Sheaf theory for stacks in manifolds and twisted cohomology for $S^1$–gerbes

Abstract

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 $ℝ$ along the structure map of a $U\left(1\right)$ gerbe over a smooth manifold $X$ is an object of the derived category of sheaves on $X$. Our main result shows that it is isomorphic in this derived category to a sheaf of twisted de Rham complexes.