nLab
skyscraper sheaf

Contents

Idea

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

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

Definition

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

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

Remarks

  • The skyscraper sheaf skysc x(S)skysc_x(S) is the direct image of SS under the geometric morphism x:SetSh(X)x : Set \to Sh(X) which defines the point of a topos given by xXx \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)