The Extended Haagerup fusion categories
Pinhas Grossman, Scott Morrison, David Penneys, Emily Peters, and Noah Snyder

In this paper we construct two new fusion categories and  many new subfactors related to the exceptional Extended Haagerup subfactor.

The Extended Haagerup subfactor has two even parts $\cE\cH_1$ and $\cE\cH_2$.
These fusion categories are mysterious and are the only known fusion categories which appear to be unrelated to finite groups, quantum groups, or Izumi quadratic categories.
One key technique which has previously revealed hidden structure in fusion categories is to study all other fusion categories in the Morita equivalence class, and hope that one of the others is easier to understand.
In this paper we show that there are exactly four categories ($\cE\cH_1, \cE\cH_2, \cE\cH_3, \cE\cH_4$) in the Morita equivalence class of Extended Haagerup, and that there is a unique Morita equivalence between each pair.
The existence of $\cE\cH_3$ and $\cE\cH_4$ gives a number of interesting new subfactors.
Neither $\cE\cH_3$ nor $\cE\cH_4$ appears to be easier to understand than the Extended Haaerup subfactor, providing further evidence that Extended Haagerup does not come from known constructions.
We also find several interesting intermediate subfactor lattices related to Extended Haagerup.

The method we use to construct $\cE\cH_3$ and $\cE\cH_4$ is interesting in its own right and gives a general computational recipe for constructing fusion categories in the Morita equivalence class of a subfactor.  We show that pivotal module $\rm C^*$ categories over a given subfactor correspond exactly to realizations of that subfactor planar algebra as a planar subalgebra of a graph planar algebra.  This allows us to construct $\cE\cH_3$ and $\cE\cH_4$ by realizing the Extended Haagerup subfactor planar algebra inside the graph planar algebras of two new graphs.  This technique also answers a long-standing question of Jones: which graph planar algebras contain a given subfactor planar algebra?

§ INTRODUCTION

The Extended Haagerup subfactor gives a Morita equivalence between two unitary fusion categories called $\cE\cH_1$ and $\cE\cH_2$.  The goal of this paper is to find all fusion categories Morita equivalent to these fusion categories.  We find two new fusion categories, seven new subfactors (along with their duals and reduced subfactors), and several interesting new intermediate subfactor lattices.

There are exactly two further fusion categories in the Morita equivalence class of $\mathcal{EH}_1$ and $\mathcal{EH}_2$, which we call $\mathcal{EH}_3$ and $\mathcal{EH}_4$.  Between any two of these four fusion categories, there is exactly one Morita equivalence.

For every choice of simple object in each of these Morita equivalences, we get a subfactor.  In addition to the original $7$-supertransitive Extended Haagerup subfactor, we get two new $3$-supertransitive subfactors: one is self-dual and comes from the Morita auto-equivalence of $\cE\cH_3$ and the other comes from the Morita equivalence between $\cE\cH_3$ and $\cE\cH_4$.  Their principal graphs are:

\begin{tikzpicture}[baseline=0mm, scale=.7]
\draw (0,0) -- (3,0);
\draw (3,0) -- (4,.5);
\draw (3,0) -- (4,-.5);
\draw (4,.5) -- (5,1);
\draw (4,.5) -- (5,.5);
\draw (4,.5) -- (5,0);
\draw (4,-.5) -- (5,0);
\draw (4,-.5) -- (5,-1);
\draw (5,.5) -- (6,.5);
\draw (5,0) -- (6,0);
\draw (5,0) -- (6,-.5);
\draw (5,-1) -- (6,-1);
\draw (6,0) -- (7,0);
\draw (6,-1) -- (7,-1);

\filldraw              (0,0) node [above] {$*$} circle (1mm);
\filldraw [fill=white] (1,0) circle (1mm);
\filldraw              (2,0) circle (1mm);
\filldraw [fill=white] (3,0) circle (1mm);
\filldraw              (4,.5) circle (1mm);
\filldraw              (4,-.5) circle (1mm);
\filldraw [fill=white] (5,1) circle (1mm);
\filldraw [fill=white] (5,.5) circle (1mm);
\filldraw [fill=white] (5,0) circle (1mm);
\filldraw [fill=white] (5,-1) circle (1mm);
\filldraw              (6,.5) circle (1mm);
\filldraw              (6,0) circle (1mm);
\filldraw              (6,-.5) circle (1mm);
\filldraw              (6,-1) circle (1mm);
\filldraw [fill=white] (7,0) circle (1mm);
\filldraw [fill=white] (7,-1) circle (1mm);
\end{tikzpicture}
\hspace{.25in}
\text{and}
\hspace{.25in}
\left(\begin{tikzpicture}[baseline=-1mm, scale=.35]
\draw (0,0) -- (2,0);
\draw (2,0) -- (4,0);
\draw (4,0) -- (6,0);
\draw (6,0) -- (8,1.5);
\draw (6,0) -- (8,.5);
\draw (6,0) -- (8,-.5);
\draw (6,0) -- (8,-1.5);

\draw (8,-.5) -- (10,0);
\draw (8,-.5) -- (10,-1);
\draw (8,-.5) -- (10,-2);
\draw (8,-1.5) -- (10,-1);
\draw (8,-1.5) -- (10,-2);

\draw (10,0) -- (12,.5);
\draw (10,0) -- (12,-.5);

\filldraw (0,0) node [above] {$*$} circle (2mm);
\filldraw [fill=white] (2,0) circle (2mm) ;
\filldraw  (4,0) circle (2mm);
\filldraw [fill=white] (6,0) circle (2mm) ;
\filldraw  (8,1.5) circle (2mm);
\filldraw  (8,.5) circle (2mm);
\filldraw  (8,-.5) circle (2mm);
\filldraw  (8,-1.5) circle (2mm);
\filldraw [fill=white] (10,0) circle (2mm) ;
\filldraw [fill=white] (10,-1) circle (2mm) ;
\filldraw [fill=white] (10,-2) circle (2mm) ;
\filldraw  (12,.5) circle (2mm);
\filldraw  (12,-.5) circle (2mm);
\end{tikzpicture}
\;,\hspace{.1in}
\begin{tikzpicture}[baseline=-1mm, scale=.35]
\draw (0,0) -- (2,0);
\draw (2,0) -- (4,0);
\draw (4,0) -- (6,0);

\draw (6,0) -- (8,1.5);
\draw (6,0) -- (8,.5);
\draw (6,0) -- (8,-.5);
\draw (6,0) -- (8,-1.5);

\draw (8,1.5) -- (10,1);
\draw (8,.5) -- (10,0);
\draw (8,-.5) -- (10,0);
\draw (8,-.5) -- (10,-1);
\draw (8,-1.5) -- (10,-1);

\draw (10,1) -- (12,.5);
\draw (10,1) -- (12,-.5);
\draw (10,0) -- (12,.5);
\draw (10,-1) -- (12,-.5);

\filldraw (0,0) node [above] {$*$} circle (2mm);
\filldraw [fill=white] (2,0) circle (2mm) ;
\filldraw  (4,0) circle (2mm) ;
\filldraw [fill=white] (6,0) circle (2mm) ;
\filldraw  (8,1.5) circle (2mm);
\filldraw  (8,.5) circle (2mm);
\filldraw  (8,-.5) circle (2mm);
\filldraw  (8,-1.5) circle (2mm);
\filldraw [fill=white] (10,1) circle (2mm) ;
\filldraw [fill=white] (10,0) circle (2mm) ;
\filldraw [fill=white] (10,-1) circle (2mm) ;
\filldraw  (12,.5) circle (2mm);
\filldraw  (12,-.5) circle (2mm);
\end{tikzpicture}\right).

The structures of $\mathcal{EH}_3$ and $\mathcal{EH}_4$ are explained in more detail in Section <ref>.  Neither appears to be easily understood using any general techniques, but we encourage the reader to look for a new way to construct them which could give a better understanding of the Extended Haagerup subfactor.

