The Stacks project

Remark 7.33.4. In fact, let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has a final object $X$ and fibre products. Let $p = u: \mathcal{C} \to \textit{Sets}$ be a functor such that

  1. $u(X) = \{ *\} $ a singleton, and

  2. for every pair of morphisms $U \to W$ and $V \to W$ with the same target the map $u(U \times _ W V) \to u(U) \times _{u(W)} u(V)$ is surjective.

  3. for every covering $\{ U_ i \to U\} $ the map $\coprod u(U_ i) \to u(U)$ is surjective.

Then, in general, $p$ is not a point of $\mathcal{C}$. An example is the category $\mathcal{C}$ with two objects $\{ U, X\} $ and exactly one non-identity arrow, namely $U \to X$. We endow $\mathcal{C}$ with the trivial topology, i.e., the only coverings are $\{ U \to U\} $ and $\{ X \to X\} $. A sheaf $\mathcal{F}$ is the same thing as a presheaf and consists of a triple $(A, B, A \to B)$: namely $A = \mathcal{F}(X)$, $B = \mathcal{F}(U)$ and $A \to B$ is the restriction mapping corresponding to $U \to X$. Note that $U \times _ X U = U$ so fibre products exist. Consider the functor $u = p$ with $u(X) = \{ *\} $ and $u(U) = \{ *_1, *_2\} $. This satisfies (1), (2), and (3), but the corresponding stalk functor ( is the functor

\[ (A, B, A \to B) \longmapsto B \amalg _ A B \]

which isn't exact. Namely, consider $(\emptyset , \{ 1\} , \emptyset \to \{ 1\} ) \to (\{ 1\} , \{ 1\} , \{ 1\} \to \{ 1\} )$ which is an injective map of sheaves, but is transformed into the noninjective map of sets

\[ \{ 1\} \amalg \{ 1\} \longrightarrow \{ 1\} \amalg _{\{ 1\} } \{ 1\} \]

by the stalk functor.

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