\subsection{Subfactors give `bi-oidal categories'.}
\label{sec:bi-oidal}

If we were going to attempt the impossible, and show you how to produce a
braided tensor category from a subfactor, we'd try this in two steps:
describe a tensor category first, and then give it a braiding. There are
difficulties at both steps. First, subfactors don't exactly give tensor
categories, in the usual sense, but something we'll call a `bi-oidal
category', and define in this section. (A common synonym for `tensor
category' is `monoidal category', %% Noah warns that some people implicitly tack on more axioms as some as they say tensor -- abelian, perhaps even symmetric.
and our made up name is intended as an analogue for this.) Second, once
we've adapted the notion of `braided' to bi-oidal categories, we still
might not know how to put such a braided structure on the category.
However, there's always something we can try,
%% mention motivated by the Jones polynomial, or something about temperley-lieb, yada??
which we describe in \S \ref{sec:braidings}, and explain why it actually
works for the $D_4$ subfactor in \S \ref{sec:back-to-d4}.

\subsubsection{What is a subfactor?}
\label{sec:what-is-a-subfactor}%
You should happily skip ahead to \S \ref{sec:bimodules} if any of this
sounds familiar!

\todo{Write this!}

\subsubsection{Bimodules for von Neumann algebras}
\label{sec:bimodules}%
Understanding bimodules for von Neumann algebras, and their tensor
products, can be tricky. Essentially there are two different versions of
the theory, depending whether one asks for bimodules to be Hilbert
spaces, or just vector spaces. Happily, the Hilbert space version is the
difficult one, and for our purposes we can get away with using the
simpler `algebraic' version, where we just ask for vector spaces. The
expense, however, is having to refer to some papers with some
high-powered analysis for a few results.

Thus, if $A$ and $B$ are von Neumann algebras, the category
$\abimod{A}{B}$ of `algebraic' $A-B$ bimodules has as objects all vector
spaces $X$ with commuting actions of $A$ and $B^\text{op}$, and as
morphisms all linear maps commuting with both actions. (In the Hilbert
space version, we'd unsurprisingly ask for for Hilbert spaces and bounded
linear maps.) There's an unnecessary looking prime in this notation
simply because in the next section we'll pass to a smaller, more useful
subcategory which we'll want to be able to refer to without the prime.

Given three von Neumann algebras $A, B$ and $C$, and two bimodules $X \in
\abimod{A}{B}$, and $Y \in \abimod{B}{C}$, we can form the tensor product
$X \tensorover{B} Y \in \abimod{A}{C}$, defined easily\footnote{It's
worth saying `easily' here, because the corresponding tensor product for
the Hilbert space version of bimodules is complicated; it's Connes'
fusion. See \cite{??} for details.} as
$$X \tensorover{B} Y = \quotient{X \tensorover{\Complex} Y}{\setc{xb \tensor y - x \tensor by}{x\in X, y\in Y, b\in
B}}.$$

\subsubsection{The bi-oidal category of bimodules for a subfactor}%
When we're looking at a $II_1$ subfactor $N \subset M$, we might think
about the four categories $\abimod{N}{N}$, $\abimod{N}{M}$,
$\abimod{M}{N}$ and $\abimod{M}{M}$, and take their disjoint union as
categories, which we'll call $\asbimod{N}{M}$ (soon we'll define a
smaller subcategory --- the prime is there in the name just because we'll
end up being more interested in the subcategory, and so reserve the name
for it). On $\asbimod{N}{M}$ we have a `partially defined' tensor
product. Each object, being in one of the four categories, has two
`labels'; the algebra it's a left module over is its `left label', and
the algebra it's a right module over is its `right label'. We can take
the tensor product of $X$ and $Y$ exactly if the right label of $X$
agrees with the left label of $Y$, in which case the tensor product just
means the algebraic tensor product over that algebra. When the labels
don't agree, there simply isn't a tensor product.

This structure is the first example of what we're going to call a
bi-oidal category, so we'll give the definition now.

\begin{defn}
\newcommand{\AC}{\leftidx{_A}{\C}{}}
\newcommand{\BC}{\leftidx{_B}{\C}{}}
\newcommand{\XC}{\leftidx{_X}{\C}{}}
\newcommand{\YC}{\leftidx{_Y}{\C}{}}
\newcommand{\XCY}{\leftidx{_X}{\C}{_Y}}
\newcommand{\YCZ}{\leftidx{_Y}{\C}{_Z}}
\newcommand{\XCZ}{\leftidx{_X}{\C}{_Z}}
\newcommand{\CA}{\leftidx{}{\C}{_A}}
\newcommand{\CB}{\leftidx{}{\C}{_B}}
\newcommand{\CX}{\leftidx{}{\C}{_X}}
\newcommand{\CY}{\leftidx{}{\C}{_Y}}
A bi-oidal category $\C$ with labels $A$ and $B$ is a category in which
\begin{itemize}
\item each object $\mathcal{O} \in \C$ has a `left label' and a `right label',
each either $A$ or $B$, often indicated by notation like
$\bimodule{A}{\mathcal{O}}{B}$,
\item and with $\XCY$ denoting the full subcategory of objects with left label
$X$ and right label $Y$, there are a `tensor product over a label'
functors
\begin{equation*}
\tensorover{Y} : \XCY \times \YCZ \To \XCZ
\end{equation*}
which are associative whenever this makes sense, that is
\begin{equation*}
\tensorover{X} \compose (\Id_{\CX} \times \tensorover{Y}) =
\tensorover{Y} \compose (\tensorover{Y} \times \Id_{\YC})
\end{equation*}
on the sub-category $\CX \times \XCY \times \YC$ of $\C \times \C \times
\C$,
\item
and there are objects $\bimodule{A}{A}{A}$ and $\bimodule{B}{B}{B}$ which
are identities for the corresponding tensor products, that is, for $X =
A$ or $B$, $U \in XC$ and $V \in CX$,
\begin{align*}
X \tensorover{X} U & = U, \\
V \tensorover{X} X & = V.
\end{align*}
\end{itemize}
\end{defn}

\todo{Does that make any sense? Is it missing something?}

\subsubsection{The bimodules generated by $M$.}%
The category $\asbimod{N}{M}$ has a profusion of objects, and we're about
to cut it down. As an example, the algebra $M$, considered as a vector
space with commuting left and right actions of $N$ and $M$ (by
multiplication!), is a bimodule. Being a $II_1$ subfactor, $M$ has a
unique normalized trace, and hence an inner product ($\pairing{x}{y} =
\tr{y^* x}$), and we can complete $M$ with respect to this inner product
to obtain a Hilbert space called $L^2(M)$. This is still a bimodule, and
hence is also an object in our category. In some sense, however, it's too
closely related to the original bimodule $M$ for our purposes, without
actually being isomorphic in the category. Our next definition avoids
this sort of problem, amongst several others! %% some cites?

Now pick out two objects in this category, namely the algebra $M$
considered as an $N-M$ bimodule, written $\bimodule{N}{M}{M}$, and the
algebra $M$ considered as an $M-N$ bimodule, written
$\bimodule{M}{M}{N}$.

\todo{continue...}

\subsubsection{Bi-oidal categories with duals}
\todo{Mention sphericalness. Define $\altTensor$.}

\subsubsection{Temperley-Lieb}
\label{sec:temperley-lieb}%
The simplest example of a bi-oidal category is the (perhaps familiar!)
shaded Temperley-Lieb category.

\todo{Finish this..., having read emily's section on temperley-lieb as a
planar algebra above.}

\subsubsection{The principal graphs of a singly generated bi-oidal category}
\todo{}
