quality:correction, information:classification, information:filtering, structural:summarization
Structure sheaf

Sometimes, it can be useful to use an alternative approach to endow a manifold with a C -structure. Here k = 1, 2,..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The structure sheaf of M, denoted C, is a sort of functor that defines, for each open set U ⊂ M, an algebra C (U) of continuous functions U → R. A structure sheaf C is said to give M the structure of a C manifold of dimension n provided that, for any p ∈ M, there exists a neighborhood U of p and n functions x,..., x ∈ C (U) such that the map f = (x,..., x) : U → R is a homeomorphism onto an open set in R, and such that C | is the pullback of the sheaf of k -times continuously differentiable functions on R.

In particular, this latter condition means that any function h in C (V), for V, can be written uniquely as h (x) = H (x (x),..., x (x)), where H is a k -times differentiable function on f (V) (an open set in R). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on R, and a fortiori this is sufficient to characterize the differential structure on the manifold.