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 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.