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categories of Polish groups and Polish spaces
0.1 Introduction
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 leftinvariant; (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 $\mathbb{R}$ 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.
0.2 Category of Polish groups
Definition 0.2.
The category of Polish groups $\mathcal{P}$ 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 $\Pi$ on $G_{P}$.
Remark 0.2.
$\mathcal{P}$ is obviously a subcategory of $\mathcal{T}_{{grp}}$ the category of topological groups; moreover, $\mathcal{T}_{{grp}}$ is a subcategory of $\mathcal{T}_{{{\mathbb{G}}}}$ the category of topological groupoids and topological groupoid homomorphisms.
Mathematics Subject Classification
54H05 no label found28A05 no label found28A12 no label found28C15 no label found Forums
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