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*-algebras. Indeed, Sims and Williams reestablished Renault's equivalence theorem by realizing C*(L) as the linking algebra for C*(G) and C* (H). Since the proof that Morita equivalence preserves exactness for C*-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.