On orthogonal factorization systems and double categories

TL;DR

This paper establishes a deep connection between orthogonal factorization systems and double $ abla$-categories, showing a full embedding and an (un)straightening equivalence that enhances understanding of their structures.

Contribution

It proves a full embedding of orthogonal factorization systems into double $ abla$-categories and establishes an (un)straightening equivalence for these structures.

Findings

Orthogonal factorization systems fully embed into double $ abla$-categories.

An (un)straightening equivalence is established for double $ abla$-categories.

The equivalence restricts to op-Gray fibrations and curved orthofibrations.

Abstract

We prove that the $\infty$-category of orthogonal factorization systems embeds fully faithfully into the $\infty$-category of double $\infty$-categories. Moreover, we prove an (un)straightening equivalence for double $\infty$-categories, which restricts to an (un)straightening equivalence for op-Gray fibrations and curved orthofibrations of orthogonal factorization systems.