Definition 13.9.1. Let $\mathcal{A}$ be an additive category. Let $f : K^\bullet \to L^\bullet$ be a morphism of complexes of $\mathcal{A}$. The cone of $f$ is the complex $C(f)^\bullet$ given by $C(f)^ n = L^ n \oplus K^{n + 1}$ and differential

$d_{C(f)}^ n = \left( \begin{matrix} d^ n_ L & f^{n + 1} \\ 0 & -d_ K^{n + 1} \end{matrix} \right)$

It comes equipped with canonical morphisms of complexes $i : L^\bullet \to C(f)^\bullet$ and $p : C(f)^\bullet \to K^\bullet [1]$ induced by the obvious maps $L^ n \to C(f)^ n \to K^{n + 1}$.

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