Cartesian Factorization Systems and Grothendieck Fibrations

TL;DR

This paper explores the relationship between Grothendieck fibrations and cartesian factorization systems, providing a new perspective and constructions that deepen understanding of their categorical structures.

Contribution

It introduces a definition of cartesian factorization systems capturing key features of Grothendieck fibrations and compares various related 2-categories, offering new insights and constructions.

Findings

Defines a cartesian factorization system with 2-of-3 property

Establishes correspondence between Grothendieck fibrations and factorization systems

Provides a fiberwise opposite construction for fibrations

Abstract

Every Grothendieck fibration gives rise to a vertical/cartesian orthogonal factorization system on its domain. We define a cartesian factorization system to be an orthogonal factorization in which the left class satisfies 2-of-3 and is closed under pullback along the right class. We endeavor to show that this definition abstracts crucial features of the vertical/cartesian factorization system associated to a Grothendieck fibration, and give comparisons between various 2-categories of factorization systems and Grothendieck fibrations to demonstrate this relationship. We then give a construction which corresponds to the fiberwise opposite of a Grothendieck fibration on the level of cartesian factorization systems. Apart from the final double categorical results, this paper is entirely review of previously established material. It should be read as an expository note.