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

becomes surjective after sheafification.

** 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 $\mathop{Mor}\nolimits (h_ U^\# , \mathcal{F}) \to \prod _ i \mathop{Mor}\nolimits (h_{U_ i}^\# , \mathcal{F})$ follows directly from the sheaf property of $\mathcal{F}$.
$\square$

Your email address will not be published.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.

## Comments (1)

Comment #982 by Johan Commelin on