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%%% Owner: bci1
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\begin{document}
\begin{definition}
The \emph{\htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $Rep(\grp)$ of \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}} has \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} the representations of a \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} $\grp$, and as \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} the \emph{intertwiners} $i : \rho_j \longrightarrow \rho_k$ that are (\htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html}) bundle morphisms $i:E \longrightarrow E$ over the \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} $M$ so that $\rho_k(g) \circ i = i \circ \rho_j$. Because representations are \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\rho: \grp \longrightarrow {\bf Vect}$, an itertwiner $i$ is in fact a \htmladdnormallink{natural transformation}{http://planetphysics.us/encyclopedia/VariableCategory2.html} between two such functors that are \htmladdnormallink{groupoid representations}{http://planetphysics.us/encyclopedia/GroupoidRepresentations.html} of $\grp$, in this case implemented {\it via} the vector bundle morphisms $i: E \longrightarrow E$.
\end{definition}
\end{document}
