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\begin{document}

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\section*{fusion category}

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

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_weak_hopf_algebras}{Relation to weak Hopf algebras}\dotfill \pageref*{relation_to_weak_hopf_algebras} \linebreak
\noindent\hyperlink{relation_to_pivotal_and_spherical_categories}{Relation to pivotal and spherical categories}\dotfill \pageref*{relation_to_pivotal_and_spherical_categories} \linebreak
\noindent\hyperlink{RelationtoTQFT}{Relation to extended 3d TQFT}\dotfill \pageref*{RelationtoTQFT} \linebreak
\noindent\hyperlink{suggestions}{Suggestions}\dotfill \pageref*{suggestions} \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}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{fusion category} is a [[rigid monoidal category|rigid]] [[semisimple category|semisimple]] [[linear category|linear]] ([[Vect]]-[[enriched category|enriched]]) [[monoidal category]] (``[[tensor category]]''), with only [[finite number|finitely]] many [[decategorification|isomorphism classes]] of [[simple objects]], such that the [[endomorphism]]s of the unit object form just the [[ground field]] $k$.

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

The name ``fusion category'' comes from the central examples of structures whose canonical [[tensor product]] is called a ``fusion product'', notably [[representation]]s of [[loop group]]s and of [[Hopf algebra]]s and of [[vertex operator algebra]]s.

Some of the easiest examples are:

\begin{itemize}%
\item Representations of a finite group or (\href{https://ncatlab.org/nlab/show/supergroup#finite_supergroups}{finite super-group})
\item For a given finite group $G$ and a 3-cocycle on $G$ with values in (the multiplicative group of units of) a field $k$ (an element of $H^3(G,k^\times)$), take $G$-graded vector spaces with the cocycle as associator.

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

\hypertarget{relation_to_weak_hopf_algebras}{}\subsubsection*{{Relation to weak Hopf algebras}}\label{relation_to_weak_hopf_algebras}

Under [[Tannaka duality]], every fusion category $C$ arises as the [[representation category]] of a [[weak Hopf algebra]]. (\hyperlink{Ostrik}{Ostrik})

\hypertarget{relation_to_pivotal_and_spherical_categories}{}\subsubsection*{{Relation to pivotal and spherical categories}}\label{relation_to_pivotal_and_spherical_categories}

Fusion categories were first systematically studied by [[Etingof]], Nikshych and [[Ostrik]] in \href{http://arxiv.org/abs/math/0203060}{On fusion categories}. This paper listed many examples and proved many properties of fusion categories. One of the important conjectures made in that paper was the following:

\begin{theorem}
\label{}\hypertarget{}{}
Every fusion category admits a [[pivotal structure]].

\end{theorem}
Providing a certain condition is satisfied, a pivotal structure on a fusion category can be shown to correspond to a `twisted' monoidal natural endotransformation of the identity functor on the category, where the twisting is given by the [[pivotal symbols]].

\hypertarget{RelationtoTQFT}{}\subsubsection*{{Relation to extended 3d TQFT}}\label{RelationtoTQFT}

Given the data of a [[fusion category]] one can build a 3d [[extended TQFT]] by various means. This is explained by the fact, see below, that fusion categories are (probably precisely) the [[fully dualizable object]]s in the [[3-category]] $MonCat$ of [[monoidal categories]]. By the [[homotopy hypothesis]] this explains how they induce [[3d TQFT]]s.

\begin{defn}
\label{MonCat}\hypertarget{MonCat}{}
Write $MonCat_{bim}$ for the [[(infinity,n)-category|(infinity,3)-category]] which has as

\begin{itemize}%
\item objects [[monoidal categories]],


\item morphism [[bimodule]]s of these,


\item and so on.



\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
With its natural [[tensor product]], $MonCat$ is a [[symmetric monoidal (infinity,n)-category|symmetric monoidal (infinity,3)-category]].

\end{prop}
\begin{prop}
\label{FusionCategoriesAreFullyDualizable}\hypertarget{FusionCategoriesAreFullyDualizable}{}
A monoidal category which is fusion is [[fully dualizable object|fully dualizable]] in the [[(infinity,n)-category|(infinity,3)-category]] $MonCat_{bim}$, def. \ref{MonCat}.

\end{prop}
This is due to (\hyperlink{DSPS13}{Douglas \& Schommer-Pries \& Snyder 13}).

\begin{remark}
\label{}\hypertarget{}{}
Via the [[cobordism theorem]] prop. \ref{FusionCategoriesAreFullyDualizable} implies that fusion categories encode [[extended TQFTs]] on 3-dimensional [[framed manifold|framed]] [[cobordisms]], while their $O(3)$-[[homotopy fixed points]] encode extended 3d TQFTs on general (not framed) cobordisms.

