A monoidal Grothendieck construction for $\infty$-categories

Abstract

We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal $\infty$-category $\mathbf C$, we endow the $\infty$-category of coCartesian fibrations over $\mathbf C$ with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the $\infty$-category of functors from $\mathbf C$ to $\mathbf{Cat}_\infty$. In fact, we do this over any $\infty$-operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an $\infty$-category $\mathbf C$ its category of coCartesian fibrations on the one hand, and its category of functors to $\mathbf{Cat}_\infty$ on the other hand.