Factorization systems in $\infty$-categories

TL;DR

This paper generalizes the concept of factorization systems from ordinary categories to $inf$-categories, providing new descriptions and characterizations using presheaves, Segal conditions, and bicategories of spans.

Contribution

It introduces a framework for factorization systems in $inf$-categories, including their description via presheaves and Segal conditions, and explores their relation to distributive laws and bicategories.

Findings

Categories with factorization systems are characterized as presheaves satisfying Segal conditions.

Distributive laws between monads are studied in the context of categories with factorization systems.

Categories with factorization systems are characterized as distributive laws in the bicategory of spans.

Abstract

We extend the notion of a factorization system in a category to the realm of $\infty$-categories. To this end, we provide a description of the category of $\infty$-categories with factorization systems as the category of presheaves of spaces on a certain category that satisfy a version of the Segal condition. Additionally, we study the notion of distributive laws between monads and, more generally, lax functors in the context of categories with factorization systems. In particular, we also characterize categories with factorization systems as distributive laws in the bicategory of spans.