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:

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

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

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

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

## Comments (3)

Comment #7968 by Haohao Liu on

Comment #8193 by Stacks Project on

Comment #8681 by Haohao Liu on