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Further Remarks on the “Smith is Huq” Condition

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Abstract

We compare the Smith is Huq condition (SH) with three commutator conditions in semi-abelian categories: first an apparently weaker condition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be stronger than (SH). This leads to the even stronger condition that weighted commutators in the sense of Gran, Janelidze and Ursini are independent of the chosen weight, which is known to be false for groups but turns out to be true in any two-nilpotent semi-abelian category.

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Authors and Affiliations

  1. Departamento de Matemática, Escola Superior de Tecnologia e Gestão, Centro para o Desenvolvimento Rápido e Sustentado do Produto, Instituto Politécnico de Leiria, 2411-901, Leiria, Portugal

    Nelson Martins-Ferreira

  2. CMUC, Department of Mathematics, University of Coimbra, 3001–454, Coimbra, Portugal

    Nelson Martins-Ferreira & Tim Van der Linden

  3. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.02, 1348, Louvain-la-Neuve, Belgium

    Tim Van der Linden

Authors
  1. Nelson Martins-Ferreira
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  2. Tim Van der Linden
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Correspondence to Nelson Martins-Ferreira.

Additional information

The first author was supported by IPLeiria/ESTG-CDRSP and Fundação para a Ciência e a Tecnologia (grants SFRH/BPD/4321/2008, PTDC/MAT/120222/2010 and PTDC/EME-CRO/120585/2010).

The second author is a Research Associate of the Fonds de la Recherche Scientifique–FNRS. His research was supported by Centro de Matemática da Universidade de Coimbra and by Fundação para a Ciência e a Tecnologia (grants SFRH/BPD/38797/2007 and PTDC/MAT/120222/2010). He wishes to thank CMUC and IPLeiria for their kind hospitality during his stays in Coimbra and in Leiria.

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Martins-Ferreira, N., Van der Linden, T. Further Remarks on the “Smith is Huq” Condition. Appl Categor Struct 23, 527–541 (2015). https://doi.org/10.1007/s10485-014-9368-5

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