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, $F_{\mathcal G}$ , 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).

"quantum fundamental groupoid" is owned by bci1.
(view preamble)