Lemma 7.22.4. Let $\mathcal{C}$ and $\mathcal{D}$ be sites and let $v : \mathcal{D} \to \mathcal{C}$ be a continuous functor. Assume $v$ has a left adjoint $u : \mathcal{C} \to \mathcal{D}$. Then
In particular, $v$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$.
Proof. Let $U$ be an object of $\mathcal{C}$ and let $\{ V_ j \to u(U)\} _{j \in J}$ be a covering in $\mathcal{D}$. Then $\{ v(V_ j) \to v(u(U))\} _{i \in I}$ is a covering in $\mathcal{C}$. Via the adjunction map $U \to v(u(U))$ we can base change this to a covering $\{ W_ j \to U\} _{j \in J}$ with $W_ j = v(V_ j) \times _{v(u(U))} U$. Denoting $p_ j : W_ j \to v(V_ j)$ the first projection, we obtain maps
where the second arrow is the adjunction map. This determines a morphism $\{ u(W_ j) \to u(U)\} \to \{ V_ j \to u(U)\} $ of families of maps with fixed target, showing that $u$ is indeed cocontinuous. The other statements are immediate from Lemmas 7.22.1 and 7.22.2. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)