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fundamental groupoid functors (Topic)

Quantum Fundamental Groupoid

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

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $F_{{\mathsf{G}}}: \textbf{Top} \to {\mathsf{G}}_2$, where Top is the category of topological spaces and ${\mathsf{G}}_2$ 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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See Also: category theory

Also defines:  quantum fundamental groupoid, fundamental groupoid functor, quantum groupoid homomorphism, category of quantum groupoids
Keywords:  quantum fundamental groupoid, fundamental groupoid functor, QFG, category of quantum groupoids

Cross-references: rigged Hilbert spaces, functor representation, spin, graphs, spin foam, spin networks, spin foams, topological, category, general definition, functor categories, Hilbert space bundles, quantum categories, homomorphisms, category of Hilbert space, functor
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This is version 2 of fundamental groupoid functors, born on 2009-02-18, modified 2009-02-18.
Object id is 540, canonical name is FundamentalGroupoidFunctors.
Accessed 1519 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
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