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.
http://www.tac.mta.ca/tac/volumes/24/17/24-17.dvi
http://www.tac.mta.ca/tac/volumes/24/17/24-17.ps
http://www.tac.mta.ca/tac/volumes/24/17/24-17.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/17/24-17.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/17/24-17.ps