Definition 4.35.
A morphism of logoi \(\phi^*\colon T \to S\) is called étale if there exists an object \(X \in T\) and an equivalence of logoi \(S \simeq T_{/X}\) such that the composite
\[T \xrightarrow{\phi^*} S \simeq T_{/X}\]
is the functor \(U \mapsto U \times X\). A morphism \(\phi_*\colon S \to T\) of topoi is étale if the corresponding morphism \(\phi^*\colon T \to S\) is étale. We denote by
\[\Logos^{\et} \; \subseteq \; \Logos \qquadtext{ and } \Topos^{\et} \; \subseteq \; \Topos\]
the wide subcategories spanned by étale morphisms.