The Stacks project

Lemma 7.10.15. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. Denote $\theta ^2 : \mathcal{F} \to \mathcal{F}^\# $ the canonical map of $\mathcal{F}$ into its sheafification. Let $U$ be an object of $\mathcal{C}$. Let $s \in \mathcal{F}^\# (U)$. There exists a covering $\{ U_ i \to U\} $ and sections $s_ i \in \mathcal{F}(U_ i)$ such that

  1. $s|_{U_ i} = \theta ^2(s_ i)$, and

  2. for every $i, j$ there exists a covering $\{ U_{ijk} \to U_ i \times _ U U_ j\} $ of $\mathcal{C}$ such that the pullbacks of $s_ i$ and $s_ j$ to each $U_{ijk}$ agree.

Conversely, given any covering $\{ U_ i \to U\} $, elements $s_ i \in \mathcal{F}(U_ i)$ such that (2) holds, then there exists a unique section $s \in \mathcal{F}^\# (U)$ such that (1) holds.

Proof. Omitted. $\square$


Comments (0)

There are also:

  • 8 comment(s) on Section 7.10: Sheafification

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Your email address will not be published.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