PlanetPhysics/Category of Borel Spaces

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.

The \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} 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 $$1_{(X_b; \mathcal{B}(X_b))}$$.