PlanetPhysics/Quantum Fundamental Groupoid

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Fundamental Groupoid Functors in Quantum Theories
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 $${\mathbb{H}}_B$$.

Let us briefly recall the description of quantum fundamental groupoids in a quantum functor category, $${\Q}_F$$: The quantum fundamental groupoid, QFG is defined by a functor $$F_{\Q}: \mathbb{H}_B \to {\Q}_G$$, where $${\Q}_G$$ is the category of quantum groupoids and their homomorphisms.

Fundamental Groupoid Functors
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.

Specific Example of QFG
One can provide a physically relevant example of QFG as spin foams, or functors of spin networks; more precise the spin foams were defined as functors between spin network categories that realize dynamic transformations on the spin space. 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 representation of CW-complexes on `rigged' Hilbert spaces, that are called Frech\'et nuclear spaces.