fundamental quantum groupoidFundamentalQuantumGroupoid2009-02-13 02:12:422009-02-13 02:14:29Definitionquantum groupoid homomorphismfundamental groupoidquantum fundamental groupoidfunctor categorynatural transformationsspin networksspin foams%PlanetPhysicsdefalut, and almost certainly you want these
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\def\D{\mathsf{D}}\begin{definition}
A \emph{fundamental quantum groupoid} $F_{\Q}$ is defined as a functor
$F_{\Q}: \H_B \to {\Q}_G$, where ${\H}_B$ is the category of Hilbert space bundles, and ${\Q}_G$ is the category of \emph{quantum groupoids} and their \emph{homomorphisms}.
\end{definition}
\subsubsection{Fundamental groupoid functors and functor categories}
The natural setting for the definition of a \emph{quantum fundamental groupoid} $F_{\Q}$
is in one of the functor categories-- that of \PMlinkname{fundamental groupoid functors}{FundamentalGroupoidFunctor},
$F_{\grp}$, and their \PMlinkname{natural transformations}{NaturalTransformation} defined in the context of quantum categories of quantum spaces ${\Q}$ represented by Hilbert space bundles or `rigged' Hilbert (or Frech\'et) spaces ${\H}_B$.
Other related \emph{functor categories} are those specified with the general definition of the \emph{fundamental groupoid functor}, $F_{\grp}: \textbf{Top} \to \grp_2$, where \textbf{Top} is the category of topological spaces and $\grp_2$ is the \PMlinkname{groupoid category}{GroupoidCategory}.
\begin{example}
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).
\end{example}