The Stacks project

Definition 13.33.1. 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 $K = \text{hocolim} K_ n$.

Comments (2)

Comment #11038 by on

TeX style comment: I'm seeing that throughout derived.tex the homotopy colimit is TeX'ed as \text{hocolim}. However, for better spacing, I think \operatorname{hocolim} should be preferable: the former gives and the latter .

Comment #11039 by on

The morphism corresponds to , right? I haven't been able to find in the Stacks Project what's the convention to denote systems of objects over the natural numbers. In Categories, Sect. 4.21, only systems over preordered sets are considered. From the equation in Remark 13.33.2 one inferes it must be .

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View Definition 13.33.1 as pdf