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.
This paper presents a new version of boundary on coarse spaces. The space of ends functor maps coarse metric spaces to uniform topological spaces and coarse maps to uniformly continuous maps.
This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful. This study provides a Künneth formula for twisted coarse cohomology. We obtain the Gromov boundary of a hyperbolic proper geodesic metric space as a quotient of its Higson corona.
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