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# frame groupoid

###### Definition 0.1.

Let $\mathcal{G}$ be a groupoid, defined as usual by a category in which all morphisms are invertible, with the *structure maps* $s,t:G_{1}\longrightarrow G_{0}$, and $u:G_{0}\longrightarrow G_{1}$. Given a vector bundle $q:E\longrightarrow G_{0}$, the *frame groupoid* is defined as

$\Phi(E)=s,t:\phi(E)\longrightarrow G_{0}$ |

, with $\phi(E)$ being the set of all vector space isomorphisms $\eta:E_{x}\longrightarrow E_{y}$ over all pairs $(x,y)\in{G_{0}}^{2}$, also with the usual conditions for the structure maps of the groupoid.

###### Definition 0.2.

Let $G$ be a group and $V$ a vector space. A *group representation* is then defined as a homomorphism

$h:G\longrightarrow End(V),$ |

with $End(V)$ being the group of endomorphisms $e:V\longrightarrow V$ of the vector space $V$.

Note:
With the notation used above, let us consider $q:E\longrightarrow G_{0}$ to be a vector bundle. Then, consider a
*group representation*– which was here defined as the representation $R_{G}$ of a group $G$ via the group action on the vector space $V$, or as the homomorphism $h:G\longrightarrow End(V)$, with $End(V)$ being the group of endomorphisms of the vector space $V$. The generalization of group representations to the representations of groupoids then occurs naturally by considering the groupoid action on a vector bundle $q:E\longrightarrow G_{0}$. Therefore, the frame groupoid enters into the definition of groupoid representations.

## Mathematics Subject Classification

55N33*no label found*55N20

*no label found*55P10

*no label found*22A22

*no label found*20L05

*no label found*18B40

*no label found*55U40

*no label found*

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