Could not include topos theory - contents
A sheaf of sets on (the category of open subsets of) a topological space is flabby (flasque) if for any open subset , the restriction morphism is onto. Equivalently, for any open the restriction 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 is flabby for every sufficiently small open subset, then is flabby. Given a continuous map and a flabby sheaf on , the direct image sheaf is also flabby. Any exact sequence of sheaves of abelian groups in which 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 and are flabby, then is flabby.
An archetypal example is the sheaf of all set-theoretic (not necessarily continuous) sections of a bundle ; 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 defined by
where denotes the stalk of at point .