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