\subsection{Translating back and forth: spherical categories and planar algebras.}
Having read this far, you'll have noticed the rapidly widening gap
between the perspective of \S \ref{sec:recipe}, which described the
$\dfour$ planar algebra, and \S \ref{sec:bi-oidal}, which explained how
to associate a bi-oidal category to a subfactor. This section reunites
these perspectives, by explaining the translation back and forth between
a singly generated bi-oidal category, with a spherical structure, and a
shaded planar algebra.

The fundamental idea is that spherical categories are essentially the
same things as a `general planar algebra' (i.e., not necessarily with the
shadings we asked for in the definition of a shaded planar algebra, and
certainly not with the dimension and positivity restrictions in the
definition of \S \ref{sec:planar-algebras} of a subfactor planar
algebra). For each specialization you might make in the definition of a
spherical category, there is a corresponding potential variation in the
definition of a planar algebra. For the translation in its most general
setting, see \cite[\S~5]{westbury}. For our purposes, however, we're more
interested in picking out the class of spherical categories that
corresponds exactly to the notion of shaded planar algebra used in the
theory of subfactors. As a result we restrict our attention first to
bi-oidal categories, to match the shading requirements on the planar
algebra side, and secondly, more significantly, to singly generated
bi-oidal categories. The description of this more specific translation
differs somewhat from that in \cite{westbury}, as we'll construct a
planar algebra from a category making use of only a certain sub-category,
and conversely reconstruct the category in two steps; recovering the
sub-category, and then the entire original category by taking its Karoubi
envelope, defined below. \scott{I don't much like this paragraph; it
sounded better when Noah said it -- could you try a partial rewrite?}

\todo{put in something about this being familiar to analysts; thinking
about projections all the time!}

\subsubsection{From spherical bi-oidal categories to shaded planar algebras}
Given a spherical bi-oidal category $\C$, equipped with a generating
object $\bimodule{A}{V}{B}$ with left label $A$ and right label $B$, we
construct the shaded planar algebra $\pa(\C)$ as follows:
\begin{defn}
The shaded planar algebra $\pa(\C)$ has disc spaces $\pa(\C)_k =
\End{C}{\altTensor^k V}$. Given a shaded planar tangle $T$ with $r$
internal discs with $2 k_1, 2 k_2, \ldots, 2 k_r$ marked points, and $2
k_0$ marked points on the external disc, we construct the linear map
$$\pa(\C)(T)  : \pa(\C)_{k_1} \tensor \pa(\C)_{k_2} \tensor \cdots \tensor
\pa(\C)_{k_r} \To \pa(\C)_{k_0}$$ by first choosing an up-to-isotopy
representative of $T$ in the manner illustrated in Figure
\ref{fig:rectangular-tangle}, which we call a `rectangular planar
tangle'. Given elements $f_1, \ldots, f_r$ of the appropriate
endomorphism spaces, we form a composition in the category $\C$, turning
cups and caps into copairing and pairing maps as usual, translating
vertical stacking into composition, and horizontal juxtaposition into
tensor product.
\end{defn}

\begin{figure}[ht]
$$\mathfig{0.3}{SampleTangle} \rightsquigarrow \mathfig{0.5}{translation/rectangular}$$%
\caption{Turning a planar tangle into a `rectangular planar tangle'. We
ask that there is a horizontal strip midway up the rectangle, containing
all of the internal boxes, possibly with vertical strands interspersed,
and that above and below this strip we see only caps and cups.}%
\label{fig:rectangular-tangle}%
\end{figure}

We demonstrate this composition explicitly for the tangle $T$ from Figure
\ref{fig:rectangular-tangle} in Figure \ref{fig:composition}. The
resulting composition is then an endomorphism in the appropriate disc
space. That the construction is independent of the choice of rectangular
planar tangle follows from the axioms for a spherical category. The
resulting map is a linear map, simply because composition and tensor
product in the category are all appropriately multilinear.

\begin{figure}[ht]
$$\mathfig{0.8}{translation/composition}$$%
\caption{Reading from bottom to top, we compose a copairing or pairing
map, for each cup or cap respectively, with the endomorphism}
\label{fig:composition}%
\end{figure}

Notice that this construction didn't depend explicitly on the whole
category $\C$, but just on a certain subcategory,
$\restrict{\C}{\altTensor V}$, the full subcategory in which the only
objects are the alternating tensor powers of the generating object $V$.

\subsubsection{From shaded planar algebras to spherical bi-oidal categories, step 1.}
For the reverse construction, we show how to construct a category
$\mathcal{C}'(\pa)$ from a shaded planar algebra, in which the only
objects are alternating tensor powers of a `formal' tensor generating
object $V$, and then, after defining a Karoubi envelope in \S
\ref{sec:karoubi}, explain in \S \ref{sec:step2} why the category
$\C(\pa) = \Kar{\mathcal{C}'(\pa)}$ is really what we want. That section
also proves that the two constructions in either direction really are
inverses.

The bi-oidal category $\mathcal{C}'(\pa)$ has 

\todo{}

\input{text/karoubi}

\subsubsection{From shaded planar algebras to spherical bi-oidal categories, step 2.}
\todo{}

\todo{Explain that the constructions are inverses.} \todo{Really, we'd
explain what morphisms of shaded planar algebras are, and explain that
the constructions give an equivalence of categories!}
