A synthetic construction of universal cocartesian fibrations

TL;DR

This paper presents a model-independent method for constructing universal cocartesian fibrations in higher category theory, establishing a straightening equivalence and analyzing descent properties.

Contribution

It introduces a new directed join construction and demonstrates cocartesian fibrations descend along localisations, advancing the theory of $( abla,1)$-categories.

Findings

Established a straightening equivalence for cocartesian fibrations.

Proved cocartesian fibrations descend along localisations.

Introduced a directed join construction for functor images.

Abstract

We give a model-independent construction of directed univalent cocartesian fibrations of $(\infty,1)$-categories, and prove a straightening equivalence against such fibrations. The key step is showing that cocartesian fibrations descend along localisations, which we accomplish by analysing mapping spaces of localisations. Along the way we introduce a directed version of the join construction, giving a sequential colimit description of the full image of any functor.