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\section*{A-infinity-category}

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

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{ordinary_linear_categories}{Ordinary linear $A_\infty$-categories}\dotfill \pageref*{ordinary_linear_categories} \linebreak
\noindent\hyperlink{examples_and_remarks}{Examples (and remarks)}\dotfill \pageref*{examples_and_remarks} \linebreak
\noindent\hyperlink{more_general_categories}{More general $A_\infty$-categories}\dotfill \pageref*{more_general_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{for_categories_in_the_sense_of_homological_algebra}{For $A_\infty$-categories in the sense of homological algebra}\dotfill \pageref*{for_categories_in_the_sense_of_homological_algebra} \linebreak
\noindent\hyperlink{for_categories_in_the_wider_sense}{For $A_\infty$-categories in the wider sense}\dotfill \pageref*{for_categories_in_the_wider_sense} \linebreak


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

An \emph{$A_\infty$-category} is a kind of [[category]] in which the associativity condition on the [[composition]] of [[morphism]]s is relaxed ``up to higher coherent homotopy''.

The ``A'' is for Associative and the ``${}_\infty$'' indicates that associativity is relaxed up to higher homotopies without bound on the degree of the homotopies.

In the most widespread use of the word $A_\infty$-categories are \emph{linear} categories in that they have [[hom-object]]s that are [[chain complex]]es. These are really models/presentations for [[stable (∞,1)-category|stable (∞,1)-categories]].

If the composition in the linear $A_\infty$-category does happen to be strictly associative it becomes the same as a [[dg-category]]. In fact, every linear $A_\infty$-category is $A_\infty$-equivalent to a [[dg-category]]. In this way, we have that $A_\infty$-categories related to [[dg-category|dg-categories]] as models for [[stable (∞,1)-category|stable (∞,1)-categories]] in roughly the same way as [[quasi-category|quasi-categories]] relate to [[simplicially enriched category|simplicially enriched categories]] as models for [[(∞,1)-category|(∞,1)-categories]]: the former is the general incarnation, while the latter is a [[semi-strict infinity-category|semi-strictified]] version.

\hypertarget{ordinary_linear_categories}{}\subsubsection*{{Ordinary linear $A_\infty$-categories}}\label{ordinary_linear_categories}

In what is strictly speaking a restrictive sense -- which is however widely and conventionally understood in [[homological algebra]] as the standard notion of $A_\infty$-category (see references below) -- the [[hom-space]]s of an $A_\infty$-category are taken to be linear spaces, i.e. [[module]]s over some [[ring]] or [[field]], and in fact [[chain complex]]es of such modules.

Therefore an $A_\infty$-category in this standard sense of [[homological algebra]] is a category which is in some way [[homotopical enrichment|homotopically enriched]] over a [[category of chain complexes]] $Ch$. Since a category which is [[enriched category|enriched]] in the ordinary sense of [[enriched category theory]] is a [[dg-category]], there is a close relation between $A_\infty$-categories and [[dg-category|dg-categories]].

$A_\infty$-categories in this linear sense are a [[horizontal categorification]] of the notion of [[A-infinity-algebra]]. As such they are to [[A-infinity-algebra]]s as [[Lie infinity-algebroid]]s are to [[L-infinity-algebra]]s. For this point of view see \hyperlink{KonsevichSoibelman08}{Konsevich-Soibelman 08}.

\begin{defn}
\label{}\hypertarget{}{}
A category $C$ such that

\begin{enumerate}%
\item for all $X,Y$ in $Ob(C)$ the [[hom-set|Hom-set]]s $Hom_C(X,Y)$ are finite dimensional [[chain complex]]es of $\mathbf{Z}$-graded modules


\item for all objects $X_1,...,X_n$ in $Ob(C)$ there is a family of linear composition maps (the higher compositions) $m_n : Hom_C(X_0,X_1) \otimes Hom_C(X_1,X_2) \otimes \cdots \otimes Hom_C(X_{n-1},X_n) \to Hom_C(X_0,X_n)$ of degree $n-2$ (homological grading convention is used) for $n\geq1$


\item $m_1$ is the differential on the chain complex $Hom_C(X,Y)$


\item $m_n$ satisfy the quadratic $A_\infty$-associativity equation for all $n\geq0$.