These 3d TQFTs hence arise from similar algebraic data as the [[Turaev-Viro model]] and the [[Reshetikhin-Turaev construction]], however there are various slight differences. See (\hyperlink{DSPS13}{Douglas \& Schommer-Pries \& Snyder 13, p. 5}).

\end{remark}
\hypertarget{suggestions}{}\subsection*{{Suggestions}}\label{suggestions}

Here are three things such that it'd be awesome if they were sorted out on this page:

\begin{enumerate}%
\item Kuperberg's theorem saying that [[abelian category|abelian]] semisimple implies [[linear category|linear]] over some field. \href{http://arxiv.org/abs/math/0209256}{Finite, connected, semisimple, rigid tensor categories are linear}


\item Some correct version of the claim that abelian semisimple is the same as [[idempotent complete category|idempotent complete]] and nondegenerate. \href{http://mathoverflow.net/questions/245/are-abelian-nondegenerate-tensor-categories-semisimple}{Math Overflow question}


\item Good notation distinguishing [[simple object|simple]] versus [[absolutely simple object|absolutely simple]] (is $End(V) = k$ or just $V$ has no nontrivial proper subobjects).



\end{enumerate}
Together 1 and 2 let you go between the two different obvious notions of semisimple which seem a bit muddled here when I clicked through the links.

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

\begin{itemize}%
\item \textbf{fusion category}

[[fusion ring]], [[Frobenius-Perron dimension]]


\item [[pivotal category]]


\item [[spherical category]]


\item [[Deligne's theorem on tensor categories]]


\item [[graded fusion category]]



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

Some classical references include

\begin{itemize}%
\item [[P. Etingof]], D. Nikshych and V. Ostrik, \href{http://arxiv.org/abs/math/0203060}{On fusion categories}.


\item [[Pavel Etingof]] and [[Damien Calaque]], \href{http://arxiv.org/abs/math/0401246}{Lectures on tensor categories}.


\item [[Michael Müger]], \href{http://arxiv.org/abs/0804.3587}{Tensor categories: A selective guided tour}.


\item [[Pavel Etingof]], D. Nikshych and V. Ostrik, \emph{Fusion categories and homotopy theory} , Quantum Topology, 1(2010), 209-273. (Earlier version available as \href{http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.3140v2.pdf}{ArXiv:0909.3140}



\end{itemize}
A review is also in chapter 6 of

\begin{itemize}%
\item [[Bruce Bartlett]], \href{http://arxiv.org/abs/0901.3975}{On unitary 2-representations of finite groups and topological quantum field theory}.

\end{itemize}
The [[Tannaka duality]] to [[weak Hopf algebras]] is discussed in

\begin{itemize}%
\item Takahiro Hayashi, \emph{A canonical Tannaka duality for finite seimisimple tensor categories} (\href{http://arxiv.org/abs/math/9904073}{arXiv:math/9904073})


\item [[Victor Ostrik]], \emph{Module categories, weak Hopf algebras and modular invariants} (\href{http://arxiv.org/abs/math/0111139}{arXiv:math/0111139})



\end{itemize}
The relation to [[3d TQFT]] is clarified via the [[cobordism hypothesis]] in

\begin{itemize}%
\item [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], \emph{The Structure of Fusion Categories via 3D TQFTs} (\href{https://sites.google.com/site/chrisschommerpriesmath/Home/recent-and-upcoming-talks/UPennTalk.pdf?attredirects=0}{talk pdf})


\item [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], \emph{Dualizable tensor categories} (\href{http://arxiv.org/abs/1312.7188}{arXiv:1312.7188})



\end{itemize}
and for the case of [[modular tensor categories]] in

\begin{itemize}%
\item [[Bruce Bartlett]], [[Christopher Douglas]], [[Chris Schommer-Pries]], [[Jamie Vicary]], \emph{Modular categories as representations of the 3-dimensional bordism 2-category} (\href{http://arxiv.org/abs/1509.06811}{arXiv:1509.06811})

\end{itemize}
For some discussion see

\begin{itemize}%
\item Math Overflow, \emph{\href{http://mathoverflow.net/questions/6180/why-are-fusion-categories-interesting}{Why are fusion categories interesting?}} .

\end{itemize}
[[!redirects fusion categories]]



\end{document}