The Stacks project

Example 7.21.3. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subspace. Recall that we have sites $X_{Zar}$ and $U_{Zar}$, see Example 7.6.4. Recall that we have the functor $u : X_{Zar} \to U_{Zar}$ associated to $j$ which is continuous and gives rise to a morphism of sites $U_{Zar} \to X_{Zar}$, see Example 7.14.2. This also gives a morphism of topoi $(j_*, j^{-1})$. Next, consider the functor $v : U_{Zar} \to X_{Zar}$, $V \mapsto v(V) = V$ (just the same open but now thought of as an object of $X_{Zar}$). This functor is cocontinuous. Namely, if $v(V) = \bigcup _{j \in J} W_ j$ is an open covering in $X$, then each $W_ j$ must be a subset of $U$ and hence is of the form $v(V_ j)$, and trivially $V = \bigcup _{j \in J} V_ j$ is an open covering in $U$. We conclude by Lemma 7.21.1 above that there is a morphism of topoi associated to $v$

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

given by ${}_ sv$ and $(v^ p\ )^\# $. We claim that actually $(v^ p\ )^\# = j^{-1}$ and that ${}_ sv = j_*$, in other words, that this is the same morphism of topoi as the one given above. Perhaps the easiest way to see this is to realize that for any sheaf $\mathcal{G}$ on $X$ we have $v^ p\mathcal{G}(V) = \mathcal{G}(V)$ which according to Sheaves, Lemma 6.31.1 is a description of $j^{-1}\mathcal{G}$ (and hence sheafification is superfluous in this case). The equality of ${}_ sv$ and $j_*$ follows by uniqueness of adjoint functors (but may also be computed directly).

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