Stalk (sheaf) - Motivation and Definition

Motivation and Definition

Sheaves are defined on open sets, but the underlying topological space X consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point x of X. Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of x, the behavior of the sheaf on that small neighborhood should be the same as the behavior of at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.

The precise definition is as follows: the stalk of at x, usually denoted, is:

Here the direct limit is indexed over all the open sets containing x, with order relation induced by inclusion (, if ). By definition (or universal property) of the direct limit, an element of the stalk is an equivalence class of elements, where two such sections and are considered equivalent if the restrictions of the two sections coincide on some neighborhood of x.