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\begin{document}

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\section*{braided monoidal 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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{in_terms_of_higher_monoidal_structure}{In terms of higher monoidal structure}\dotfill \pageref*{in_terms_of_higher_monoidal_structure} \linebreak
\noindent\hyperlink{the_2category_of_braided_monoidal_categories}{The 2-category of braided monoidal categories}\dotfill \pageref*{the_2category_of_braided_monoidal_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{tannaka_duality}{Tannaka duality}\dotfill \pageref*{tannaka_duality} \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}

Intuitively speaking, a braided monoidal category is a category with a tensor product and an isomorphism called the `braiding' which lets us `switch' two objects in a tensor product like $x \otimes y$. Thus the tensor product is ``commutative'' in a sense, but not as coherently commutative as in a [[symmetric monoidal category]].

A braided monoidal category is a special case of the notion of [[braided pseudomonoid]] in a [[braided monoidal 2-category]].

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

\begin{defn}
\label{BraidedMonoidalCategory}\hypertarget{BraidedMonoidalCategory}{}
A \textbf{braided monoidal category}, or (``braided [[tensor category]]'', but see there), is a [[monoidal category]] $\mathcal{C}$ equipped with a [[natural isomorphism]]

\begin{displaymath}
B_{x,y} : x \otimes y \to y \otimes x
\end{displaymath}
called the \textbf{[[braiding]]}, such that the following two kinds of [[commuting diagram|diagrams commute]] for all [[objects]] involved (called the \textbf{hexagon identities} encoding the compatibility of the braiding with the [[associator]] for the tensor product):

\begin{displaymath}
\itexarray{
   (x \otimes y) \otimes z 
   &\stackrel{a_{x,y,z}}{\to}&
   x \otimes (y \otimes z)
   &\stackrel{B_{x,y \otimes z}}{\to}&
   (y \otimes z) \otimes x
   \\
   \downarrow^{B_{x,y}\otimes Id}
   &&&&
   \downarrow^{a_{y,z,x}}
   \\
   (y \otimes x) \otimes z
   &\stackrel{a_{y,x,z}}{\to}&
   y \otimes (x \otimes z)
   &\stackrel{Id \otimes B_{x,z}}{\to}&
   y \otimes (z \otimes x)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
   x \otimes (y \otimes z) 
   &\stackrel{a^{-1}_{x,y,z}}{\to}&
   (x \otimes y) \otimes z
   &\stackrel{B_{x \otimes y, z}}{\to}&
   z \otimes (x \otimes y)
   \\
   \downarrow^{Id \otimes B_{y,z}}
   &&&&
   \downarrow^{a^{-1}_{z,x,y}}
   \\
   x \otimes (z \otimes y)
   &\stackrel{a^{-1}_{x,z,y}}{\to}&
   (x \otimes z) \otimes y
   &\stackrel{B_{x,z} \otimes Id}{\to}&
   (z \otimes x) \otimes y
  }
  \,,
\end{displaymath}
where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the [[associator]] of $\mathcal{C}^\otimes$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
If the braiding in def. \ref{BraidedMonoidalCategory} ``squares'' to the identity in that $B_{y,x} \circ B_{x,y} = id_{x \otimes y}$, then the braided monoidal category is called a \emph{[[symmetric monoidal category]]}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Intuitively speaking, the first hexagon identity in def. \ref{BraidedMonoidalCategory} says we may braid $x$ past $y \otimes z$ `all at once' or in two steps. The second hexagon identity says that we may braid $x \otimes y$ past $z$ all at once or in two steps.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
From these axioms in def. \ref{BraidedMonoidalCategory}, it follows that the braiding is compatible with the left and right [[unitors]] $l_x : I \otimes x \to x$ and $r_x : x \otimes I \to x$. That is to say, for all objects $x$ the diagram

\begin{displaymath}
\itexarray{
    I \otimes x &&\stackrel{B_{I,x}}{\to}&& x \otimes I
    \\
    & {}_{l_x}\searrow && \swarrow_{r_x}  
    \\
    && x
  }
\end{displaymath}
commutes.

\end{remark}
\hypertarget{in_terms_of_higher_monoidal_structure}{}\subsubsection*{{In terms of higher monoidal structure}}\label{in_terms_of_higher_monoidal_structure}

In terms of the language of [[k-tuply monoidal n-categories]] a braided monoidal category is a \emph{doubly monoidal 1-category} .

Accordingly, by [[delooping]] twice, it may be identified with a [[tricategory]] with a single [[object]] and a single 1-[[morphism]].

However, unlike the definition of a monoidal category as a [[bicategory]] with one object, this identification is not trivial; a doubly-degenerate tricategory is literally a category with two monoidal structures that [[interchange law|interchange]] up to isomorphism. It requires the [[Eckmann-Hilton argument]] to deduce an equivalence with braided monoidal categories.

A commutative [[monoid]] is the same as a monoid [[internalization|in the category of]] monoids. Similarly, a braided monoidal category is equivalent to a monoidal-category object (that is, a [[pseudomonoid]]) in the monoidal 2-category of monoidal categories. This result goes back to the \href{http://maths.mq.edu.au/~street/JS1.pdf}{1986 paper by Joyal and Street}. (There is also a notion of [[braided pseudomonoid]] that specializes directly in [[Cat]] to braided monoidal categories.)

