The Stacks project

Remark 89.7.12. 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 89.5.2 (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
\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 89.7.9 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 ( 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 (

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 ( 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 89.7.9. 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 ( Using $\underline{R} = \widehat{\underline{R}|_{\mathcal{C}_\Lambda }}$ (see Remark 89.7.11) and unwinding the definitions of $\Phi $ and $\Psi $ we conclude that $\iota (\xi )$ is isomorphic to the completion of $\underline{\xi }$.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06HC. Beware of the difference between the letter 'O' and the digit '0'.