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%%% Primary Title: fundamental groupoid functors in quantum theories
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%%% Owner: bci1
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\begin{document}
\subsection{Fundamental Groupoid Functors in Quantum Theories}
The natural setting for the definition of a \htmladdnormallink{quantum fundamental groupoid}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid4.html} $F_{\Q}$ is in one of the \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} categories-- that of \htmladdnormallink{fundamental groupoid functors}{http://planetphysics.us/encyclopedia/FundamentalGroupoidFunctor.html},
$F_{\grp}$, and their \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/NaturalTransformation.html} defined in the context of \htmladdnormallink{quantum categories}{http://planetphysics.us/encyclopedia/QuantumCategories.html} of quantum spaces ${\Q}$ represented by \htmladdnormallink{Hilbert space bundles}{http://planetphysics.us/encyclopedia/HilbertBundle.html} or `rigged' Hilbert (or Frech\'et) spaces ${\H}_B$.
Let us briefly recall the description of quantum fundamental groupoids in a quantum \htmladdnormallink{functor category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, ${\Q}_F$:
\begin{definition}
The \emph{quantum fundamental groupoid}, QFG is defined by a functor
$F_{\Q}: \H_B \to {\Q}_G$, where ${\Q}_G$ is the \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of \emph{\htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html}} and their \emph{\htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}}.
\end{definition}
\subsubsection{Fundamental Groupoid Functors}
Other related functor categories are those specified with the \htmladdnormallink{general definition}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of the \emph{fundamental groupoid functor}, $F_{\grp}: \textbf{Top} \to \grp_2$, where \textbf{Top} is the
category of \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} spaces and $\grp_2$ is the \htmladdnormallink{groupoid category}{http://planetphysics.us/encyclopedia/GroupoidCategory.html}.
\subsubsection{Specific Example of QFG}
One can provide a physically relevant example of QFG as \htmladdnormallink{spin foams}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}, or functors of \htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}; more precise the spin foams were defined as functors between spin network categories that realize \htmladdnormallink{dynamic}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} transformations on the \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} space. Thus, because spin networks (or \htmladdnormallink{graphs}{http://planetphysics.us/encyclopedia/Cod.html}) are specialized one-dimensional
CW-complexes whose cells are linked quantum spin states their
quantum fundamental groupoid is defined as a \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of CW-complexes on `\htmladdnormallink{rigged' Hilbert spaces}{http://planetphysics.us/encyclopedia/I3.html}, that are called \emph{Frech\'et nuclear spaces}.
\end{document}
