nLab
E-infinity algebra

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Definition

Given a symmetric monoidal (infinity,1)-category CC, an E E_\infty-algebra in CC is synonymous with a commutative monoid in CC. Most often CC 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 E_\infty-algebra is an algebra over an operad for an E-∞ operad.

Realizations

In chain complexes

E 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,)H^*(X,\mathbb{Z}) of a topological space is a graded-commutative algebra over the integers, but the the singular cochain complex C *(X,)C^*(X,\mathbb{Z}) is not: instead, it is an E E_\infty-algebra.

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

d(x 1y)=xy(1) |x||y|yx.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 E_\infty-algebra knows everything about the weak homotopy type of XX. In fact a stronger statement holds:

Theorem

Finite type nilpotent spaces XX and YY are weakly homotopy equivalent if and only if the E E_\infty-algebras C *(X,)C^*(X,\mathbb{Z}) and C *(Y,)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 kk-fold loop space for all kk \in \mathbb{N} admits the structure of an E E_\infty-space.

In simplicial sets

This is in BergerMoerdijk I, BergerMoerdijk II.

In spectra

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

In \infty-stacks

See Ek-Algebras.

In (,n)(\infty,n)-categories

See symmetric monoidal (∞,n)-category.

References

\bibitem{MCO} James E. McClure?, Jeffrey H. Smith, Multivariable cochain operations and little nn-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 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 5, 2019 at 18:54:37. See the history of this page for a list of all contributions to it.