User:Stca74/Sheaf/Example

A detailed example: A constant sheaf on a two point space

Constant presheaf on a two-point discrete space

Two-point discrete topological space

Let X be the topological space consisting of two points p and q with the discrete topology. X has four open sets: ∅, {p}, {q}, {p,q}. The nine non-trivial inclusions of the open sets of X are shown in the chart.

A presheaf on X chooses a set for each of the four open sets of X and a restriction map for each of the nine inclusions. The constant presheaf with value Z, which we will denote F, is the presheaf which chooses all four sets to be Z, the integers, and all restriction maps to be the identity. F is a functor, hence a presheaf, because it is constant. Each of the restriction maps is injective, so F is a separated presheaf. F satisfies the gluing axiom, but it is not a sheaf because it fails the normalization axiom. A similar presheaf G which satisfies the normalization axiom is constructed as follows. Let G(∅) = 0, where 0 is a one-element set. On all non-empty sets, give G the value Z. For each inclusion of open sets, G returns either the unique map to 0, if the larger set is empty, or the identity map on Z.

Intermediate step for the constant sheaf

Notice that as a consequence of the normalization, anything involving the empty set is boring. This is true for any presheaf satisfying the normalization axiom, and in particular for any sheaf.

G is a separated presheaf which satisfies the normalization axiom, but it fails the gluing axiom. {p,q} is covered by the two open sets {p} and {q}, and these sets have empty intersection. A section on {p} or on {q} is an element of Z, that is, it is a number. Choose a section m over {p} and n over {q}, and assume that m ≠ n. Because m and n restrict to the same element 0 over ∅, the gluing axiom requires the existence of a unique section s on G({p, q}) which restricts to m on {p} and n on {q}. But because the restriction map from {p, q} to {p} is the identity, s = m, and similarly s = n, so m = n, a contradiction.

Constant sheaf on a two-point topological space

G({p, q}) is too small to carry information about both {p} and {q}. To enlarge it so that it satisfies the gluing axiom, let H({p, q}) = Z ⊕ Z. Let π₁ and π₂ be the two projection maps Z ⊕ Z → Z. Define H({p} ⊆ {p, q}) = π₁ and H({q} ⊆ {p, q}) = π₂. For the remaining open sets and inclusions, let H equal G. H is a sheaf called the constant sheaf on X with value Z. Because Z is a ring and all the restriction maps are ring homomorphisms, H is a sheaf of commutative rings.

In general, for any set S and any topological space X there is a constant presheaf F which has F(U) = S for all U and all restriction maps equal to the identity. F is never a sheaf because it fails the normalization axiom. Some authors take a slightly different definition of a constant presheaf analogous to G above. They define the constant presheaf to have G(U) = S for all nonempty U and all restriction maps between nonempty sets equal to the identity. G(∅) is taken to be a one element set, and restriction maps involving the empty set are taken to be the unique map to the one element set. In this case, G is always a separated presheaf, and G is a sheaf if and only if the topological space is irreducible. The argument that it is not a sheaf is analogous to the situation above.

There is also always a constant sheaf with value S, and it is usually denoted ${\underline {S}}$. We let ${\underline {S}}(U)$ be the set of all functions from U to S which are constant on each connected component. In other words, if U has a single connected component, then ${\underline {S}}(U)$ is S. If U has two connected components, then ${\underline {S}}(U)$ is S × S; one factor of S is the section over one component, and the other factor is the section over the other component. Restriction corresponds to restriction of functions. It can be checked that this makes ${\underline {S}}$ a sheaf. More generally, if S is an object in a concrete category C which has all set-indexed products, then we define the constant sheaf ${\underline {S}}$ to be the sheaf which takes an open set U to the set of all functions U → S which are constant on the connected components of U. For example, this can always be done with Z to get the constant sheaf ${\underline {\mathbf {Z} }}$; this is the same as the sheaf H in the example above. If C is a category such as the category of groups or the category of commutative rings, this will give a sheaf of groups or a sheaf of commutative rings, respectively.