At first it was thought that all fusion categories might come from groups or quantum groups at roots of unity.
The first “exotic” examples which appeared to be unrelated to finite groups or quantum groups came from Haagerup's small index classification program <cit.>, namely the Haagerup and Asaeda–Haagerup subfactors constructed in <cit.> and the Extended Haagerup subfactor constructed in <cit.> (after numerical evidence for existence was given by <cit.>).
However, with time, the first two of these three examples have been seen to be related to Izumi quadratic categories.
Izumi generalized the Haagerup subfactor to a possibly infinite family of quadratic $3^G$ subfactors <cit.>.
Recently in <cit.>, Grossman-Izumi-Snyder found all fusion categories Morita equivalent to the Asaeda-Haagerup fusion categories, and discovered that one is an Izumi quadratic category.  The only remaining fusion categories which appear unrelated to groups, quantum groups at roots of unity, or Izumi quadratic categories are the Extended Haagerup fusion categories.
Unlike in the Asaeda–Haagerup case where one of the new categories was a quadratic category, neither $\cE\cH_3$ nor $\cE\cH_4$ has nontrivial invertible objects and so neither can be a quadratic category.

The proof of the main theorem has two parts.  On the one hand we need to limit the possible fusion categories and Morita equivalences, and on the other hand we need to construct the remaining possibilities.  The former is an application of the techniques introduced in <cit.>, using combinatorial restrictions for compatible fusion rules for the hypothetical fusion categories and bimodule categories.

We construct $\cE\cH_3$ and $\cE\cH_4$ using a general graph planar algebra <cit.> technique for finding module categories over any fusion category where we have a good skein theoretic description.
This technique can be viewed as a generalization of the Ocneanu cell technique for $SU(n)$ (<cit.>, <cit.>, <cit.>) to arbitrary tensor categories with good skein theoretic descriptions.
From our combinatorial calculation we know that there is at most one module category over each of $\cE\cH_1$ and $\cE\cH_2$ whose dual can be $\cE\cH_3$ (or $\cE\cH_4$).
So if we can construct a module category with the correct fusion rules, we will have a construction of $\cE\cH_3$ (or $\cE\cH_4$) as the commutant category.

We can package  $\cE\cH_1$ and $\cE\cH_2$ together with their Morita equivalence into a single multifusion category which is the Extended Haagerup planar algebra.  The fusion rules for tensoring simple objects in a module with the single strand in Extended Haagerup give a bipartite graph $\Gamma$.
We prove a graph planar algebra embedding theorem for module categories,
which shows such module categories for this planar algebra correspond (up to gauging and graph automorphism) to embeddings of the Extended Haagerup subfactor planar algebra inside the graph planar algebra of $\Gamma$.     This generalizes the original graph planar algebra embedding theorem <cit.> which only applied to the principal graphs.

Suppose $\cP_\bullet$ is a finite depth subfactor planar algebra. Let $\scrC$ denote the unitary multifusion category of projections in $\cP_\bullet$, with distinguished object $X=\id_{1,+}\in\cP_{1,+}$ and the standard unitary pivotal structure with respect to $X$. There is an equivalence between:

* Planar $\dag$-algebra embeddings $\cP_\bullet \to \cG\cP\cA(\Gamma)_\bullet$, where $\Gamma$ is a finite connected bipartite graph, and

* indecomposable finitely semisimple pivotal left $\scrC$-module $\Cstar$ categories $\cM$ whose fusion graph with respect to $X$ is $\Gamma$.

This theorem answers a long-standing question of Vaughan Jones: given a finite depth subfactor planar algebra $\cP_\bullet$, determine all bipartite graphs $\Gamma$ for which $\cP_\bullet$ embeds in $\cG\cP\cA(\Gamma)_\bullet$.

By the skein theoretic description of Extended Haagerup in <cit.>, in order to construct a map from Extended Haagerup into a graph planar algebra, we need only specify a number for each loop of length $16$ and check a large number of linear and quadratic equations in these numbers.

The Extended Haagerup planar algebra can be embedded into the graph planar algebras of each of the following bipartite graphs:

\Gamma_3 =
\begin{tikzpicture}[baseline, scale=.7]
\draw (-2,-1) -- (0,0);
\draw (-2,1)--(0,0);
\draw (0,0)--(3,0);
\draw (3,0)--(4,.5);
\draw (3,0)--(5,-1);
\filldraw [fill]  (-2,-1) circle (1mm);
\filldraw [fill]  (-2,1) circle (1mm);
\filldraw [fill=white] (-1,-.5) circle (1mm);
\filldraw [fill=white] (-1,.5) circle (1mm);
\filldraw [fill]  (0,0) circle (1mm);
\filldraw [fill=white] (1,0) circle (1mm);
\filldraw [fill]  (2,0) circle (1mm);
\filldraw [fill=white] (3,0) circle (1mm);
\filldraw [fill]  (4,-.5) circle (1mm);
\filldraw [fill]  (4,.5) circle (1mm);
\filldraw [fill=white] (5,-1) circle (1mm);
\end{tikzpicture}
\hspace{.1in} \text{and} \hspace{.1in}
\Gamma_4 =
\begin{tikzpicture}[baseline, scale=.7]
\draw (-1,-.5) -- (0,0);
\draw (-1,.5)--(0,0);
\draw (0,0)--(5,0);
\draw (5,0)--(6,.5);
\draw (5,0)--(6,-.5);
\draw (3,0)--(3,-1);
\filldraw [fill=white] (-1,-.5) circle (1mm);
\filldraw [fill=white] (-1,.5) circle (1mm);
\filldraw [fill]  (0,0) circle (1mm);
\filldraw [fill=white] (1,0) circle (1mm);
\filldraw [fill]  (2,0) circle (1mm);
\filldraw [fill=white] (3,0) circle (1mm);
\filldraw [fill]  (4,0) circle (1mm);
\filldraw [fill=white] (5,0) circle (1mm);
\filldraw [fill]  (6,-.5) circle (1mm);
\filldraw [fill]  (6,.5) circle (1mm);
\filldraw [fill] (3,-1) circle (1mm);
\end{tikzpicture}

Thus, the existence of $\cE\cH_3$ and $\cE\cH_4$ is now a corollary to Theorems <ref> and <ref>.

From our complete classification of module categories, we see that there are exactly four graphs whose graph planar algebras take maps from the Extended Haagerup planar algebra: the principal and dual principal graphs for the original subfactor, and the two graphs in the previous theorem.
(One may think of these embeddings as giving four independent constructions of the Extended Haagerup subfactor planar algebra.)

Theorem <ref> also connects the results of <cit.> and <cit.> to complete the classification of graph planar algebra embeddings for the Haagerup planar algebra.
In the last section of <cit.>, three embeddings of the Haagerup planar algebra into graph planar algebras were found, corresponding to the two principal graphs and the `broom' graph.  However, it was not proven there could not be others.
The main result of <cit.> shows there are exactly three module categories over the Haagerup subfactor planar algebra.  Thus we have:

The Haagerup subfactor planar algebra embeds into $\cG\cP\cA(\Gamma)_{\bullet}$ if and only if $\Gamma$ is one of the following:

\begin{align*}
\begin{tikzpicture}[baseline, scale=.7]
\draw (0,0)--(3,0);
\draw (3,0)--(4,.5) -- (6,.5);
\draw (3,0)--(4,-.5) -- (6,-.5);
\filldraw [fill]  (0,0) circle (1mm);
\filldraw [fill=white] (1,0) circle (1mm);
\filldraw [fill]  (2,0) circle (1mm);
\filldraw [fill=white] (3,0) circle (1mm);
\filldraw [fill]  (4,-.5) circle (1mm);
\filldraw [fill]  (4,.5) circle (1mm);
\filldraw [fill=white] (5,-.5) circle (1mm);
\filldraw [fill=white] (5,.5) circle (1mm);
\filldraw [fill] (6,-.5) circle (1mm);
\filldraw [fill] (6,.5) circle (1mm);
\end{tikzpicture}
\text{(Haagerup principal graph)}
\\
\begin{tikzpicture}[baseline, scale=.7]
\draw (0,0)--(3,0);
\draw (3,0)--(4,.5);
\draw (3,0)--(4,-.5) -- (5,0);
\draw (4,-.5) -- (5,-1);
\filldraw [fill=white]  (0,0) circle (1mm);
\filldraw [fill] (1,0) circle (1mm);
\filldraw [fill=white]  (2,0) circle (1mm);
\filldraw [fill] (3,0) circle (1mm);
\filldraw [fill=white]  (4,-.5) circle (1mm);
\filldraw [fill=white]  (4,.5) circle (1mm);
\filldraw [fill] (5,0) circle (1mm);
\filldraw [fill] (5,-1) circle (1mm);
\end{tikzpicture}
\text{(dual principal graph)}
\\
\begin{tikzpicture}[baseline, scale=.7]
\draw (1,0)--(4,0);
\draw (4,.5)--(3,0)--(4,-.5);
\filldraw [fill=white] (1,0) circle (1mm);
\filldraw [fill]  (2,0) circle (1mm);
\filldraw [fill=white] (3,0) circle (1mm);
\filldraw [fill]  (4,-.5) circle (1mm);
\filldraw [fill]  (4,.5) circle (1mm);
\filldraw [fill] (4,0) circle (1mm);
\end{tikzpicture}
\text{(`broom' graph)}
\end{align*}

Here, the two unshaded vertices of the `broom' graph correspond to the third $\cH_2$-module from <cit.>.

