Cocartesian fibrations and straightening internal to an $\infty$-topos

TL;DR

This paper extends the theory of cartesian and cocartesian fibrations to categories internal to an $topos$, establishing a straightening equivalence within this advanced mathematical framework.

Contribution

It introduces the definition and analysis of fibrations between internal categories in an $topos$ and proves a straightening equivalence in this setting.

Findings

Established a straightening equivalence for internal fibrations.

Defined cartesian and cocartesian fibrations within an $topos$.

Extended classical fibration theory to a higher topos context.

Abstract

We define and study cartesian and cocartesian fibrations between categories internal to an $\infty$-topos and prove a straightening equivalence in this context.