symmetric monoidal (∞,1)-category of spectra
Given a 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. 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.
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 September 2, 2017 at 21:56:41. See the history of this page for a list of all contributions to it.