Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smith Ideals of Operadic Algebras in Monoidal Model Categories

David White, Donald Yau

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.

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 -category approach to encoding the homotopy theory of algebras over Sigma-cofibrant operads that are not necessarily admissible.

Paper Structure

This paper contains 26 sections, 34 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Model Structures on the Arrow Category
  3. Monoidal Model Categories
  4. Arrow Categories
  5. Injective Model Structure
  6. Projective Model Structure
  7. Smith Ideals for Operads
  8. Operads, Algebras, and Bimodules
  9. Arrow Category of Operadic Algebras
  10. Operadic Smith Ideals
  11. Operadic Smith Ideals and Morphisms of Operadic Algebras
  12. Homotopy Theory of Smith Ideals for Operads
  13. Admissibility of Operads
  14. Admissibility of Operads in the Arrow Category
  15. Quillen Adjunction Between Operadic Smith Ideals and Algebra Morphisms
  16. ...and 11 more sections

Key Result

Theorem A

Suppose $\mathsf{M}$ is a sufficiently nice stable monoidal model category, and $\mathcal{O}$ is a $\mathfrak{C}$-colored operad in $\mathsf{M}$ such that cofibrant Smith $\mathcal{O}$-ideals are also entrywise cofibrant in the arrow category of $\mathsf{M}$ with the projective model structure. Then induced by the cokernel and the kernel.

Theorems & Definitions (104)

  • Theorem A
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • Definition 2.2.1
  • Definition 2.2.2
  • Theorem 2.3.1
  • proof
  • ...and 94 more