Smith Ideals of Operadic Algebras in Monoidal Model Categories

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.