Lemma 7.38.6. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $\{ (q_ i, v_ i)\} _{i\in I}$ be a conservative family of points of $\mathcal{D}$. If each functor $u_ i = v_ i \circ u$ defines a point of $\mathcal{C}$, then $u$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$.
Proof. Denote $p_ i$ the stalk functor (7.32.1.1) on $\textit{PSh}(\mathcal{C})$ corresponding to the functor $u_ i$. We have
\[ (f^{-1}\mathcal{F})_{q_ i} = (u_ s\mathcal{F})_{q_ i} = (u_ p\mathcal{F})_{q_ i} = \mathcal{F}_{p_ i} \]
The first equality since $f^{-1} = u_ s$, the second equality by Lemma 7.32.5, and the third by Lemma 7.34.1. Hence if $p_ i$ is a point, then pulling back by $f$ and then taking stalks at $q_ i$ is an exact functor. Since the family of points $\{ q_ i\} $ is conservative, this implies that $f^{-1}$ is an exact functor and we see that $f$ is a morphism of sites by Definition 7.14.1. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)