Definition 13.23.2. Let $\mathcal{A}$ be an abelian category with enough injectives. A *resolution functor*^{1} for $\mathcal{A}$ is given by the following data:

for all $K^\bullet \in \mathop{\mathrm{Ob}}\nolimits (K^{+}(\mathcal{A}))$ a bounded below complex of injectives $j(K^\bullet )$, and

for all $K^\bullet \in \mathop{\mathrm{Ob}}\nolimits (K^{+}(\mathcal{A}))$ a quasi-isomorphism $i_{K^\bullet } : K^\bullet \to j(K^\bullet )$.

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