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Torsion Theories and Coverings of V-Groups

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Abstract

For a commutative, unital and integral quantale V, we generalize to V-groups the results developed by Gran and Michel for preordered groups. We first of all show that, in the category V-\(\mathsf {Grp}\) of V-groups, there exists a torsion theory whose torsion and torsion-free subcategories are given by those of indiscrete and separated V-groups, respectively. It turns out that this torsion theory induces a monotone-light factorization system that we characterize, and it is then possible to describe the coverings in V-\(\mathsf {Grp}\). We next classify these coverings as internal actions of a Galois groupoid. Finally, we observe that the subcategory of separated V-groups is also a torsion-free subcategory for a pretorsion theory whose torsion subcategory is the one of symmetric V-groups. As recently proved by Clementino and Montoli, this latter category is actually not only coreflective, as it is the case for any torsion subcategory, but also reflective.

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Acknowledgements

The author warmly thanks Marino Gran, Maria Manuel Clementino and the anonymous referees for an accurate checking and their useful comments on a preliminary version of the paper. She is also grateful to Maria Manuel Clementino for suggesting the generalization studied in this article.

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  1. Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium

    Aline Michel

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  1. Aline Michel
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Correspondence to Aline Michel.

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Communicated by Maria Manuel Clementino.

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The author’s research is funded by a FRIA doctoral grant of the Communauté française de Belgique

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Michel, A. Torsion Theories and Coverings of V-Groups. Appl Categor Struct 30, 659–684 (2022). https://doi.org/10.1007/s10485-021-09670-w

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