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We introduce a notion of weakly Mal'cev category, and show that: (a) every internal reflexive graph in a weakly Mal'tsev category admits at most one multiplicative graph structure in the sense of [10] (see also [11]), and such a structure always makes it an internal category; (b) (unlike the special case of Mal'tsev categories) there are weakly Mal'tsev(More)
1] proved that all topological models of a semi-abelian variety admit semidirect products in the categorical sense introduced by D. Bourn and G. Janelidze [2]. Using some techniques developed in [3], we can show that, in any semi-abelian variety, the semidirect product of two objects X and B always appears as a subset of a certain cartesian product built(More)
Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an internal(More)
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