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quantum fundamental groupoid (Definition)
Definition 0.1   A quantum fundamental groupoid $F_{\mathcal Q}$ is defined as a functor $F_{\mathcal Q}: \H _B \to {\mathcal Q}_G$, where ${\H }_B$ is the category of Hilbert space bundles, and ${\mathcal Q}_G$ is the category of quantum groupoids and their homomorphisms.

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, $F_{\mathcal G}$, and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathcal Q}$ represented by Hilbert space bundles or rigged Hilbert (also called Frechét) spaces ${\H }_B$.

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.

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).



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Cross-references: rigged Hilbert spaces, functor representation, spin, graphs, spin foam, spin networks, spin foams, topological, category, fundamental groupoid functor, general definition, functor categories, Hilbert space bundles, quantum categories, homomorphisms, category of quantum groupoids, category of Hilbert space, functor
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This is version 1 of quantum fundamental groupoid, born on 2009-05-09.
Object id is 734, canonical name is QuantumFundamentalGroupoid4.
Accessed 318 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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