Definition:Stalk of Presheaf

Definition
Let $X$ be a topological space.

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

Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $X$.

Let $x\in X$.

The stalk $\mathcal F_x$ of $\mathcal F$ at $x$ is the inductive limit:
 * $\displaystyle\varinjlim_{U \ni x} \mathcal F(U)$

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