PlanetPhysics/Quantum Fundamental Groupoid 4

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A quantum fundamental groupoid   is defined as a functor , where  is the category of Hilbert space bundles, and  is the category of quantum groupoids and their homomorphisms.

Fundamental groupoid functors and functor categories[edit | edit source]

The natural setting for the definition of a quantum fundamental groupoid is in one of the functor categories-- that of fundamental groupoid functors, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}} , and their natural transformations defined in the context of quantum categories of quantum spaces represented by Hilbert space bundles or rigged Hilbert (also called Frech\'et) spaces .

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}: '''Top''' \to \grp_2} , where Top is the category of topological spaces and Failed to parse (unknown function "\grp"): {\displaystyle \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).