Étale groupoids, derived categories, and operations. (Q2785085)

A \(G\)-sheaf on an étale groupoid \(G\rightrightarrows G_0\) is a sheaf on the space \(G_0\) equipped with a continuous right \(G\)-action. The author discusses the construction of the derived category of \(G\)-sheaves. For a morphism \(f\) between étale groupoids, operations \(f^*\) and \(f_*\) are defined and the cohomology of \(G\) is formulated. The author explains that Morita equivalent étale groupoids have equivalent derived categories. The operations \(f_!\) and \(f^!\) related to compactly supported homology and cohomology are discussed. In the conclusion, the author explains the embedding category to describe the derived category of étale groupoid in a combinatorial way, and presents the derived category by the cohomology of the classifying space of \(G\). This fact is used in the classification of extensions of étale groupoids by the author [K-Theory 28, 207--258 (2003; Zbl 1042.58008)].NEWLINENEWLINEFor the entire collection see [Zbl 0972.00029].