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[parent] category of Polish groups (Definition)
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).

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: $ \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.



"category of Polish groups" is owned by bci1.
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See Also: category, metrizable, category of Borel spaces, topological group, measure space, Polish space

Other names:  subcategory of {\bf Grp }
Also defines:  Polish group homomorphism, metrizable topological groups
Keywords:  category, Polish groups, Polish group homomorphism

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Cross-references: homomorphisms, topological groupoid, topological groupoids, category, subcategory, group homomorphisms, morphisms, objects, basis, neighborhood, countable, identity, Hausdorff, iff, topological group, left-invariant, metric, compatible, groups, metrizable, Polish groups, topological space, completely metrizable, separable, Polish space
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This is version 7 of category of Polish groups, born on 2008-09-21, modified 2008-10-20.
Object id is 11068, canonical name is CategoryOfPolishGroups.
Accessed 357 times total.

Classification:
AMS MSC: 28C15 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures on topological spaces )
 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)
 28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets)
 54H05 (General topology :: Connections with other structures, applications :: Descriptive set theory )
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