Orbifolds, sheaves and groupoids
Orbifolds were defined by \textit{I. Satake} [Proc. Natl. Acad. Sci. USA 42, 359-363 (1956; Zbl 0074.18103)] and have been studied by \textit{Thurston} [Notes (Minnesota, 1992)]. An orbifold is a space \(M\), together with an atlas of charts consisting of an open set \(\widetilde U\) of \(\mathbb{R}^n\), for some \(n>0\), an action of a finite group of \(C^\infty\)-automorphisms \(G\) on \(\widetilde U\) and a homeomorphism of \(\widetilde U/G\) onto an open set \(\widetilde U\) of \(M\). Any orbit space of a proper action of a discrete group on a manifold has an orbifold structure, so moduli spaces are orbifolds. The leaf space of any foliation with compact leaves and finite holonomy also has an orbifold structure. The authors associate to each orbifold a category of sheaves. This category of sheaves characterises the orbifold and vice versa. The category of sheaves is also the category of sheaves on a groupoid of germs of diffeomorphisms and so the orbifold is completely determined by this groupoid. These results can be used to obtain change of base formulae for sheaf cohomology of orbifolds and to identify the fundamental groupoid of the orbifold in terms of that of the classifying space of the associated groupoid.
scientific article; zbMATH DE number 1059650