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The Stacks project

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[1] 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.


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