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

Taichi Uemura

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arXiv · 2506.10431

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

Taichi Uemura

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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.

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Taichi Uemura. 2025-06-12. Colimits in the $\infty$-category of $\infty$-topoi and \'etale morphisms. https://arxiv.org/abs/2506.10431

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