topos theory

# Contents

## Definition

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

## Properties

Flabbiness is a local property: if $F|_U$ is flabby for every sufficiently small open subset, then $F$ is flabby. Given a continuous map $f:X\to Y$ and a flabby sheaf $F$ on $X$, the direct image sheaf $f_* F : V\mapsto F(f^{-1}V)$ is also flabby. Any exact sequence of sheaves of abelian groups $0\to F_1\to F_2\to F_3\to 0$ in which $F_1$ 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 $F_1$ and $F_2$ are flabby, then $F_3$ is flabby.

## Examples

An archetypal example is the sheaf of all set-theoretic (not necessarily continuous) sections of a bundle $E\to X$; 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 $C^0(X,F)$ defined by

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

where $F_x$ denotes the stalk of $F$ at point $x$.

## References

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 (89.204.137.79)