# PlanetPhysics/Fundamental Quantum Groupoid

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A fundamental quantum groupoid  ${\displaystyle F_{\mathbb {Q} }}$ is defined as a functor ${\displaystyle F_{\mathbb {Q} }:\mathbb {H} _{B}\to {\mathbb {Q} }_{G}}$, where ${\displaystyle {\mathbb {H} }_{B}}$ is the category of Hilbert space bundles, and ${\displaystyle {\mathbb {Q} }_{G}}$ is the category of quantum groupoids and their homomorphisms.


#### Fundamental Groupoid Functors and Functor Categories

The natural setting for the definition of a quantum fundamental groupoid ${\displaystyle F_{\mathbb {Q} }}$  is in one of the functor categories-- that of fundamental groupoid functors, $\displaystyle F_{\grp}$ , and their natural transformations defined in the context of quantum categories of quantum spaces ${\displaystyle {\mathbb {Q} }}$  represented by Hilbert space bundles or rigged' Hilbert (or Frech\'et) spaces ${\displaystyle {\mathbb {H} }_{B}}$ .

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