nLab soft sheaf



Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




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

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


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 VV. Therefore every flabby sheaf is soft, because flabbiness gives the extension from open subsets. Fine sheaves are always soft.


Standard references are Tohoku and

  • Roger Godement, Topologie Algébrique et Théorie des Faisceaux. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13 Hermann, Paris 1958.
category: sheaf theory

Last revised on April 1, 2019 at 10:01:32. See the history of this page for a list of all contributions to it.