PlanetPhysics/Fundamental Groupoid Functors

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Quantum Fundamental GroupoidEdit

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 categoriesEdit

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 (or 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).