In the category of groups, split extensions have a lot of interesting properties. One of them is the fact that the category of split extensions is equivalent to the category of group actions.
Unfortunately, this property doesn't hold in the category of monoids. Nevertheless, it was proved in [1], that there exists such an equivalence if the split extensions are « Schreier extensions ». In this talk I will explain this construction of Schreier split extensions and their properties. Also motivated by a recent result on split extensions of unitary magmas [2] I will then consider split extensions in the category of Hopf algebras. What do we need to assume on the split extensions, in order to have an equivalence with the category of actions of Hopf algebras?

[1] D. Bourn, N. Martins-Ferreira, A. Montoli, M. Sobral, Schreier split epimorphisms in monoids and semirings, Textos de Matematica Série B, Departamento de Matematica da Universidade de Coimbra, vol. 45 (2014).
[2] M. Gran, G. Janelidze and M. Sobral, Split extensions and semidirect products of unitary magmas, arXiv:1906.02310 (2019).