Peri-abelian categories and the universal central extension condition

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 $K\le X$, 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).

arXiv comment14 pages, final version accepted for publication

References in corpus (6)

• A note on the "Smith is Huq" condition
• On the second cohomology group in semi-abelian categories
• Universal central extensions in semi-abelian categories
• The ternary commutator obstruction for internal crossed modules
• On the normality of Higgins commutators
• The cohomological comparison arising from the associated abelian object

Cited by in corpus (4)

• On the normality of Higgins commutators
• The Yoneda isomorphism commutes with homology
• Algebraic logoi
• Categorical-algebraic aspects of Heyting semilattices