Sheaf (mathematics)

In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property, as applied to a function.

Stalks of a sheaf at a point and germs of functions

Fix a point x of X. We would like to study the behavior of F near the point x. In analytical terms, we would like to somehow take the limit as we get nearer and nearer to the point x. The corresponding concept is to take the direct limit of F(N) as N runs over the open neighbourhoods of x ordered by inclusion (in categorical terminology, this is an example of a colimit). We denote this limit by Fx and call it the stalk of F at x. If F is a C-valued sheaf on X, then the stalk Fx is an object of C, for C a category such as the category of abelian groups or the category of commutative rings.

Related Topics:
Analytical - Direct limit - Colimit - Category of abelian groups - Category of commutative rings

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For any open set U containing x there is a morphism from F(U) to Fx. If C is a concrete category, then applying this morphism to an element f in F(U) gives an element of Fx called the germ of f at x.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This corresponds to the notion of germ of a function used elsewhere in mathematics. Intuitively, the germ of the function f at x describes the local behavior of f at the point x; it is a kind of 'ghost' of f, looked at only very near x. See also the detailed example given at local ring.

Related Topics:
Germ of a function - Local ring

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For some sheaves, germs behave well, and can give good local information; the germ of an analytic function around a point determines the function in a small neighbourhood of the point, using its power series expansion. However, some sheaves do not behave well; the germ of a smooth function at any point does not determine the function in any small neighbourhood of the point. As an example, take any bump function. Its local behavior on the interval where it is one is that of a constant function, but knowing that a bump function is the constant one near a given point does not tell you where the function begins to decay; from its local behavior, you cannot even conclude that it is a bump function!

Related Topics:
Analytic function - Power series - Smooth function - Bump function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~