The Stacks project

Remark 88.7.7. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda $. We claim that there is a canonical equivalence

\[ can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \longrightarrow \mathcal{F}. \]

Namely, let $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$ and let $(A, \xi _ n, f_ n)$ be an object of $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda }(A)$. Since $A$ is Artinian there is a minimal $m \in \mathbf{N}$ such that $\mathfrak m_ A^ m = 0$. Then $can$ sends $(A, \xi _ n, f_ n)$ to $\xi _ m$. This functor is an equivalence of categories cofibered in groupoids by Categories, Lemma 4.35.8 because it is an equivalence on all fibre categories by Lemma 88.7.4 and the fact that the $\mathfrak m_ A$-adic topology on a local Artinian ring $A$ comes from the zero ideal. We will frequently identify $\mathcal{F}$ with a full subcategory of $\widehat{\mathcal{F}}$ via a quasi-inverse to the functor $can$.


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