Cosmological Unstraightening

TL;DR

This paper generalizes Lurie's unstraightening equivalence from quasi-categories to all $ abla$-cosmoi of $( abla,1)$-categories, establishing a biequivalence of fibrations and presheaves across various models.

Contribution

It extends the unstraightening construction to all $ abla$-cosmoi, including complete Segal spaces and $1$-complicial sets, via a cosmological biequivalence.

Findings

Established biequivalence of fibrations and presheaves for all $ abla$-cosmoi.

Lifted the unstraightening construction to a cosmological biequivalence.

Unified different models of $( abla,1)$-categories under a common framework.

Abstract

The unstraightening construction due to Lurie establishes an equivalence between presheaves and fibrations, using one prominent model of $(\infty,1)$-categories, namely quasi-categories. In this work we generalize this result by proving that for all $\infty$-cosmoi of $(\infty,1)$-categories in the sense of Riehl and Verity, which includes quasi-categories but also complete Segal spaces or $1$-complicial sets, their corresponding notions of fibrations and presheaves are biequivalent $\infty$-cosmoi via a natural zig-zag of cosmological biequivalences. The major idea that makes this possible is a lift of the quasi-categorical unstraightening construction to a cosmological biequivalence.