Section <ref> summarizes the combinatorial structure of the four Extended Haagerup fusion categories and the Morita equivalences between them.  In particular, we describe the fusion rules for each fusion category, and give the principal and dual principal graphs for all the subfactors coming from small objects in the bimodule categories.  This section concludes with a table of all lattices of intermediate subfactors coming from these fusion categories, which can be read off from the fusion rules of the bimodule categories following <cit.>.  There are several particularly interesting examples: a (3,3)-supertransitive non-commuting but cocommuting quadrilateral with indices $(7.0283\ldots, 8.0283\ldots)$, a (2,2)-supertransitive non-commuting but cocommuting quadrilateral with indices $(13.3305\ldots, 14.3305\ldots)$, and a hexagonal intermediate subfactor lattice whose lower and upper inclusions are the $7$-supertransitive index $4.3772\ldots$ extended Haagerup subfactor, and whose middle inclusions are $2$-supertransitive with index $13.3305 \ldots$  The first of these quadrilaterals is the smallest known example of index above $4$ in Class II of the Grossman-Izumi classification <cit.> of highly supertransitive non-commuting quadrilaterals.  It is striking that the smallest examples in Class III and Class IV are respectively the Asaeda–Haagerup and Haagerup subfactors.

The main goal of Section <ref> is to prove Theorem <ref>.
We begin by recalling some key background about module categories, bimodule categories, Brauer–Picard groupoids, and the Maximal Atlas.
This background is used throughout the paper.  We also recall  the relationship between module categories and functors $\scrC \rightarrow \End(M)$.  We then relate $\End(M)$ to a version of graph planar algebra.
We hope this exposition will make graph planar algebra techniques accessible to readers with a background in tensor categories.
In particular, we prove a purely algebraic analogue of Theorem <ref> directly from the action map $\scrC \rightarrow \End(M)$.
In order to adapt this simple algebraic argument to prove Theorem <ref>, we recall some technical background on the definition and classification of unitary pivotal structures from <cit.>, and the correct unitary pivotal analogues of module categories (analogous to <cit.> in the unitary non-pivotal case and <cit.> in the non-unitary pivotal case).
Note that the unitary pivotal version of this result is essential to our main results, because the characterization of maps out of the Extended Haagerup planar algebra in <cit.> relies on positive definiteness in order to check more complicated skein relations based on only a few simple skein relations.

In Section <ref> we show that there are at most four fusion categories in the Extended Haagerup Morita equivalence class and exactly one Morita equivalence between each pair of categories in this class.  Furthermore we determine all the fusion rules between objects in each of these potential fusion categories and bimodule categories.  This closely follows the techniques introduced in <cit.> by computing compatible fusion rules for the hypothetical fusion categories and bimodule categories.

In Section <ref> we recall and slightly modify the characterization of maps out of the Extended Haagerup planar algebra proved in <cit.>.  We then prove Theorem <ref>, again following the outline of <cit.>.

In the first appendix we give an alternate construction of $\cE\cH_3$ by directly constructing a Q-system on $1 \oplus f^{(6)}$ in the Extended Haagerup planar algebra using its explicit embedding into the graph planar algebra of its principal graph.
This approach does not work for $\cE\cH_4$ because the smallest Q-system yielding $\cE\cH_4$ lives in too large a box space for computer calculations to be feasible.
In the second appendix we outline one promising skein theoretic approach which could give a more natural description of $\cE\cH_3$ and $\cE\cH_4$.  These appendices are not intended to appear in the published version.

Throughout the paper we will use the notation $\scrC(X \rightarrow Y)$ to denote the morphisms between objects $X$ and $Y$ in the category $\scrC$.

§.§ Acknowledgements

This work was completed at the 2016 and 2017 AIM SQuaRE “Classifying fusion categories.”
The authors would like to thank AIM for their hospitality.
NS and DP would like to thank André Henriques and Corey Jones for helpful conversations.  In particular, André had an immense impact on the ideas and results in Section <ref>.
PG was supported by ARC grants DP140100732 and DP170103265.
SM was supported by Discovery
Projects ‘Subfactors and symmetries’ DP140100732 and ‘Low dimensional categories’ DP160103479,
and a Future Fellowship ‘Quantum symmetries’ FT170100019 from the Australian Research Council.
DP was supported by NSF DMS grants 1500387/1655912 and 1654159.
EP was supported by NSF DMS grant 1501116.
NS was supported by NSF DMS grant 1454767.

§ FACTS ABOUT THE EXTENDED HAAGERUP FUSION CATEGORIES

In this section, we describe the Extended Haagerup fusion categories from Theorem <ref>.
The logic here is somewhat convoluted; the statements of this section logically depend on the later sections (and we're careful not to use the statements here in those sections!). We have decided to put this summary first in order to make the structure of these new fusion categories as accessible as possible.

Recall that by the main results of this paper, there are exactly four unitary fusion categories $\cE\cH_1$, $\cE\cH_2$, $\cE\cH_3$, and $\cE\cH_4$ in the Extended Haagerup Morita equivalence class, and between any two of these four, there is exactly one Morita equivalence.
Furthermore, because the center of the Extended Haagerup fusion category has no nontrivial invertible objects, there is a unique composition for each tensor product of bimodules
{}_\mathcal{A} \mathcal{K} {}_\mathcal{B} \boxtimes_{\mathcal{B}} {}_\mathcal{B} \mathcal{L} {}_\mathcal{C}
\cong
{}_\mathcal{A} \mathcal{M} {}_\mathcal{C}.

For $1\leq i,j\leq 4$, we denote by $\cE\cH_{ij}$ the unique invertible $\cE\cH_i - \cE\cH_j$
bimodule category, and notice that $\cE\cH_{ii} = \cE\cH_i$.
One may view $\cE\cH = (\cE\cH_{ij})_{i,j=1}^4$ as a single $4\times 4$ unitary multifusion category.

We interpret the fusion ring $EH:=K_0(\cE\cH)$ for $\cE\cH$ as a single ring whose basis consists of the disjoint union of a set of representatives of simple objects $\Irr(\cE\cH_{ij})$ for each $\cE\cH_{ij}$.
We denote by $EH_{ij}$ the span of the set of representatives of simple objects in $\Irr(\cE\cH_{ij})$, and notice that $EH_{ii} = K_0(\cE\cH_i)$.
Of course the products of objects which are not composable are declared to be zero; that is, $EH$ is faithfully graded by the standard system of matrix units for $M_4(\bbC)$.

Within each $\cE\cH_{ij}$ we order simple objects by increasing dimension, so $O_{ij}^k$ denotes the $k$th smallest simple object in $\cE\cH_{ij}$ (or, abusing notation, the corresponding basis element in the Grothendieck ring).
When there are duplicate dimensions the ties are broken arbitrarily.

We describe the fusion ring $EH$ in the Mathematica notebook , which is a wrapper for the data file , both of which are bundled with the arXiv sources of this article.
Therein, we supply a 6-dimensional tensor $T$ whose $(i,j,k,x,y,z)$-entry is the coefficient of
$z$-th basis element of $EH_{ik}$ in the product of the
$x$-th basis element of $EH_{ij}$
and the
$y$-th basis element of $EH_{jk}$.
On the level of categories, these coefficients are the dimensions of $\text{Hom} $ spaces between simple objects and tensor products of pairs of simple objects.
That is,

O_{ij}^x \otimes O_{jk}^y \cong \bigoplus_z T(i,j,k,x,y,z) O_{ik}^z.

