Sheaf (mathematics)
Sheaves on manifolds

On an n {\displaystyle n} -dimensional C k {\displaystyle C^{k}} -manifold M {\displaystyle M}, there are a number of important sheaves, such as the sheaf of j {\displaystyle j} -times continuously differentiable functions O M j {\displaystyle {\mathcal {O}}_{M}^{j}} (with j ≤ k {\displaystyle j\leq k}). Its sections on some open U {\displaystyle U} are the C j {\displaystyle C^{j}} -functions U → R {\displaystyle U\to \mathbb {R} }. For j = k {\displaystyle j=k}, this sheaf is called the structure sheaf and is denoted O M {\displaystyle {\mathcal {O}}_{M}}. The nonzero C k {\displaystyle C^{k}} functions also form a sheaf, denoted O X × {\displaystyle {\mathcal {O}}_{X}^{\times }}. Differential forms (of degree p {\displaystyle p}) also form a sheaf Ω M p {\displaystyle \Omega _{M}^{p}}. In all these examples, the restriction morphisms are given by restricting functions or forms.

The assignment sending U {\displaystyle U} to the compactly supported functions on U {\displaystyle U} is not a sheaf, since there is, in general, no way to preserve this property by passing to a smaller open subset. Instead, this forms a cosheaf, a dual concept where the restriction maps go in the opposite direction than with sheaves. However, taking the dual of these vector spaces does give a sheaf, the sheaf of distributions.
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