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# category of Borel spaces

###### Definition 0.1.

The *category of Borel spaces* $\mathbb{B}$ has, as its objects, all Borel spaces $(X_{b};\mathcal{B}(X_{b}))$, and as its morphisms the Borel morphisms $f_{b}$ between Borel spaces; the Borel morphism composition is defined so that it preserves the Borel structure determined by the $\sigma$-algebra of Borel sets.

###### Remark 0.1.

The *category of (standard) Borel G-spaces* $\mathbb{B}_{G}$ is defined in a similar manner to
$\mathbb{B}$, with the additional condition that Borel G-space morphisms commute with
the *Borel actions* $a:G\times X\to X$ defined as Borel functions
(or Borel-measurable maps). Thus, $\mathbb{B}_{G}$ is a subcategory of $\mathbb{B}$; in its turn,
$\mathbb{B}$ is a subcategory of $\mathbb{T}op$–the category of topological spaces and continuous
functions.

The category of rigid Borel spaces can be defined as above with the additional condition that the only automorphism $f:X_{b}\to X_{b}$ (bijection) is the identity $1_{{(X_{b};\mathcal{B}(X_{b}))}}$.

## Mathematics Subject Classification

54H05*no label found*28A05

*no label found*28A12

*no label found*28C15

*no label found*

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