categories of Polish groups and Polish spaces
Definition 0.1
Let us recall that a
Polish space is a separable, completely metrizable topological space, and
that
Polish groups

are metrizable (topological) groups whose topology is Polish, and thus they admit a compatible
metric

which is left-invariant; (a topological group

is
metrizable iff

is Hausdorff, and the identity

of

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.
Definition 0.2
The
category of Polish groups

has, as its objects, all Polish groups

and, as its morphisms
the group homomorphisms

between Polish groups, compatible with the
Polish topology
on

.
Remark 0.2

is obviously a subcategory of

the category of topological groups; moreover,

is a subcategory of

-the category of topological
groupoids
and topological
groupoid homomorphisms.
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As of this snapshot date, this entry was owned by bci1.