Definition 0.1. Let us recall that a Polish space is a separable, completely metrizable topological space, and that Polish groups G_P are metrizable (topological) groups whose topology is Polish, and thus they admit a compatible metric d which is left-invariant; (a topological group G_T is metrizable iff G_T is Hausdorff, and the identity e of G_T has a countable neighborhood basis).

Remark 0.1.

Polish spaces can be classified up to a (Borel) isomorphism according to the following provable results:

• “All uncountable Polish spaces are Borel isomorphic to ℝ equipped with the standard topology;”

  This also implies that all uncountable Polish space have the cardinality of the continuum.

• “Two Polish spaces are Borel isomorphic if and only if they have the same cardinality.”

Furthermore, the subcategory of Polish spaces that are Borel isomorphic is, in fact, a Borel groupoid.

Definition 0.2. The category of Polish groups 𝒫 has, as its objects, all Polish groups G_P and, as its morphisms the group homomorphisms g_P between Polish groups, compatible with the Polish topology Π on G_P .

Remark 0.2. 𝒫 is obviously a subcategory of 𝒯_grp the category of topological groups; moreover, 𝒯_grp is a subcategory of 𝒯_𝔾 -the category of topological groupoids and topological groupoid homomorphisms.