nLab E-infinity monoid in a symmetric monoidal model category

Idea

An $E_\infty$-monoid (or E-infinity monoid object) in a symmetric monoidal model category $C$ is an algebra over an operad over the E-infinity operad. Assuming that $C$ is cofibrantly generated, there is a model structure on $E_\infty$-monoids, given by the model structure on algebras over an operad over the E-infinity operad (which is cofibrant). This model category presents the (infinity,1)-category of commutative monoids in a symmetric monoidal (infinity,1)-category (in the symmetric monoidal (infinity,1)-category presented by $C$).

Rectification

In some symmetric monoidal model categories, $E_\infty$-monoids can be rectified to (strictly) commutative monoids in a symmetric monoidal model category. See there for more.

Related concepts

• model structure on commutative monoids in a symmetric monoidal model category

• model structure on monoids in a monoidal model category

• model structure on algebras over an operad

• model structure on algebras over a monad

• model structure on simplicial T-algebras

References

• Jacob Lurie, Higher Algebra.