Definition:Étalé Space of Presheaf

Definition
Let $X$ be a topological space.

Let $\mathbf C$ be a category.

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

The étalé space of $\mathcal F$ is the pair $\struct {\map {\operatorname {\acute Et} } {\mathcal F}, \pi}$ where:
 * $\map {\operatorname {\acute Et} } {\mathcal F}$ is the disjoint union $\displaystyle \bigsqcup_{x \mathop \in X} \mathcal F_x$ of stalks of $\mathcal F$
 * $\pi: \map {\operatorname {\acute Et} } {\mathcal F} \to X$ is the canonical projection.

Also denoted as
The étalé space of $\mathcal F$ is also denoted $\map {\operatorname {Sp\acute e} } {\mathcal F}$.

Also see

 * Definition:Section of Étalé Space
 * Definition:Sheafification