Projective Resolution Exists Iff Enough Projectives

Theorem
Let $\AA$ be an abelian category.

Then $\AA$ has enough projectives any object in $\AA$ has a projective resolution.

Proof
Suppose $\AA$ has enough projectives.

Let $X$ be an object in $\AA$.

Then there is an epimorphism $\varepsilon : P_0 \to X$ for some projective object $P_0$.

In particular
 * $P_0 \to X \to 0$

is exact at $X$.

Since $\AA$ is abelian it has kernels.

Thus $\varepsilon$ has a kernel $K \to P_0$.

Since $\AA$ has enough projectives, there is an epimorphism $\tilde d_1 : P_1 \to K$.

Define $d_1 : P_1 \to K \to P_0$ by composition.

By Uniqueness of Epi-Mono Factorization and since this is an epi-mono factorization of $d_1$, $K$ is the image of $d_1$.

Hence
 * $P_1 \to P_0 \to X \to 0$

is exact at $P_0$.

By induction there is a projective resolution of $X$.

Conversely suppose, that any object $X$ admits a projective resolution.

Then the augmentation map $\varepsilon: P_0 \to X$ is an epimorphism by a projective object $P_0$.

Hence $\AA$ has enough projectives.