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

Contents
### Context

#### Model category theory

**model category**, model $\infty$-category

**Definitions**

**Morphisms**

**Universal constructions**

**Refinements**

**Producing new model structures**

**Presentation of $(\infty,1)$-categories**

**Model structures**

*for $\infty$-groupoids*

for ∞-groupoids

*for equivariant $\infty$-groupoids*

*for rational $\infty$-groupoids*

*for rational equivariant $\infty$-groupoids*

*for $n$-groupoids*

*for $\infty$-groups*

*for $\infty$-algebras*

*general $\infty$-algebras*

*specific $\infty$-algebras*

*for stable/spectrum objects*

*for $(\infty,1)$-categories*

*for stable $(\infty,1)$-categories*

*for $(\infty,1)$-operads*

*for $(n,r)$-categories*

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

#### Monoidal categories

**monoidal categories**

## With symmetry

## With duals for objects

## With duals for morphisms

## With traces

## Closed structure

## Special sorts of products

## Semisimplicity

## Morphisms

## Internal monoids

## Examples

## Theorems

## In higher category theory

#### Higher 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.