On straightening for Segal spaces

TL;DR

This paper offers an alternative proof of the straightening-unstraightening equivalence for $( abla, 1)$-categories and extends it to all higher categorical dimensions using combinatorial methods involving double categories.

Contribution

It introduces a new combinatorial approach to extend the straightening-unstraightening correspondence to higher categories, providing explicit constructions.

Findings

Provides an alternative proof of the straightening-unstraightening equivalence.

Extends the correspondence to all higher categorical dimensions.

Uses combinatorial results on fibrations between double categories.

Abstract

The straightening-unstraightening correspondence of Grothendieck--Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$-categories and diagrams of $(\infty, 1)$-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening-unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories.