Any completely regular space embeds into a compact space. But suppose G is a topological group and X is a completely regular G-space. There is a largest G-map α_X: X → Y where Y is compact and α_X has dense image, but α_X need not be an embedding. Recently, Pestov has constructed an example of a topological group G and non-trivial flow X for which α_X is the map to a singleton.

In this talk, we consider automorphism groups of categorical metric structures, which include the Urysohn sphere, the unit sphere of the Banach lattice L_p, and the unit sphere of the Hilbert space L₂. We show that if G is the group of automorphisms of a categorical metric structure X, then α_X is the embedding of X into the space of 1-types over X.

(Joint work with Itai Ben Yaacov)