Lemma 7.27.2. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Assume $\mathcal{C}$ has products of pairs of objects. Then

1. the functor $j_ U$ has a continuous right adjoint, namely the functor $v(X) = X \times U / U$,

2. the functor $v$ defines a morphism of sites $\mathcal{C}/U \to \mathcal{C}$ whose associated morphism of topoi equals $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

3. we have $j_{U*}\mathcal{F}(X) = \mathcal{F}(X \times U/U)$.

Proof. The functor $v$ being right adjoint to $j_ U$ means that given $Y/U$ and $X$ we have

$\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(Y, X) = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}/U}(Y/U, X \times U/U)$

which is clear. To check that $v$ is continuous let $\{ X_ i \to X\}$ be a covering of $\mathcal{C}$. By the third axiom of a site (Definition 7.6.2) we see that

$\{ X_ i \times _ X (X \times U) \to X \times _ X (X \times U)\} = \{ X_ i \times U \to X \times U\}$

is a covering of $\mathcal{C}$ also. Hence $v$ is continuous. The other statements of the lemma follow from Lemmas 7.22.1 and 7.22.2. $\square$

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Your email address will not be published.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 03CE. Beware of the difference between the letter 'O' and the digit '0'.