nLab E-infinity operad



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Higher algebra



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


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


For every E E_\infty-operad PP, all the spaces P nP_n are contractible.

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

PComm P \to Comm

and for each nn this is by assumption a weak homotopy equivalence

P nComm n=* P_n \to Comm_n = *

of topological spaces.

The only extra condition on an operad PP with contractible operation spaces to be E E_\infty is that it is in addition cofibrant . This imposes the condition that the action of the symmetric group Σ n×P nP n\Sigma_n \times P_n \to P_n in each degree is free .


Last revised on December 3, 2016 at 15:53:38. See the history of this page for a list of all contributions to it.