Equivalence and Exact Groupoids

Given two locally compact Hausdorff groupoids $G$ and $H$ and a $(G,H)$-equivalence $Z$, one can construct the associated linking groupoid $L$. This is reminiscent of the linking algebra for Morita equivalent $C^{\ast }$-algebras. Indeed, Sims and Williams reestablished Renault’s equivalence theorem by realizing $C^{\ast }(L)$ as the linking algebra for $C^{\ast }(G)$ and $C^{\ast }(H)$. Since the proof that Morita equivalence preserves exactness for $C^{\ast }$-algebras depends on the linking algebra, the linking groupoid should serve the same purpose for groupoid exactness and equivalence. We exhibit such a proof here.