model category

Model structures

for ∞-groupoids

for $(\infty,1)$-sheaves / $\infty$-stacks

Higher algebra

higher algebra

universal algebra

Contents

Idea

An $E_\infty$-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 $E_\infty$-operad is an E-∞ algebra.

Definition

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

Properties

For every $E_\infty$-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 $Comm_n = *$ there is a unique morphism of operads (necessarily respecting the action of the symmetric group)

$P \to Comm$

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

$P_n \to Comm_n = *$

of topological spaces.

The only extra condition on an operad $P$ with contractible operation spaces to be $E_\infty$ 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 $E_\infty$-operad is the Barratt-Eccles operad.

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

Revised on August 13, 2012 10:30:33 by Beren Sanders (96.251.14.54)