category of Borel spaces

Definition 0.1   A 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))}$ .



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