Sheaf

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.