# Is every sheaf a subsheaf of a flasque sheaf?

Call a sheaf flasque if for all open sets $U \subset V$, the restriction map$$\mathcal{F}(V) \to \mathcal{F}(U)$$is surjective. Is every sheaf a subsheaf of a flasque sheaf?

• Yes, do you know what the étalé space of a sheaf is? – user1971 Oct 10 '15 at 19:40
• The search phrase here is "Godement resolution". – Hoot Oct 10 '15 at 21:31
• or 'sheaf of discontinuous sections.' – Tomo Nov 7 '15 at 4:29

Let$$\mathcal{G} = \prod_{p \in X} \mathcal{F}_p.$$Then $\mathcal{F}$ is a subsheaf of $\mathcal{G}$ and $\mathcal{G}$ is easily seen to be flasque.
• It may be a simple question, but how can you prove that $\mathcal{F}$ is a subsheaf of $\mathcal{G}$? I'm new to sheaf theory and I need to prove that there exists a sheaf monomorphism from any sheaf $\mathcal{G}$ to a flasque sheaf, which I expect to be this $\mathcal{G}$. If so, I guess it is equivalent to your statement. I could open a new question but it has to risk to be quite similar to this one, so... Thanks for your help :) – MoebiusCorzer Jan 18 '16 at 16:49