fundamental groupoid functors

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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