The Stacks project

Example 7.21.4. This example is a slight generalization of Example 7.21.3. Let $f : X \to Y$ be a continuous map of topological spaces. Assume that $f$ is open. Recall that we have sites $X_{Zar}$ and $Y_{Zar}$, see Example 7.6.4. Recall that we have the functor $u : Y_{Zar} \to X_{Zar}$ associated to $f$ which is continuous and gives rise to a morphism of sites $X_{Zar} \to Y_{Zar}$, see Example 7.14.2. This also gives a morphism of topoi $(f_*, f^{-1})$. Next, consider the functor $v : X_{Zar} \to Y_{Zar}$, $U \mapsto v(U) = f(U)$. This functor is cocontinuous. Namely, if $f(U) = \bigcup _{j \in J} V_ j$ is an open covering in $Y$, then setting $U_ j = f^{-1}(V_ j) \cap U$ we get an open covering $U = \bigcup U_ j$ such that $f(U) = \bigcup f(U_ j)$ is a refinement of $f(U) = \bigcup V_ j$. We conclude by Lemma 7.21.1 above that there is a morphism of topoi associated to $v$

\[ \mathop{\mathit{Sh}}\nolimits (X) \longrightarrow \mathop{\mathit{Sh}}\nolimits (Y) \]

given by ${}_ sv$ and $(v^ p\ )^\# $. We claim that actually $(v^ p\ )^\# = f^{-1}$ and that ${}_ sv = f_*$, in other words, that this is the same morphism of topoi as the one given above. For any sheaf $\mathcal{G}$ on $Y$ we have $v^ p\mathcal{G}(U) = \mathcal{G}(f(U))$. On the other hand, we may compute $u_ p\mathcal{G}(U) = \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} \mathcal{G}(V) = \mathcal{G}(f(U))$ because clearly $(f(U), U \subset f^{-1}(f(U)))$ is an initial object of the category $\mathcal{I}_ U^ u$ of Section 7.5. Hence $u_ p = v^ p$ and we conclude $f^{-1} = u_ s = (v^ p\ )^\# $. The equality of ${}_ sv$ and $f_*$ follows by uniqueness of adjoint functors (but may also be computed directly).

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