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 autohomeomorphisms of X, equipped with a suitable topology. This result is based on previous joint work with F. Borceux and A. Montoli [1], where this result is proved for quasilocally compact groups X, and a completely topological approach to the problem, obtained jointly with F. Cagliari [4]. [1] F. Borceux, M.M. Clementino, A. Montoli, On the representability of actions for topological algebras, Textos de Matemática, Série B, Vol. 46 (2014), pp. 4166. [2] F. Borceux, G.M. Kelly, G. Janelidze, On the representability of actions in a semiabelian category, Theory Appl. Categ. 14 (2005), 85104. [3] F. Borceux, G.M. Kelly, G. Janelidze, Internal object actions, Comment. Math. Univ. Carolinae 46 82005), 235255. [4] F. Cagliari, M.M. Clementino, Topological groups have representable actions, in preparation.
