soft sheaf


Topos Theory

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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 Godement’s book.

category: sheaf theory

Revised on March 6, 2013 19:50:09 by Zoran Škoda (