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Let $\mathcal F$ a sheaf of a category $C$ over some space $X$. There is an injection $$\mathcal F \hookrightarrow \Pi_{x\in X} \ \mathcal F_x.$$ It seems like if the category $C$ has enough injectives, then we can inject each stalk $\mathcal F_x$in a injective object $I_x \in C$ and thus we get an injection of sheaves: $$\mathcal F \hookrightarrow \Pi_{x\in X}\ \mathcal F_x\hookrightarrow \Pi_{x\in X} \ I_x.$$

Is this true? Further question, do the categories one usually works with have enough injectives? For example Abelian groups, $\mathcal O_X$-modules etc...?

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Yes, it is called the Godement resolution. For abelian groups there are always enough injective (because injective $\Leftrightarrow$ divisible). For quasi-coherent $\mathcal O_X$-modules, this is true if $X$ is noetherian. This is not true for coherent sheaves however, so this is why we need quasi-coherent sheaves.

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