# nLab E-infinity algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

An $E_\infty$-algebra is an algebra over an operad for an E-∞ operad.

## Realizations

### In chain complexes

$E_\infty$-algebras in chain complexes are equivalent to those in abelian simplicial group.

For details on this statement see monoidal Dold-Kan correspondence and operadic Dold-Kan correspondence.

### In topological spaces

###### Theorem (May recognition theorem)

A connected space of the homotopy type of a CW-complex with a non-degenrate basepoint that has the homotopy type of a $k$-fold loop space for all $k \in \mathbb{N}$ admits the structure of an $E_\infty$-space.

### In simplicial sets

###### Theorem

The model structure on algebras over an operad over E-∞ operads in Top and in sSet are Quillen equivalent.

This is in BergerMoerdijk I, BergerMoerdijk II.

### In spectra

An $E_\infty$-algebra in spectra is an E-∞ ring.

See Ek-Algebras.

## References

• Martin Markl, Steve Shnider, Jim Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.

In the context of (infinity,1)-operads $E_\infty$-algebras are discussed in

A systematic study of model category structures on operads and their algebras is in

The induced model structures and their properties on algebras over operads are discussed in

Revised on October 18, 2014 12:41:37 by Urs Schreiber (185.26.182.37)