A sheaf on a topological space is a presheaf whose local sections can be uniquely glued. More precisely, for every open covering of an open set , it satisfies the following conditions:

1. If two sections have equal images under all restriction maps to , then .

2. If sections agree on every intersection , then there is a section whose image under the restriction map to each is .

For example, let be an algebraic variety over an algebraically closed field . If denotes the ring of regular functions from to , then with the usual restriction maps, is a sheaf called the sheaf of regular functions on .

The sheaf of continuous real-valued functions on any topological space and the sheaf of smooth real-valued functions on a smooth manifold are defined similarly.

See also

Coherent Sheaf, Local System, Locally Free Sheaf, Presheaf, Restriction Map, Sheaf of Planes, Sheaf Section, Star

Portions of this entry contributed by José Gallardo Alberni

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References

Godement, R. Topologie Algébrique et Théorie des Faisceaux. Paris, France: Hermann, 1958.Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.Iyanaga, S. and Kawada, Y. (Eds.). "Sheaves." §377 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1171-1174, 1980.

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Sheaf

Cite this as:

Weisstein, Eric W., with contributions by José Gallardo Alberni. "Sheaf." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Sheaf.html

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