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 (http://planetmath.org/PolishSpacesUpToBorelIsomorphism):

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“All uncountable Polish spaces are Borel isomorphic to $\mathrm{R}$ equipped with the standard topology;”
This also implies that all uncountable Polish space have the cardinality of the continuum^{}.

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“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 $\mathrm{\Pi}$ 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^{}.
Title  categories of Polish groups and Polish spaces 
Canonical name  CategoriesOfPolishGroupsAndPolishSpaces 
Date of creation  20130322 18:25:04 
Last modified on  20130322 18:25:04 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  19 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 54H05 
Classification  msc 28A05 
Classification  msc 28A12 
Classification  msc 28C15 
Synonym  subcategory of topological groupoid category 
Related topic  Category 
Related topic  Metrizable 
Related topic  CategoryOfBorelSpaces 
Related topic  PolishSpacesUpToBorelIsomorphism 
Related topic  TopologicalGroup2 
Related topic  MeasureSpace 
Related topic  PolishSpace 
Defines  Polish group homomorphism 
Defines  metrizable topological groups 