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The Stacks project

Remark 90.7.12. Let R be an object of \widehat{\mathcal{C}}_\Lambda . It defines a functor \underline{R}: \widehat{\mathcal{C}}_\Lambda \to \textit{Sets} as described in Remarks 90.5.2 (12). As usual we identify this functor with the associated cofibered set. If \mathcal{F} is a cofibered category over \mathcal{C}_\Lambda , then there is an equivalence of categories

90.7.12.1
\begin{equation} \label{formal-defos-equation-formal-objects-maps} \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \underline{R}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \longrightarrow \widehat{\mathcal{F}}(R). \end{equation}

It is given by the composition

\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \underline{R}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \xrightarrow {\Phi } \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow {\sim } \widehat{\mathcal{F}}(R)

where \Phi is as in Remark 90.7.9 and the second equivalence comes from the 2-Yoneda lemma (the cofibered analogue of Categories, Lemma 4.41.2). Explicitly, the equivalence sends a morphism \varphi : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F} to the formal object (R, \varphi (R \to R/\mathfrak {m}_ R^ n), \varphi (f_ n)) in \widehat{\mathcal{F}}(R), where f_ n : R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n is the projection.

Assume a choice of pushforwards for \mathcal{F} has been made. Given any \xi \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{F}}(R)) we construct an explicit \underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F} which maps to \xi under (90.7.12.1). Namely, say \xi = (R, \xi _ n, f_ n). An object \alpha in \underline{R}|_{\mathcal{C}_\Lambda } is the same thing as a morphism \alpha : R \to A of \widehat{\mathcal{C}}_\Lambda with A Artinian. Let m \in \mathbf{N} be minimal such that \mathfrak m_ A^ m = 0. Then \alpha factors through a unique \alpha _ m : R/\mathfrak m_ R^ m \to A and we can set \underline{\xi }(\alpha ) = \alpha _{m, *}\xi _ m. We omit the description of \underline{\xi } on morphisms and we omit the proof that \underline{\xi } maps to \xi via (90.7.12.1).

Assume a choice of pushforwards for \widehat{\mathcal{F}} has been made. In this case the proof of Categories, Lemma 4.41.2 gives an explicit quasi-inverse

\iota : \widehat{\mathcal{F}}(R) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \underline{R}, \widehat{\mathcal{F}})

to the 2-Yoneda equivalence which takes \xi to the morphism \iota (\xi ) : \underline{R} \to \widehat{\mathcal{F}} sending f \in \underline{R}(S) = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }(R, S) to f_*\xi . A quasi-inverse to (90.7.12.1) is then

\widehat{\mathcal{F}}(R) \xrightarrow {\iota } \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \underline{R}, \widehat{\mathcal{F}}) \xrightarrow {\Psi } \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \underline{R}|_{\mathcal{C}_\Lambda }, \mathcal{F})

where \Psi is as in Remark 90.7.9. Given \xi \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{F}}(R)) we have \Psi (\iota (\xi )) \cong \underline{\xi } where \underline{\xi } is as in the previous paragraph, because both are mapped to \xi under the equivalence of categories (90.7.12.1). Using \underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda }} (see Remark 90.7.11) and unwinding the definitions of \Phi and \Psi we conclude that \iota (\xi ) is isomorphic to the completion of \underline{\xi }.


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