Actions of a group B on a group X correspond bijectively to split extensions of B with kernel X, or to semidirect products of X and B, or to group homomorphisms from B to the group Aut(X) of automorphisms of X. This last property of Aut(X) is usually identified as representability of actions, or classification of split extensions. In the first part of this talk, using results from [3, 4, 5], we describe split extensions for (topological) semiabelian algebras as presented in [2]. In the second part of the talk, following [1], we investigate the existence of split extensions classifiers for topological groups and some other algebraic structures. [1] F. Borceux, M.M. Clementino, A. Montoli, On the representability of actions for topological algebras, Preprint 1419, Department of Mathematics, University of Coimbra, 2014. [2] M.M. Clementino, A. Montoli, L. Sousa, Semidirect products of (topological) semiabelian algebras, J. Pure Appl. Algebra 219 (2015), 183197. [3] J.R.A. Gray, N. MartinsFerreira, On algebraic and more general categories whose split epimorphisms have underlying product projections, Appl. Categorical Structures, available online, DOI 10.1007/s1048501393365. [4] E.B. Inyangala, Semidirect products and crossed modules in varieties of right Omegaloops, Theory Appl. Categories 25 (2011), 426435. [5] G. Metere, A. Montoli, Semidirect products of internal groupoids, J. Pure Appl. Algebra 214 (2010), 18541861.
