symmetric monoidal (∞,1)-category of spectra
Given a stable symmetric monoidal (infinity,1)-category $C$, an $E_\infty$-algebra in $C$ is synonymous with a commutative monoid in $C$. Most often $C$ is the (infinity,1)-category of chain complexes over a field or commutative ring, or the stable (infinity,1)-category of spectra (see also E-infinity ring).
In terms of operads, an $E_\infty$-algebra is an algebra over an operad for an E-∞ operad.
$E_\infty$-algebras in chain complexes are equivalent to those in abelian simplicial groups.
For details on this statement see monoidal Dold-Kan correspondence and operadic Dold-Kan correspondence.
The singular cohomology $H^*(X,\mathbb{Z})$ of a topological space is a graded-commutative algebra over the integers, but the the singular cochain complex $C^*(X,\mathbb{Z})$ is not: instead, it is an $E_\infty$-algebra.
A concrete construction of an $E_\infty$-algebra structure on singular cochains, and, more generally, simplicial cochains of a simplicial set can be found in McClure and Smith. Their work gives a precise description of the involved sequence operad, which is an $E_\infty$-operad, as well as its action on simplicial cochains, which involves a generalization of Steenrod’s cup-$i$ products. Recall that the cup-1 product controls the noncommutativity of the ordinary cup product:
It can be defined using a formula similar to the one used for the cup product, and the higher operations also have similar nature.
Remarkably, for simply connected spaces of finite type this $E_\infty$-algebra knows everything about the weak homotopy type of $X$. In fact a stronger statement holds:
Finite type nilpotent spaces $X$ and $Y$ are weakly homotopy equivalent if and only if the $E_\infty$-algebras $C^*(X,\mathbb{Z})$ and $C^*(Y,\mathbb{Z})$ are quasi-isomorphic.
This was proved by Mandell in 2003.
A connected space of the homotopy type of a CW-complex with a non-degenerate basepoint that has the homotopy type of a $k$-fold loop space for all $k \in \mathbb{N}$ admits the structure of an $E_\infty$-space.
The model structure on algebras over an operad over E-∞ operads in Top and in sSet are Quillen equivalent.
This is in BergerMoerdijk I, BergerMoerdijk II.
An $E_\infty$-algebra in spectra is an E-∞ ring.
See Ek-Algebras.
See symmetric monoidal (∞,n)-category.
James E. McClure, Jeffrey H. Smith, Multivariable cochain operations and little $n$-cubes_,arXiv, Journal of the AMS 16:3 (2003), 681–704.
Mike Mandell, Cochains and homotopy type, Publ. Math. IHES (2006) 103: 213–246. (arXiv)
In the context of (infinity,1)-operads $E_\infty$-algebras are discussed in
A systematic study of model category structures on operads and their algebras is in
The induced model structures and their properties on algebras over operads are discussed in
Last revised on October 24, 2023 at 05:24:24. See the history of this page for a list of all contributions to it.