Semi-abelian categories, cocommutative Hopf algebras and Hopf braces

In this talk I’ll present some recent results on the interactions between Hopf algebra theory and semi-abelian categories. Semi-abelian categories 1 have played a central role in categorical algebra during the last 25 years. After recalling the motivation for studying them, some basic concepts and a few examples, we shall explain why the category of cocommutative Hopf algebras is semi-abelian 2. We shall then turn our attention to (cocommutative) Hopf braces 3, that extend cocommutative Hopf algebras and can be seen as a Hopf-theoretic generalization of skew braces4, that are useful to study solutions of the Yang-Baxter equation. In the recent article with Andrea Sciandra 5 we have investigated the exactness properties of the category of Hopf braces, and some natural constructions therein. First, we show that cocommutative Hopf braces form a semi-abelian category, that is also strongly protomodular. When the base field is algebraically closed and has zero characteristic one can find an interesting torsion theory therein, whose torsion-free subcategory is equivalent to the variety of skew braces, which turns out to be also a localization. Finally, we provide some explicit descriptions of the categorical commutator and of the central extensions of Hopf braces, that are likely to be useful for some new applications in non-abelian (co)homology theory. References 1 G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002) 367-386 2 M. Gran, F. Sterck and J. Vercruysse, A semi-abelian extension of a theorem by Takeuchi, J. Pure Appl. Algebra 223 (2019) 4171-4190 3 I. Angiono, C. Galindo, L. Vendramin, Hopf braces and Yang-Baxter operators, Proc. Amer. Math. Soc. 145 (2017) 1981-1995 4 L. Guarnieri, L. Vendramin, Skew braces and the Yang-Baxter equation. Math. Comp. 86 (2017) 2519-2534 5 M. Gran and A. Sciandra, Hopf braces and semi-abelian categories, preprint, arXiv:2411.19238 (2024)