The Stacks project

Definition 89.7.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. The category $\widehat{\mathcal{F}}$ of formal objects of $\mathcal{F}$ is the category with the following objects and morphisms.

  1. A formal object $\xi = (R, \xi _ n, f_ n)$ of $\mathcal{F}$ consists of an object $R$ of $\widehat{\mathcal{C}}_\Lambda $, and a collection indexed by $n \in \mathbf{N}$ of objects $\xi _ n$ of $\mathcal{F}(R/\mathfrak m_ R^ n)$ and morphisms $f_ n : \xi _{n + 1} \to \xi _ n$ lying over the projection $R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n$.

  2. Let $\xi = (R, \xi _ n, f_ n)$ and $\eta = (S, \eta _ n, g_ n)$ be formal objects of $\mathcal{F}$. A morphism $a : \xi \to \eta $ of formal objects consists of a map $a_0 : R \to S$ in $\widehat{\mathcal{C}}_\Lambda $ and a collection $a_ n : \xi _ n \to \eta _ n$ of morphisms of $\mathcal{F}$ lying over $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$, such that for every $n$ the diagram

    \[ \xymatrix{ \xi _{n + 1} \ar[r]_{f_ n} \ar[d]_{a_{n + 1}} & \xi _ n \ar[d]^{a_ n} \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n } \]


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Your email address will not be published.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 06H3. Beware of the difference between the letter 'O' and the digit '0'.