Model Structures on Commutative Monoids in General Model Categories

Abstract

We provide conditions on a monoidal model category $\mathcal{M}$ so that the category of commutative monoids in $\mathcal{M}$ inherits a model structure from $\mathcal{M}$ in which a map is a weak equivalence or fibration if and only if it is so in $\mathcal{M}$. We then investigate properties of cofibrations of commutative monoids, rectification between $E_{\infty }$-algebras and commutative monoids, the relationship between commutative monoids and monoidal Bousfield localization functors, when the category of commutative monoids can be made left proper, and functoriality of the passage from a commutative monoid $R$ to the category of commutative $R$-algebras. In the final section we provide numerous examples of model categories satisfying our hypotheses.

Version 2 adds material about rectification between strict commutative monoids and $E_{\infty }$-algebras, adds material about lifting Quillen equivalences to categories of commutative monoids, and adds several new examples: simplicial presheaves, diagram categories, and commutative Smith ideals of ring spectra

References in corpus (5)

• Higher Topos Theory
• Derived Algebraic Geometry III: Commutative Algebra
• Homotopy theory of non-symmetric operads
• Overcategories and undercategories of model categories
• Cofibrancy of operadic constructions in positive symmetric spectra

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