Stalk of a sheaf (${\mathcal{F}}_x$)

The stalk ${\mathcal{F}}_x$ is the direct limit of $\mathcal{F}(U)$ over neighborhoods $U$ of $x$. A morphism of sheaves is an isomorphism or forms an exact sequence exactly when it does so on every stalk.

A skyscraper sheaf supported at $x$ has a prescribed nonzero stalk at $x$ and zero stalks elsewhere.