# PlanetPhysics/Fundamental Quantum Groupoid

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Afundamental quantum groupoidis defined as a functor , where is the category of Hilbert space bundles, and is the category ofquantum groupoidsand theirhomomorphisms.

#### 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).