nLab
E-infinity algebra

Definition

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

Realizations

In chain complexes

$E_{\mathrm{\infty }}$-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 $E_{\mathrm{\infty }}$-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 $E_{\mathrm{\infty }}$-algebra in spectra is an E-∞ ring.

In $\mathrm{\infty }$-stacks

See Ek-Algebras.

References

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

• 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 $E_{\mathrm{\infty }}$-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)