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\begin{document}

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\section*{A-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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\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{rectification}{Rectification}\dotfill \pageref*{rectification} \linebreak
\noindent\hyperlink{in_topological_space}{In Topological space}\dotfill \pageref*{in_topological_space} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{rectification_2}{Rectification}\dotfill \pageref*{rectification_2} \linebreak
\noindent\hyperlink{in_spectra}{In spectra}\dotfill \pageref*{in_spectra} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \emph{$A_\infty$-algebra} is a [[monoid]] internal to a [[homotopical category]] such that the [[associativity]] law holds not as an equation, but only up to higher [[coherent]] [[homotopy]].

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

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{$A_\infty$-algebra} is an [[algebra over an operad]] over an [[A-∞ operad]].

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

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

Let here $\mathcal{E}$ be the [[category of chain complexes]] $\mathcal{Ch}_\bullet$. Notice that often in the literature this choice of $\mathcal{E}$ is regarded as default and silently assumed.

An $A_\infty$-algebra in chain complexes is concretely the following data.

A chain $A_\infty$-algebra is the structure of a degree 1 [[coderivation]]

\begin{displaymath}
D : T^c V \to T^c V
\end{displaymath}
on the reduced tensor coalgebra $T^c V = \oplus_{n\geq 1} V^{\otimes n}$ (with the standard noncocommutative coproduct, see [[differential graded Hopf algebra]]) over a [[graded vector space]] $V$ such that

\begin{displaymath}
D^2 = 0
  \,.
\end{displaymath}
Coderivations on free coalgebras are entirely determined by their ``value on cogenerators'', which allows one to decompose $D$ as a sum:

\begin{displaymath}
D = D_1 + D_2 + D_3 + \cdots
\end{displaymath}
with each $D_k$ specified entirely by its action

\begin{displaymath}
D_k : V^{\otimes k} \to V
  \,.
\end{displaymath}
which is a map of degree $2-k$ (or can be alternatively understood as a map $D_k : (V[1])^{\otimes k}\to V[1]$ of degree $1$).

Then:

\begin{itemize}%
\item $D_1 : V\to V$ is the \emph{differential} with $D_1^2 = 0$;


\item $D_2 : V^{\otimes 2} \to V$ is the \emph{product} in the algebra;


\item $D_3 : V^{\otimes 3} \to V$ is the \emph{associator} which measures the failure of $D_2$ to be associative;


\item $D_4 : V^{\otimes 4} \to V$ is the \emph{pentagonator} (or so) which measures the failure of $D_3$ to satisfy the pentagon identity;


\item and so on.



\end{itemize}
One can also allow $D_0$, in which case one talks about weak $A_\infty$-algebras, which are less understood.

There is a resolution of the operad $\mathrm{Ass}$ of associative algebras (as operad on the category of chain complexes) which is called the $A_\infty$-operad; the algebras over the $A_\infty$-[[A-infinity operad|operad]] are precisely the $A_\infty$-algebras.

A \textbf{morphism of $A_\infty$-algebras} $f : A\to B$ is a collection $\lbrace f_n\rbrace_{n\geq 1}$ of maps

\begin{displaymath}
f_n : (A[1])^{\otimes n}\to B[1]
\end{displaymath}
of degree $0$ satisfying

\begin{displaymath}
\sum_{0\leq i+j\leq n} f_{i+j+1}\circ(1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j})
= \sum_{i_1+\ldots+i_r=n} D_r\circ (f_{i_1}\otimes\ldots f_{i_r}).
\end{displaymath}
For example, $f_1\circ D_1 = D_1\circ f_1$.

\hypertarget{rectification}{}\paragraph*{{Rectification}}\label{rectification}

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Kadeishvili (1980), Merkulov (1999))}

If $A$ is a [[dg-algebra]], regarded as a strictly associative $A_\infty$-algebra, its [[chain homology and cohomology|chain cohomology]] $H^\bullet(A)$, regarded as a [[chain complex]] with trivial differentials, naturally carries the structure of an $A_\infty$-algebra, unique up to isomorphism, and is weakly equivalent to $A$ as an $A_\infty$-algebra.

