On products of skeleta

Given a symmetric monoidal $\mathrm{\infty }$-category $\mathcal{E}$, compatible with finite colimits, we show that the functor sending a simplicial object in $\mathcal{E}$ to its skeletal filtration is canonically lax symmetric monoidal. This monoidal structure is the analogue of the one induced by the Eilenberg-Zilber homomorphism from the Dold-Kan correspondence. To accomplish this, we establish some new results around $\mathcal{O}$-promonoidal $\mathrm{\infty }$-categories for any $\mathrm{\infty }$-operad $\mathcal{O}$; most notably, we show that it is possible to localize $\mathcal{O}$-promonoidal $\mathrm{\infty }$-categories in the same way one localizes symmetric monoidal $\mathrm{\infty }$-categories.