model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
symmetric monoidal (∞,1)-category of spectra
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.
The definition of $E_\infty$-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 $E_\infty$-operad, in that its ∞-algebras over an (∞,1)-operad are E-∞ algebras;
when presenting $(\infty,1)-Operad$ 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 $E_\infty$-operad is (any) cofibrant resolution of Comm in the standard model structure on operads over the model structure on topological spaces.
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)
and for each $n$ this is by assumption a weak homotopy equivalence
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 .
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 $E_\infty$.
Last revised on December 3, 2016 at 15:53:38. See the history of this page for a list of all contributions to it.