The Stacks project

Lemma 7.14.6. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be continuous. Assume all the categories $(\mathcal{I}_ V^ u)^{opp}$ of Section 7.5 are filtered. Then $u$ defines a morphism of sites $\mathcal{D} \to \mathcal{C}$, in other words $u_ s$ is exact.

Proof. Since $u_ s$ is the left adjoint of $u^ s$ we see that $u_ s$ is right exact, see Categories, Lemma 4.24.6. Hence it suffices to show that $u_ s$ is left exact. In other words we have to show that $u_ s$ commutes with finite limits. Because the categories $\mathcal{I}_ Y^{opp}$ are filtered we see that $u_ p$ commutes with finite limits, see Categories, Lemma 4.19.2 (this also uses the description of limits in $\textit{PSh}$, see Section 7.4). And since sheafification commutes with finite limits as well (Lemma 7.10.14) we conclude because $u_ s = \# \circ u_ p$. $\square$


Comments (0)

There are also:

  • 1 comment(s) on Section 7.14: Morphisms of sites

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Your email address will not be published.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 00X5. Beware of the difference between the letter 'O' and the digit '0'.