On Congruence Modular Varieties and Gumm Categories

DOI10.1080/00927872.2021.2006679arXiv2103.12387MaRDI QIDQ6363500

Dominique Bourn

Publication date: 23 March 2021

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 $PEE$ (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.

Mathematics Subject Classification ID

Equational logic, Mal'tsev conditions (08B05) Subalgebras, congruence relations (08A30) Epimorphisms, monomorphisms, special classes of morphisms, null morphisms (18A20) Theories (e.g., algebraic theories), structure, and semantics (18C10) Structured objects in a category (group objects, etc.) (18C40) Protomodular categories, semi-abelian categories, Mal'tsev categories (18E13)

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