Peri-abelian categories and the universal central extension condition

James R.A. Gray ^[1] ; Tim Van der Linden ^[2]
1. [1] Stellenbosch University
  
  Stellenbosch University
  
  Stellenbosch, Sudáfrica
2. [2] Université Catholique de Louvain
  
  Université Catholique de Louvain
  
  Arrondissement de Nivelles, Bélgica
Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 7 (July 2015), 2015, págs. 2506-2520
Idioma: inglés
DOI: 10.1016/j.jpaa.2014.09.013
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Resumen
- We study the relation between Bourn's notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular, we show that a semi-abelian category is peri-abelian if and only if for each normal subobject K◁XK◁X, the Higgins commutator of K with itself coincides with the normalisation of the Smith commutator of the denormalisation of K with itself. We show that if a category is peri-abelian, then the condition (UCE), which was introduced and studied by Casas and the second author, holds for that category. In addition, we show, using amongst other things a result by Cigoli, that all categories of interest in the sense of Orzech are peri-abelian and therefore satisfy the condition (UCE).