# PlanetPhysics/Category of Representations

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The \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} $\displaystyle Rep(\grp)$
of representations} has objects the representations of a groupoid $\displaystyle \grp$
, and as morphisms the intertwiners  ${\displaystyle i:\rho _{j}\longrightarrow \rho _{k}}$ that are (vector) bundle morphisms ${\displaystyle i:E\longrightarrow E}$ over the manifold ${\displaystyle M}$ so that ${\displaystyle \rho _{k}(g)\circ i=i\circ \rho _{j}}$. Because representations are functors $\displaystyle \rho: \grp \longrightarrow {\mathbf Vect}$
, an itertwiner ${\displaystyle i}$ is in fact a natural transformation between two such functors that are groupoid representations of $\displaystyle \grp$
, in this case implemented {\it via} the vector bundle morphisms ${\displaystyle i:E\longrightarrow E}$.