Definition:Étalé Space of Presheaf

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Definition

Let $X$ be a topological space.

Let $\mathbf C$ be a category.

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

The étalé space of $\FF$ is the pair $\struct {\map {\operatorname {\acute Et} } \FF, \pi}$ where:

$\map {\operatorname {\acute Et} } \FF$ is the disjoint union $\ds \bigsqcup_{x \mathop \in X} \FF_x$ of stalks of $\FF$

$\pi: \map {\operatorname {\acute Et} } \FF \to X$ is the canonical projection.

Topology on Étalé Space

The topology on $\map {\operatorname {\acute Et} } \FF$ is the final topology with respect to the sections associated to elements of $\map \FF U$ with $U \subseteq S$ open.

Also denoted as

The étalé space of $\FF$ is also denoted $\map {\operatorname {Sp\acute e} } \FF$.

Also see

• Definition:Section of Étalé Space
• Definition:Sheafification