Lemma 85.12.2. In Situation 85.3.3.

A sheaf $\mathcal{F}$ of sets or abelian groups is cartesian if and only if the maps $(f_{\delta ^ n_ j})^{-1}\mathcal{F}_{n - 1} \to \mathcal{F}_ n$ are isomorphisms.

An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only if the maps $(f_{\delta ^ n_ j})^{-1}K_{n - 1} \to K_ n$ are isomorphisms.

If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$ a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules is cartesian if and only if the maps $(f_{\delta ^ n_ j})^*\mathcal{F}_{n - 1} \to \mathcal{F}_ n$ are isomorphisms.

If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$ an object $K$ of $D(\mathcal{O})$ is cartesian if and only if the maps $L(f_{\delta ^ n_ j})^*K_{n - 1} \to K_ n$ are isomorphisms.

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