In this section, we gather information on all the Extended Haagerup fusion categories, including fusion rules, the simplest Q-systems, and intermediate subfactor lattices.

Our convention for principal graphs of subfactors and fusion graphs of fusion categories is that we always tensor on the right.
In particular, the fusion graph for $X$ has $\dim(\cE\cH(A\otimes X \to B))$ oriented edges between simples $A$ and $B$.
Later, in <ref>, we will discuss fusion graphs for left module categories, which use the opposite convention.

§.§ Structure of EH1

The fusion rules for $\cE\cH_1$ (which is the dual even half of the Extended Haagerup subfactor) are given by

$\otimes$     $\jw{2}$     $\jw{4}$     $\jw{6}$     $P'$     $Q'$
$\jw{2}  $     $1 \oplus  \jw{2} \oplus  \jw{4}  $     $ \jw{2} \oplus  \jw{4} \oplus  \jw{6}  $     $ \jw{4} \oplus  \jw{6} \oplus  P' \oplus  Q'  $     $ \jw{6} \oplus  2 P' \oplus  Q'  $     $ \jw{6} \oplus  P' $
$ \jw{4}  $     $ \jw{2} \oplus  \jw{4} \oplus  \jw{6}  $     $\begin{array}{c} 1 \oplus  \jw{2} \oplus  \jw{4} \\  \oplus  \jw{6} \oplus  P' \oplus  Q' \end{array}$     $\begin{array}{c} \jw{2} \oplus  \jw{4} \oplus  2 \jw{6} \\ \oplus  3 P' \oplus  Q'  \end{array}$     $ \begin{array}{c} \jw{4} \oplus  3 \jw{6} \\ \oplus  3 P' \oplus  2 Q'  \end{array}$     $\begin{array}{c} \jw{4} \oplus  \jw{6} \\ \oplus  2 P' \oplus  Q' \end{array}$
$\jw{6}$     $\jw{4} \oplus  \jw{6} \oplus  P' \oplus  Q'  $     $\begin{array}{c} \jw{2} \oplus  \jw{4} 2 \jw{6} \\ \oplus  3 P' \oplus  Q'  \end{array}$     $\begin{array}{c} 1 \oplus \jw{2} \oplus  2 \jw{4}  \\ \oplus  4 \jw{6} \oplus  5 P' \oplus  3 Q'  \end{array}$     $\begin{array}{c} \jw{2} \oplus  3 \jw{4} \oplus  5 \jw{6}  \\ \oplus  6 P' \oplus  3 Q' \end{array} $     $\begin{array}{c} \jw{2} \oplus  \jw{4} \oplus  3 \jw{6} \\ \oplus  3 P' \oplus  2 Q' \end{array}$
$P'$     $ \jw{6} \oplus  2 P' \oplus  Q' $     $\begin{array}{c} \jw{4} \oplus  3 \jw{6} \\ \oplus  3 P' \oplus  2 Q'\end{array}$     $\begin{array}{c} \jw{2} \oplus  3 \jw{4} \oplus  5 \jw{6} \\ \oplus  6 P' \oplus  3 Q'  \end{array}$     $\begin{array}{c} 1 \oplus  2 \jw{2} \oplus  3 \jw{4} \\ \oplus  6 \jw{6} \oplus  7 P' \oplus  4 Q'   \end{array}$     $ \begin{array}{c} \jw{2} \oplus  2 \jw{4} \oplus  3 \jw{6} \\ \oplus  4 P' \oplus  2 Q' \end{array}$
$Q'  $     $ \jw{6} \oplus  P'  $     $\begin{array}{c} \jw{4} \oplus  \jw{6} \\ \oplus  2 P' \oplus  Q'  \end{array}$     $\begin{array}{c} \jw{2} \oplus  \jw{4} \oplus  3 \jw{6} \\ \oplus  3 P' \oplus  2 Q' \end{array}$     $\begin{array}{c} \jw{2} \oplus  2 \jw{4} \oplus  3 \jw{6} \\ \oplus  4 P' \oplus  2 Q' \end{array}$     $\begin{array}{c}1 \oplus  \jw{4} \oplus  2 \jw{6}  \\ \oplus  2 P' \oplus  Q' \end{array}$

Here we have given more informative names to the objects, corresponding to those used in <cit.> rather than merely naming them $O_{11}^x$.
The dimensions of the objects $(\jw{2}, \jw{4}, \jw{6}, P', Q')$ are roughly $(3.4, 7.0, 13.3, 16.0, 8.7)$, and this determines the ordering used in the $O_{11}^x$ notation.
In particular $O_{11}^1 = \mathbf{1}$, $O_{11}^2 = f^{(2)}$, $O_{11}^3 = f^{(4)}$, $O_{11}^4 = Q'$ (as it has the next smallest dimension), $O_{11}^5 = f^{(6)}$, and $O_{11}^6 = P'$.

A Morita equivalence between $\scrC$ and $\cD$ gives a braided equivalence $Z(\scrC) \cong Z(\cD)$.  Any such $\cD$ is of the form $A$-mod for $A$ a Lagrangian algebra in $Z(\scrC)$  <cit.>.  In general there might be several Lagrangian algebras yielding a given $\cD$, but in our case since the Brauer–Picard group is trivial there is a unique Lagrangian algebra for each $\cD$.  Using the notation of <cit.> for the objects in the center $Z(\cE\cH)$, the Lagrangian algebra giving $\cE\cH_1$ has underlying object:

$$\omega_0\oplus \omega_1\oplus \omega_2 \oplus \alpha_1 \oplus \alpha_2 \oplus \alpha_3.$$

§.§ Structure of EH2

The fusion rules for $\cE\cH_2$ (which is the principal even half of the Extended Haagerup subfactor) are given by

$\otimes$      $\jw{2}$     $\jw{4}$     $\jw{6}$     $P$     $Q$     $A$     $B$

$\jw{2}$     $1\oplus  \jw{2}\oplus \jw{4}$     $\jw{2}\oplus \jw{4}\oplus \jw{6}$     $\jw{4} \oplus  W$     $B \oplus  W$     $A\oplus  W$     $Q$     $P$

$\jw{4}$     $\jw{2} \oplus  \jw{4} \oplus  \jw{6}$     $\begin{array}{c}1 \oplus  \jw{2} \oplus  \\ \jw{4} \oplus  W \end{array}$     $\begin{array}{c} \jw{2} \oplus  \jw{4} \oplus  \\ A \oplus  B \oplus  2 W \end{array}$     $\jw{4} \oplus  A \oplus  2 W$     $\jw{4} \oplus  B \oplus  2 W$     $\jw{6} \oplus  P$     $\jw{6} \oplus  Q$

$\jw{6}$     $\jw{4} \oplus  W$     $\begin{array}{c}\jw{2} \oplus  \jw{4} \oplus  \\ A \oplus  B \oplus  2 W \end{array}$     $1 \oplus  W \oplus  Z$     $\jw{6} \oplus  Q \oplus  Z$     $\jw{6} \oplus  P \oplus  Z$     $\jw{4} \oplus  B \oplus  W$     $\jw{4} \oplus  A \oplus  W$

$P$     $A \oplus  W$     $\jw{4} \oplus  B \oplus  2 W$     $\jw{6} \oplus  Q \oplus  Z$     $1 \oplus  P \oplus  Z$     $\jw{6} \oplus  Z$     $\jw{2} \oplus  A \oplus  W$     $\jw{4} \oplus  W$

$Q$     $B \oplus  W$     $ \jw{4} \oplus  A \oplus  2 W$     $\jw{6} \oplus  P \oplus  Z$     $\jw{6} \oplus  Z$     $1 \oplus  Q \oplus  Z$     $\jw{4} \oplus  W$     $\jw{2} \oplus  B \oplus  W$

$A$     $P$     $\jw{6} \oplus  Q$     $ \jw{4} \oplus  B \oplus  W$
$\jw{4} \oplus  W$     $\jw{2} \oplus  A \oplus  W$
$\jw{6}$     $1 \oplus  P$

$B$     $Q$     $\jw{6} \oplus  P$     $\jw{4} \oplus  A \oplus  W$
$\jw{2} \oplus  B \oplus  W$     $\jw{4} \oplus  W$
$1 \oplus  Q$     $\jw{6}$

using the abbreviations $W  = \jw{6}+P+Q$ and $Z  =A + B + \jw{2} + 2 \jw{4} + 3 \jw{6} + 3 P + 3 Q$.

