The Stacks project

Definition 85.13.1. In Situation 85.3.3. A simplicial system of the derived category consists of the following data

  1. for every $n$ an object $K_ n$ of $D(\mathcal{C}_ n)$,

  2. for every $\varphi : [m] \to [n]$ a map $K_\varphi : f_\varphi ^{-1}K_ m \to K_ n$ in $D(\mathcal{C}_ n)$

subject to the condition that

\[ K_{\varphi \circ \psi } = K_\varphi \circ f_\varphi ^{-1}K_\psi : f_{\varphi \circ \psi }^{-1}K_ l = f_\varphi ^{-1} f_\psi ^{-1}K_ l \longrightarrow K_ n \]

for any morphisms $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$ of $\Delta $. We say the simplicial system is cartesian if the maps $K_\varphi $ are isomorphisms for all $\varphi $. Given two simplicial systems of the derived category there is an obvious notion of a morphism of simplicial systems of the derived category.

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