quantum fundamental groupoid

Definition 0.1.

A quantum fundamental groupoid F𝒬 is defined as a functorMathworldPlanetmath F𝒬:𝐇B→𝒬G, where 𝐇B is the categoryMathworldPlanetmath of Hilbert spaceMathworldPlanetmath bundles, and 𝒬G is the category of locally compact quantum groupoidsPlanetmathPlanetmathPlanetmath and their homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

0.1 Fundamental groupoid functors and functor categories

The natural setting for the definition of a quantum fundamental groupoid F𝒬 is in one of the functor categoriesPlanetmathPlanetmath– that of fundamental groupoid functors (http://planetmath.org/FundamentalGroupoidFunctor), F𝒒, and their natural transformations (http://planetmath.org/NaturalTransformation) defined in the context of quantum categoriesPlanetmathPlanetmathPlanetmath of quantum spaces 𝒬 represented by Hilbert space bundles or rigged Hilbert (also called FrechΓ©t) spaces 𝐇B.

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, F𝒒:𝐓𝐨𝐩→𝒒2, where Top is the category of topological spaces and 𝒒2 is the groupoid categoryPlanetmathPlanetmath (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-complexesMathworldPlanetmath whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a functor representation of CW-complexes on rigged Hilbert spacesMathworldPlanetmathPlanetmath (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