Lemma 4.58. ([Uemura 2025, Lemma 5.13])
Let \(\{C_i\}_{i\in I}\) be a small family of logoi. Then every projection
\[\prod_{j\in I} C_j \longrightarrow C_i\]
is an étale morphism.
Proof
Let \(D=\prod_{j\in I}C_j\). Define an object \(x\in D\) by taking \(x_i=*\) and \(x_j=\emptyset\) for \(j\neq i\). Then
\[D_{/x} \simeq \prod_{j\in I}(C_j)_{/x_j} \simeq C_i,\]
since the slice over the initial object is terminal. Under this equivalence, the canonical étale functor \(D\to D_{/x}\) is the projection to \(C_i\).References
- Taichi Uemura. Colimits in the ∞-category of ∞-topoi and étale morphisms. 2025.