The Stacks project

Remark 88.7.9. We claim the completion functor of Remark 88.7.8 and the restriction functor $|_{\mathcal{C}_\Lambda } : \text{Cof}(\widehat{\mathcal{C}}_\Lambda ) \to \text{Cof}(\mathcal{C}_\Lambda )$ of Remarks 88.5.2 (15) are “2-adjoint” in the following precise sense. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\text{Cof}(\mathcal{C}_\Lambda ))$ and let $\mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\text{Cof}(\widehat{\mathcal{C}}_\Lambda ))$. Then there is an equivalence of categories

\[ \Phi : \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(\mathcal{G}, \widehat{\mathcal{F}}) \]

To describe this equivalence, we define canonical morphisms $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda }}$ and $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ as follows

  1. Let $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda ))$ and let $\xi $ be an object of the fiber category $\mathcal{G}(R)$. Choose a pushforward $\xi \to \xi _ n$ of $\xi $ to $R/\mathfrak m_ R^ n$ for each $n \in \mathbf{N}$, and let $f_ n : \xi _{n + 1} \to \xi _ n$ be the induced morphism. Then $\mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda }}$ sends $\xi $ to $(R, \xi _ n, f_ n)$.

  2. This is the equivalence $can : \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ of Remark 88.7.7.

Having said this, the equivalence $\Phi : \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \to \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(\mathcal{G}, \widehat{\mathcal{F}})$ sends a morphism $\varphi : \mathcal{G}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ to

\[ \mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda }} \xrightarrow {\widehat{\varphi }} \widehat{\mathcal{F}} \]

There is a quasi-inverse $\Psi : \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \mathcal{G}, \widehat{\mathcal{F}}) \to \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F})$ to $\Phi $ which sends $\psi : \mathcal{G} \to \widehat{\mathcal{F}}$ to

\[ \mathcal{G}|_{\mathcal{C}_\Lambda } \xrightarrow {\psi |_{\mathcal{C}_\Lambda }} \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \to \mathcal{F}. \]

We omit the verification that $\Phi $ and $\Psi $ are quasi-inverse. We also do not address functoriality of $\Phi $ (because it would lead into 3-category territory which we want to avoid at all cost).


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