# category of Borel spaces

###### Definition 0.1.

The *category of Borel spaces* $\mathrm{\pi \x9d\x94\u0389}$ has, as its objects, all Borel spaces^{} $({X}_{b};\mathrm{\beta \x84\neg}\beta \x81\u2019({X}_{b}))$, 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 $\mathrm{{\rm O}\x83}$-algebra^{} of Borel sets.

###### Remark 0.1.

The *category of (standard) Borel G-spaces* ${\mathrm{\pi \x9d\x94\u0389}}_{G}$ is defined in a similar manner to
$\mathrm{\pi \x9d\x94\u0389}$, with the additional condition that Borel G-space morphisms commute with
the *Borel actions* $a:G\Gamma \x97X\beta \x86\x92X$ defined as Borel functions (http://planetmath.org/BorelGroupoid)
(or Borel-measurable maps). Thus, ${\mathrm{\pi \x9d\x94\u0389}}_{G}$ is a subcategory of $\mathrm{\pi \x9d\x94\u0389}$; in its turn,
$\mathrm{\pi \x9d\x94\u0389}$ is a subcategory of $\mathrm{\pi \x9d\x95\x8b}\beta \x81\u2019o\beta \x81\u2019p$βthe 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^{} $f:{X}_{b}\beta \x86\x92{X}_{b}$ (bijection) is the identity^{} ${1}_{({X}_{b};\mathrm{\beta \x84\neg}\beta \x81\u2019({X}_{b}))}$.

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 |