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

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

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

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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