Peri-abelian categories and the universal central extension condition

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).