It's well-known (and fairly easy to prove) that over any topological space $X$, if a sheaf $\mathcal{F}\in\mathfrak{Ab}(X)$ is injective, then it must also be flasque. However, I've never seen mentioned that this is an equivalence, so I suspect there must be some example of a sheaf $\mathcal{F}\in\mathfrak{Ab}(X)$ that is flasque without being injective. However, it's relatively difficult to prove that a sheaf is injective, so I don't see how to construct such an example.



Is there such a thing as a non-injective flasque abelian sheaf?
Just take a non-injective abelian group and a point in your space, and form the skyscraper sheaf at that point with stalk the abelian group.