\end{enumerate}
\end{defn}
$m_1$ and $m_2$ will be [[chain complex|chain map]]s but the compositions $m_i$ of higher order are not chain maps, nevertheless they are [[Massey product]]s.

The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of $A_\infty$-categories and $A_\infty$-functors. Many features of $A_\infty$-categories and $A_\infty$-functors come from the fact that they form a symmetric closed [[multicategory]], which is revealed in the language of [[comonad|comonads]].

From a higher dimensional perspective $A_\infty$-categories are weak $\omega$-categories with all morphisms invertible. $A_\infty$-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.

\hypertarget{examples_and_remarks}{}\subsubsection*{{Examples (and remarks)}}\label{examples_and_remarks}

\begin{itemize}%
\item Every [[dg-category]] may be regarded as a special case when there are no higher maps (trivial homotopies) of an $A_\infty$-category.


\item Every $A_\infty$-category is $A_\infty$-equivalent to a [[dg-category]].

\begin{itemize}%
\item This is a corollary of the $A_\infty$-categorical [[Yoneda lemma]].


\item beware that this statement does not imply that the notion of $A_\infty$-categories is obsolete (see section 1.8 in Bespalov et al.): in practice it is often easier to work with a given naturally arising $A_\infty$-category than constructing its equivalent [[dg-category]]

\begin{itemize}%
\item for instance when dealing with a [[Fukaya A-infinity-category|Fukaya]] $A_\infty$-category;


\item or when dealing with various constructions on dg-categories, for instance certain quotients,that naturally yield directly $A_\infty$-categories instead of dg-categories.



\end{itemize}


\end{itemize}

\item The [[path space ]] of a topological space $X$


\item The [[Fukaya category]] $Fuk(X)$ of a topological space $X$ -- a [[Calabi-Yau A-∞ category]]


\item $A_\infty$-[[A-infinity-algebra|algebras]] are the $A_\infty$-categories with one object.

\begin{itemize}%
\item For example, the delooping $\mathbf{B}\Omega{X}$ of [[loop space]] $\Omega{X}$ of a topological space $X$

\end{itemize}


\end{itemize}
\hypertarget{more_general_categories}{}\subsubsection*{{More general $A_\infty$-categories}}\label{more_general_categories}

In the widest sense, $A_\infty$-category may be used as a term for a category in which the [[composition]] operation constitutes an algebra over an [[operad]] which resolves in some sense the associative operad $Ass$.

One should be aware, though, that this use of the term is not understood by default in the large body of literature concerned with the above linear notion.

A less general but non-linear definition is fairly straight forward in any category in which there is a notion of \emph{homotopy} with the usual properties.

\begin{udefn}
An $A_\infty$-category is a [[category over an operad|category over]] the $A_\infty$-[[A-infinity operad|operad]]: e.g. the free resolution in the context of dg-operads of the linear associative operad.

\end{udefn}
\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item [[Trimble n-category|Trimble n-categories]];


\item also the classical notion of [[bicategory]] can be interpreted as an $A_\infty$-category in [[Cat]] for a suitable Cat-operad.



\end{itemize}
\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[A-∞ space]], [[associahedron]]


\item [[A-∞ algebra]]


\item [[stable (∞,1)-category]]



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

\hypertarget{for_categories_in_the_sense_of_homological_algebra}{}\subsubsection*{{For $A_\infty$-categories in the sense of homological algebra}}\label{for_categories_in_the_sense_of_homological_algebra}

For a short and precise introduction see

\begin{itemize}%
\item B. Keller, \emph{Introduction to $A_\infty$-algebras and modules} (, ) and Addendum (), Homology, Homotopy and Applications 3 (2001), 1-35;


\item B. Keller, \emph{$A_\infty$ algebras, modules and functor categories}, (, ).



