Sheaf (mathematics)

In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property, as applied to a function.

Examples

In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sections are very important examples. Suppose E and X are topological spaces and π : E → X is a continuous map. For every open set U in X, let F(U) be the set all continuous maps f : U → E such that π(f(x)) = x for all x in U. Such a function f is called a section of π. It is not difficult to check that F is a sheaf of sets on X. In fact, every sheaf of sets on X is essentially of this type, for very special maps π; see below.

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Given a sheaf F on X, the elements of F(X) are also called the global sections, a terminology motivated by the previous example.

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Further examples:

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• Any fiber bundle gives rise to a sheaf of sets, by taking sections.
• Ringed spaces are sheaves of commutative rings; especially important are the locally ringed spaces where all stalks (see below) are local rings.
• Schemes are special locally ringed spaces important in algebraic geometry; sheaves of modules are important in the associated theory.