PlanetPhysics/Category of Borel Spaces

A category of Borel spaces $𝔹$ has, as its [[../TrivialGroupoid/|objects]], all [[../BorelSpace/|Borel spaces]] $(X_b;ℬ(X_b))$, and as its [[../TrivialGroupoid/|morphisms]] the [[../InvariantBorelSet/|Borel morphisms]] $f_b$ between Borel spaces; the Borel morphism [[../Cod/|composition]] is defined so that it preserves the Borel structure determined by the $\sigma$-algebra of [[../InvariantBorelSet/|Borel sets]].

The \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} of standard Borel [[../TopologicalGSpace/|G-spaces]]} ${𝔹}_G$ is defined in a similar manner to $𝔹$, with the additional condition that [[../InvariantBorelSet/|Borel G-space]] morphisms [[../Commutator/|commute]] with the [[../InvariantBorelSet/|Borel actions]] $a:G\times X\to X$ defined as [[../BorelGroupoid/|Borel functions]] (or [[../InvariantBorelSet/|Borel-measurable maps]]). Thus, ${𝔹}_G$ is a subcategory of $𝔹$; in its turn, $𝔹$ is a subcategory of $𝕋op$--the category of [[../CoIntersections/|topological]] spaces and continuous [[../Bijective/|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 [[../Cod/|identity]] $1_{(X_b;ℬ(X_b))}$.