\end{itemize}
and for a [[Fukaya category]]-oriented introduction see chapter 1 in

\begin{itemize}%
\item P. Seidel, \emph{Fukaya category and Picard-Lefschetz theory},

\end{itemize}


A very detailed treatment of $A_\infty$-categories is a recent book

\begin{itemize}%
\item Yu. Bespalov, [[Volodymyr Lyubashenko]], O. Manzyuk, \emph{Pretriangulated $A_\infty$-categories}, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 (\href{http://www.math.ksu.edu/~lub/cmcMonad.gz}{ps.gz})

\begin{itemize}%
\item notice: the ps.gz file has different page numbers than the printed version, but the numbering of sections and formulae is final. Errata to published version are \href{http://www.math.ksu.edu/~lub/cmcMoCor.pdf}{here}.

\end{itemize}

\item Oleksandr Manzyk, \emph{A-infinity-bimodules and Serre A-infinity-functors}, dissertation \href{https://kluedo.ub.uni-kl.de/files/1910/dissertation.pdf}{pdf}, \href{https://kluedo.ub.uni-kl.de/files/1910/dissertation.djvu}{djvu}; \emph{Serre $A_\infty$ functors}, talk at Categories in geometry and math. physics, Split 2007, slides, \href{http://www.irb.hr/korisnici/zskoda/manzyukslides.pdf}{pdf}, work with [[Volodymyr Lyubashenko]]



\end{itemize}
The relation of $A_\infty$-categories to [[differential graded algebra]]s is emphasized in the introduction of

\begin{itemize}%
\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Notes on $A_\infty$-algebras, $A_\infty$-categories and non-commutative geometry. I}, in [[Anton Kapustin]], [[Maximilian Kreuzer]], [[Karl-Georg Schlesinger]] (eds.) \emph{Homological mirror symmetry -- New developments and perspectives}, Springer 2008 (\href{http://arxiv.org/abs/math/0606241}{arXiv:math/0606241}, \href{https://www.springer.com/de/book/9783540680291}{doi:10.1007/978-3-540-68030-7})

\end{itemize}
which mostly discusses just [[A-infinity-algebra]]s, but points out a generalizations to $A_\infty$-categories, see the overview on \href{http://arxiv.org/PS_cache/math/pdf/0606/0606241v2.pdf#page=3}{p. 3}

Essentially the authors say that an $A_\infty$-category should be a non(-graded-)commutative [[dg-manifold]]/[[L-infinity-algebroid]].

More [[category theory]] and [[homotopy theory]] of $A_\infty$-categories is discussed in

\begin{itemize}%
\item Kenji Lef\`e{}vre-Hasegawa, \emph{Sur les A-infini cat\'e{}gories} (\href{http://arxiv.org/abs/math/0310337}{arXiv:math/0310337})


\item [[Bruno Valette]], \emph{Homotopy theory of homotopy algebras} (\href{http://math.unice.fr/~brunov/HomotopyTheory.pdf}{pdf})



\end{itemize}
\hypertarget{for_categories_in_the_wider_sense}{}\subsubsection*{{For $A_\infty$-categories in the wider sense}}\label{for_categories_in_the_wider_sense}

If one understands $A_\infty$-category as ``operadically defined higher category'', then relevant references would include:

\begin{itemize}%
\item Eugenia Cheng, \emph{Comparing operadic definitions of $n$-category} (\href{http://arxiv.org/abs/0809.2070}{arXiv})

\end{itemize}
With operads modeled by [[dendroidal sets]], [[n-categories]] for low $n$ viewed as objects with an $A-\infty$-composition operation are discussed in section 5 of

\begin{itemize}%
\item Andor Lucacs, \emph{Cyclic Operads, Dendroidal Structures, Higher Categories} (\href{http://igitur-archive.library.uu.nl/dissertations/2011-0211-200314/lukacs.pdf}{pdf})

\end{itemize}
and

\begin{itemize}%
\item Andor Lucacs, \emph{Dendroidal weak 2-categories} (\href{http://de.arxiv.org/abs/1304.4278}{arXiv:1304.4278})

\end{itemize}
See also the references at \emph{[[model structure on algebras over an operad]]}.

[[!redirects A-infinity category]] [[!redirects A-infinity categories]] [[!redirects A-∞ category]] [[!redirects A-∞ categories]] [[!redirects A? category]] [[!redirects A? categories]] [[!redirects A∞-category]] [[!redirects A∞-categories]] [[!redirects A-∞-category]] [[!redirects A-∞-categories]]



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