead sequences ⋮ On the ``Smith is Huq condition in \(S\)-protomodular categories ⋮ Algebras with representable representations ⋮ Mal’tsev and Lawvere functors ⋮ A characterisation of Lie algebras amongst anti-commutative algebras ⋮ A decomposition formula for the weighted commutator ⋮ A push forward construction and the comprehensive factorization for internal crossed modules ⋮ Peri-abelian categories and the universal central extension condition ⋮ Unnamed Item ⋮ A comonadic interpretation of Baues-Ellis homology of crossed modules ⋮ Universal central extensions of internal crossed modules via the non-abelian tensor product ⋮ Fibered aspects of Yoneda's regular span ⋮ On the ``three subobjects lemma and its higher-order generalisations ⋮ Further remarks on the ``Smith is Huq condition ⋮ Compatible actions of Lie algebras ⋮ Abstractly constructed prime spectra ⋮ Abelian extensions and crossed modules of Hom-Lie algebras ⋮ On the capability of Hom-Lie algebras

Cites Work

• Relative commutator theory in semi-abelian categories
• The ``closed subgroup theorem for localic herds and pregroupoids
• Mal'cev varieties
• Normal subobjects and abelian objects in protomodular categories
• A categorical approach to commutator theory
• On the second cohomology group in semi-abelian categories
• COMMUTATOR, NILPOTENCY, AND SOLVABILITY IN CATEGORIES
• \(3\times 3\) lemma and protomodularity
• Semi-abelian categories
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