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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)

For a given category B we are interested in studying internal categorical structures in B. This work is the starting point, where we consider reflexive graphs and precategories (i.e., for the purpose of this note, a sim-plicial object truncated at level 2). We introduce the notions of reflexive graph and precategory relative to split epimorphisms. We study… (More)

- M M Clementino, F Borceux, J Gray, N Martins-Ferreira, Sandra Mantovani, Beppe Metere
- 2013

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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