In the first appendix we give an alternate construction of $\cE\cH_3$ by constructing a $Q$-system on $1+\jw{6}$ in $\cE\cH_2$ whose dual category is $\cE\cH_3$.  This construction is viable because
$$\dim(\Hom(\jw{6}\otimes \jw{6} \to \jw{6})) = 4$$
is not too large.

The dimensions of $(\jw{2}, \jw{4}, \jw{6}, P, Q, A, B)$ are roughly $(3.4, 7.0, 13.3, 12.3, 12.3, 3.7, 3.7)$, which again determines the ordering in the $O_{22}^x$ naming.

Since $A$ and $B = \overline{A}$ have small dimension we also record the fusion graph for tensoring with $A$ and the principal graph for the corresponding subfactor. The fusion graph is:

%Fusion graph for tensoring on the right with $A$, checked 7/2/18
\begin{tikzpicture}[baseline=-1mm, scale=.65]
\node (1) at (0,0)  {\scriptsize{$1$}};
\node (A) at (2,-2)  {\scriptsize{$A$}};
\node (B) at (2,2)  {\scriptsize{$B$}};
\node (6) at (4,0)  {\scriptsize{$\jw{6}$}};
\node (4) at (6,0)  {\scriptsize{$\jw{4}$}};
\node (2) at (8,0)  {\scriptsize{$\jw{2}$}};
\node (Q) at (6,2)  {\scriptsize{$Q$}};
\node (P) at (6,-2)  {\scriptsize{$P$}};
\draw[->] (1) -- (A);
\draw[->] (A) -- (6);
\draw[->] (6) -- (B);
\draw[->] (B) -- (1);
\draw[<->] (6) -- (4);
\draw[<->] (6) -- (P);
\draw[<->] (6) -- (Q);
\draw[->] (P) -- (2);
\draw[->] (2) -- (Q);
\draw[->] (Q) -- (4);
\draw[->] (4) -- (P);
\draw[->] (B) -- (Q);
\draw[->] (P) -- (A);
\draw[->] (6) to [out=250,in=290,looseness=8] (6);
\draw[->] (P) to [out=300,in=330,looseness=8] (P);
\draw[->] (Q) to [out=30,in=60,looseness=8] (Q);
\draw[<->] (P) to [out=60,in=300] (Q);
\end{tikzpicture}.

The corresponding subfactor $1 \subset A\overline{A}$ has index roughly $13.3$.
It has the following principal and dual principal graph.
Notice that every simple appears twice — once as an even vertex and once as an odd vertex.

\begin{equation} \label{eq:S13}
%Graph for alternating subfactor associated $A, \bar A$, checked 7/2/18
\begin{tikzpicture}[baseline=-1mm, scale=.65]
\draw (0,0) -- (2,0);
\draw (2,0) -- (4,0);

\draw (4,0) -- (6,0);
\draw (4,0) -- (2,2);
\draw (4,0) -- (4,2);

\draw (6,0) -- (8,0);
\draw (6,0) -- (6,2);
\draw (6,0) -- (8,2);

\draw(8,0) -- (10,0);

\draw(2,2) -- (2,4);
\draw(4,2) -- (2,4);

\draw(6,2) -- (8,4);
\draw(8,2) -- (8,4);

\draw(2,2) to [out=30, in=150] (8,2);
\draw(2,2) to [out=-30, in=-150] (6,2);
\draw(4,2) to [out=-30, in=-150] (8,2);
\draw(4,2) -- (6,2);

\draw(2,1) -- (4,0);
\draw(8,1) -- (6,0);

\draw(4,2) -- (4,4);
\draw(6,2) -- (6,4);

\filldraw (0,0) node [above] {$*$} node [below] {$1$} circle (1mm);
\filldraw [fill=white] (2,0) node [below] {$A$} circle (1mm);
\filldraw (4,4) node [left] {$A$} circle (1mm);
\filldraw (4,0) node [below right] {$P$} circle (1mm);
\filldraw (8,1) node [right] {$f^{(2)}$} circle (1mm);
\filldraw [fill=white] (6,0) node [below left] {$Q$} circle (1mm);
\filldraw [fill=white] (2,1) node [left] {$f^{(2)}$} circle (1mm);
\filldraw (8,0) node [below] {$B$} circle (1mm);
\filldraw [fill=white] (6,4) node [right] {$B$} circle (1mm);
\filldraw [fill=white] (10,0) node [below] {$1$} circle (1mm);;
\filldraw [fill=white] (2,2) node [left] {$P$} circle (1mm);
\filldraw [fill=white] (4,2) node [left] {$f^{(6)}$} circle (1mm);
\filldraw (6,2) node [right] {$\jw{6}$} circle (1mm);
\filldraw (8,2) node [right] {$Q$} circle (1mm);
\filldraw (2,4) node [left] {$\jw{4}$} circle (1mm);
\filldraw [fill=white] (8,4) node [right] {$f^{(4)}$} circle (1mm);
\end{tikzpicture}
\end{equation}

The Extended Haagerup subfactor planar algebra constructed in <cit.> (see <ref>) provides a Morita equivalence between $\cE\cH_1$ and $\cE\cH_2$.
The generating object $X$ in the Morita equivalence is the smallest object $O_{12}^1$ in the unique invertible bimodule category between these two fusion categories.
The principal graphs are

\left(
\begin{tikzpicture}[baseline=-1mm, scale=.5]
\draw (0,0) -- (7,0);
\draw (7,0)--(7.7,.7)--(9.7,.7);
\draw (7,0)--(7.7,-.7)--(9.7,-.7);

\filldraw              (0,0) node [above] {$1$} circle (1mm);
\filldraw [fill=white] (1,0) node {} circle (1mm);
\filldraw              (2,0) node [above] {$\jw{2}$}  circle (1mm);
\filldraw [fill=white] (3,0) node {}  circle (1mm);
\filldraw              (4,0) node [above] {$\jw{4}$}  circle (1mm);
\filldraw [fill=white] (5,0) node {}  circle (1mm);
\filldraw              (6,0) node [above] {$\jw{6}$}  circle (1mm);
\filldraw [fill=white] (7,0) node {}  circle (1mm);

\filldraw              (7.7,.7) node [above] {$P$}  circle (1mm);
\filldraw [fill=white] (8.7,.7) node [above] {}  circle (1mm);
\filldraw              (9.7,.7) node [above] {$A$}  circle (1mm);
\filldraw              (7.7,-.7) node [below] {$Q$}  circle (1mm);
\filldraw [fill=white] (8.7,-.7) node [above] {}  circle (1mm);
\filldraw              (9.7,-.7) node [below] {$B$}  circle (1mm);
\end{tikzpicture}
\,,\,
\begin{tikzpicture}[baseline=-1mm, scale=.5]
\draw (0,0) -- (7,0);
\draw (7,0)--(7.7,.7);
\draw (7,0)--(7.7,-.7);
\draw (7.7,.7)--(8.4,1.4);
\draw (7.7,.7)--(8.4,0);

\filldraw              (0,0) node [above] {$1$} circle (1mm);
\filldraw [fill=white] (1,0) node {} circle (1mm);
\filldraw              (2,0) node [above] {$\jw{2}$}  circle (1mm);
\filldraw [fill=white] (3,0) node {}  circle (1mm);
\filldraw              (4,0) node [above] {$\jw{4}$}  circle (1mm);
\filldraw [fill=white] (5,0) node {}  circle (1mm);
\filldraw              (6,0) node [above] {$\jw{6}$}  circle (1mm);
\filldraw [fill=white] (7,0) node {}  circle (1mm);

\filldraw              (7.7,.7) node [above] {$P'$}  circle (1mm);
\filldraw              (7.7,-.7) node [below] {$Q'$}  circle (1mm);
\filldraw [fill=white] (8.4,1.4) node [above] {}  circle (1mm);
\filldraw [fill=white] (8.4,0) node [above] {}  circle (1mm);
\end{tikzpicture}
\right).

