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Colimits in the $\infty$-category of $\infty$-topoi and \'etale morphisms
Taichi Uemura
https://arxiv.org/abs/2506.10431 https://…

Colimits in the $\infty$-category of $\infty$-topoi and étale morphisms
We provide an alternative proof of Lurie's result that the wide subcategory of the $\infty$-category of $\infty$-topoi spanned by the étale morphisms is closed under small colimits. Our proof is based on a new characterization of étale 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.