Internal groupoids in semi-abelian categories

Talk by Marino Gran (Université catholique de Louvain), at the Antwerp Algebra Colloquium on May 21, 2021.

Since their introduction twenty years ago, semi-abelian categories [1] have attracted a lot of interest, as they are useful to study some fundamental exactness properties the categories of groups, Lie algebras, compact groups and crossed modules have in common. In this talk I shall explain some simple ideas of this area of categorical algebra, with a special emphasis on the role of internal groupoids in semi-abelian categories. These structures are closely connected to commutators, central extensions and non-abelian homology. The category of groupoids in a semi-abelian category contains various interesting non-abelian torsion theories. It can also be seen as the exact completion of its subcategory of equivalence relations, as it follows from a general characterization of the semi-localizations of semi-abelian categories [2]. A couple of results concerning the internal structures in the semi-abelian category of cocommutative Hopf algebras will also be considered [3].

References:
[1] G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002) 367-386.
[2] M. Gran and S. Lack, Semi-localizations of semi-abelian categories, J. Algebra 454 (2016) 206-232.
[3] M. Gran, F. Sterck and J. Vercruysse, A semi-abelian extension of a theorem by Takeuchi, J. Pure Appl. Algebra 223 (2019) 4171-4190.