Sheaf of infinitely differentiable functions
This article describes a sheaf on a manifold (the manifold may possess some additional structure in terms of which the sheaf is defined)
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Let be a differential manifold. The sheaf of differentiable functions of is defined as follows:
- To every open set, we associate the ring of all differentiable functions (-functions) from that open set to the real numbers (the ring structure arises from pointwise operations)
- The restriction map is simply function restriction
In fact, a differential manifold is completely characterized by its sheaf of differentiable functions. In other words, given a topological manifold and the sheaf of differentiable functions arising from some choice of differential structure on it, the differential manifold structure can be recovered from the sheaf.