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,

$\mathcal{F}_ x = \mathop{\mathrm{colim}}\nolimits _{x\in U} \mathcal{F}(U)$

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. $\square$

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