Hopf formulae for homology of skew braces
(2024) Category Theory and Categorical Algebra — Location: University of Milan (9.September.2024)
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- The variety SKB of skew (left) braces [1] contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. Since SKB is a variety of Ω-groups and therefore a semi-abelian category, one can investigate it from the point of view of categorical algebra, a good example in this direction being the study of the commutators of ideals [2]. In this joint work with Thomas Letourmy and Leandro Vendramin [3] we apply the methods of non-abelian homological algebra [4] to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety SKB of skew braces to each of the three subvarieties mentioned above. These results are useful to describe the terms appearing in the Stallings- Stammbach five-term exact sequence associated with any exact sequence of skew braces, and to prove a new result concerning central series. References [1] L. Guarnieri and L. Vendramin, Skew braces and the Yang-Baxter equations, Math. Comp. 86 (307) (2015) 2519-2534. [2] D. Bourn, A. Facchini and M. Pompili, Aspects of the category SKB of skew braces, Comm. in Algebra 51(5) (2022) 2129-2143. [3] M. Gran, T. Letourmy and L. Vendramin, Hopf formulae for homology of skew braces, preprint (2024). [4] T. Everaert, M. Gran, and T. Van der Linden, Higher Hopf Formulae for Homology via Galois Theory, Adv. Math. 217 (2008) 2231-2267. 1
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Gran, M. (2024). Hopf formulae for homology of skew braces. Category Theory and Categorical Algebra, University of Milan. https://hdl.handle.net/2078.5/248008