# 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