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.

The formal definition

To define sheaves we will proceed in two steps. The first step is to introduce the concept of a presheaf, which captures the idea of associating local information to a topological space. The second step is to introduce an additional axiom, called the gluing axiom or the sheaf axiom, which captures the idea of gluing local information to get global information.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Definition of a presheaf

Suppose X is a topological space, and C is a category (often, this is the category of sets, the category of Abelian groups, the category of commutative rings, or the category of modules over a fixed ring). A presheaf F of objects in C on the space X is given by the following data:

Related Topics:
Category - Set - Abelian group - Commutative ring - Modules - Ring

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• for every open set U in X, an object F(U) in C
• for every inclusion of open sets V ⊂ U, a morphism res_U,V : F(U) → F(V) in the category C. This is called the restriction morphism. The restriction morphism is required to satisfy two properties:
• for every open set U in X, we have res_U,U = id_F(U), i.e., the restriction from U to U is the identity morphism on F(U).
• given any three open sets W ⊂ V ⊂ U, we have res_V,W ○ res_U,V = res_U,W, i.e. the restriction from U to V and then to W is the same as the restriction from U directly to W.

This definition can be expressed naturally in terms of category theory. First we define the category of open sets on X to be the category TopX whose objects are the open sets of X and whose morphisms are inclusions. TopX is then the category corresponding to the partial order ⊂ on the open sets of X. A C-presheaf on X is then a contravariant functor from TopX to C.

Related Topics:
Category theory - Open set - Contravariant functor

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If F is a C-valued presheaf on X, and U is an open subset of X, then F(U) is called the sections of F over U. (This is by analogy with sections of fiber bundles; see below) If C is a concrete category, then each element of F(U) is called a section. F(U) is also often denoted Γ(U,F).

Related Topics:
Fiber bundle - Concrete category

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The gluing axiom

See main article gluing axiom for a higher-level discussion

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Sheaves are presheaves on which sections over small open sets can be glued to give sections over larger open sets. Here the gluing axiom will be given in a form that requires C to be a concrete category.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Let U be the union of the collection of open sets {Ui}. For each Ui, choose a section fi on Ui. We say that the fi are compatible if for any i and j,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:resUi,Ui∩Uj(fi) = resUj,Ui∩Uj(fj).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Intuitively speaking, if the fi represent functions, this says that any two compatible functions agree where they overlap. The sheaf axiom says that we can produce from the fi a unique section f over U whose restriction to each Ui is fi, i.e., resU,Ui(f)=fi. Sometimes this is split into two axioms, one guaranteeing existence, and the other guaranteeing uniqueness.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~