The Stacks project

Coverings become surjective after sheafification.

Lemma 7.12.4. Let $\mathcal{C}$ be a site. If $\{ U_ i \to U\} _{i \in I}$ is a covering of the site $\mathcal{C}$, then the morphism of presheaves of sets

\[ \coprod \nolimits _{i \in I} h_{U_ i} \to h_ U \]

becomes surjective after sheafification.

Proof. By Lemma 7.11.2 above we have to show that $\coprod \nolimits _{i \in I} h_{U_ i}^\# \to h_ U^\# $ is an epimorphism. Let $\mathcal{F}$ be a sheaf of sets. A morphism $h_ U^\# \to \mathcal{F}$ corresponds to a section $s \in \mathcal{F}(U)$. Hence the injectivity of $\mathop{\mathrm{Mor}}\nolimits (h_ U^\# , \mathcal{F}) \to \prod _ i \mathop{\mathrm{Mor}}\nolimits (h_{U_ i}^\# , \mathcal{F})$ follows directly from the sheaf property of $\mathcal{F}$. $\square$


Comments (3)

Comment #982 by on

Suggested slogan: Coverings become surjective after sheafification.

Comment #8673 by Figo on

Is here ?

Comment #9390 by on

Yes, if the two coproducts are for presheaves (on the left) and for sheaves (on the right).


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Your email address will not be published.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