The Stacks project

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

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

  2. 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.

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

Comments (2)

Comment #2344 by Katharina on

Why does the stalk functor only have to be left exact and not exact?

Comment #2413 by on

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).

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  • 1 comment(s) on Section 7.32: Points

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