PlanetPhysics/Fundamental Quantum Groupoid

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A fundamental quantum groupoid $$F_{\Q}$$ is defined as a functor $$F_{\Q}: \mathbb{H}_B \to {\Q}_G$$, where $${\mathbb{H}}_B$$ is the category of Hilbert space bundles, and $${\Q}_G$$ 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 $$F_{\Q}$$ is in one of the functor categories-- that of fundamental groupoid functors, $$F_{\grp}$$, and their natural transformations defined in the context of quantum categories of quantum spaces $${\Q}$$ represented by Hilbert space bundles or `rigged' Hilbert (or Frech\'et) spaces $${\mathbb{H}}_B$$.

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $$F_{\grp}: Top \to \grp_2$$, where Top  is the category of topological spaces and $$\grp_2$$ is the groupoid category.

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 Frech\'et nuclear spaces).