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Concept
Constant sheaf
Summary
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 non-empty 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 in some (e.g. when is the , or ).
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 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.
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