The Stacks project

Definition 85.14.1. In Situation 85.3.3. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. A simplicial system of the derived category of modules consists of the following data

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

  2. for every $\varphi : [m] \to [n]$ a map $K_\varphi : Lf_\varphi ^*K_ m \to K_ n$ in $D(\mathcal{O}_ n)$

subject to the condition that

\[ K_{\varphi \circ \psi } = K_\varphi \circ Lf_\varphi ^*K_\psi : Lf_{\varphi \circ \psi }^*K_ l = Lf_\varphi ^* Lf_\psi ^*K_ l \longrightarrow K_ n \]

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 of modules.

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