Lemma 5.49.
Consider topoi \(\Uu\) and \(\Zz\) and an accessible left exact functor \(F\colon \Uu \to \Zz\).
The associated recollement \(\Xx\) is a topos,
The functor \(j_*\colon \Uu \to \Xx\) is an open immersion of topoi,
The functor \(i_*\colon \Zz \to \Xx\) is a closed immersion of topoi.
Proof
The comma category \(\Xx\) is accessible, and the results of Proposition B.1 show that it admits all small colimits and finite limits. Hence \(\Xx\) is presentable. Moreover, the forgetful functor
\[(j^*,i^*)\colon \Xx\longrightarrow \Uu\times\Zz\]
is cocontinuous, left exact, and conservative. The Giraud conditions for \(\Xx\) may therefore be checked after applying this functor, where they follow from the corresponding conditions in \(\Uu\) and \(\Zz\). Thus \(\Xx\) is a topos.Let \[V:=(*,\emptyset,\emptyset\to F(*))\in\Xx.\]
This object is \((-1)\)-truncated. Base change along \(V\to *\) identifies \(\Xx_{/V}\) with \(\Uu\): a morphism \((U,Z,\varphi)\to V\) forces \(Z\iso\emptyset\), and is then determined by \(U\). Under this equivalence, the inverse-image functor of the étale morphism \(\Xx_{/V}\to\Xx\) identifies with \(j^*\), and its right adjoint identifies with \(j_*\). Thus \(j_*\) is an open immersion by Lemma 5.45.Finally, an object \((U,Z,\varphi)\) belongs to the closed complement of this open immersion precisely when \[j^*(U,Z,\varphi)=U\iso *.\]
Since \(F(*)\iso *\), such an object is uniquely of the form \((*,Z,Z\to *)=i_*(Z)\). Hence \(i_*\) identifies \(\Zz\) with the closed complement of \(\Uu\), and is therefore a closed immersion.