The Stacks project

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

\[ \xymatrix{ A \ar[r] & B \\ P \ar@{-->}[u] \ar[ru] & } \]

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View Definition 12.28.1 as pdf