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\begin{document}

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\section*{Calabi-Yau category}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{1categorical}{1-categorical}\dotfill \pageref*{1categorical} \linebreak
\noindent\hyperlink{categorical}{$(\infty,1)$-categorical}\dotfill \pageref*{categorical} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{FromCYVarieties}{From Calabi-Yau varieties}\dotfill \pageref*{FromCYVarieties} \linebreak
\noindent\hyperlink{from_symplectic_manifolds}{From symplectic manifolds}\dotfill \pageref*{from_symplectic_manifolds} \linebreak
\noindent\hyperlink{from_string_topology}{From string topology}\dotfill \pageref*{from_string_topology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classification_of_2d_tqft}{Classification of 2d TQFT}\dotfill \pageref*{classification_of_2d_tqft} \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}

The notion of \emph{Calabi-Yau category} is a [[horizontal categorification]] of that of [[Frobenius algebra]] -- a \emph{Frobenius [[algebroid]]} . Their name derives from the fact that the definition of Calabi-Yau categories have been originally studied as an abstract version of the [[derived category]] of [[coherent sheaves]] on a [[Calabi-Yau manifold]].

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

\hypertarget{1categorical}{}\subsubsection*{{1-categorical}}\label{1categorical}

A \textbf{Calabi-Yau category} is a [[Vect]]-[[enriched category]] $C$ equipped for each object $c \in C$ with a [[trace]]-like map

\begin{displaymath}
Tr_C : C(c,c) \to k
\end{displaymath}
to the ground field, such that for all objects $d \in C$ the induced pairing

\begin{displaymath}
\langle -,-\rangle_{c,d} :
  C(c,d) \otimes C(d,c) \to k
\end{displaymath}
given by

\begin{displaymath}
\langle f,g \rangle = Tr(g \circ f)
\end{displaymath}
is symmetric and non-degenerate.

Question: this 1-categorical definition seems to allow for different Frobenius structures on the endomorphism algebras of isomorphic objects. Would it be better to define it as a dinatural transformation from the Hom-functor to the constant functor with value the ground field $k$?

A Calabi-Yau category with a single object is the same (or rather: is equivalently the [[pointed object|pointed]] [[monoid]] [[delooping]]) of a [[Frobenius algebra]].

\hypertarget{categorical}{}\subsubsection*{{$(\infty,1)$-categorical}}\label{categorical}

A \textbf{Calabi-Yau $A_\infty$-category} of \emph{dimension} $d \in \mathbb{N}$ is an [[A-∞ category]] $C$ equipped with, for each pair $a,b$ of [[object]]s, a morphism of [[chain complex]]es

\begin{displaymath}
\langle -,-\rangle_{a,b}
  : 
  C(a,b) \otimes C(b,a)
  \to
  k[d]
\end{displaymath}
such that

\begin{enumerate}%
\item this is non-degenerate and is symmetric in that

\begin{displaymath}
\langle - , - \rangle_{a,b} =  \langle - , - \rangle_{b,a} \circ \sigma_{a,b}
\end{displaymath}
for $\sigma_{a,b} : C(a,b)\otimes C(b,a) \to C(b,a) \otimes C(a,b)$ the symmetry [[isomorphism]] of the [[symmetric monoidal category]] of [[chain complex]]es;


\item this is cyclically invariant in that for all elements $(\alpha_i)$ is the respective hom-complexes we have

\begin{displaymath}
\langle
    m_{n-1}(\alpha_0 \otimes \cdots \otimes \alpha_{n-2}),
    \alpha_{n-1} 
  \rangle
  =
  (-1)^{(n+1)+ |\alpha_0| \sum_{i = 1}^{n-1}|\alpha_i|}
  \langle
    m_{n-1}(\alpha_1 \otimes \cdots \otimes \alpha_{n-2}),
    \alpha_0
  \rangle
  \,.
\end{displaymath}


\end{enumerate}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{FromCYVarieties}{}\subsubsection*{{From Calabi-Yau varieties}}\label{FromCYVarieties}

\begin{itemize}%
\item Let $X$ be a smooth projective [[Calabi-Yau variety]] of dimension $d$. Write $D^b(X)$ for the bounded [[derived category]] of that of [[coherent sheaves]] on $X$.

