The Stacks project

Lemma 7.25.8. Let $\mathcal{C}$ be a site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram

\[ \xymatrix{ \mathcal{C}/V \ar[rd]_{j_ V} \ar[rr]_ j & & \mathcal{C}/U \ar[ld]^{j_ U} \\ & \mathcal{C} & } \]

of continuous and cocontinuous functors. The functor $j : \mathcal{C}/V \to \mathcal{C}/U$, $(a : W \to V) \mapsto (f \circ a : W \to U)$ is identified with the functor $j_{V/U} : (\mathcal{C}/U)/(V/U) \to \mathcal{C}/U$ via the identification $(\mathcal{C}/U)/(V/U) = \mathcal{C}/V$. Moreover we have $j_{V!} = j_{U!} \circ j_!$, $j_ V^{-1} = j^{-1} \circ j_ U^{-1}$, and $j_{V*} = j_{U*} \circ j_*$.

Proof. The commutativity of the diagram is immediate. The agreement of $j$ with $j_{V/U}$ follows from the definitions. By Lemma 7.21.2 we see that the following diagram of morphisms of topoi
\begin{equation} \label{sites-equation-relocalize} \vcenter { \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) \ar[rd]_{j_ V} \ar[rr]_ j & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[ld]^{j_ U} \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) & } } \end{equation}

is commutative. This proves that $j_ V^{-1} = j^{-1} \circ j_ U^{-1}$ and $j_{V*} = j_{U*} \circ j_*$. The equality $j_{V!} = j_{U!} \circ j_!$ follows formally from adjointness properties. $\square$

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Your email address will not be published.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