On some categorical properties of uniform spaces of probability measures

Abstract

We deal with the functor P β u: Unif → Unif of uniform spaces of probability measures, defined by Sadovnichy (1994). We show that there is a unique natural transformation T: S ∘ P gb u → P ∘ S, where S: Unif → c Unif 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 T u 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 P β u ∘ U = U ∘ P β .