On products of skeleta
Liam Keenan, Maximilien Péroux
TL;DR
This work develops a higher-categorical analogue of the Eilenberg–Zilber and Dold–Kan frameworks by embedding skeletal filtrations of simplicial objects into an operadic, promonoidal setting. It introduces and exploits $\mathscr{O}$-promonoidal $\infty$-categories and Day convolution to construct a canonical lax symmetric monoidal structure on skeletal filtrations, with a key localization theory that preserves promonoidal structure. The main achievement is that the skeletal filtration functor $\mathrm{sk}_*^{\mathscr{E}}$ is lax symmetric monoidal and recovers the classical Eilenberg–Zilber map in the stable setting (e.g., spectra), while yielding multiplicative structures on associated spectral sequences. The framework clarifies how promonoidal localization and Day convolution govern coherences, enabling a systematic translation of skeletal filtrations into multiplicative, operadic structures with potential applications to higher algebraic models and spectral sequence theory.
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.