# PlanetPhysics/Quantum Fundamental Groupoid

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

### Fundamental Groupoid Functors in Quantum TheoriesEdit

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 .

Let us briefly recall the description of quantum fundamental groupoids in a quantum functor category, :
The *quantum fundamental groupoid* , QFG is defined by a functor
, where is the category of *quantum groupoids* and their *homomorphisms*.

#### Fundamental Groupoid FunctorsEdit

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.

#### Specific Example of QFGEdit

One can provide a physically relevant example of QFG as spin foams, or functors of spin networks; more precise the spin foams were defined as functors between spin network categories that realize dynamic transformations on the spin space. 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 representation of CW-complexes on `rigged' Hilbert spaces, that are called *Frech\'et nuclear spaces* .