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

Raffael Stenzel

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

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arxiv 2403.01167 v1 pith:YN5IQVZS submitted 2024-03-02 math.CT

Raffael Stenzel This is my paper · ORCID

classification math.CT

keywords inftyunstraighteningbiequivalencecategorycosmosesequivalenceluriequasi-categories

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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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Cosmological Unstraightening
math.CT 2025-05 accept novelty 7.0 of 10

The straightening-unstraightening equivalence between presheaves and fibrations lifts to a biequivalence of infinity-cosmoi for all models of (infinity,1)-categories.

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