nLab
E-infinity operad

Idea

An $E_{\mathrm{\infty }}$-operad is a topological operad that is a homotopy theoretic resolution of Comm, the operad for commutative monoids: an algebra over an operad over an $E_{\mathrm{\infty }}$-operad is an E-∞ algebra.

Definition

The definition of $E_{\mathrm{\infty }}$-operads depends a bit on which presentation of the (∞,1)-category of (∞,1)-operads one uses:

• abstractly, in the (∞,1)-category of (∞,1)-operads, the operad Comm itself already is an $E_{\mathrm{\infty }}$-operad, in that its ∞-algebras over an (∞,1)-operad are E-∞ algebras;

• when presenting $(\mathrm{\infty },1)\mathrm{Operad}$ by the model structure on operads for topological operads, forming the homotopy-algebras over any operad means forming the ordinary algebras over an operad for any of its cofibrant resolutions. Therefore one say: an $E_{\mathrm{\infty }}$-operad is (any) cofibrant resolution of Comm in the standard model structure on operads over the model structure on topological spaces.

Properties

For every $E_{\mathrm{\infty }}$-operad $P$, all the spaces $P_n$ are contractible.

In fact, every topological operad $P$ for which $P_n\simeq *$ for all $n\in \mathbb{N}$ is weakly equivalent to Comm: because ${\mathrm{Comm}}_n=*$ there is a unique morphism of operads (necessarily respecting the action of the symmetric group)

$$P\to \mathrm{Comm}$$P \to Comm

and for each $n$ this is by assumption a weak homotopy equivalence

$$P_n\to {\mathrm{Comm}}_n=*$$P_n \to Comm_n = *

of topological spaces.

The only extra condition on an operad $P$ with contractible operation spaces to be $E_{\mathrm{\infty }}$ is that it is in addition cofibrant . This imposes the condition that the action of the symmetric group ${\Sigma }_n\times P_n\to P_n$ in each degree is free .

Examples

• In some sense the universal model for an $E_{\mathrm{\infty }}$-operad is the Barratt-Eccles operad.

• The little k-cubes operad for $k\to \mathrm{\infty }$ is $E_{\mathrm{\infty }}$.

Related concepts

• A-∞ operad

• E-∞ ring