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 $S\in \mathrm{Set}$ a set, the skyscraper sheaf ${\mathrm{skyscr}}_x(S)\in \mathrm{Sh}(X)$ in the category of sheaves on the category of open subsets $\mathrm{Op}(X)$ of $X$ supported at $x$ with value $S$ is the sheaf of sets given by the assignment

$${\mathrm{skysc}}_x(S):(U\subset X)\mapsto \{\begin{array}{cc}S& \mathrm{if}x\in U\\ *& \mathrm{otherwise}\end{array}$$skysc_x(S) : (U \subset X) \mapsto \left\{ \array{ S & if x \in U \\ {*} & otherwise } \right.

Remarks

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