A categorical genealogy for the congruence distributive property (Q2724146)

In an earlier paper [J. Algebra 177, No.~3, 647-657 (1995; Zbl 0843.08004)], \textit{M. C.~Pedicchio} studied the `congruence distributive property' of universal algebra in the context of Barr-exact categories with coequalizers. In particular, she showed that in this context the congruence distributive property for a category \(\mathcal C\) is equivalent to the assertion that every internal groupoid in \(\mathcal C\) is an equivalence relation. The latter assertion makes sense in any category with finite limits; in the present paper, the author considers its significance in Mal'cev categories (that is, categories with finite limits in which every reflexive relation is an equivalence relation) from the point of view of the fibration of pointed objects. He shows that, in such a category, the condition on internal groupoids holds if and only if there are no nontrivial internal groups in the fibres of the fibration of pointed objects; using this result, he extends some of Pedicchio's results to a more general setting.