quantum fundamental groupoid

Definition 0.1.

A quantum fundamental groupoid $F_{\mathcal{Q}}$ is defined as a functor $F_{\mathcal{Q}}:\mathbf{H}_{B}\to{\mathcal{Q}}_{G}$ , where $\mathbf{H}_{B}$ is the category of Hilbert space bundles, and ${\mathcal{Q}}_{G}$ 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 $F_{\mathcal{Q}}$ is in one of the functor categories– that of fundamental groupoid functors (http://planetmath.org/FundamentalGroupoidFunctor), $F_{\mathcal{G}}$ , and their natural transformations (http://planetmath.org/NaturalTransformation) defined in the context of quantum categories of quantum spaces ${\mathcal{Q}}$ represented by Hilbert space bundles or rigged Hilbert (also called Frechét) spaces $\mathbf{H}_{B}$ .

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $F_{\mathcal{G}}:\textbf{Top}\to\mathcal{G}_{2}$ , where Top is the category of topological spaces and $\mathcal{G}_{2}$ 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 Frechét 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	Frechét spaces