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.

Introduction

Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (that is depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as open sets, analytic functions, manifolds, and so on.

Related Topics:
Topology - Algebraic geometry - Differential geometry - Open set - Analytic function - Manifold

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For a typical example, consider a topological space X, and for every open set U in X, let F(U) be the set of all continuous functions U → R. If V is an open subset of U, then the functions on U can be restricted to V, and we get a map F(U) → F(V). "Gluing" describes the following process: suppose the Ui are given open sets with union U, and for each i we are given an element fi ∈ F(Ui), a continuous function fi : Ui → R. If these functions agree where they overlap, then we can glue them together in a unique way to form a continuous function f : U → R which agrees with all the given fi. The collection of the sets F(U) together with the restriction maps F(U) → F(V) then form a sheaf of sets on X. Furthermore, each F(U) is a commutative ring and the restriction maps are ring homomorphisms, making F a sheaf of rings on X.

Related Topics:
Continuous - Commutative ring - Ring homomorphism

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For a very similar example, consider a differentiable manifold X, and for every open set U of X, let F(U) be the set of differentiable functions U → R. Here too, gluing works and we obtain a sheaf of rings on X. Another sheaf on X assigns to every open set U of X the vector space of all differentiable vector fields defined on U. Restriction and gluing of vector fields works like that of functions, and we obtain a sheaf of vector spaces on the manifold X.

Related Topics:
Manifold - Vector space - Vector field

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