categories of Polish groups and Polish spaces

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 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:

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

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.



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