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\begin{document}

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\section*{E-infinity algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{realizations}{Realizations}\dotfill \pageref*{realizations} \linebreak
\noindent\hyperlink{in_chain_complexes}{In chain complexes}\dotfill \pageref*{in_chain_complexes} \linebreak
\noindent\hyperlink{in_topological_spaces}{In topological spaces}\dotfill \pageref*{in_topological_spaces} \linebreak
\noindent\hyperlink{in_simplicial_sets}{In simplicial sets}\dotfill \pageref*{in_simplicial_sets} \linebreak
\noindent\hyperlink{in_spectra}{In spectra}\dotfill \pageref*{in_spectra} \linebreak
\noindent\hyperlink{in_stacks}{In $\infty$-stacks}\dotfill \pageref*{in_stacks} \linebreak
\noindent\hyperlink{in_categories}{In $(\infty,n)$-categories}\dotfill \pageref*{in_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a [[symmetric monoidal (infinity,1)-category]] $C$, an \textbf{$E_\infty$-algebra} in $C$ is synonymous with a [[commutative monoid in a symmetric monoidal (infinity,1)-category|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]].

\hypertarget{realizations}{}\subsection*{{Realizations}}\label{realizations}

\hypertarget{in_chain_complexes}{}\subsubsection*{{In chain complexes}}\label{in_chain_complexes}

$E_\infty$-algebras in [[chain complex|chain complexes]] are equivalent to those in abelian [[simplicial group|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 cohomology|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 \cite{MCO}. Their work gives a precise description of the involved \textbf{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]]:

\begin{displaymath}
d(x\cup_1 y)=x\cup y-(-1)^{|x|\cdot|y|}y\cup x.
\end{displaymath}
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:

\begin{utheorem}
Finite type [[nilpotent spaces]] $X$ and $Y$ are [[weak equivalence|weakly homotopy equivalent]] if and only if the $E_\infty$-algebras $C^*(X,\mathbb{Z})$ and $C^*(Y,\mathbb{Z})$ are quasi-isomorphic.

\end{utheorem}
This was proved by \hyperlink{Mandell}{Mandell} in 2003.

\hypertarget{in_topological_spaces}{}\subsubsection*{{In topological spaces}}\label{in_topological_spaces}

\begin{utheorem}
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.

\end{utheorem}
\hypertarget{in_simplicial_sets}{}\subsubsection*{{In simplicial sets}}\label{in_simplicial_sets}

\begin{utheorem}
The [[model structure on algebras over an operad]] over [[E-∞ operad]]s in [[Top]] and in [[sSet]] are [[Quillen equivalence|Quillen equivalent]].

\end{utheorem}
This is in \hyperlink{BergerMoerdijkHomotopy}{BergerMoerdijk I}, \hyperlink{BergerMoerdijkAlgebras}{BergerMoerdijk II}.

\hypertarget{in_spectra}{}\subsubsection*{{In spectra}}\label{in_spectra}

An $E_\infty$-algebra in [[spectra]] is an [[E-∞ ring]].

\hypertarget{in_stacks}{}\subsubsection*{{In $\infty$-stacks}}\label{in_stacks}

See [[Ek-Algebras]].

\hypertarget{in_categories}{}\subsubsection*{{In $(\infty,n)$-categories}}\label{in_categories}

See \emph{[[symmetric monoidal (∞,n)-category]]}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[A-∞ algebra]]


\item [[Hopf E-∞ algebra]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Peter May]], \emph{The geometry of iterated loop spaces} (\href{http://www.math.uchicago.edu/~may/BOOKS/gils.pdf}{pdf})

\end{itemize}
$\backslash$bibitem\{MCO\} [[James E. McClure]], [[Jeffrey H. Smith]], \emph{Multivariable cochain operations and little $n$-cubes\_}, \href{https://arxiv.org/abs/math/0106024}{arXiv}, \href{https://doi.org/10.1090/S0894-0347-03-00419-3}{Journal of the AMS 16:3 (2003), 681–704}.

\begin{itemize}%
\item [[Mike Mandell]], \emph{Cochains and homotopy type}, Publ. Math. IHES (2006) 103: 213--246. (\href{https://arxiv.org/abs/math/0311016}{arXiv})

\end{itemize}
\begin{itemize}%
\item [[Martin Markl]], Steve Shnider, [[Jim Stasheff]], \emph{Operads in algebra, topology and physics}, Math. Surveys and Monographs \textbf{96}, Amer. Math. Soc. 2002.

\end{itemize}
In the context of [[(infinity,1)-operad]]s $E_\infty$-algebras are discussed in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Ek-Algebras]]}

\end{itemize}
A systematic study of [[model category]] structures on operads and their algebras is in

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{Axiomatic homotopy theory for operads} Comment. Math. Helv. 78 (2003), 805--831. (\href{http://arxiv.org/abs/math/0206094}{arXiv:math/0206094})

\end{itemize}
The induced model structures and their properties on [[algebras over operads]] are discussed in

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{Resolution of coloured operads and rectification of homotopy algebras} (\href{http://arxiv.org/abs/math/0512576}{arXiv:math/0512576})

\end{itemize}
[[!redirects E-infinity-algebra]] [[!redirects E-infinity algebras]] [[!redirects E-infinity-algebras]]

[[!redirects E-∞ algebra]] [[!redirects E-∞ algebras]]

[[!redirects E-∞-algebra]] [[!redirects E-∞-algebras]]



\end{document}