Some remarks on pullbacks in Gumm categories

TL;DR

This paper extends properties of pullbacks from Mal'tsev to Gumm categories, showing that certain algebraic structures like congruence modular varieties exhibit these properties, simplifying proofs of known results in Galois theory.

Contribution

It generalizes pullback properties from Mal'tsev to Gumm categories, providing a new proof that central and normal extensions coincide in this context.

Findings

Pullback properties extend from Mal'tsev to Gumm categories.

Congruence modular varieties are exactly Gumm categories.

Central and normal extensions coincide in Galois structures of Gumm categories.

Abstract

We extend some properties of pullbacks which are known to hold in a Mal'tsev context to the more general context of Gumm categories. The varieties of universal algebras which are Gumm categories are precisely the congruence modular ones. These properties lead to a simple alternative proof of the known property that central extensions and normal extensions coincide for any Galois structure associated with a Birkhoff subcategory of an exact Goursat category.