When is a sheaf isomorphic to a constant sheaf?

Question

If I have a sheaf $\mathcal{F}$ in a topological space when can I say that it's isomorphic to some constant sheaf? This is when can I say that exists an isomorphism from $\mathcal{F}$ to some constant sheaf?

Answer 1

Well, a necessary condition is that $\mathcal{F}_x$ is constant for each $x$, but you probably know that it is not enough. The traditionnal way to prove that $\mathcal{F}$ is constant is to construct a morphism of sheaves $$\mathcal{F} \to M$$ (here $M$ is the constant sheaf associated to an abelian group $M$) and to prove that, on the stalks, this map gives you isomosphisms $$\mathcal{F}_x \simeq M.$$ But it is important to notice that you need a global morphism.