Geodesic
Smith ideals of operadic algebras in monoidal model categories
White, David1 ; Yau, Donald2
1 Department of Mathematics and Computer Science, Denison University, Granville, OH, United States
2 Department of Mathematics, The Ohio State University at Newark, Newark, OH, United States
Algebraic and Geometric Topology, Tome 24 (2024) no. 1, pp. 341-392

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/agt.2024.24.341
Keywords: operadic algebras, Smith ideals, monoidal model categories

White, David 1 ; Yau, Donald 2

1 Department of Mathematics and Computer Science, Denison University, Granville, OH, United States
2 Department of Mathematics, The Ohio State University at Newark, Newark, OH, United States 
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White, David; Yau, Donald. Smith ideals of operadic algebras in monoidal model categories. Algebraic and Geometric Topology, Tome 24 (2024) no. 1, pp. 341-392. doi: 10.2140/agt.2024.24.341

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