Sheaf theory for stacks in manifolds and twisted cohomology for ^1hBHgerbes

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(1) 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.