
Definition 12.24.5. Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has functorial injective embeddings if there exists a functor

$J : \mathcal{A} \longrightarrow \text{Arrows}(\mathcal{A})$

such that

1. $s \circ J = \text{id}_\mathcal {A}$,

2. for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the morphism $J(A)$ is injective, and

3. for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the object $t(J(A))$ is an injective object of $\mathcal{A}$.

We will denote such a functor by $A \mapsto (A \to J(A))$.

Comment #807 by OS on

I suggest to add that $J$ should be an additive functor. (This seems to be meant judging from the later results.)

The same comment applies to the definition of "functorial projective surjections", tag 013F; otherwise module categories would have them.

(It is clear what is meant but I could not find the definition of the category $\text{Arrows}(\mathcal{A})$.)

Comment #809 by OS on

I apologize for the previous comment. It now seems to me that many of the functorial injective embeddings constructed later on are not at all additive. So maybe it is good to mention explicitly that $J$ is not assumed to be additive...

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