On products of skeleta

TL;DR

The paper proves that the skeletal filtration functor in symmetric monoidal $ abla$-categories is canonically lax symmetric monoidal, extending the structure of the Dold-Kan correspondence and developing new localization results for $ ext{O}$-promonoidal $ abla$-categories.

Contribution

It introduces a canonical lax symmetric monoidal structure on skeletal filtrations and establishes localization techniques for $ ext{O}$-promonoidal $ abla$-categories.

Findings

The skeletal filtration functor is canonically lax symmetric monoidal.

Localization of $ ext{O}$-promonoidal $ abla$-categories is possible.

New results on $ ext{O}$-promonoidal $ abla$-categories for any $ abla$-operad.

Abstract

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.