nLab soft sheaf

Definition

Given a sheaf $F$ of sets over a topological space $X$, the section over an arbitrary (= not necessarily open) subset $V\subset X$ is a continuous section of the corresponding etale space restricted to $V$.

A sheaf $F$ of sets (or of abelian groups) over a paracompact Hausdorff space $X$ is soft if for any closed subset $V\subset X$, every section of $F$ over $V$ can be extended to the whole $X$.

Properties

The local extension of germs to the open neighborhoods of points by paracompactness gives rise to an extension of the section to an open neighborhood of the whole set $V$. Therefore every flabby sheaf is soft, because flabbiness gives the extension from open subsets. Fine sheaves are always soft.

Related concepts

• sheaf

• abelian sheaf cohomology

  • soft sheaf

  • fine sheaf

  • flabby sheaf

References

Standard references:

• Tohoku

• Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

category: sheaf theory