We first deal with the case of abelian groups as a model for the general case.
Lemma 6.12.1. Let X be a topological space. Let \mathcal{F} be a presheaf of abelian groups on X. There exists a unique structure of an abelian group on \mathcal{F}_ x such that for every U \subset X open, x\in U the map \mathcal{F}(U) \to \mathcal{F}_ x is a group homomorphism. Moreover,
holds in the category of abelian groups.
Proof. We define addition of a pair of elements (U, s) and (V, t) as the pair (U \cap V, s|_{U\cap V} + t|_{U \cap V}). The rest is easy to check. \square
What is crucial in the proof above is that the partially ordered set of open neighbourhoods is a directed set (Categories, Definition 4.21.1). Namely, the coproduct of two abelian groups A, B is the direct sum A \oplus B, whereas the coproduct in the category of sets is the disjoint union A \amalg B, showing that colimits in the category of abelian groups do not agree with colimits in the category of sets in general.
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