nLab skyscraper sheaf

Idea

A skyscraper sheaf is a sheaf supported at a single point.

This is not unlike the Dirac $\delta$-distribution.

Definition

For $X$ a topological space, $x \in X$ a point of $X$ and $S \in Set$ a set, the skyscraper sheaf $skyscr_x(S) \in Sh(X)$ in the category of sheaves on the category of open subsets $Op(X)$ of $X$ supported at $x$ with value $S$ is the sheaf of sets given by the assignment

$$skysc_x(S) : (U \subset X) \mapsto \left\{ \array{ S & if \; x \in U \\ {*} & otherwise } \right.$$

Remarks

• The skyscraper sheaf $skysc_x(S)$ is the direct image of $S$ under the geometric morphism $x : Set \to Sh(X)$ which defines the point of a topos given by $x \in X$ (see there for more details on this perspective).

References

• James Milne, section 6 of Lectures on Étale Cohomology

category: sheaf theory