Definition 13.31.1. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n, f_ n)$ be a system of objects of $\mathcal{D}$. We say an object $K$ is a derived colimit, or a homotopy colimit of the system $(K_ n)$ if the direct sum $\bigoplus K_ n$ exists and there is a distinguished triangle

$\bigoplus K_ n \to \bigoplus K_ n \to K \to \bigoplus K_ n[1]$

where the map $\bigoplus K_ n \to \bigoplus K_ n$ is given by $1 - f_ n$ in degree $n$. If this is the case, then we sometimes indicate this by the notation $K = \text{hocolim} K_ n$.

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