Lemma 5.45.
Let \(j_*\colon \Xx_{/U} \to \Xx\) be an étale morphism of topoi, \(U \in \Xx\). Then \(j_*\) is an open immersion if and only if \(U\) is \((-1)\)-truncated.
Proof
We saw in Proposition 4.38 that the base change of \(j_*\) along itself is given by the functor \((\Xx_{/U})_{/U \times U} \to \Xx_{/U}\). This is an equivalence if and only if the object \((U \times U) \in \Xx_{/U}\) is terminal, i.e. if the projection map \(\pr_1\colon U \times U \to U\) is an isomorphism. This in turn is equivalent to the diagonal \(\Delta\colon U \to U \times U\) being an isomorphism, which by definition means that \(U\) is \((-1)\)-truncated.