PlanetPhysics/Category of Representations

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