flabby sheaf


Topos Theory

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In higher category theory




A sheaf FF of sets on (the category of open subsets of) a topological space XX is flabby (flasque) if for any open subset UXU\subset X, the restriction morphism F(X)F(U)F(X)\to F(U) is onto. Equivalently, for any open UVXU\subset V\subset X the restriction F(V)F(U)F(V)\to F(U) is surjective. In mathematical literature in English, the original French word flasque is still often used instead of flabby here.


Flabbiness is a local property: if F| UF|_U is flabby for every sufficiently small open subset, then FF is flabby. Given a continuous map f:XYf:X\to Y and a flabby sheaf FF on XX, the direct image sheaf f *F:VF(f 1V)f_* F : V\mapsto F(f^{-1}V) is also flabby. Any exact sequence of sheaves of abelian groups 0F 1F 2F 300\to F_1\to F_2\to F_3\to 0 in which F 1F_1 is exact, is also an exact sequence in the category of presheaves (the exactness for stalks implies exactness for groups of sections over any fixed open set). As a corollary, if F 1F_1 and F 2F_2 are flabby, then F 3F_3 is flabby.


An archetypal example is the sheaf of all set-theoretic (not necessarily continuous) sections of a bundle EXE\to X; regarding that every sheaf over a topological space is the sheaf of sections of an etale space, every sheaf can be embedded into a flabby sheaf C 0(X,F)C^0(X,F) defined by

U xUF x U \mapsto \prod_{x\in U} F_x

where F xF_x denotes the stalk of FF at point xx.


Flabby sheaves were probably first studied in Tohoku, where flabby resolutions were also considered. A classical reference is Godement’s monograph.

category: sheaf theory

Revised on November 24, 2013 08:13:57 by Urs Schreiber (