The Stacks project

Remark 90.7.5. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids. Suppose that for each $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$ we are given a filtration $\mathcal{I}_ R$ of $R$ by ideals. If $\mathcal{I}_ R$ induces the $\mathfrak m_ R$-adic topology on $R$ for all $R$, then one can define a category $\widehat{\mathcal{F}}_\mathcal {I}$ by mimicking the definition of $\widehat{\mathcal{F}}$. This category comes equipped with a morphism $\widehat{p}_\mathcal {I} : \widehat{\mathcal{F}}_\mathcal {I} \to \widehat{\mathcal{C}}_\Lambda $ making it into a category cofibered in groupoids such that $\widehat{\mathcal{F}}_\mathcal {I}(R)$ is isomorphic to $\widehat{\mathcal{F}}_{\mathcal{I}_ R}(R)$ as defined above. The categories cofibered in groupoids $\widehat{\mathcal{F}}_\mathcal {I}$ and $\widehat{\mathcal{F}}$ are equivalent, by using over an object $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$ the equivalence of Lemma 90.7.4.