Abelian group of continuous sections

Question

I'm trying to prove that the sheafification of a presheaf is indeed a sheaf. By definition, $\mathcal{F}^+(U)=\text{set of all continuous sections }s:U\rightarrow|\mathcal{F}|$. Part of the proof I'm suppose to show that $\mathcal{F}^+(U)$ is an Abelian group. Can someone tell me the operations that makes $s$ an Abelian group? It doesn't make sense to add or multiply them.

Answer 1

I presume $| \mathscr{F} |$ denotes the espace étalé of the presheaf $\mathscr{F}$. If $\mathscr{F}$ is a presheaf of abelian groups (or groups, or modules, etc.), then the fibres of $| \mathscr{F} |$ have the same structure, and thus the set of sections of $| \mathscr{F} |$ inherits a "pointwise" structure.

To show that the fibres of $| \mathscr{F} |$ have the desired structure, recall that the elements of the fibre of $| \mathscr{F} |$ over $X$ are equivalence classes of elements of $\coprod_U \mathscr{F} (U)$, where $U$ varies over the open neighbourhoods of $x$. Since the intersection of any finite number open neighbourhoods of $x$ is again an open neighbourhood of $x$, given any finite number of elements of the fibre, we can pick representatives from the same $\mathscr{F} (U)$ and then perform the required operation on them as elements of $\mathscr{F} (U)$. Of course, one has to check well-definedness etc. but this is all straightforward.