A braided monoidal category is equivalently a category that is equipped with the structure of an [[algebra over an operad|algebra over]] the [[little cubes operad|little 2-cubes operad]].

Details are in example 1.2.4 of

\begin{itemize}%
\item [[Jacob Lurie]], $\mathbb{E}[k]$-[[Ek-Algebras|Algebras]]

\end{itemize}
\hypertarget{the_2category_of_braided_monoidal_categories}{}\subsubsection*{{The 2-category of braided monoidal categories}}\label{the_2category_of_braided_monoidal_categories}

There is a [[strict 2-category]] BrMonCat with:

\begin{itemize}%
\item braided monoidal categories as objects,
\item [[braided monoidal functor|braided monoidal functors]] as morphisms,
\item [[braided monoidal natural transformation|braided monoidal natural transformations]] as 2-morphisms.

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

\begin{itemize}%
\item Any [[symmetric monoidal category]] is a braided monoidal category.


\item The monoidal category $R$-$\mathcal{GMod}$ of [[graded modules]] over a [[commutative ring]] $R$ (with the usual [[tensor product of modules|tensor product of graded modules]]) can be made into a braided monoidal category in multiple ways, each one given by an invertible element $u$ of the base ring. Infact, all braidings on $R$-$\mathcal{GMod}$ arise in this way (as in \href{JoyalStreet86}{Joyal, Street}). The braiding $B^u_{V,W} : V \otimes W \to W \otimes V$, is defined as

\begin{displaymath}
x \otimes y \mapsto u^{|x||y|} y \otimes x,
\end{displaymath}
where $|x|$ and $|y|$ denote the degrees. It's evident that the resulting braided monoidal category is symmetric if and only if $u^2 = 1$.


\item The category of [[crossed G-set|crossed G-sets]].



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

\hypertarget{tannaka_duality}{}\subsubsection*{{Tannaka duality}}\label{tannaka_duality}

[[!include structure on algebras and their module categories - table]]

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

\begin{itemize}%
\item [[k-tuply monoidal (n,r)-category]]


\item [[monoidal category]], [[monoidal (∞,1)-category]]


\item \textbf{braided monoidal category}, [[braided monoidal (∞,1)-category]]

\begin{itemize}%
\item [[braided 2-group]], [[braided ∞-group]]


\item [[braided monoidal 2-category]]



\end{itemize}

\item [[symmetric monoidal category]], [[symmetric monoidal (∞,1)-category]]


\item [[closed monoidal category]] , [[closed monoidal (∞,1)-category]]


\item [[2-fold monoidal category]], [[duoidal category]]



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

Exposition of basics of [[monoidal categories]] and [[categorical algebra]]:

\begin{itemize}%
\item \emph{[[geometry of physics -- categories and toposes]]}, Section 2: \emph{\href{geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra}{Basic notions of categorical algebra}}

\end{itemize}
The original papers on braided monoidal categories are by [[Andre Joyal|Joyal]] and [[Ross Street|Street]]. The published version does not completely supersede the \emph{Macquarie Math Reports} version, which has some nice extra results:

\begin{itemize}%
\item [[André Joyal]] and [[Ross Street]], \href{http://maths.mq.edu.au/~street/JS1.pdf}{Braided monoidal categories}, \emph{Macquarie Math Reports} \textbf{860081} (1986).


\item [[André Joyal]] and [[Ross Street]], \emph{Braided tensor categories} , Adv. Math. \textbf{102} (1993), 20--78.



\end{itemize}
Around the same time the same definition was also proposed independently by [[Lawrence Breen]] in a letter to [[Pierre Deligne]]:

\begin{itemize}%
\item [[Lawrence Breen]], \emph{Une lettre \`a{} P. Deligne au sujet des $2$-cat\'e{}gories tress\'e{}es} (1988) (\href{http://www.math.univ-paris13.fr/~breen/deligne.pdf}{pdf})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[nLab:Pavel Etingof]], Shlomo Gelaki, Dmitri Nikshych, [[nLab:Victor Ostrik]], chapter 2 of \emph{Tensor categories}, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (\href{http://www-math.mit.edu/~etingof/egnobookfinal.pdf
}{pdf})

\end{itemize}
For a review of definitions of braided monoidal categories, braided monoidal functors and braided monoidal natural transformations, see:

\begin{itemize}%
\item [[John Baez]], \href{http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf}{Some definitions everyone should know}.

\end{itemize}
For an elementary introduction to braided monoidal categories using [[string diagram]]s, see:

\begin{itemize}%
\item [[John Baez]] and [[Mike Stay]], \href{http://math.ucr.edu/home/baez/rosetta.pdf}{Physics, topology, logic and computation: a Rosetta Stone}, to appear in \emph{New Structures in Physics}, ed. Bob Coecke.

\end{itemize}
Eventually we should include all these diagrams here, along with the definition of braided monoidal functor and braided monoidal natural transformation! Can anyone help out?

[[!redirects braided monoidal category]] [[!redirects braided monoidal categories]] [[!redirects braided category]] [[!redirects braided categories]] [[!redirects braided tensor category]] [[!redirects braided tensor categories]]

[[!redirects hexagon identity]] [[!redirects hexagon identities]]



\end{document}