Notice all even vertices are self dual except for $\overline{A} \cong B$.

The Lagrangian algebra giving $\cE\cH_2$ has underlying object:

$$\omega_0\oplus \omega_1\oplus \omega_2 \oplus 2\alpha_1 \oplus \alpha_2.$$

§.§ Structure of EH3

The fusion rules for $\cE\cH_3$ are as follows.

$Y_ 1$

$Y_ 2$

$1 {\oplus} Y_ 2$

$V {\oplus} X$

$\overline{X} {\oplus} W$

$V {\oplus} W {\oplus} Y_ 2$

$U {\oplus} V {\oplus} W {\oplus} Y_ 1$

$1 {\oplus} Y_ 1$

$\overline{X} {\oplus} V$

$W {\oplus} X$

$V {\oplus} W {\oplus} Y_ 1$

$U {\oplus} V {\oplus} W {\oplus} Y_ 2$

$Y_ 1$

$\overline{X} {\oplus} V$

$1 {\oplus} V {\oplus} W {\oplus} Y_ 1$

$U {\oplus} V {\oplus} W$

$U {\oplus} V {\oplus} W {\oplus} Y_ 2$

\overline{X} {\oplus} U {\oplus} 2 V {\oplus}
\\
2 W {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

U {\oplus} 2 V {\oplus} 3 W {\oplus}
\\
X {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

$Y_ 2$

$V {\oplus} X$

$U {\oplus} V {\oplus} W$

$1 {\oplus} V {\oplus} W {\oplus} Y_ 2$

$U {\oplus} V {\oplus} W {\oplus} Y_ 1$

U {\oplus} 2 V {\oplus} 2 W {\oplus}
\\
X {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

\overline{X} {\oplus} U {\oplus} 2 V {\oplus}
\\
3 W {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

$\overline{X} {\oplus} W$

$W {\oplus} X$

$U {\oplus} V {\oplus} W {\oplus} Y_ 2$

$U {\oplus} V {\oplus} W {\oplus} Y_ 1$

1 {\oplus} U {\oplus} V {\oplus}
\\
W {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

$U {\oplus} 2 V {\oplus} 3 W {\oplus} Y_ 1 {\oplus} Y_ 2$

\overline{X} {\oplus} U {\oplus} 3 V {\oplus}
\\
3 W {\oplus} X {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

$V {\oplus} W {\oplus} Y_ 1$

$V {\oplus} W {\oplus} Y_ 2$

U {\oplus} 2 V {\oplus} 2 W {\oplus}
\\
X {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

\overline{X} {\oplus} U {\oplus} 2 V {\oplus}
\\
2 W {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

$U {\oplus} 2 V {\oplus} 3 W {\oplus} Y_ 1 {\oplus} Y_ 2$

1 {\oplus} \overline{X} {\oplus} 2 U {\oplus} 4 V {\oplus}
\\
5 W {\oplus} X {\oplus} 2 Y_ 1 {\oplus} 2 Y_ 2
\end{array}$

\overline{X} {\oplus} 3 U {\oplus} 5 V {\oplus}
\\
6 W {\oplus} X {\oplus} 2 Y_ 1 {\oplus} 2 Y_ 2
\end{array}$

$U {\oplus} V {\oplus} W {\oplus} Y_ 2$

$U {\oplus} V {\oplus} W {\oplus} Y_ 1$

\overline{X} {\oplus} U {\oplus} 2 V {\oplus}
\\
3 W {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

U {\oplus} 2 V {\oplus} 3 W {\oplus}
\\
X {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

\overline{X} {\oplus} U {\oplus} 3 V {\oplus}
\\
3 W {\oplus} X {\oplus} Y_ 1 {\oplus} Y_ 2
\end{array}$

\overline{X} {\oplus} 3 U {\oplus} 5 V {\oplus}
\\
6 W {\oplus} X {\oplus} 2 Y_ 1 {\oplus} 2 Y_ 2
\end{array}$

1 {\oplus} \overline{X} {\oplus} 3 U {\oplus} 6 V {\oplus}
\\
7 W {\oplus} X {\oplus} 3 Y_ 1 {\oplus} 3 Y_ 2
\end{array}$

The dimensions of $(X,\overline{X}, Y_1, Y_2, U, V, W)$ are approximately $(2.6, 2.6, 6.0, 6.0, 7.0, 13.3, 15.9)$.  They are listed in the order $O_{33}^2$, $O_{33}^3$, etc.

The fusion graph for $X\in \cE\cH_3$ is given by

%Fusion graph for tensoring on the right with $X$, checked 7/2/18
\begin{tikzpicture}[baseline=-1mm, scale=.6]
\node (1) at (0,0)  {\scriptsize{$1$}};
\node (X) at (2,-2)  {\scriptsize{$X$}};
\node (Xb) at (2,2)  {\scriptsize{$\overline{X}$}};
\node (U) at (4,0)  {\scriptsize{$U$}};
\node (W) at (6,0)  {\scriptsize{$W$}};
\node (V) at (8,0)  {\scriptsize{$V$}};
\node (Y1) at (6,2)  {\scriptsize{$Y_1$}};
\node (Y2) at (6,-2)  {\scriptsize{$Y_2$}};
\draw[->] (1) -- (X);
\draw[->] (X) -- (U);
\draw[->] (U) -- (Xb);
\draw[->] (Xb) -- (1);
\draw[<->] (U) -- (W);
\draw[->] (Xb) -- (Y1);
\draw[->] (Y1) -- (W);
\draw[<->] (W) -- (V);
\draw[<-] (V) -- (Y2);
\draw[->] (W) -- (Y2);
\draw[->] (V) -- (Y1);
\draw[->] (Y2) -- (X);
\draw[->] (W) to [out=300,in=330,looseness=8] (W);
\draw[->] (V) to [out=300,in=330,looseness=8] (V);
\end{tikzpicture}

The principal graph of the subfactor corresponding to the algebra object $X\otimes \overline{X} \in \cE\cH_3$ is given below.
It has index roughly $7.0$.
Notice that each simple of $\cE\cH_3$ appears twice — once as an even vertex and once as an odd vertex.

\begin{equation} \label{eq:S7}
%Graph for alternating subfactor associated $X, \bar X$, not checked 7/2/18
\begin{tikzpicture}[baseline=-1mm, scale=.6]
\draw (0,0) -- (7,0);
\draw (3,0)--(3,3);
\draw (4,0)--(4,3);
\draw (2,1)--(5,1);

\filldraw              (0,0) node [above] {$*$} node [below] {$1$} circle (1mm);
\filldraw [fill=white] (1,0) circle (1mm) node [below] {$X$};
\filldraw              (2,0) circle (1mm) node [below] {$Y_2$};
\filldraw [fill=white] (3,0) circle (1mm) node [below] {$V$};
\filldraw              (4,0) circle (1mm) node [below] {$V$};
\filldraw [fill=white] (5,0) circle (1mm) node [below] {$Y_1$};
\filldraw              (6,0) circle (1mm) node [below] {$\overline{X}$};
\filldraw [fill=white] (7,0) circle (1mm) node [below] {$1$};

\filldraw [fill=white] (2,1) circle (1mm) node [left] {$Y_2$};
\filldraw              (3,1) circle (1mm) node [below left] {$W$};
\filldraw [fill=white] (4,1) circle (1mm) node [below right] {$W$};;
\filldraw              (5,1) circle (1mm) node [right] {$Y_1$};

\filldraw [fill=white] (3,2) circle (1mm) node [left] {$U$};
\filldraw              (4,2) circle (1mm) node [right] {$U$};

\filldraw              (3,3) circle (1mm) node [left] {$X$};
\filldraw [fill=white] (4,3) circle (1mm) node [right] {$\overline{X}$};
\end{tikzpicture}
\,.
\end{equation}

This paper establishes that $\cE\cH_3$ is a categorification of the above ring, but we don't know that this is the only such categorification.  Thus in order to construct the Extended Haagerup subfactor from an alternative proposed construction of $\cE\cH_3$ one would need to do additional work.
Since the even parts of the Extended Haagerup subfactor are the only categorifications of their fusion rings, it would be enough to construct a categorification of the $\cE\cH_3$ fusion ring plus an algebra structure on $1\oplus U$, and check that the fusion ring of the commutant category corresponding to the algebra $1\oplus U$ is the fusion ring of $\cE\cH_2$.

