# category of Borel spaces

###### 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 (http://planetmath.org/BorelGroupoid) (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 $1_{(X_{b};\mathcal{B}(X_{b}))}$.

 Title category of Borel spaces Canonical name CategoryOfBorelSpaces Date of creation 2013-03-22 18:25:01 Last modified on 2013-03-22 18:25:01 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 16 Author bci1 (20947) Entry type Definition Classification msc 54H05 Classification msc 28A05 Classification msc 28A12 Classification msc 28C15 Synonym category of measure spaces Related topic Category Related topic BorelSpace Related topic BorelGSpace Related topic BorelMorphism Related topic CategoryOfPointedTopologicalSpaces Related topic CategoryOfSets Related topic CategoryOfPolishGroups Related topic IndexOfCategories Defines composition of Borel morphisms Defines category of rigid Borel spaces