nLab E-infinity algebra

Definition

Given a stable 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.

A concrete construction of an $E_\infty$-algebra structure on singular cochains, and, more generally, simplicial cochains of a simplicial set can be found in McClure and Smith. Their work gives a precise description of the involved sequence operad, which is an $E_\infty$-operad, as well as its action on simplicial cochains, which involves a generalization of Steenrod’s cup-$i$ products. Recall that the cup-1 product controls the noncommutativity of the ordinary cup product:

$$d(x\cup_1 y)=x\cup y-(-1)^{|x|\cdot|y|}y\cup x.$$

It can be defined using a formula similar to the one used for the cup product, and the higher operations also have similar nature.

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.

In $\infty$-stacks

See Ek-Algebras.

In $(\infty,n)$-categories

See symmetric monoidal (∞,n)-category.

Related concepts

• A-∞ algebra

• Hopf E-∞ algebra

References

• Peter May, The geometry of iterated loop spaces (pdf)

• James E. McClure, Jeffrey H. Smith, Multivariable cochain operations and little $n$-cubes_,arXiv, Journal of the AMS 16:3 (2003), 681–704.

• 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

• Jacob Lurie, Ek-Algebras

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

• Clemens Berger, Ieke Moerdijk, Axiomatic homotopy theory for operads Comment. Math. Helv. 78 (2003), 805–831. (arXiv:math/0206094)

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

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (arXiv:math/0512576)