category of Borel spaces

Definition 0.1.

The category of Borel spaces $𝔹$ has, as its objects, all Borel spaces $(X_b;ℬ(X_b))$, 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 ${𝔹}_G$ is defined in a similar manner to$𝔹$, with the additional condition that Borel G-space morphisms commute withthe Borel actions $a:G\times X\to X$ defined as Borel functions (http://planetmath.org/BorelGroupoid)(or Borel-measurable maps). Thus, ${𝔹}_G$ is a subcategory of $𝔹$; in its turn,$𝔹$ is a subcategory of $𝕋op$–the category of topological spaces and continuousfunctions.

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