Тұрақты шоғырлар және топологиялық кеңістіктер

In mathematics, the constant sheaf on a topological space associated to a set is a sheaf of sets on whose stalks are all equal to It is denoted by or The constant presheaf with value is the presheaf that assigns to each open subset of the value , and all of whose restriction maps are the identity map The constant sheaf associated to is the sheafification of the constant presheaf associated to This sheaf identifies with the sheaf of locally constant valued functions on
In certain cases, the set may be replaced with an object in some category (e. g. when is the category of abelian groups, or commutative rings). Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology.

Let be a topological space, and a set. The sections of the constant sheaf over an open set may be interpreted as the continuous functions , where is given the discrete topology. If is connected, then these locally constant functions are constant. If is the unique map to the one point space and is considered as a sheaf on , then the inverse image is the constant sheaf on The sheaf space of is the projection map (where is given the discrete topology).

Let be the topological space consisting of two points and with the discrete topology. has four open sets: The five non trivial inclusions of the open sets of are shown in the chart. A presheaf on chooses a set for each of the four open sets of and a restriction map for each of the nine inclusions (five non trivial inclusions and four trivial ones). The constant presheaf with value , which we will denote , is the presheaf that chooses all four sets to be , the integers, and all restriction maps to be the identity. is a functor, hence a presheaf, because it is constant. satisfies the gluing axiom, but it is not a sheaf because it fails the local identity axiom on the empty set. This is because the empty set is covered by the empty family of sets: Vacuously, any two sections of over the empty set are equal when restricted to any set in the empty family. The local identity axiom would therefore imply that any two sections of over the empty set are equal, but this is not true. A similar presheaf that satisfies the local identity axiom over the empty set is constructed as follows. Let , where 0 is a one element set. On all non empty sets, give the value For each inclusion of open sets, returns either the unique map to 0, if the smaller set is empty, or the identity map on
Notice that as a consequence of the local identity axiom for the empty set, all the restriction maps involving the empty set are boring. This is true for any presheaf satisfying the local identity axiom for the empty set, and in particular for any sheaf. is a separated presheaf (that is, satisfies the local identity axiom), but unlike it fails the gluing axiom. is covered by the two open sets and , and these sets have empty intersection. A section on or on is an element of , that is, it is a number. Choose a section over and over , and assume that Because and restrict to the same element 0 over , the gluing axiom requires the existence of a unique section on that restricts to on and on But because the restriction map from to is the identity, , and similarly , so , a contradiction. is too small to carry information about both and To enlarge it so that it satisfies the gluing axiom, let Let and be the two projection maps Define and For the remaining open sets and inclusions, let equal is a sheaf called the constant sheaf on with value Because is a ring and all the restriction maps are ring homomorphisms, is a sheaf of commutative rings.