The Stacks project

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. the inclusion functor $\mathcal{A}' \to \mathcal{A}$ is exact,

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

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

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

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

Proof. Omitted. $\square$

Comments (4)

Comment #7968 by Haohao Liu on

A tiny suggestion: it may be better to define "abelian subcategory".

Comment #8681 by Haohao Liu on

To Comment #8193: Then for a topological space , the category of abelian sheaves on is an abelian subcategory of the category of abelian presheaves on in the comment's sense. In general, the inclusion functor is not exact, so is not an abelian subcategory of in the sense of p.7 of "Introduction to Homological Algebra" by Weibel.

Comment #9386 by on

OK, thanks for the comment on this. I have added the condition that we want to be exact in the statement of the lemma. See here.

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