Proposition 7.33.3. Let $\mathcal{C}$ be a site. Assume that finite limits exist in $\mathcal{C}$. (I.e., $\mathcal{C}$ has fibre products, and a final object.) A point $p$ of such a site $\mathcal{C}$ is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that

1. $u$ commutes with finite limits, and

2. if $\{ U_ i \to U\}$ is a covering, then $\coprod _ i u(U_ i) \to u(U)$ is surjective.

Proof. Suppose first that $p$ is a point (Definition 7.32.2) given by a functor $u$. Condition (2) is satisfied directly from the definition of a point. By Lemma 7.32.3 we have $(h_ U)_ p = u(U)$. By Lemma 7.32.5 we have $(h_ U^\# )_ p = (h_ U)_ p$. Thus we see that $u$ is equal to the composition of functors

$\mathcal{C} \xrightarrow {h} \textit{PSh}(\mathcal{C}) \xrightarrow {{}^\# } \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \xrightarrow {()_ p} \textit{Sets}$

Each of these functors is left exact, and hence we see $u$ satisfies (1).

Conversely, suppose that $u$ satisfies (1) and (2). In this case we immediately see that $u$ satisfies the first two conditions of Definition 7.32.2. And its stalk functor is exact, because it is a left adjoint by Lemma 7.32.5 and it commutes with finite limits by Lemma 7.33.2. $\square$

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Your email address will not be published.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