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 agroup 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 (http://planetmath.org/GroupoidRepresentation4).