# quantum fundamental groupoid

###### Definition 0.1.

A *quantum fundamental groupoid* ${F}_{\mathrm{\pi \x9d\x92\neg}}$ is defined as a functor^{}
${F}_{\mathrm{\pi \x9d\x92\neg}}:{\mathrm{\pi \x9d\x90\x87}}_{B}\beta \x86\x92{\mathrm{\pi \x9d\x92\neg}}_{G}$, where ${\mathrm{\pi \x9d\x90\x87}}_{B}$ is the category^{} of Hilbert space^{} bundles, and ${\mathrm{\pi \x9d\x92\neg}}_{G}$ is the category of locally compact quantum groupoids^{} and their homomorphisms^{}.

## 0.1 Fundamental groupoid functors and functor categories

The natural setting for the definition of a quantum fundamental groupoid ${F}_{\mathrm{\pi \x9d\x92\neg}}$
is in one of the functor categories^{}β that of fundamental groupoid functors (http://planetmath.org/FundamentalGroupoidFunctor),
${F}_{\mathrm{\pi \x9d\x92\u2019}}$, and their natural transformations (http://planetmath.org/NaturalTransformation) defined in the context of quantum categories^{} of quantum spaces $\mathrm{\pi \x9d\x92\neg}$ represented by Hilbert space bundles or rigged Hilbert (also called FrechΓ©t) spaces ${\mathrm{\pi \x9d\x90\x87}}_{B}$.

Other related functor categories are those specified with the general definition of the *fundamental groupoid functor*, ${F}_{\mathrm{\pi \x9d\x92\u2019}}:\text{\pi \x9d\x90\x93\pi \x9d\x90\xa8\pi \x9d\x90\xa9}\beta \x86\x92{\mathrm{\pi \x9d\x92\u2019}}_{2}$, where Top is the
category of topological spaces and ${\mathrm{\pi \x9d\x92\u2019}}_{2}$ is the groupoid category^{} (http://planetmath.org/GroupoidCategory).

###### Example 0.1.

A specific example of a quantum fundamental groupoid can be given for spin foams of spin networks, with a spin foam defined as a functor between spin network categories. Thus, because spin networks or graphs are specialized
one-dimensional CW-complexes^{} whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a functor representation of CW-complexes on rigged Hilbert spaces^{} (also called FrechΓ©t nuclear spaces).

Title | quantum fundamental groupoid |

Canonical name | QuantumFundamentalGroupoid |

Date of creation | 2013-03-22 18:15:53 |

Last modified on | 2013-03-22 18:15:53 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 62 |

Author | bci1 (20947) |

Entry type | Definition |

Classification | msc 55Q05 |

Classification | msc 55U40 |

Classification | msc 20L05 |

Synonym | quantum spacetimes homotopy theory |

Related topic | FundamentalGroupoidFunctor |

Related topic | SpinNetworksAndSpinFoams |

Related topic | QuantumGroupoids2 |

Related topic | GroupoidCategory |

Related topic | SpinNetworksAndSpinFoams |

Related topic | FunctorCategories |

Related topic | FundamentalGroupoid |

Related topic | HomotopyGroups |

Related topic | HigherDimensionalAlgebraHDA |

Related topic | GeneralizedVanKampenTheoremsHigherDimensional |

Related topic | Quantu |

Defines | category of quantum groupoids |

Defines | FrechΓ©t spaces |