Proof of a conjecture of A. Haefliger
An étale groupoid is a topological groupoid whose source and target maps are local homeomorphisms. The main examples of these groupoids are the Haefliger groupoid \(\Gamma^{q}\) and the holonomy groupoid of a foliation. Each étale groupoid \(G\) defines an abelian category on \(G\)-sheaves. If \(A\) is an arbitrary \(G\)-sheaf, we can consider a natural sheaf cohomology \(H^{*} (G,A)\) with coefficients in \(A\) and also a sheaf \(\widetilde{A}\) on a classifying space \(BG\) of \(G\). In this paper, the author clarifies the above concepts and proves the following result conjectured by Haefliger: There exists an isomorphism \[ H^{*} (G,A) \cong H^{*} (BG,\widetilde{A}) \] natural in \(G\) and \(A\). Finally, the factorial properties of \(H^{*} (G,A)\) are outlined in an appendix.
scientific article; zbMATH DE number 1133062