The Stacks project

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

\[ P : \mathcal{A} \longrightarrow \text{Arrows}(\mathcal{A}) \]

such that

  1. $t \circ P = \text{id}_\mathcal {A}$,

  2. for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the morphism $P(A)$ is surjective, and

  3. for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the object $s(P(A))$ is an projective object of $\mathcal{A}$.

We will denote such a functor by $A \mapsto (P(A) \to A)$.


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