Sheaves have several applications in topology and especially in algebraic
and differential geometry.
First, geometric structures such as that of a differentiable manifold or
a scheme can be expressed in terms of a sheaf of rings on the space.
In such contexts several geometric constructions such as vector bundles or
divisors are naturally specified in terms of sheaves.
Second, sheaves provide the framework for a very general cohomology theory,
which encompasses also the "usual" topological cohomology theories
such as singular cohomology.
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