  To a coarse structure we associate a Grothendieck topology which is
determined by coarse covers. A coarse map between coarse spaces gives rise to a
morphism of Grothendieck topologies. This way we define sheaves and sheaf
cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on
the coarse category: if two coarse maps are close they induce the same map in
cohomology. There is a coarse version of a Mayer-Vietoris sequence and for
every inclusion of coarse spaces there is a coarse version of relative
cohomology. Cohomology with constant coefficients can be computed using the
number of ends of a coarse space.
