skyscraper sheaf

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

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

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

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

- 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).

- James Milne, section 6 of
*Lectures on Étale Cohomology*

category: sheaf theory

Revised on July 17, 2014 11:02:18
by Urs Schreiber
(82.136.246.44)