Torsion theories and Galois coverings of topological groups

Abstract

For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups.

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

“…Thanks to descent theory it is possible to give the characterization of the¯ 1 -coverings, that generalizes the one in the exact case: a regular epimorphism f : A → B in C is a¯ 1 -covering if and only if there is a centralizing relation on R[ f ] and ∇ A . The fact that C is required to be regular but not necessarily exact allows us to include as new example the category C = Th(Top) of topological Mal'cev algebras, where the effective descent morphisms are exactly the open surjective homomorphisms ( [14,26]). The second application concerns the following adjunction…”

Effective descent morphisms in star-regular categories

“…Accordingly, the equivalence relation (R, τ R ) is the kernel pair of its coequaliser. Hence, the category T(Top) is semi-effective star-regular and, by Theorem 3.2, every regular epimorphism is an effective descent morphism in T(Top) (this result is known, see [11] for instance, although the proof presented here is different).…”

A Torsion Theory in the Category of Cocommutative Hopf Algebras

“…As it follows from the results in [7] the reflector I in the adjunction where η Y is the Y -component of the unit of the adjunction and f lies in the subcategory F. The adjunction is then admissible in the sense of categorical Galois theory [17]: this opens the way to further investigations in the direction of semi-abelian homology [12]. The fact that the torsion theory is hereditary and Hopf K,coc a homological category implies that the corresponding Galois coverings are precisely those regular epimorphisms f : A → B in Hopf K,coc with the property that the kernel Hker(f ) is in F (by applying Theorem 4.5 in [14]). This fact is crucial to describe generalized Hopf formulae for homology, as explained in [13].…”