Definition 7.32.2. Let $\mathcal{C}$ be a site. A point $p$ of the site $\mathcal{C}$ is given by a functor $u : \mathcal{C} \to \textit{Sets}$ such that

For every covering $\{ U_ i \to U\} $ of $\mathcal{C}$ the map $\coprod u(U_ i) \to u(U)$ is surjective.

For every covering $\{ U_ i \to U\} $ of $\mathcal{C}$ and every morphism $V \to U$ the maps $u(U_ i \times _ U V) \to u(U_ i) \times _{u(U)} u(V)$ are bijective.

The stalk functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}_ p$ is left exact.

OK, yes we could change the definition and say this, but I wanted to write down minimal conditions. The discussion following the definition shows that indeed the stalk functor is exact (as it defines a point of the topos).

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Your email address will not be published.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.

## Comments (2)

Comment #2344 by Katharina on

Comment #2413 by Johan on