A group of continuous self-maps on a topological groupoid (Q1640001)

Let \(G\) be a topological groupoid, and let \(G^{2}\) denote the groupoid of all composable pairs in \(G\times G\). It is proved in this work that the group of bisections of \(G^{2}\) is isomorphic to the group of units in the monoid \(S_{G} = \{ f:G \to G \;: \;f \text{ is continuous, and } (x,f(x)) \in G^{2} \text{ for every } x \in G \}\). In addition, it is proved further that the group of transitive bisections of \(G\) can be embedded in \(S_{G}\).