Definition 13.34.1. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n, f_ n)$ be an inverse system of objects of $\mathcal{D}$. We say an object $K$ is a derived limit, or a homotopy limit of the system $(K_ n)$ if the product $\prod K_ n$ exists and there is a distinguished triangle

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

where the map $\prod K_ n \to \prod K_ n$ is given by $(k_ n) \mapsto (k_ n - f_{n+1}(k_{n + 1}))$. If this is the case, then we sometimes indicate this by the notation $K = R\mathop{\mathrm{lim}}\nolimits K_ n$.

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