Lax Colimits and Free Fibrations in $\infty$-Categories

Abstract

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