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 $1_{(X_b; \mathcal{B}(X_b))}$ .