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Smith ideals of operadic algebras in monoidal model categories

David White and Donald Yau

Algebraic & Geometric Topology 24 (2024) 341–392

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 morphisms induced by the cokernel and the kernel. For symmetric spectra, this applies to the commutative operad and all Σ–cofibrant operads. For chain complexes over a field of characteristic zero and the stable module category, this Quillen equivalence holds for all operads. We end with a comparison between the semi-model category approach and the –category approach to encoding the homotopy theory of algebras over Σ–cofibrant operads that are not necessarily admissible.

operadic algebras, Smith ideals, monoidal model categories
Mathematical Subject Classification
Primary: 18C20, 18G65, 18M75, 55P43, 55U35
Received: 12 September 2021
Revised: 28 August 2022
Accepted: 31 October 2022
Published: 18 March 2024
David White
Department of Mathematics and Computer Science
Denison University
Granville, OH
United States
Donald Yau
Department of Mathematics
The Ohio State University at Newark
Newark, OH
United States

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