fundamental groupoid functorsFundamentalGroupoidFunctors2009-02-18 15:16:502026-09-08 02:39:41Topicsection at topquantum fundamental groupoidfundamental groupoid functorquantum groupoid homomorphismcategory of quantum groupoidsquantum fundamental groupoidfundamental groupoid functorQFGcategory of quantum groupoids% PhysicsLibrary object 540 -- FundamentalGroupoidFunctors
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\subsection{Quantum Fundamental Groupoid}
\begin{definition}
A \emph{quantum fundamental groupoid} $F_{\mathcal Q}$ is defined as a functor
\[
F_{\mathcal Q}: \mathcal H_B \longrightarrow \mathcal Q_G,
\]
where $\mathcal H_B$ is the category of Hilbert space bundles, and
$\mathcal Q_G$ is the category of quantum groupoids and their homomorphisms.
\end{definition}
\subsection{Fundamental groupoid functors and functor categories}
The natural setting for the definition of a quantum fundamental groupoid
$F_{\mathcal Q}$ is in one of the functor categories---that of
\PMlinkname{fundamental groupoid functors}{FundamentalGroupoidFunctor},
$F_{\mathsf G}$, and their
\PMlinkname{natural transformations}{NaturalTransformation}, defined in the
context of quantum categories of quantum spaces $\mathcal Q$ represented by
Hilbert space bundles or ``rigged'' Hilbert (or Fr\'echet) spaces
$\mathcal H_B$.
Other related functor categories are those specified with the general definition
of the \emph{fundamental groupoid functor},
\[
F_{\mathsf G}: \mathbf{Top} \longrightarrow \mathsf G_2,
\]
where $\mathbf{Top}$ is the category of topological spaces and $\mathsf G_2$ 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 Fr\'echet nuclear spaces).
\end{example}