Weakly Mal'cev categories

N. Martins-Ferreira

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 Janelidze, and such a structure always makes it an internal category; (b) (unlike the special case of Mal'tsev categories) there are weakly Mal'tsev categories in which not every internal category is an internal groupoid. We also give a simplified characterization of internal groupoids among internal categories in this context.

Keywords: Admissible reflexive graph, multiplicative graph, internal category, internal groupoid, weakly Mal'cev category, naturally weakly Mal'cev category, Mal'cev variety of universal algebras

2000 MSC: Primary 18E05; Secondary 18B40

Theory and Applications of Categories, Vol. 21, 2008, No. 6, pp 91-117.


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