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Erschienen in:
2016 | OriginalPaper | Buchkapitel
3. Sheaves
verfasst von : Torsten Wedhorn
Erschienen in: Manifolds, Sheaves, and Cohomology
Verlag: Springer Fachmedien Wiesbaden
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Abstract
In this chapter we introduce one of the central notions in this book: sheaves. They are the main tool to keep track systematically of locally defined data attached to an open subset of a topological space. Giving such data are compatible with restriction to smaller open subsets yields the notions of a presheaf. A sheaf is then a presheaf where such local data, which are given on an open covering \((U_{i})_{i}\) and are compatible with restriction to pairwise intersections \(U_{i}\cap U_{j}\), can be glued to a global datum. One basic example is to attach to an open set U the \(\mathbb{R}\)-algebra of continuous functions \(U\to\mathbb{R}\): For an open covering \((U_{i})_{i}\) of X giving a continuous function on X is the same as giving continuous functions f
i
on all U
i
such that \(f_{i}{}_{|}{}_{U_{i}\cap U_{j}}=f_{j}{}_{|}{}_{U_{i}\cap U_{j}}\) for all \(i,j\).