Definition:Stalk of Presheaf

Definition

Let $T = \struct{S, \tau}$ be a topological space.

Let $\mathbf C$ be a category which has all small inductive limits.

Let $\FF$ be a $\mathbf C$-valued presheaf on $T$.

Let $x \in S$.

The stalk $\FF_x$ of $\FF$ at $x$ is the inductive limit:

$\ds \varinjlim_{U \ni x} \map \FF U$

over all open $U \subseteq S$ containing $x$.