In recent years, many new applications of categorical Galois theory 1 have emerged in various algebraic contexts. In particular, this approach has led to new descriptions of the fundamental group in terms of generalized commutators in categories such as that of compact groups2, crossed modules, and skew braces 3, among others. These categories are semi-abelian 4, and thus share many structural properties with the categories of groups and Lie algebras. The category of cocommutative Hopf algebras is also semi-abelian 5, raising the question of whether these homological methods can be applied to study such structures as well. In this talk, I will first review some properties of semi-abelian categories and recall some simple examples. I will then show that the answer to the above question is affirmative: it is possible to establish new Hopf-type formulae for the homology of cocommutative Hopf algebras, as recently observed in joint work with Andrea Sciandra 6. References: 1 G. Janelidze, Galois groups, abstract commutators and Hopf formulae, Appl. Categ. Structures (2008) 653-668 2 T. Everaert and M. Gran, Protoadditive functors, derived torsion theories and homology, J. Pure and Applied Algebra 219 (2015) 3629-3676 3 M. Gran, T. Letourmy and L. Vendramin, Hopf formulae for homology of skew braces, preprint, arXiv:2409.18056 (2024) 4 G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002) 367-386 5 M. Gran, F. Sterck and J. Vercruysse, A semi-abelian extension of a theorem by Takeuchi, J. Pure Appl. Algebra 223 (2019) 4171-4190 6 M. Gran and A. Sciandra, Hopf formulae for cocommutative Hopf algebras, preprint, in preparation (2025)
Gran et al. (Wed,) studied this question.