A note on coCartesian fibrations

TL;DR

This paper establishes the properness of (co)Cartesian fibrations and demonstrates a compatible straightening-unstraightening equivalence over the nerve of a small category, advancing the theoretical framework of higher category theory.

Contribution

It proves properness of (co)Cartesian fibrations and constructs a compatible straightening-unstraightening equivalence for bases given by nerves of small categories.

Findings

Properness of (co)Cartesian fibrations established

Straightening-unstraightening equivalence constructed

Compatibility with cartesian products demonstrated

Abstract

We prove properness of (co)Cartesian fibrations as well as a straightening and unstraightening equivalence, which is compatible with cartesian products, when the base is the nerve of a small category.