PlanetPhysics/Fundamental Groupoid Functors

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Quantum Fundamental Groupoid

A quantum fundamental groupoid $F_{\mathbb{Q}}$ is defined as a [[../TrivialGroupoid/|functor]] $F_{\mathbb{Q}}:{ℍ}_B\to {\mathbb{Q}}_G$, where ${ℍ}_B$ is the [[../CategoryOfHilbertSpaces/|category of Hilbert space]] bundles, and ${\mathbb{Q}}_G$ is the category of quantum groupoids and their [[../TrivialGroupoid/|homomorphisms]].

Fundamental groupoid functors and functor categories

The natural setting for the definition of a quantum fundamental groupoid $F_{\mathbb{Q}}$ is in one of the functor categories-- that of [[../FundamentalGroupoidFunctor/|fundamental groupoid functors]], Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}} , and their [[../NaturalTransformation/|natural transformations]] defined in the context of [[../QuantumCategories/|quantum categories]] of quantum spaces $\mathbb{Q}$ represented by [[../HilbertBundle/|Hilbert space bundles]] or `rigged' Hilbert (or Frech\'et) spaces ${ℍ}_B$.

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

A specific example of a quantum fundamental groupoid can be given for [[../SimplicialCWComplex/|spin foams]] of [[../SimplicialCWComplex/|spin networks]], with a [[../SimplicialCWComplex/|spin foam]] defined as a functor between spin network categories. Thus, because spin networks or [[../Cod/|graphs]] are specialized one-dimensional CW-complexes whose cells are linked quantum [[../QuarkAntiquarkPair/|spin]] states, their quantum fundamental groupoid is defined as a [[../CategoryOfLogicAlgebras/|functor representation]] of CW-complexes on `[[../I3/|rigged' Hilbert spaces]] (also called Frech\'et nuclear spaces).