Remark 90.7.12 (06HC)—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 $\underline{\xi }$.

The Stacks project

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