Is epimorphism preserved under taking sections? Let $\phi:\mathscr{F}\rightarrow\mathscr{G}$ be en epimorphism of sheaves (say, of rings) on $X$. Is it true that $\phi(U):\mathscr{F}(U)\rightarrow\mathscr{G}(U)$ is an epimorphism of rings for every open subset $U\subset X$ ?
 A: No. A counterexample (specialised from Gunning/Rossi, Analytic Functions of Several Complex Variables, Chapter VI):
Let $D$ an open interval in $\mathbb{R}$, and $\mathscr{F}$ the sheaf of germs of constant (complex valued) functions ($\mathscr{F}_z \cong \mathbb{C}$ for all $z \in D$). Let $\mathscr{K}$ be the subsheaf of $\mathscr{F}$ with stalk $0$ in two points $a < b$, and stalk $\mathbb{C}$ everywhere else. Let $\mathscr{G}$ be the quotient sheaf $\mathscr{F}/\mathscr{K}$. $\mathscr{G}$ has stalk $0$ above all points except $a$ and $b$, where it has stalk $\mathbb{C}$.
We have a short exact sequence
$$0 \to \mathscr{K} \to \mathscr{F} \to \mathscr{G} \to 0$$
of sheaves, but $\Gamma(U,\,\mathscr{F}) \to \Gamma(U,\,\mathscr{G})$ is not surjective for any connected $U$ containing both $a$ and $b$, since a section of $\mathscr{G}$ can take different values in $a$ and $b$, but a section of $\mathscr{F}$ must be constant.
If you want to avoid $\{0\}$-rings, consider
$$0 \to \mathscr{K} \oplus 0 \to \mathscr{F} \oplus \mathscr{F} \to \mathscr{G} \oplus \mathscr{F} \to 0.$$
For sheaves above paracompact Hausdorff spaces $D$, a short exact sequence
$$0 \to \mathscr{R} \to \mathscr{S} \to \mathscr{T} \to 0$$
induces a short exact sequence
$$0 \to \Gamma(D,\, \mathscr{R}) \to \Gamma(D,\,\mathscr{S}) \to \Gamma(D,\,\mathscr{T}) \to 0$$
for example if $\mathscr{R}$ is a soft sheaf (any section over a closed subset $K \subset D$ can be extended to a section over all of $D$).
A: The failure of this is the starting point for one of the main tools of algebraic geometry: Sheaf cohomology (see MO/38966 for a nice motivation). A basic example is the exponentiel sequence. Complex logarithms exist locally, but not globally.
