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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 $F$.

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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)

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    $\begingroup$ For another perspective on the first part, every epimorphism being split means that the Axiom of Choice holds internally which, via Diaconescu's theorem, would mean that these sheaf categories would be Boolean toposes, but most sheaf categories aren't Boolean. $\endgroup$ – Derek Elkins Feb 19 at 18:38

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