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.

30 pages, comments welcome ! v2: Improved exposition following a referee report, published in Nagoya Mathematical Journal