Recently on glancing through Hartshorne's description
of Cartier divisors I started pondering the definition of
sheafification which led me to a question I can't answer. Neither
can I find a discussion in the standard texts.
First let's set up some notation.
Let $\mathcal{F}$ be a presheaf (let's say of sets)
on the topological space $X$.
It has a sheafification $\mathcal{F}^+$. We want to describe
the global sections of $\mathcal{F}^+$.
A global section of $\mathcal{F}^+$ consists
of a map $f:x\mapsto f_x$ where $f_x\in\mathcal{F}_x$,
the stalk of $\mathcal{F}$ at the point $x\in X$. This map
$f$ also obeys a local compatibility condition: there is an open
covering $(U_i)_{i\in I}$ of $X$, and sections $g_i\in\mathcal{F}(U_i)$
with the property that whenever $x\in U_i$ then $f_x$
equals the germ of $g_i$ at the point $x$. Let's call such a covering
$(U_i)$ and sections $(g_i)$ a representing system of sections for
the global section. (Is there a standard term for this concept?)
My question is this:
For each global section of $\mathcal{F}^+$
is there always a representing system of sections for it having the
stronger compatibility property that
$$g_i|_{U_i\cap U_j}=g_j|_{U_i\cap U_j}\in\mathcal{F}(U_i\cap U_j)$$
for all $i$, $j\in I$?
If not, is there some reasonable condition on the presheaf $\mathcal{F}$
that will guarantee this?
Another motivation is to find a good "pointless" description of
the sheafification functor (in the sense of "pointless topology" or locale
theory). The definitions of presheaf and sheaf only use the complete
lattice structure on the collection of open sets of $X$ and so are
thoroughly "pointless", but the usual description of the sheafification
functor uses the definitely "pointy" notion of stalk.
Best Answer
The following is a counterexample for $\mathcal{F}$ a presheaf of abelian groups (or sets, if you like).
Let $X=\lbrace a,b,c,d\rbrace$ with nontrival opens given by $\lbrace a \rbrace,\lbrace b \rbrace,U=\lbrace a,b,c \rbrace,V=\lbrace a,b,d \rbrace, U\cap V$.
Define the presheaf $\mathcal{F}$ by
$\mathcal{F}(\lbrace a \rbrace)=\mathcal{F}(\lbrace b \rbrace)=\mathbb{Z}/2\mathbb{Z}$,
$\mathcal{F}(U)=\mathcal{F}(V)=\mathcal{F}(U\cap V)=\mathcal{F}(X)=\mathbb{Z}$,
with the obvious restriction maps.
Then $\mathcal{F}^+(X)=\lbrace (x,y)\in\mathbb{Z}\oplus\mathbb{Z}| x\equiv y\text{ (mod 2)}\rbrace$, since the germs at $a$ and $b$ are determined by those at $c$ and $d$, and the only restrictions on $c$ and $d$ are that they give the same germs at $a$ and $b$.
Consider $(0,2)\in\mathcal{F}^+(X)$. The germs at $c$ and $d$ cannot come from a common section of $\mathcal{F}(X)$. Any system of sections which does not include $X$ in the cover must include both $U$ and $V$, having sections 0 and 2, respectively. Of course these do not agree when restricted to $U\cap V$. QED
This construction relies crucially on the fact that the presheaf is not separated (i.e. gluing is not unique). If the presheaf $\it{is}$ separated, the condition described in the question is clearly satisfied.
This construction was shown to me by Paul Balmer.