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 category. Under condition (III), every star-multiplicative graph can be extended (uniquely) to a multiplicative graph, a problem raised by G. Janelidze in the semiabelian context.
When the three conditions hold, internal groupoids have a simple description, that, in the semiabelian context, correspond to the notion of internal crossed module, in the sense of Janelidze.
Keywords: Internal category, internal groupoid, reflexive graph, multiplicative graph, star-multiplicative graph, jointly epic pair, admissible pair, jointly epic split extension, split short five lemma, pointed protomodular
2000 MSC: 18D35
Theory and Applications of Categories,
Vol. 23, 2010,
No. 9, pp 170-198.
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