The Stacks project

Lemma 8.10.3. Let $\mathcal{C}$ be a site. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of fibred categories over $\mathcal{C}$. Then $F$ is a continuous and cocontinuous functor between the structure of sites inherited from $\mathcal{C}$. Hence $F$ induces a morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{Y})$ with $f_* = {}_ sF = {}_ pF$ and $f^{-1} = F^ s = F^ p$. In particular $f^{-1}(\mathcal{G})(x) = \mathcal{G}(F(x))$ for a sheaf $\mathcal{G}$ on $\mathcal{Y}$ and object $x$ of $\mathcal{X}$.

Proof. We first prove that $F$ is continuous. Let $\{ x_ i \to x\} _{i \in I}$ be a covering of $\mathcal{X}$. By Categories, Definition 4.32.9 the functor $F$ transforms strongly cartesian morphisms into strongly cartesian morphisms, hence $\{ F(x_ i) \to F(x)\} _{i \in I}$ is a covering of $\mathcal{Y}$. This proves part (1) of Sites, Definition 7.13.1. Moreover, let $x' \to x$ be a morphism of $\mathcal{X}$. By Categories, Lemma 4.32.13 the fibre product $x_ i \times _ x x'$ exists and $x_ i \times _ x x' \to x'$ is strongly cartesian. Hence $F(x_ i \times _ x x') \to F(x')$ is strongly cartesian. By Categories, Lemma 4.32.13 applied to $\mathcal{Y}$ this means that $F(x_ i \times _ x x') = F(x_ i) \times _{F(x)} F(x')$. This proves part (2) of Sites, Definition 7.13.1 and we conclude that $F$ is continuous.

Next we prove that $F$ is cocontinuous. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ and let $\{ y_ i \to F(x)\} _{i \in I}$ be a covering in $\mathcal{Y}$. Denote $\{ U_ i \to U\} _{i \in I}$ the corresponding covering of $\mathcal{C}$. For each $i$ choose a strongly cartesian morphism $x_ i \to x$ in $\mathcal{X}$ lying over $U_ i \to U$. Then $F(x_ i) \to F(x)$ and $y_ i \to F(x)$ are both a strongly cartesian morphisms in $\mathcal{Y}$ lying over $U_ i \to U$. Hence there exists a unique isomorphism $F(x_ i) \to y_ i$ in $\mathcal{Y}_{U_ i}$ compatible with the maps to $F(x)$. Thus $\{ x_ i \to x\} _{i \in I}$ is a covering of $\mathcal{X}$ such that $\{ F(x_ i) \to F(x)\} _{i \in I}$ is isomorphic to $\{ y_ i \to F(x)\} _{i \in I}$. Hence $F$ is cocontinuous, see Sites, Definition 7.20.1.

The final assertion follows from the first two, see Sites, Lemmas 7.21.1, 7.20.2, and 7.21.5. $\square$


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Your email address will not be published.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 06NW. Beware of the difference between the letter 'O' and the digit '0'.