# Contents

## 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 ${\mathrm{skyscr}}_{x}\left(S\right)\in \mathrm{Sh}\left(X\right)$ in the category of sheaves on the category of open subsets $\mathrm{Op}\left(X\right)$ of $X$ supported at $x$ with value $S$ is the sheaf of sets given by the assignment

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

## Remarks

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

## References

category: sheaf theory

Revised on November 22, 2013 04:08:09 by Urs Schreiber (82.169.114.243)