We are able to list all Q-systems in $\cE\cH_3$.  The Q-system corresponding to $O_{23}^1$ has relatively small dimension and its underlying object is $1 \oplus U$.  The dual algebra in this case is $1 \oplus f^{(6)}$ in $\cE\cH_2$.
The index of this subfactor is approximately $14.3$, and the principal graphs are:

\begin{equation}\label{eq:S14}
%Graph for subfactor associated $\cE\cH_{22},\cE\cH_{23}$, tensoring on right, checked 7/2/18
\begin{tikzpicture}[baseline=-1mm, scale=.7]
\draw (0,0) -- (2,0);
\draw (2,0) -- (4,0);
\draw (4,0) -- (6,1.5);
\draw (4,0) -- (6,.5);
\draw (4,0) -- (6,-.5);
\draw (4,0) -- (6,-1.5);
\draw (6,1.5)--(8,2.5);
\draw (6,.5)--(8,2.5);
\draw (6,-.5)--(8,2.5);
\draw (6,-1.5)--(8,2.5);
\draw (6,1.5)--(8,1.5);
\draw (6,.5)--(8,1.5);
\draw (6,-.5)--(8,1.5);
\draw (6,-1.5)--(8,1.5);
\draw (6,-.5)--(8,.5);
\draw (6,-.5)--(8,-.5);
\draw (6,-1.5)--(8,-.5);
\draw (6,-1.5)--(8,-1.5);
\draw (6,-1.5)--(8,-2.5);

\filldraw              (0,0) node [above] {$*$} node [below] {$1$} circle (1mm);
\filldraw [fill=white] (2,0) circle (1mm) ;
\filldraw  (4,0) circle (1mm) node [below] {$f^{(6)}$};
\filldraw [fill=white] (6,1.5) circle (1mm) ;
\filldraw [fill=white] (6,.5) circle (1mm) ;
\filldraw [fill=white] (6,-.5) circle (1mm) ;
\filldraw [fill=white] (6,-1.5) circle (1mm) ;
\filldraw  (8,2.5) circle (1mm) node [right] {$P$} ;
\filldraw  (8,1.5) circle (1mm) node [right] {$Q$} ;
\filldraw  (8,.5) circle (1mm) node [right] {$f^{(2)}$} ;
\filldraw  (8,-.5) circle (1mm) node [right] {$f^{(4)}$} ;
\filldraw  (8,-1.5) circle (1mm) node [right] {$A$} ;
\filldraw  (8,-2.5) circle (1mm) node [right] {$B$} ;
\end{tikzpicture}
\hspace{1in}
\begin{tikzpicture}[baseline=-1mm, scale=.7]
\draw (0,0) -- (2,0);
\draw (2,0) -- (4,0);
\draw (4,0) -- (6,1.5);
\draw (4,0) -- (6,.5);
\draw (4,0) -- (6,-.5);
\draw (4,0) -- (6,-1.5);

\draw (6,1.5)--(8,2.5);

\draw (6,.5)--(8,1.5);

\draw (6,1.5)--(8,.5);
\draw (6,-.5)--(8,.5);

\draw (6,.5)--(8,-.5);
\draw (6,-.5)--(8,-.5);

\draw (6,-.5)--(8,-1.5);
\draw (6,-1.5)--(8,-1.5);

\draw[double] (6,-1.5)--(8,-2.5);
\draw (6,-.5)--(8,-2.5);
\draw (6,.5)--(8,-2.5);
\draw (6,1.5)--(8,-2.5);

\filldraw (0,0) node [above] {$*$} node [below] {$1$} circle (1mm);
\filldraw [fill=white] (2,0) circle (1mm) ;
\filldraw  (4,0) circle (1mm) node [below] {$V$};
\filldraw [fill=white] (6,1.5) circle (1mm) ;
\filldraw [fill=white] (6,.5) circle (1mm) ;
\filldraw [fill=white] (6,-.5) circle (1mm) ;
\filldraw [fill=white] (6,-1.5) circle (1mm) ;
\filldraw  (8,2.5) circle (1mm) node [right] {$X$} ;
\filldraw  (8,1.5) circle (1mm) node [right] {$\overline{X}$} ;
\filldraw  (8,.5) circle (1mm) node [right] {$Y_2$} ;
\filldraw  (8,-.5) circle (1mm) node [right] {$Y_1$} ;
\filldraw  (8,-1.5) circle (1mm) node [right] {$U$} ;
\filldraw  (8,-2.5) circle (1mm) node [right] {$W$} ;
\end{tikzpicture}
\,.
\end{equation}

The Lagrangian algebra in $Z(\cE\cH)$ giving $\cE\cH_3$ has underlying object:

$$\omega_0\oplus \omega_1\oplus \omega_2 \oplus 2\alpha_1 \oplus \alpha_2.$$

Note that this is the same underlying object as the Lagrangian algebra corresponding to $\cE\cH_2$; the algebra structures must be different.

§.§ Structure of EH4

We now turn to describing some of the combinatorial structure of $\cE\cH_4$.  The fusion rules for $\cE\cH_4$ are as follows.

$K_ 1$

$K_ 2$

$G {\oplus} K_ 1 {\oplus} K_ 2 {\oplus} L$

1 {\oplus} G {\oplus}
\\
H {\oplus} K_ 2 {\oplus} L
\end{array}$

H {\oplus} K_ 2 {\oplus}
\\
L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z}
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

1 {\oplus} G {\oplus}
\\
H {\oplus} K_ 1 {\oplus} L
\end{array}$

$G {\oplus} K_ 1 {\oplus} K_ 2 {\oplus} L$

H {\oplus} K_ 1 {\oplus}
\\
L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z}
\end{array}$

H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

H {\oplus} K_ 1 {\oplus}
\\
L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

H {\oplus} K_ 2 {\oplus}
\\
L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

1 {\oplus} G {\oplus} H {\oplus}
\\
K_ 1 {\oplus} K_ 2 {\oplus} L
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z}
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z}
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

1 {\oplus} G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} 2 H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 3 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

$K_ 1$

H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z}
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z}
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

1 {\oplus} G {\oplus} 2 H {\oplus} K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

2 G {\oplus} 2 H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 3 L {\oplus} \overline{Z} {\oplus} 2 Z
\end{array}$

$K_ 2$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

1 {\oplus} G {\oplus} 2 H {\oplus} 2 K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

2 G {\oplus} 2 H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 3 L {\oplus} 2 \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 2 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

G {\oplus} 2 H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 3 L {\oplus} \overline{Z} {\oplus} Z
\end{array}$

2 G {\oplus} 2 H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 3 L {\oplus} 2 \overline{Z} {\oplus} Z
\end{array}$

2 G {\oplus} 2 H {\oplus} 2 K_ 1 {\oplus}
\\
2 K_ 2 {\oplus} 3 L {\oplus} \overline{Z} {\oplus} 2 Z
\end{array}$

1 {\oplus} 2 G {\oplus} 3 H {\oplus} 3 K_ 1 {\oplus}
\\
3 K_ 2 {\oplus} 4 L {\oplus} 2 \overline{Z} {\oplus} 2 Z
\end{array}$

The dimensions of the objects $(Z, \overline{Z}, G, H, K_1, K_2, L)$ are approximately $(6.3, 6.3, 7.0, 8.6, 9.6, 9.6, 13.3)$.  These are already in increasing order, so the objects $O_{44}^x$ appear in this order.

