# nLab E-infinity-ring

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Idea

An $E_\infty$-ring is a commutative monoid in the stable (∞,1)-category of spectra. Sometimes this is called a commutative ring spectrum. An E-∞ algebra in spectra.

This means that an $E_\infty$-ring is an A-∞ ring that is commutative up to coherent higher homotopies. $E_\infty$-rings are the analogue in higher algebra of the commutative rings in ordinary algebra.

In terms of model categories, and $E_\infty$-rings may be modeled as ordinary commutative monoids with respect to the symmetric monoidal smash product of spectra, a fact sometimes referred to as “brave new algebra”.

## Examples

A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

## References

• Peter May with contributions by Frank Quinn, Nigel Ray and Jorgen Tornehave, $E_\infty$-Ring spaces and $E_\infty$ ring spectra (pdf)

• Moritz Groth, A short course on infinity-categories, pdf.

Discussion of a Blakers-Massey theorem for $E_\infty$-rings is in

Revised on July 25, 2014 10:17:56 by Urs Schreiber (89.204.135.196)