A synthetic construction of universal cocartesian fibrations

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.

31 pages