## 3.12 Abelian categories and injectives

The following lemma applies to the category of modules over a sheaf of rings on a site.

Lemma 3.12.1. Suppose given a big category $\mathcal{A}$ (see Categories, Remark 4.2.2). Assume $\mathcal{A}$ is abelian and has enough injectives. See Homology, Definitions 12.5.1 and 12.27.4. Then for any given set of objects $\{ A_ s\} _{s\in S}$ of $\mathcal{A}$, there is an abelian subcategory $\mathcal{A}' \subset \mathcal{A}$ with the following properties:

1. $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ is a set,

2. $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ contains $A_ s$ for each $s \in S$,

3. $\mathcal{A}'$ has enough injectives, and

4. an object of $\mathcal{A}'$ is injective if and only if it is an injective object of $\mathcal{A}$.

Proof. Omitted. $\square$

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