Proposition 4.57. ([Uemura 2025, Corollary 5.11 and Lemma 5.12])
For every logos \(C\), the functor
\[C\catop \longrightarrow \Logos^{\et}_{C/}, \qquad X \longmapsto (C \to C_{/X})\]
is an equivalence. Under this equivalence, the inclusion \(\Logos^{\et}_{C/}\hookrightarrow \Logos_{C/}\) preserves limits.
Proof
Essential surjectivity and full faithfulness are the slice classification of étale morphisms from Proposition 4.38, translated to logoi. Explicitly, an étale morphism \(\phi^*\colon C\to S\) is equivalent to \(C\to C_{/\phi_{\sharp}(*)}\) by Proposition 4.56.For preservation of limits, let \(X_{\bullet}\colon I\to C\) be a diagram with colimit \(X\). Descent gives
\[C_{/X} \simeq \lim_i C_{/X_i}\]
in \(\Logos_{C/}\). Passing to \(C\catop\), this says exactly that \(X\mapsto C_{/X}\) preserves the relevant limits.References
- Taichi Uemura. Colimits in the ∞-category of ∞-topoi and étale morphisms. 2025.