# nLab E-infinity monoid in a symmetric monoidal model category

Contents

model category

## Model structures

for ∞-groupoids

### for $(\infty,1)$-sheaves / $\infty$-stacks

#### Monoidal categories

monoidal categories

## In higher category theory

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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.

## References

Created on March 11, 2015 at 13:15:32. See the history of this page for a list of all contributions to it.