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
\[ \xymatrix{ A \ar[r] \ar[d] & B \ar@{-->}[ld] \\ J & } \]
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
in $\mathcal{A}$ is split.
We have $\mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(B, I) = 0$ for all $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$.
\[ 0 \to I \to A \to B \to 0 \]
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 $\mathcal{A}$.
\[ J : \mathcal{A} \longrightarrow \text{Arrows}(\mathcal{A}) \]
We will denote such a functor by $A \mapsto (A \to J(A))$.
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