Lemma 5.47.
Given an open immersion \(j\colon \Uu \hookrightarrow \Xx\), the closed complement \(\Zz := \Xx / \Uu\) is a topos, and the inclusion \(i\colon \Zz \hookrightarrow \Xx\) is a morphism of topoi.
Proof
By Lemma 5.45, we may identify \(\Uu\) with the slice \(\Xx_{/U}\) for some \((-1)\)-truncated object \(U\) of \(\Xx\). Given \(X \in \Xx\), we then have \(X \in \Zz\) if and only if the projection map \(X \times U \to U\) is an isomorphism. Equivalently, given an object \(Y \in \Xx\), every morphism \(Y \to U\) admits a unique lift \(Y \to U \times X\). We conclude that the subcategory \(\Zz = \Xx / \Uu\) consists precisely of those objects that are local with respect to the following class of morphisms:
\[S_U := \{\,\emptyset \to V \mid \text{there exists a map } V \to U\,\}.\]
Choose a small set of generators \(\mathcal G\) of \(\Xx\). The objects \(g\colon G\to U\), with \(G\in\mathcal G\) and \(g\) ranging through all morphisms to \(U\), form a small set of generators of the slice \(\Xx_{/U}\). It therefore suffices to impose locality with respect to the corresponding set \[S_U^0:=\{\,\emptyset\to G\mid G\in\mathcal G, g\colon G\to U\,\},\]
Indeed, every object of \(\Xx_{/U}\) is a colimit of these generators, and mapping out of this colimit identifies locality with respect to \(S_U^0\) with locality with respect to the entire class \(S_U\). Consequently, Proposition A.15 gives a left adjoint \(\Xx\to\Zz\) to the inclusion.It remains to show that this left adjoint preserves finite limits, or equivalently that the strong saturation of \(S_U^0\), which agrees with the strong saturation of \(S_U\), is a congruence. The class \(S_U\) is closed under base change, so \((S_U)^s=(S_U)^m\) by Proposition 5.32. Moreover, its morphisms are monomorphisms, whence \((S_U)^m=(S_U)^c\) by Corollary 5.43. We deduce that the localization \(\Xx\to\Zz\) is left exact, exhibiting \(\Zz\) as a topos and its inclusion into \(\Xx\) as a morphism of topoi.