On products of skeleta

2025/10/21 by Keenan, Liam, Péroux, Maximilien
#18G31 #18M05 #18N55 #18N60 #55P42. Secondary: 18N40 #55T05 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Primary: 18D60

paper · doi:10.48550/arxiv.2510.18961

Abstract

Given a symmetric monoidal ∞-category \mathscrE, compatible with finite colimits, we show that the functor sending a simplicial object in \mathscrE 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 \mathscrO-promonoidal ∞-categories for any ∞-operad \mathscrO; most notably, we show that it is possible to localize \mathscrO-promonoidal ∞-categories in the same way one localizes symmetric monoidal ∞-categories.

Citations

• Mapping spaces between operads in relation to bimodules
• Global homotopy theory via partially lax limits
• Parametrized higher category theory II: Universal constructions
• Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause
• A monoidal Dold-Kan correspondence for comodules
• The reductive Borel-Serre compactification as a model for unstable algebraic K-theory
• Spanier-Whitehead duality for topological coHochschild homology
• Coalgebras in the Dwyer-Kan localization of a model category
• Topological cyclic homology
• K-theoretic Tate-Poitou duality and the fiber of the cyclotomic trace
• Dualizing cartesian and cocartesian fibrations
• Dwyer-Kan localization revisited
• HZ-algebra spectra are differential graded algebras

Related

• A duality between monads and monadic morphisms
• Lax monoidality for products of enriched higher categories
• ∞-operads as symmetric monoidal ∞-categories
• Lax monoidal adjunctions, two-variable fibrations and the calculus of mates
• A monoidal Grothendieck construction for ∞-categories