Examples
Because sheaves encode exactly the data needed to pass between local and global situations, there are many examples of sheaves occurring throughout mathematics. Here are some additional examples of sheaves:
- Any continuous map of topological spaces determines a sheaf of sets. Let f : Y → X be a continuous map. We define a sheaf Γ(Y/X) on X by setting Γ(Y/X)(U) equal to the sections U → Y, that is, Γ(Y/X)(U) is the set of all functions s : U → Y such that fs = idU. Restriction is given by restriction of functions. This sheaf is called the sheaf of sections of f, and it is especially important when f is the projection of a fiber bundle onto its base space. Notice that if the image of f does not contain U, then Γ(Y/X)(U) is empty. For a concrete example, take X = C \ {0}, Y = C, and f(z) = exp(z). Γ(Y/X)(U) is the set of branches of the logarithm on U.
- Fix a point x in X and an object S in a category C. The skyscraper sheaf over x with stalk S is the sheaf Sx defined as follows: If U is an open set containing x, then Sx(U) = S. If U does not contain x, then Sx(U) is the terminal object of C. The restriction maps are either the identity on S, if both open sets contain x, or the unique map from S to the terminal object of C.
Read more about this topic: Sheaf (mathematics)
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