Torsion theories and Galois theories in preordered groups
(2023)
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- Authors
- Supervisors
- Gran, Marino
- Abstract
- In 2002 G. Janelidze, L. Márki and W. Tholen introduced semi-abelian categories. These categories provide an axiomatic context where it is possible to deal with several concepts and properties that groups, rings and Lie algebras have in common. Semi-abelian categories have been intensively studied during the last 20 years, yielding many interesting results in algebra. In parallel, category theorists have also realized that there are some other interesting categories which share a lot of important properties with groups but which are not semi-abelian. The category of preordered groups, for instance, is one of them. In this thesis, the exactness properties of preordered groups are thoroughly investigated. More precisely, we focus on the study, in this particular setting, of torsion (and pretorsion) theories and of categorical Galois theory. We prove that there is a torsion theory in preordered groups which gives rise to an absolute Galois structure and a monotone-light factorization system. We also observe that there exists a pretorsion theory whose torsion-free subcategory is the same as for the torsion theory. We then characterize the trivial and central extensions with respect to the induced absolute Galois structure. It turns out that all these results hold more generally in any category of V-groups (for V a suitable quantale) hence, in particular, in the categories of Lawvere metric groups, Lawvere ultrametric groups, probabilistic metric groups, etc. Finally, we provide an extension of the well-known categorical Galois theory of groups induced by the abelianization functor. We show that both central and normal extensions do actually coincide in this context, and we characterize them algebraically.
- Affiliations
Citations
Michel, A. (2023). Torsion theories and Galois theories in preordered groups. https://hdl.handle.net/2078.5/102415