Internal profunctors and commutator theory;
applications to extensions classification and categorical Galois Theory

Dominique Bourn

We clarify the relationship between internal profunctors and connectors on pairs (R,S) of equivalence relations which originally appeared in our new profunctorial approach of the Schreier-Mac Lane extension theorem. This clarification allows us to extend this Schreier-Mac Lane theorem to any exact Mal'cev category with centralizers. On the other hand, still in the Mal'cev context and in respect to the categorical Galois theory associated with a reflection I, it allows us to produce the faithful action of a certain abelian group on the set of classes (up to isomorphism) of I-normal extensions having a given Galois groupoid.

Keywords: Mal'cev categories, centralizers, profunctor, Schreier-Mac Lane extension theorem, internal groupoid, Galois groupoid

2000 MSC: 18G50, 18D35, 18B40, 20J15, 08C05

Theory and Applications of Categories, Vol. 24, 2010, No. 17, pp 451-488.

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