Definition 12.27.1. Let $\mathcal{A}$ be an abelian category. An object $J \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ is called *injective* if for every injection $A \hookrightarrow B$ and every morphism $A \to J$ there exists a morphism $B \to J$ making the following diagram commute

## 12.27 Injectives

Here is the obligatory characterization of injective objects.

Lemma 12.27.2. Let $\mathcal{A}$ be an abelian category. Let $I$ be an object of $\mathcal{A}$. The following are equivalent:

The object $I$ is injective.

The functor $B \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(B, I)$ is exact.

Any short exact sequence

\[ 0 \to I \to A \to B \to 0 \]in $\mathcal{A}$ is split.

We have $\mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(B, I) = 0$ for all $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$.

**Proof.**
Omitted.
$\square$

Lemma 12.27.3. Let $\mathcal{A}$ be an abelian category. Suppose $I_\omega $, $\omega \in \Omega $ is a set of injective objects of $\mathcal{A}$. If $\prod _{\omega \in \Omega } I_\omega $ exists then it is injective.

**Proof.**
Omitted.
$\square$

Definition 12.27.4. Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has *enough injectives* if every object $A$ has an injective morphism $A \to J$ into an injective object $J$.

Definition 12.27.5. Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has *functorial injective embeddings* if there exists a functor

such that

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

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

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 $A \mapsto (A \to J(A))$.

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