Lurie's Unstraightening as a weak biequivalence of $\infty$-cosmoses

Raffael Stenzel

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arXiv · 2403.01167

Lurie's Unstraightening as a weak biequivalence of $\infty$-cosmoses

Raffael Stenzel

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Abstract

We give a direct proof of the fact that Lurie's Unstraightening functor induces an equivalence between the strict $(\infty,2)$-category of indexed quasi-categories and the strict $(\infty,2)$-category of fibered quasi-categories over any given quasi-categorical base. We conclude that Unstraightening preserves simplicial cotensors up to a (strictly) natural homotopy equivalence, and thus gives rise to an accordingly weakened notion of cosmological biequivalence between the two underlying $\infty$-cosmoses.

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Raffael Stenzel. 2024-03-02. Lurie's Unstraightening as a weak biequivalence of $\infty$-cosmoses. https://arxiv.org/abs/2403.01167

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