Lemma 3.12.1 (0010)—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:

the inclusion functor $\mathcal{A}' \to \mathcal{A}$ is exact,
$\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ 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 (4)

Comment #7968 by Haohao Liu on January 13, 2023 at 03:59

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

Comment #8193 by Stacks Project on March 02, 2023 at 19:15

It is a subcategory which is also an abelian category.

Comment #8681 by Haohao Liu on September 27, 2023 at 02:21

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

Comment #9386 by Stacks project on June 05, 2024 at 09:46

OK, thanks for the comment on this. I have added the condition that we want $\mathcal{A}' \to \mathcal{A}$ to be exact in the statement of the lemma. See here.

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