\end{theorem}
More details are at \emph{[[Kadeishvili's theorem]]}.

\begin{remark}
\label{}\hypertarget{}{}
This theorem provides a large supply of examples of $A_\infty$-algebras: there is a natural $A_\infty$-algebra structure on all cohomologies such as

\begin{itemize}%
\item [[de Rham cohomology]]


\item [[Hochschild cohomology]]



\end{itemize}
etc.

\end{remark}
\hypertarget{in_topological_space}{}\subsubsection*{{In Topological space}}\label{in_topological_space}

An $A_\infty$-algebra in [[Top]] is also called an \emph{[[A-∞ space]]} .

\hypertarget{examples}{}\paragraph*{{Examples}}\label{examples}

Every [[loop space]] is canonically an [[A-∞ space]]. (See there for details.)

\hypertarget{rectification_2}{}\paragraph*{{Rectification}}\label{rectification_2}

\begin{theorem}
\label{}\hypertarget{}{}
Every $A_\infty$-space is [[weak homotopy equivalence|weakly homotopy equivalent]] to a topological [[monoid]].

\end{theorem}
This is a classical result by (\hyperlink{Stasheff}{Stasheff}, \hyperlink{BoardmanVogt}{BoardmanVogt}). It follows also as a special case of the more general result on rectification in a [[model structure on algebras over an operad]] (see there).

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

See [[ring spectrum]] and [[algebra spectrum]].

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

\begin{itemize}%
\item \textbf{$A_\infty$-algebra}, [[A-∞-category]]

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


\item [[curved A-∞ algebra]]



\end{itemize}

\item [[A-n algebra]]


\item [[E-∞ algebra]]


\item [[L-∞ algebra]], .



\end{itemize}
[[!include k-monoidal table]]

[[!include deformation quantization - table]]

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

A survey of $A_\infty$-algebras in chain complexes is in

\begin{itemize}%
\item [[Bernhard Keller]], \emph{A brief introduction to $A_\infty$-algebras} (\href{http://people.math.jussieu.fr/~keller/publ/IntroAinfEdinb.pdf}{pdf})

\end{itemize}
Classical articles on $A_\infty$-algebra in topological spaces are

\begin{itemize}%
\item [[Jim Stasheff]], \emph{Homotopy associativity of H-spaces I} , Trans. Amer. Math. Soc. 108 (1963), p. 275-292.

\end{itemize}
\begin{itemize}%
\item [[Michael Boardman]] and [[Rainer Vogt]], \emph{Homotopy invariant algebraic structures on topological spaces} , Lect. Notes Math. 347 (1973).

\end{itemize}
A brief survey is in section 1.8 of

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

\end{itemize}
The 1986 thesis of [[Alain Prouté]] explores the possibility of obtaining analogues of [[minimal model]]s for $A_\infty$ algebras. It was published in TAC much later.

\begin{itemize}%
\item [[Alain Prouté]], \emph{Alg\`e{}bres diff\'e{}rentielles fortement homotopiquement associatives ($A_\infty$-alg\`e{}bres)}, thesis, available as \href{http://www.logique.jussieu.fr/~alp/these_A_Proute-TAC.pdf}{Reprints in Theory and Applications of Categories, No. 21, 2011, pp. 1--99}

\end{itemize}
[[!redirects A-infinity-algebras]] [[!redirects A-infinity algebra]] [[!redirects A-infinity algebras]] [[!redirects A-∞ algebra]] [[!redirects A-∞ algebras]] [[!redirects A? algebra]] [[!redirects A? algebras]] [[!redirects A-∞-algebra]]

[[!redirects A-∞-algebras]]



\end{document}