Hopf algebras, Hopf braces and non-abelian homology

In this talk, I will present some recent results on the interactions between Hopf algebra theory and semi-abelian categories. Semi-abelian categories 1 play a central role in categorical and homological algebra. After recalling the motivation for studying them, along with some basic concepts and illustrative examples, we will explain why the category of cocommutative Hopf algebras is semi-abelian 2. The abelian objects within the category of cocommutative Hopf algebras form an abelian reflective subcategory, which is also closed under subalgebras and quotients. It turns out that Janelidze’s categorical Galois theory 3 can then be applied to this context, leading to the establishment of new Hopf formulae for the homology of cocommutative Hopf algebras 4. Cocommutative Hopf braces 5, which extend both cocommutative Hopf algebras and skew braces, are useful tools for studying solutions to the Yang–Baxter equation. It is therefore natural to explore similar questions in the context of Hopf braces. In a joint article with Andrea Sciandra 6, we showed that cocommutative Hopf braces form a semi-abelian category. This structural result gives rise to numerous applications, including explicit descriptions of the categorical commutator and central extensions of Hopf braces, as well as the study of a torsion theory in the category of Hopf braces, where the torsion-free subcategory is the category of skew braces. References 1 G. Janelidze, L. Márki 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 G. Janelidze, Galois groups, abstract commutators and Hopf formulae, Appl. Categ. Struct. (2008), 653–668. 4 M. Gran and A. Sciandra, Hopf formulae for cocommutative Hopf algebras, preprint, in preparation (2025). 5 I. Angiono, C. Galindo, L. Vendramin, Hopf braces and Yang–Baxter operators, Proc. Amer. Math. Soc. 145 (2017), 1981–1995. 6 M. Gran and A. Sciandra, Hopf braces and semi-abelian categories, preprint, arXiv:2411.19238 (2024).