Definition 84.12.1. In Situation 84.3.3.

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 $\varphi : [m] \to [n]$.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 $\varphi : [m] \to [n]$.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 $\varphi : [m] \to [n]$.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 $\varphi : [m] \to [n]$.

## Comments (2)

Comment #6285 by Rachel Webb on

Comment #6403 by Johan on