Example 6.4.5. Let $X$ be a topological space. For each $x \in X$ suppose given an abelian group $M_ x$. For $U \subset X$ open we set

$\mathcal{F}(U) = \bigoplus \nolimits _{x \in U} M_ x.$

We denote a typical element in this abelian group by $\sum _{i = 1}^ n m_{x_ i}$, where $x_ i \in U$ and $m_{x_ i} \in M_{x_ i}$. (Of course we may always choose our representation such that $x_1, \ldots , x_ n$ are pairwise distinct.) We define for $V \subset U \subset X$ open a restriction mapping $\mathcal{F}(U) \to \mathcal{F}(V)$ by mapping an element $s = \sum _{i = 1}^ n m_{x_ i}$ to the element $s|_ V = \sum _{x_ i \in V} m_{x_ i}$. We leave it to the reader to verify that this is a presheaf of abelian groups.

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