The Stacks project

Lemma 13.33.7. Let $\mathcal{A}$ be an abelian category. Let $L_ n^\bullet $ be a system of complexes of $\mathcal{A}$. Assume colimits over $\mathbf{N}$ exist and are exact in $\mathcal{A}$. Then the termwise colimit $L^\bullet = \mathop{\mathrm{colim}}\nolimits L_ n^\bullet $ is a homotopy colimit of the system in $D(\mathcal{A})$.

Proof. We have an exact sequence of complexes

\[ 0 \to \bigoplus L_ n^\bullet \to \bigoplus L_ n^\bullet \to L^\bullet \to 0 \]

by Lemma 13.33.6. The direct sums are direct sums in $D(\mathcal{A})$ by Lemma 13.33.5. Thus the result follows from the definition of derived colimits in Definition 13.33.1 and the fact that a short exact sequence of complexes gives a distinguished triangle (Lemma 13.12.1). $\square$

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