Abstract 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.

中文翻译：

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具有扭曲系数的粗上同调

摘要 对于一个粗略的结构，我们将一个由粗覆盖决定的 Grothendieck 拓扑联系起来。粗糙空间之间的粗糙映射产生格洛腾迪克拓扑的态射。通过这种方式，我们可以在粗糙空间上定义滑轮和滑轮上同调。我们得到层上同调是粗类别上的函子：如果两个粗映射很接近，它们会在上同调中产生相同的映射。Mayer-Vietoris 序列有一个粗略版本，并且对于每个包含的粗略空间，都有一个相对上同调的粗略版本。可以使用粗空间的端点数计算具有常数系数的上同调。