# nLab E-infinity algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

Given a symmetric monoidal (infinity,1)-category $C$, an $E_\infty$-algebra in $C$ is synonymous with a commutative monoid in $C$. Most often $C$ is the (infinity,1)-category of chain complexes over a field or commutative ring, or the stable (infinity,1)-category of spectra (see also E-infinity ring).

In terms of operads, 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 groups.

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

The singular cohomology $H^*(X,\mathbb{Z})$ of a topological space is a graded-commutative algebra over the integers, but the the singular cochain complex $C^*(X,\mathbb{Z})$ is not: instead, it is an $E_\infty$-algebra. Remarkably, for simply connected spaces of finite type this $E_\infty$-algebra knows everything about the weak homotopy type of $X$. In fact a stronger statement holds:

###### Theorem

Finite type nilpotent spaces $X$ and $Y$ are weakly homotopy equivalent if and only if the $E_\infty$-algebras $C^*(X,\mathbb{Z})$ and $C^*(Y,\mathbb{Z})$ are quasi-isomorphic.

This was proved by Mandell in 2003.

### In topological spaces

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

A connected space of the homotopy type of a CW-complex with a non-degenerate 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

• Mike Mandell, Cochains and homotopy type, Publ. Math. IHES (2006) 103: 213–246. (arXiv)
• 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

Last revised on September 2, 2017 at 21:56:41. See the history of this page for a list of all contributions to it.