A short proof of the straightening theorem

TL;DR

This paper offers a concise, self-contained proof of Lurie's straightening theorem, establishing an equivalence between cartesian fibrations over an ∞-category and contravariant functors to small ∞-categories.

Contribution

It presents a shorter, more accessible proof of the straightening theorem, simplifying the understanding of the equivalence in higher category theory.

Findings

Provides a concise proof of Lurie's straightening theorem

Clarifies the relationship between cartesian fibrations and contravariant functors

Enhances accessibility of the theorem for researchers

Abstract

We provide a short and reasonably self-contained proof of Lurie's straightening equivalence, relating cartesian fibrations over a given $\infty$-category $S$ with contravariant functors from $S$ to the $\infty$-category of small $\infty$-categories.