Then $D^b(X)$ is a CY $A_\infty$-category in a naive way:

\begin{itemize}%
\item the non-binary composition maps are all trivial;


\item the pairing is given by [[Serre duality]] (one needs also a choice of trivialization of the canonical bundle of $X$)



\end{itemize}


\end{itemize}
This is however not the morally correct CY $A_\infty$-structure associated with a Calabi-Yau. A correct choice is, for example, the [[Dolbeault cohomology|Dolbeault]] [[enhanced triangulated category|dg-enhancement]] of the derived category

(\hyperlink{Costello04}{Costello 04, 7.2}, \hyperlink{Costello05}{Costello 05, 2.2})

\hypertarget{from_symplectic_manifolds}{}\subsubsection*{{From symplectic manifolds}}\label{from_symplectic_manifolds}

The [[Fukaya category]] associated with a [[symplectic manifold]] $X$. But see \href{http://mathoverflow.net/questions/13114/are-fukaya-categories-calabi-yau-categories}{this MO discussion} for more.

\hypertarget{from_string_topology}{}\subsubsection*{{From string topology}}\label{from_string_topology}

[[string topology]]: for $X$ a [[compact space|compact]] [[simply connected]] [[orientation|oriented]] [[manifold]], its cohomology $H^{\bullet}(X)$ is naturally a Calabi-Yau $A_\infty$-category with a single object. The $A_\infty$ structure comes from the [[homological perturbation lemma]]. One could also use the dg algebra of cochains $C^\bullet(X)$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{classification_of_2d_tqft}{}\subsubsection*{{Classification of 2d TQFT}}\label{classification_of_2d_tqft}

Calabi-Yau $A_\infty$-categories classify [[TCFT]]s. This remarkable result is what actually one should expect. Indeed, TCFTs originally arose as an abstract version of the [[CFT]]s constructed from [[sigma-model]]s whose targets are [[Calabi-Yau spaces]].

[[!include 2d TQFT -- table]]

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

\begin{itemize}%
\item [[Calabi-Yau object]]

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

\begin{itemize}%
\item [[Kevin Costello]], \emph{Topological conformal field theories and Calabi-Yau categories} (\href{http://arxiv.org/abs/math/0412149}{arXiv:math/0412149})


\item [[Kevin Costello]], \emph{The Gromov-Witten potential associated to a TCFT} (\href{http://arxiv.org/abs/math/0509264}{arXiv:0509264})


\item [[Maxim Kontsevich]], [[Yan Soibelman]]. \emph{Notes on A-infinity algebras, A-infinity categories and non-commutative geometry} $<$http://arxiv.org/abs/math/0606241{\tt \symbol{62}}


\item Lee Cho, \emph{Notes on Kontsevich-Soibelman's theorem about cyclic A-infinity algebras} (\href{http://arxiv.org/abs/1002.3653}{arXiv:1002.3653})


\item [[Jacob Lurie]], section 4.2 of \emph{[[On the Classification of Topological Field Theories]]} (\href{http://arxiv.org/abs/0905.0465}{arXiv:0905.0465})



\end{itemize}
[[!redirects Calabi-Yau category]] [[!redirects Calabi-Yau categories]] [[!redirects Calabi?Yau category]] [[!redirects Calabi?Yau categories]] [[!redirects Calabi--Yau category]] [[!redirects Calabi--Yau categories]]

[[!redirects CY category]] [[!redirects CY categories]] [[!redirects CY-category]] [[!redirects CY-categories]]

[[!redirects Calabi-Yau A-infinity category]] [[!redirects Calabi-Yau A-infinity categories]] [[!redirects Calabi-Yau A-infinity-category]] [[!redirects Calabi-Yau A-infinity-categories]] [[!redirects Calabi-Yau A-∞ category]] [[!redirects Calabi-Yau A-∞ categories]] [[!redirects Calabi-Yau A-∞-category]] [[!redirects Calabi-Yau A-∞-categories]]



\end{document}