The Stacks project

Definition 85.12.1. In Situation 85.3.3.

1. A sheaf $\mathcal{F}$ of sets or of abelian groups on $\mathcal{C}_{total}$ is cartesian if the maps $\mathcal{F}(\varphi ) : f_\varphi ^{-1}\mathcal{F}_ m \to \mathcal{F}_ n$ are isomorphisms for all $\varphi : [m] \to [n]$.

2. If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$, then a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules is cartesian if the maps $f_\varphi ^*\mathcal{F}_ m \to \mathcal{F}_ n$ are isomorphisms for all $\varphi : [m] \to [n]$.

3. An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if the maps $f_\varphi ^{-1}K_ m \to K_ n$ are isomorphisms for all $\varphi : [m] \to [n]$.

4. If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$, then an object $K$ of $D(\mathcal{O})$ is cartesian if the maps $Lf_\varphi ^*K_ m \to K_ n$ are isomorphisms for all $\varphi : [m] \to [n]$.