The Stacks project

Lemma 7.32.7. Let $\mathcal{C}$ be a site.

  1. Let $p$ be a point of the site $\mathcal{C}$. Then the pair of functors $(p_*, p^{-1})$ introduced above define a morphism of topoi $\mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

  2. Let $p = (p_*, p^{-1})$ be a point of the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Then the functor $u : U \mapsto p^{-1}(h_ U^\# )$ gives rise to a point $p'$ of the site $\mathcal{C}$ whose associated morphism of topoi $(p'_*, (p')^{-1})$ is equal to $p$.

Proof. Proof of (1). By the above the functors $p_*$ and $p^{-1}$ are adjoint. The functor $p^{-1}$ is required to be exact by Definition 7.32.2. Hence the conditions imposed in Definition 7.15.1 are all satisfied and we see that (1) holds.

Proof of (2). Let $\{ U_ i \to U\} $ be a covering of $\mathcal{C}$. Then $\coprod (h_{U_ i})^\# \to h_ U^\# $ is surjective, see Lemma 7.12.4. Since $p^{-1}$ is exact (by definition of a morphism of topoi) we conclude that $\coprod u(U_ i) \to u(U)$ is surjective. This proves part (1) of Definition 7.32.2. Sheafification is exact, see Lemma 7.10.14. Hence if $U \times _ V W$ exists in $\mathcal{C}$, then

\[ h_{U \times _ V W}^\# = h_ U^\# \times _{h_ V^\# } h_ W^\# \]

and we see that $u(U \times _ V W) = u(U) \times _{u(V)} u(W)$ since $p^{-1}$ is exact. This proves part (2) of Definition 7.32.2. Let $p' = u$, and let $\mathcal{F}_{p'}$ be the stalk functor defined by Equation (7.32.1.1) using $u$. There is a canonical comparison map $c : \mathcal{F}_{p'} \to \mathcal{F}_ p = p^{-1}\mathcal{F}$. Namely, given a triple $(U, x, \sigma )$ representing an element $\xi $ of $\mathcal{F}_{p'}$ we think of $\sigma $ as a map $\sigma : h_ U^\# \to \mathcal{F}$ and we can set $c(\xi ) = p^{-1}(\sigma )(x)$ since $x \in u(U) = p^{-1}(h_ U^\# )$. By Lemma 7.32.3 we see that $(h_ U)_{p'} = u(U)$. Since conditions (1) and (2) of Definition 7.32.2 hold for $p'$ we also have $(h_ U^\# )_{p'} = (h_ U)_{p'}$ by Lemma 7.32.5. Hence we have

\[ (h_ U^\# )_{p'} = (h_ U)_{p'} = u(U) = p^{-1}(h_ U^\# ) \]

We claim this bijection equals the comparison map $c : (h_ U^\# )_{p'} \to p^{-1}(h_ U^\# )$ (verification omitted). Any sheaf on $\mathcal{C}$ is a coequalizer of maps of coproducts of sheaves of the form $h_ U^\# $, see Lemma 7.12.5. The stalk functor $\mathcal{F} \mapsto \mathcal{F}_{p'}$ and the functor $p^{-1}$ commute with arbitrary colimits (as they are both left adjoints). We conclude $c$ is an isomorphism for every sheaf $\mathcal{F}$. Thus the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{p'}$ is isomorphic to $p^{-1}$ and we in particular see that it is exact. This proves condition (3) of Definition 7.32.2 holds and $p'$ is a point. The final assertion has already been shown above, since we saw that $p^{-1} = (p')^{-1}$. $\square$


Comments (0)


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 00YA. Beware of the difference between the letter 'O' and the digit '0'.