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%%% Primary Title: fundamental quantum groupoid
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%%% Owner: bci1
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\begin{document}
\begin{definition}
A \emph{fundamental quantum groupoid} $F_{\Q}$ is defined as a \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $F_{\Q}: \H_B \to {\Q}_G$, where ${\H}_B$ is the \htmladdnormallink{category of Hilbert space}{http://planetphysics.us/encyclopedia/CategoryOfHilbertSpaces.html} bundles, and ${\Q}_G$ 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 and Functor Categories}
The natural setting for the definition of a \emph{\htmladdnormallink{quantum fundamental groupoid}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid4.html}} $F_{\Q}$
is in one of the functor 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 ${\H}_B$.
Other related \emph{\htmladdnormallink{functor categories}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}} are those specified with the \htmladdnormallink{general definition}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of the \emph{\htmladdnormallink{fundamental groupoid functor}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid.html}}, $F_{\grp}: \textbf{Top} \to \grp_2$, 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}.
\begin{example}
A specific example of a quantum fundamental groupoid can be given for \htmladdnormallink{spin foams}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html} of \htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}, with a \htmladdnormallink{spin foam}{http://planetphysics.us/encyclopedia/ComplexOfSpinNetworks.html} defined as a functor between spin network categories. Thus, because spin networks or \htmladdnormallink{graphs}{http://planetphysics.us/encyclopedia/Cod.html} are specialized
one-dimensional CW-complexes whose cells are linked quantum \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} states, their quantum fundamental groupoid is defined as a \htmladdnormallink{functor representation}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} of
CW-complexes on `\htmladdnormallink{rigged' Hilbert spaces}{http://planetphysics.us/encyclopedia/I3.html} (also called Frech\'et nuclear spaces).
\end{example}
\end{document}
