Peri-abelian categories and the universal central extension condition

James R. A. Gray; Tim Van der Linden

doi:10.1016/j.jpaa.2014.09.013

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arXiv · 1404.3067

Peri-abelian categories and the universal central extension condition

James R. A. Gray·Tim Van der Linden

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Abstract

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 $K\leq X$, 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).

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BibTeX RIS

James R. A. Gray, Tim Van der Linden. 2014-08-08. Peri-abelian categories and the universal central extension condition. https://doi.org/10.1016/j.jpaa.2014.09.013

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