On the representability of actions for topological algebras
Francis Borceux, Maria Manuel Clementino, Andrea Montoli · Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT) · 2014
The actions of a group B on a group X correspond bijectively to the group homomorphisms B ⟶ Aut(X), proving that the functor “actions on X” is representable by the group of automorphisms of X. Making the detour through pseudotopological spaces, we generalize this result to the topological case, for quasi-locally compact groups and some other algebraic structures. We investigate next the case of arbitrary topological algebras for a semi-abelian theory and prove that the representability of topological actions reduces to the preservation of coproducts by the functor Act(−,X).