On Congruence Modular Varieties and Gumm Categories

Abstract

In (B-Gran, 2004), was given a categorical formulation of the Shifting Lemma which is a characterization of the Congruence Modular Varieties among all the variety of Universal Algebra, introduced in (Gumm, 1983). Starting from a characterization of this Shifting Lemma by a property of the fibers of the fibration of points $\P\EE$ (B, 2005), on the model of what happens for Mal'tsev categories, we shall investigate three directions:\\ 1) a new one: in following the golden thread of abelian split epimorphisms naturally provided by this characterization;\\ 2) a more or less expected one: in measuring the distance between the consequences of the Shifting Lemma in the varietal context and in the much more general categorical one;\\ 3) a quite amazing phenomenon, which I should call Algebraic Crystallography, and which is described in the Introduction.