Model Structures on Commutative Monoids in General Model Categories

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.