On Congruence Modular Varieties and Gumm Categories

TL;DR

This paper explores the categorical foundations of congruence modular varieties, focusing on the Shifting Lemma, its fiber properties, and novel phenomena like Algebraic Crystallography, advancing understanding in universal algebra and category theory.

Contribution

It introduces new categorical perspectives on the Shifting Lemma, investigates fiber properties, and uncovers the phenomenon of Algebraic Crystallography in congruence modular varieties.

Findings

Characterization of the Shifting Lemma via fiber properties

Analysis of abelian split epimorphisms in this context

Discovery of Algebraic Crystallography phenomenon

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.