Let $\mathscr{F}$ be a presheaf over a topological space $X$ with values on a category $C$. We define the stalk of $\mathscr{F}$ at a point $p\in X$ as the colimit of all $\mathscr{F}(U)$ over all open sets $U$ containing $p$:
$$\mathscr{F}_p:=\varinjlim \mathscr{F}(U).$$
When $C$ is the category of sets, this has a simple description as
$$\left(\coprod_{U\ni x}\mathscr{F}(U)\right)\bigg/ \sim,$$
where $(f,U)\sim (g,V)$ if there is some open set $W\subset U\cap V$ containing $p$ such that $f|_W=g|_W$. If I understood correctly, people usually use the same description when $C$ is the category of abelian groups, for example. But isn't in this case the colimit given by the quocient of a direct sum instead?
If this same construction indeed works in the category of abelian groups, on which other categories does it also work? (Maybe in every concrete category? Every concrete category whose underlying set functor is conservative?)
Best Answer
The point here is that the forgetful functor $\mathbf{Ab\to Set}$ preserves filtered colimits, and that the colimit you're taking here is always filtered, so you get a description of that type, with the set coproduct (although you could also have a definition with direct sums)
This will work whenever your forgetful functor preserves filtered colimits, e.g. for rings, modules over a ring (or a sheaf of rings), ...