Lax colimits and free fibrations in $\infty$-categories

TL;DR

This paper develops the theory of lax and weighted colimits in $ abla$-categories, characterizes free Cartesian fibrations, and applies these to show preservation properties and presentability results in $ abla$-categories.

Contribution

It introduces a simple characterization of free Cartesian fibrations and demonstrates their applications in preservation and presentability within $ abla$-categories.

Findings

Lax colimits correspond to coCartesian fibrations.

Lax representable functors are preserved under exponentiation.

Total spaces of presentable Cartesian fibrations are presentable.

Abstract

We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of $\infty$-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between $\infty$-categories is presentable, generalizing a theorem of Makkai and Par\'e to the $\infty$-categorical setting. Lastly, in the appendix, we observe that pseudofunctors between (2,1)-categories give rise to functors between $\infty$-categories via the Duskin nerve.