Smith Ideals of Operadic Algebras in Monoidal Model Categories

TL;DR

This paper develops a homotopy theory for Smith ideals of operads in monoidal model categories, establishing Quillen equivalences between model structures for various algebraic contexts and comparing different homotopical frameworks.

Contribution

It extends Smith ideals to general operads in monoidal model categories and proves Quillen equivalences under broad conditions, including symmetric spectra and chain complexes.

Findings

Quillen equivalence between Smith ideals and algebra maps in stable monoidal model categories

Applicability to symmetric spectra, chain complexes, and stable module categories

Comparison between semi-model and ∞-category approaches to operad algebras

Abstract

Building upon Hovey's work on Smith ideals for monoids, we develop a homotopy theory of Smith ideals for general operads in a symmetric monoidal category. For a sufficiently nice stable monoidal model category and an operad satisfying a cofibrancy condition, we show that there is a Quillen equivalence between a model structure on Smith ideals and a model structure on algebra maps induced by the cokernel and the kernel. For symmetric spectra, this applies to the commutative operad and all Sigma-cofibrant operads. For chain complexes over a field of characteristic zero and the stable module category, this Quillen equivalence holds for all operads. This paper ends with a comparison between the semi-model category approach and the $\infty$-category approach to encoding the homotopy theory of algebras over Sigma-cofibrant operads that are not necessarily admissible.