Terminology: Sheaves with surjective structure maps?

Question

Is there established terminology for sheaves (on a topological space), the structure maps of which are all surjective?

I have come across some "cosheaves" with injective structure maps and would like to choose terminology in accordance to common sheaf theory terminology.

Answer 1

These might be called "flasque" or "flabby" sheaves. In the case of sheaves on a topological space, this is well-established terminology and can be found in Godement [1958]. More precisely,

Un faisceau $\mathscr{F}$ d'ensembles sur un espace $X$ est dit flasque si, pour tout ouvert $U$ de $X$, l'application de restriction $$\mathscr{F}(X) \to \mathscr{F}(U)$$ est surjective.

but it is clear that for any $U \subseteq V \subseteq X$, the restriction map $\mathscr{F} (V) \to \mathscr{F} (U)$ must also be surjective under this hypothesis.

In general, however, things are more subtle. Let $(\mathcal{E}, \mathscr{O})$ be a ringed topos. A flasque $\mathscr{O}$-module in the sense of Verdier is defined in [SGA 4, Exposé V, §4] to be an $\mathscr{O}$-module $\mathscr{F}$ such that, for all objects $U$ in $\mathcal{E}$, the sheaf cohomology group $H^q (U, \mathscr{F})$ vanishes for all $q > 0$. It is not at all obvious to me whether there is any relation between this and surjectivity of the restriction maps of $\mathscr{F}$.