For X a topological space, the presheaf F of Abelian groups (rings, ...) on X is defined such that

1. For every open subset U subset= X, an Abelian group (ring, ...) F(U), and

2. For every inclusion V subset= U of open subsets of X, a morphism of Abelian groups (rings, ...) rho_(UV):F(U)->F(V)

subject to the conditions:

1. If emptyset denotes the empty set, then F(emptyset)=0,

2. rho_(UU) is the identity map F(U)->F(U), and

3. If W subset= V subset= U are three open subsets, then rho_(UW)=rho_(VW) degreesrho_(UV).

In the language of categories, let Top(X) be the category whose objects are the open subsets of X and the only morphisms are the inclusion maps. Thus, Hom(V,U) is empty if V !subset= U and Hom(V,U) has just one element if V subset= U. Then a presheaf is a contravariant functor from the category Top(X) to the category Ab of Abelian groups (Ring of rings, ...).

As a terminology, if F is a presheaf on X, then F(U) are called the sections of the presheaf over the open set U, sometimes denoted as Gamma(U,F). The maps rho_(UV) are called the restriction maps. If s in F(U), then the notation rho_(UV)(s) is usually used instead of s|_V.

See also

Presheaf of Categories, Topological Sheaf

This entry contributed by José Gallardo Alberni

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Hartshorne, R. Algebraic Geometry. Berlin: Springer-Verlag, pp. 60-61, 1977.

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Cite this as:

Alberni, José Gallardo. "Presheaf." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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