In this paper we give a sheaf theory interpretation of the twisted cohomology of manifolds. To this end we develop a sheaf theory on smooth stacks. The derived push-forward of the constant sheaf with value along the structure map of a gerbe over a smooth manifold is an object of the derived category of sheaves on . Our main result shows that it is isomorphic in this derived category to a sheaf of twisted de Rham complexes.
"Sheaf theory for stacks in manifolds and twisted cohomology for $S^1$–gerbes." Algebr. Geom. Topol. 7 (2) 1007 - 1062, 2007. https://doi.org/10.2140/agt.2007.7.1007