Stalk (sheaf)

The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.

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 ${\displaystyle {\mathcal {F}}}$ on that small neighborhood should be the same as the behavior of ${\displaystyle {\mathcal {F}}}$ 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 ${\displaystyle {\mathcal {F}}}$ at x, usually denoted ${\displaystyle {\mathcal {F}}_{x}}$, is:

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