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# groupoid representation

Let $q:E\longrightarrow M$ be a vector bundle, with $E$ the *total space*, and $M$ a smooth manifold. Then, consider the representation $R_{G}$ of a group $G$ as an action on a vector space $V$, that is, as a homomorphism $h:G\longrightarrow End(V)$, with $End(V)$ being the group of endomorphisms of the vector space $V$. The generalization of the group representation to general representations of groupoids then occurs somewhat naturally by considering the groupoid action on a vector bundle $E\longrightarrow M$.

###### Definition 0.1.

Let $\mathcal{G}$ be a groupoid, and given a vector bundle $q:E\longrightarrow M$ consider the *frame groupoid*

$\Phi(E)=s,t:\phi(E)\longrightarrow M,$ |

with $\phi(E)$ being the set of all vector space isomorphisms
$\eta:E_{x}\longrightarrow E_{y}$ over all pairs $(x,y)\in M^{2}$, also with the associated structure maps. Then, a general *representation* $R_{d}$ of a groupoid $\mathcal{G}$ is defined as a homomorphism $R_{d}:\mathcal{G}\longrightarrow\Phi(E)$

Example 0.1: Lie groupoid representations

###### Definition 0.2.

Let $\mathcal{G}_{L}=s,t:G_{1}\longrightarrow M$ be a Lie groupoid. A *representation of a Lie groupoid* $\mathcal{G}_{L}=s,t:G_{1}\longrightarrow M$ on a vector bundle $q:E\longrightarrow M$ is defined as a smooth homomorphism (or a functor) $\rho:\mathcal{G}_{L}\longrightarrow\Phi(E)$ of Lie groupoids over $M$.

Note:
A *Lie groupoid representation* $\rho$ thus yields a functor, $R:\mathcal{G}_{L}\longrightarrow{\bf Vect},$ with ${\bf Vect}$ being the category of vector spaces and $R(x)=E_{x}$ being the fiber at each $x\in M$, as well as an isomorphism $R(g)$ for each $g:x\to y$.

## Mathematics Subject Classification

55N33*no label found*55N20

*no label found*55P10

*no label found*22A22

*no label found*20L05

*no label found*55U40

*no label found*

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