Remark 13.33.2. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n, f_ n)$ be a system of objects of $\mathcal{D}$. We may think of a derived colimit as an object $K$ of $\mathcal{D}$ endowed with morphisms $i_ n : K_ n \to K$ such that $i_{n + 1} \circ f_ n = i_ n$ and such that there exists a morphism $c : K \to \bigoplus K_ n$ with the property that

$\bigoplus K_ n \xrightarrow {1 - f_ n} \bigoplus K_ n \xrightarrow {i_ n} K \xrightarrow {c} \bigoplus K_ n[1]$

is a distinguished triangle. If $(K', i'_ n, c')$ is a second derived colimit, then there exists an isomorphism $\varphi : K \to K'$ such that $\varphi \circ i_ n = i'_ n$ and $c' \circ \varphi = c$. The existence of $\varphi$ is TR3 and the fact that $\varphi$ is an isomorphism is Lemma 13.4.3.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CRH. Beware of the difference between the letter 'O' and the digit '0'.