The Stacks project

Lemma 8.4.3. Let $p : \mathcal{S} \to \mathcal{C}$ be a stack over the site $\mathcal{C}$. Let $\mathcal{S}'$ be a subcategory of $\mathcal{S}$. Assume

  1. if $\varphi : y \to x$ is a strongly cartesian morphism of $\mathcal{S}$ and $x$ is an object of $\mathcal{S}'$, then $y$ is isomorphic to an object of $\mathcal{S}'$,

  2. $\mathcal{S}'$ is a full subcategory of $\mathcal{S}$, and

  3. if $\{ f_ i : U_ i \to U\} $ is a covering of $\mathcal{C}$, and $x$ an object of $\mathcal{S}$ over $U$ such that $f_ i^*x$ is isomorphic to an object of $\mathcal{S}'$ for each $i$, then $x$ is isomorphic to an object of $\mathcal{S}'$.

Then $\mathcal{S}' \to \mathcal{C}$ is a stack.

Proof. Omitted. Hints: The first condition guarantees that $\mathcal{S}'$ is a fibred category. The second condition guarantees that the $\mathit{Isom}$-presheaves of $\mathcal{S}'$ are sheaves (as they are identical to their counter parts in $\mathcal{S}$). The third condition guarantees that the descent condition holds in $\mathcal{S}'$ as we can first descend in $\mathcal{S}$ and then (3) implies the resulting object is isomorphic to an object of $\mathcal{S}'$. $\square$


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Your email address will not be published.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 04TU. Beware of the difference between the letter 'O' and the digit '0'.