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

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.