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A sheaf is a presheaf with "something" added allowing us to define things locally. This task is forbidden for presheaves in general. Specifically, a presheaf on a topological space is a sheaf if it satisfies the following conditions:
1. if is an open set, if is an open covering of and if is an element such that for all , then .
2. if is an open set, if is an open covering of and if we have elements for each , with the property that for each, , , then there is an element such that for all .
The first condition implies that is unique.
For example, let be a variety over a field . If denotes the ring of regular functions from to then with the usual restrictions is a sheaf which is called the sheaf of regular functions on .
In the same way, one can define the sheaf of continuous real-valued functions on any topological space, and also for differentiable functions.
REFERENCES:
Godement, R. Topologie Algébrique et Théorie des Faisceaux. Paris: 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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