On some categorical properties of uniform spaces of probability measures

We deal with the functor Pβu:Unif → Unif of uniform spaces of probability measures, defined by Sadovnichy (1994). We show that there is a unique natural transformation T: S ∘ Pgbu → P ∘ S, where S: Unif → cUnif is the functor of Samuel compactification. In our first main result (Theorem 4.3) it is established that for a uniform space (X,u) the component Tu of this natural transformation T is a homeomorphism iff u is a precompact uniformity. The second main result (Theorem 4.6) shows that there is no embedding U: Tych → Unif such that Pβu ∘ U = U ∘ Pβ.