Definition 12.28.1. Let $\mathcal{A}$ be an abelian category. An object $P \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ is called projective if for every surjection $A \rightarrow B$ and every morphism $P \to B$ there exists a morphism $P \to A$ making the following diagram commute
\[ \xymatrix{ A \ar[r] & B \\ P \ar@{-->}[u] \ar[ru] & } \]
Here is the obligatory characterization of projective objects.
Lemma 12.28.2. Let $\mathcal{A}$ be an abelian category. Let $P$ be an object of $\mathcal{A}$. The following are equivalent:
The object $P$ is projective.
The functor $B \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(P, B)$ is exact.
Any short exact sequence
in $\mathcal{A}$ is split.
We have $\mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(P, A) = 0$ for all $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$.
\[ 0 \to A \to B \to P \to 0 \]
Proof. Omitted. $\square$
Lemma 12.28.3. Let $\mathcal{A}$ be an abelian category. Suppose $P_\omega $, $\omega \in \Omega $ is a set of projective objects of $\mathcal{A}$. If $\coprod _{\omega \in \Omega } P_\omega $ exists then it is projective.
Proof. Omitted. $\square$
Definition 12.28.4. Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has enough projectives if every object $A$ has an surjective morphism $P \to A$ from an projective object $P$ onto it.
Definition 12.28.5. Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has functorial projective surjections if there exists a functor such that
$t \circ P = \text{id}_\mathcal {A}$,
for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the morphism $P(A)$ is surjective, and
for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the object $s(P(A))$ is an projective object of $\mathcal{A}$.
\[ P : \mathcal{A} \longrightarrow \text{Arrows}(\mathcal{A}) \]
We will denote such a functor by $A \mapsto (P(A) \to A)$.
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