# PlanetPhysics/Quantum Fundamental Groupoid 4

\newcommand{\sqdiagram}[9]{**Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}}**
}

Aquantum fundamental groupoidis 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 (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).