Admissible replacements for simplicial monoidal model categories

TL;DR

This paper constructs improved symmetric monoidal model categories with enhanced operadic properties using Dugger's universal model categories, and classifies stable cases via commutative ring spectra.

Contribution

It introduces a method to replace simplicial and combinatorial symmetric monoidal model categories with versions that support all operadic algebras.

Findings

Replacements admit model structures on algebras over any colored operad.

Stable symmetric monoidal model categories are classified by commutative ring spectra.

Equivalent to modules over a unique commutative ring spectrum.

Abstract

Using Dugger's construction of universal model categories, we produce replacements for simplicial and combinatorial symmetric monoidal model categories with better operadic properties. Namely, these replacements admit a model structure on algebras over any given colored operad. As an application, we show that in the stable case, such symmetric monoidal model categories are classified by commutative ring spectra when the monoidal unit is a compact generator. In other words, they are strong monoidally Quillen equivalent to modules over a uniquely determined commutative ring spectrum.