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

Raffael Stenzel

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.

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

Abstract

We give a direct proof of the fact that Lurie's Unstraightening functor induces an equivalence between the strict -category of indexed quasi-categories and the strict -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 -cosmoses.
Paper Structure (5 sections, 7 theorems, 20 equations)

This paper contains 5 sections, 7 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Corresponding results in the literature.
  3. Brief reminder of the underlying constructions
  4. Proof of the Theorem
  5. Unstraightening as a weak cosmological biequivalence

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a quasi-category. The Unstraightening functor over $\mathcal{C}$ exhibits a simplicial enrichment which induces a DK-equivalence of underlying strict $(\infty,2)$-categories. It furthermore comes equipped with a binatural transformation between the respective simplicial cotensors for $F\colon\mathfrak{C}(\mathcal{C})^{op}\rightarrow\mathbf{S}^+$ and $I\in\mathbf{S}$, which i

Theorems & Definitions (14)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • ...and 4 more