The actions of a group B on a group X correspond bijectively to the group homomorphisms from B to the group of automorphisms of X. This can be stated using representability of the functor 'actions on X' (see [2, 3]).
In this talk we show the corresponding result for topological groups: The continuous actions of a topological group B on a topological group X correspond bijectively to the continuous homomorphisms from B to the group of auto-homeomorphisms of X, equipped with a suitable topology. This result is based on previous joint work with F. Borceux and A. Montoli , where this result is proved for quasi-locally compact groups X, and a completely topological approach to the problem, obtained jointly with F. Cagliari .
 F. Borceux, M.M. Clementino, A. Montoli, On the representability of actions for topological algebras, Textos de Matemática, Série B, Vol. 46 (2014), pp. 41-66.
 F. Borceux, G.M. Kelly, G. Janelidze, On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14 (2005), 85-104.
 F. Borceux, G.M. Kelly, G. Janelidze, Internal object actions, Comment. Math. Univ. Carolinae 46 82005), 235-255.
 F. Cagliari, M.M. Clementino, Topological groups have representable actions, in preparation.