nLab
E-infinity operad

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Contents

Idea

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.

Definition

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

Properties

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 .

Examples

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