The Stacks project

Lemma 96.5.2. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $p$ and $q$ are limit preserving on objects, then so is the composition $q \circ p$.

Proof. This is formal. Let $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ be the directed limit of affine schemes $U_ i$ over $S$, let $z_ i$ be an object of $\mathcal{Z}$ over $U_ i$ for some $i$, let $x$ be an object of $\mathcal{X}$ over $U$, and let $\gamma : q(p(x)) \to z_ i|_ U$ be an isomorphism. As $q$ is limit preserving on objects there exist an $i' \geq i$, an object $y_{i'}$ of $\mathcal{Y}$ over $U_{i'}$, an isomorphism $\beta : y_{i'}|_ U \to p(x)$, and an isomorphism $\gamma _{i'} : q(y_{i'}) \to z_ i|_{U_{i'}}$ such that ( is commutative. As $p$ is limit preserving on objects there exist an $i'' \geq i'$, an object $x_{i''}$ of $\mathcal{X}$ over $U_{i''}$, an isomorphism $\beta ' : x_{i''}|_ U \to x$, and an isomorphism $\gamma '_{i''} : p(x_{i''}) \to y_{i'}|_{U_{i''}}$ such that ( is commutative. The solution is to take $x_{i''}$ over $U_{i''}$ with isomorphism

\[ q(p(x_{i''})) \xrightarrow {q(\gamma '_{i''})} q(y_{i'})|_{U_{i''}} \xrightarrow {\gamma _{i'}|_{U_{i''}}} z_ i|_{U_{i''}} \]

and isomorphism $\beta ' : x_{i''}|_ U \to x$. We omit the verification that ( is commutative. $\square$

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Your email address will not be published.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