The Stacks project

Lemma 88.7.2. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids. Then $\widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda $ is a category cofibered in groupoids.

Proof. Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda $. Let $(R, \xi _ n, f_ n)$ be an object of $\widehat{\mathcal{F}}$. For each $n$ choose a pushforward $\xi _ n \to \eta _ n$ of $\xi _ n$ along $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$. For each $n$ there exists a unique morphism $g_ n : \eta _{n + 1} \to \eta _ n$ in $\mathcal{F}$ lying over $S/\mathfrak m_ S^{n + 1} \to S/\mathfrak m_ S^ n$ such that

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

commutes (by the first axiom of a category cofibred in groupoids). Hence we obtain a morphism $(R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ lying over $R \to S$, i.e., the first axiom of a category cofibred in groupoids holds for $\widehat{\mathcal{F}}$. To see the second axiom suppose that we have morphisms $a : (R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ and $b : (R, \xi _ n, f_ n) \to (T, \theta _ n, h_ n)$ in $\widehat{\mathcal{F}}$ and a morphism $c_0 : S \to T$ in $\widehat{\mathcal{C}}_\Lambda $ such that $c_0 \circ a_0 = b_0$. By the second axiom of a category cofibred in groupoids for $\mathcal{F}$ we obtain unique maps $c_ n : \eta _ n \to \theta _ n$ lying over $S/\mathfrak m_ S^ n \to T/\mathfrak m_ T^ n$ such that $c_ n \circ a_ n = b_ n$. Setting $c = (c_ n)_{n \geq 0}$ gives the desired morphism $c : (S, \eta _ n, g_ n) \to (T, \theta _ n, h_ n)$ in $\widehat{\mathcal{F}}$ (we omit the verification that $h_ n \circ c_{n + 1} = c_ n \circ g_ n$). $\square$


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Your email address will not be published.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 06H4. Beware of the difference between the letter 'O' and the digit '0'.