This talk is based on the work [1] where we show that the category of cancellative conjugation semigroups is weakly Mal'tsev and give a characterization of all admissible diagrams there.
In the category of cancellative conjugation monoids we describe, for Schreier split epimorphisms with codomain B and kernel X, all morphisms from X to B which induce a reflexive graph, an internal category or an internal groupoid. We describe Schreier split epimorphisms in terms of external actions and consider the notions of pre-crossed semimodule, crossed semimodule and crossed module in the context of cancellative conjugation monoids. In this category we prove that a relative version of the so-called ``Smith is Huq" condition for Schreier split epimorphisms holds as well as other relative conditions.
[1] Ana Paula Garrão, Nelson Martins-Ferreira, Margarida Raposo and Manuela Sobral, "Conjugation semigroups and conjugation monoids with cancellation", Pré-publicações DMUC, 18-49.
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