The Stacks project

Lemma 7.32.9. Let $\mathcal{C}$ be a site. Let $p : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a point of the topos associated to $\mathcal{C}$. For any set $E$ there are canonical maps

\[ E \longrightarrow (p_*E)_ p \longrightarrow E \]

whose composition is $\text{id}_ E$.

Proof. There is always an adjunction map $(p_*E)_ p = p^{-1}p_*E \to E$. This map is an isomorphism when $E = \{ *\} $ because $p_*$ and $p^{-1}$ are both left exact, hence transform the final object into the final object. Hence given $e \in E$ we can consider the map $i_ e : \{ *\} \to E$ which gives

\[ \xymatrix{ p^{-1}p_*\{ *\} \ar[rr]_{p^{-1}p_*i_ e} \ar[d]_{\cong } & & p^{-1}p_*E \ar[d] \\ \{ *\} \ar[rr]^{i_ e} & & E } \]

whence the map $E \to (p_*E)_ p = p^{-1}p_*E$ as desired. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 7.32: Points

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Your email address will not be published.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 05UX. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 7.32.9 as pdf