Colimits in the $\infty$-category of $\infty$-topoi and \'etale morphisms

TL;DR

This paper offers a new proof that the subcategory of -topoi with -étale morphisms is closed under small colimits, using novel characterizations and elementary constructions without larger universes.

Contribution

It introduces a new characterization of -étale morphisms and provides an elementary construction of univalent completion, simplifying existing proofs.

Findings

-étale morphisms are closed under small colimits.

A new characterization of -étale morphisms related to univalent families.

Elementary construction of univalent completion.

Abstract

We provide an alternative proof of Lurie's result that the wide subcategory of the $\infty$-category of $\infty$-topoi spanned by the \'etale morphisms is closed under small colimits. Our proof is based on a new characterization of \'etale morphisms of $\infty$-topoi in relation to univalent families and does not rely on a larger universe. During the proof, we also give an elementary construction of univalent completion.