SAMS Subject Classification Number: 04

It is well known that a split extension of a group \(B\) with kernel \(X\) is determined up to isomorphism by a group action of \(B\) on \(X\). It turns out that an action of \(B\) on \(X\) can be presented as an algebra structure for \(X\) with respect to the monad on the category of groups, obtained from the functor sending each split extension to its kernel. This fact, discovered by D. Bourn and G. Janelidze [3], is, as explained there, a general categorical phenomenon which holds in every semi-abelian category (in the sense of G. Janelidze, L. Márki and W. Tholen [4]).

If \(X\) is a \(B\)-group (that is, \(X\) is a group equipped with an action of \(B\) on \(X\)) and \(S\) is a subgroup of \(X\), one can show that there is a largest sub-\(B\)-group of \(X\) contained in \(S\), which is called the action core of \(S\) with respect to \(X\) [2]. This fact can be stated in terms of internal object actions (whose name derives from [1] and are algebras over the above mentioned monad) and leads to the categorical definition of an action core. We call the corresponding notion for split extensions, split extension cores.

The main aim of this talk is to show that if \(\mathcal{V}\) is a semi-abelian variety of universal algebras admitting split extension cores and \(\mathbb{C}\) is a cartesian closed category with small limits, then internal \(\mathcal{V}\) algebras in \(\mathbb{C}\) admit split extension cores.

F. Borceux, G. Janelidze, and G. M. Kelly, Internal object actions, Commentationes Mathematicae Universitatis Carolinae 46(2), 235–255, 2005.

D. Bourn, A.S. Cigoli, J.R.A. Gray, T. Van der Linden, Algebraic logoi, Journal of Pure and Applied Algebra, accepted, 2022.

D. Bourn, and G. Janelidze, Protomodularity, descent and semidirect products, Theory and Applications of Categories 4(2), 37-46, 1998.

G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, Journal of Pure and Applied Algebra, 168, 367-386, 2002.