On products of skeleta

Given a symmetric monoidal $\infty$-category $\mathscr{E}$, compatible with finite colimits, we show that the functor sending a simplicial object in $\mathscr{E}$ 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 $\mathscr{O}$-promonoidal $\infty$-categories for any $\infty$-operad $\mathscr{O}$; most notably, we show that it is possible to localize $\mathscr{O}$-promonoidal $\infty$-categories in the same way one localizes symmetric monoidal $\infty$-categories.