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.
The concept generalizes in a straightforward manner to flabby sheaves on locales.
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 .
Let be a sheaf on a topological space (or locale) . Then the following statements are equivalent.
For any open subset and any section there is an open covering such that, for each , there is an extension of to (that is, a section such that ). (If is a space instead of a locale, this can be equivalently formulated as follows: For any open subset , any section , and any point , there is an open neighbourhood of and an extension of to (that is, a section such that ).)
The canonical map is final from the internal point of view, that is
Here is the object of subsingletons of .
The implication “1 2” is trivial. The converse direction uses a typical argument with Zorn's lemma, considering a maximal extension. The equivalence “” is routine, using the Kripke-Joyal semantics to interpret the internal statement. We omit details for the time being. Condition 4 is a straightforward reformulation of condition 3.
Condition 2 of the proposition is, unlike the standard definition of flabbiness given at the top of the article, manifestly local. Also the equivalence with condition 3 and condition 4 is constructively valid. Therefore one could consider to adopt condition 2 as the definition of flabbiness.
The object of subsingletons of can be interpreted as the object of "partially-defined elements" of . The sheaf is flabby if and only if any such partially-defined element can be refined to an honest element of .