A MONOIDAL GROTHENDIECK CONSTRUCTION FOR <i>∞</i> -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.