Let $f: F\to G$ be an epimorphism of (pre)sheaves of sets on a Grothendieck site. Does it admit a section? By this I mean a morphism $g: G\to F$ with $f\circ g={\rm id}$.
I'm particularly interested in the case with $G=*$ being the final sheaf, in which case, it amounts to a coherent choice of local sections of $F$, or a global section of $F$.
For sheaves, the answer is obviously no in general : the existence of a section implies the sujectivity of $f$ on global sections, which is known no to be implied by epi-ness of $f$ (other wise sheaf-cohomology would be quite dull), even if your site is a topological space.
For the special case $G=*$, this is still not true. The example I give will show that it's not true for sheaves or presheaves in general.
Indeed look at presheaves (which can be seen as sheaves with the appropriate topology) on the poset $\mathbb{N}^{op}$, and look at a sheaf $F$ which is a tree with no infinite branches, that is for all $x\in F_0$, there is $n$ such that for $m\geq n$, $f_{m0}^{-1}(x) = \emptyset$, but such that every level is nonempty (for all $n$, $F_n\neq \emptyset$). It's easy to construct examples of this kind.
Then the unique map $F\to *$ is an epimorphism (look on the stalks !), but a section would be an infinite branch of $F$, so there is no such section.
Now this example of course works for sheaves and presheaves
(Any presheaf is a sheaf if you put the trivial topology on your indexing category)