None of the objects in $\cE\cH_4$ is small enough to have a nice fusion graph.  The Q-system corresponding to $O_{34}^1$ does give a $3$-supertransitive subfactor.  The underlying object of this Q-system is $1 \oplus G$ and the dual Q-system is $1 \oplus U$ in $\cE\cH_3$, and the index is roughly $8.0$.  The principal graphs for this subfactor are given by:

\begin{align}\label{eq:S8}
%Subfactor graph checked on 7/2/18, tensoring on right.
\begin{tikzpicture}[baseline=-1mm, scale=.7]
\draw (0,0) -- (2,0);
\draw (2,0) -- (4,0);
\draw (4,0) -- (6,0);
\draw (6,0) -- (8,1.5);
\draw (6,0) -- (8,.5);
\draw (6,0) -- (8,-.5);
\draw (6,0) -- (8,-1.5);
\draw (8,-.5) -- (10,0);
\draw (8,-.5) -- (10,-1);
\draw (8,-.5) -- (10,-2);
\draw (8,-1.5) -- (10,-1);
\draw (8,-1.5) -- (10,-2);
\draw (10,0) -- (12,.5);
\draw (10,0) -- (12,-.5);
\filldraw (0,0) node [above] {$*$} node [below] {$1$} circle (1mm);
\filldraw [fill=white] (2,0) circle (1mm) ;
\filldraw  (4,0) circle (1mm) node [below] {$U$};
\filldraw [fill=white] (6,0) circle (1mm) ;
\filldraw  (8,1.5) circle (1mm) node [below] {$Y_1$} ;
\filldraw  (8,.5) circle (1mm) node [below] {$Y_2$} ;
\filldraw  (8,-.5) circle (1mm) node [below] {$W$} ;
\filldraw  (8,-1.5) circle (1mm) node [below] {$V$} ;
\filldraw [fill=white] (10,0) circle (1mm) ;
\filldraw [fill=white] (10,-1) circle (1mm) ;
\filldraw [fill=white] (10,-2) circle (1mm) ;
\filldraw  (12,.5) circle (1mm) node [right] {$X$} ;
\filldraw  (12,-.5) circle (1mm) node [right] {$\overline{X}$} ;
\end{tikzpicture}
\displaybreak[1]\\
\begin{tikzpicture}[baseline=-1mm, scale=.7]
\draw (0,0) -- (2,0);
\draw (2,0) -- (4,0);
\draw (4,0) -- (6,0);
\draw (6,0) -- (8,1.5);
\draw (6,0) -- (8,.5);
\draw (6,0) -- (8,-.5);
\draw (6,0) -- (8,-1.5);
\draw (8,1.5) -- (10,1);
\draw (8,.5) -- (10,0);
\draw (8,-.5) -- (10,0);
\draw (8,-.5) -- (10,-1);
\draw (8,-1.5) -- (10,-1);
\draw (10,1) -- (12,.5);
\draw (10,1) -- (12,-.5);
\draw (10,0) -- (12,.5);
\draw (10,-1) -- (12,-.5);
\filldraw (0,0) node [above] {$*$} node [below] {$1$} circle (1mm);
\filldraw [fill=white] (2,0) circle (1mm) ;
\filldraw  (4,0) circle (1mm) node [below] {$G$};
\filldraw [fill=white] (6,0) circle (1mm) ;
\filldraw  (8,1.5) circle (1mm) node [below] {$H$} ;
\filldraw  (8,.5) circle (1mm) node [below] {$K_1$} ;
\filldraw  (8,-.5) circle (1mm) node [below] {$L$} ;
\filldraw  (8,-1.5) circle (1mm) node [below] {$K_2$} ;
\filldraw [fill=white] (10,1) circle (1mm) ;
\filldraw [fill=white] (10,0) circle (1mm) ;
\filldraw [fill=white] (10,-1) circle (1mm) ;
\filldraw  (12,.5) circle (1mm) node [right] {$\overline{Z}$} ;
\filldraw  (12,-.5) circle (1mm) node [right] {$Z$} ;
\end{tikzpicture}
\notag
\end{align}

As with $\cE\cH_2$ and $\cE\cH_3$, the Lagrangian algebra giving $\cE\cH_4$ has underlying object:

$$\omega_0\oplus \omega_1\oplus \omega_2 \oplus 2\alpha_1 \oplus \alpha_2.$$
We thus see that this object must have three distinct algebra structures on it.

§.§ Intermediate subfactors

In this section we describe all the lattices of intermediate subfactors for subfactors coming from the objects in the Extended Haagerup bimodule categories.
These can be read directly from the fusion rules for the bimodules following <cit.>.
The relationship between bimodule fusion rules and intermediate subfactors is particularly simple for Extended Haagerup because the categories have no nontrivial outer automorphisms and no invertible objects.
In such a case intermediate subfactors correspond exactly to triples of objects $X$, $Y$, $Z$ in the bimodule categories such that $X \otimes Y \cong Z$.
The corresponding subfactors are $1 \subset X \otimes \overline{X} \subset  X \otimes (Y \otimes \overline{Y}) \otimes \overline{X} = Z \otimes \overline{Z}$.
The small subfactors $1 \subset X \otimes \overline{X}$ and $X \otimes \overline{X} \subset X \otimes (Y \otimes \overline{Y}) \otimes \overline{X}$ are irreducible exactly when $X$ and $Y$ are, and the large subfactor $1 \subset Z \otimes \overline{Z}$ is irreducible exactly when $X \otimes Y$ is simple.
Note that it is rare to have a pair of simple objects $X, Y$ such that the product $X \otimes Y$ is simple, and this explains why
irreducible subfactors typically have few intermediates.

Each row of the following table lists triples of simple objects $X = O_{ij}^x$, $Y = O_{jk}^y$, and their simple product $X \otimes Y = O_{ik}^z$.
The columns labelled `ST' and `Index' indicate the supertransitivity and index of the corresponding subfactors.
We only list one representative of each dual pair, so in the tables we always have $i<j$.
We've grouped rows together according to the identity of the large object $Z$, as these rows are all intermediate subfactors of the same large subfactor.

We recall that:

* $O_{21}^1$ and $O_{12}^1$ correspond to the Extended Haagerup subfactor and its dual,

* $O_{33}^2$ corresponds to the subfactor with principal graph shown in Equation (<ref>),

* $O_{34}^1$ and $O_{43}^1$ correspond to the subfactor with principal graph shown in Equation (<ref>),

* $O_{22}^3$ and  $O_{22}^4$ both correspond to the subfactor with principal graph shown in Equation (<ref>), and

* $O_{23}^1$ and $O_{32}^1$ correspond to the subfactor with principal graph shwon in Equation (<ref>).

Z    Index(Z)    X    ST(X)    Index(X)    Y    ST(Y)    Index(Y)
O_11^6     255.411     O_12^4    1     58.3502    O_21^1     7     4.3772

O_12^3     1     58.3502     O_21^1     7     4.3772

O_12^1     7     4.3772     O_21^3    1     58.3502

O_12^1     7     4.3772     O_21^4    1     58.3502
O_12^3     58.3502     O_12^1    7     4.3772     O_22^4     2     13.3305
O_12^4     58.3502     O_12^1    7     4.3772     O_22^3     2     13.3305
O_12^6     329.743     O_13^1     1     23.0099      O_32^1     2     14.3305

O_13^2     1     23.0099      O_32^1     2     14.3305

O_12^2     1     24.736      O_22^3     2     13.3305

O_12^2     1     24.736      O_22^4     2     13.3305

O_11^4     1     75.3318     O_12^1     7     4.3772
O_13^4     62.7274     O_12^1     7     4.3772     O_23^1     2     14.3305
O_13^5     161.72     O_13^1     1     23.0099     O_33^2     3     7.0283

O_13^2     1     23.0099     O_33^3     3     7.0283
O_13^6     262.439     O_14^1     1     32.6893     O_43^1     3     8.0283

O_14^2     1     32.6893     O_43^1     3     8.0283

O_13^3     1     37.3404     O_33^2     3     7.0283

O_13^3     1     37.3404     O_33^3     3     7.0283

O_11^2     1     11.4055     O_13^1     1     23.0099

O_11^2     1     11.4055     O_13^2     1     23.0099
O_22^6     152.041     O_22^2     1     11.4055     O_22^4     2     13.3305

O_22^3     2     13.3305     O_22^2     1     11.4055
O_22^7     152.041     O_22^2     1     11.4055     O_22^3     2     13.3305

O_22^4     2     13.3305     O_22^2     1     11.4055
O_22^8     177.702     O_22^3    2     13.3305     O_22^3     2     13.3305

O_22^4     2     13.3305     O_22^4     2     13.3305
O_23^2     100.719     O_23^1     2     14.3305     O_33^3     3     7.0283

O_21^1     7     4.3772     O_13^2     1     23.0099
O_23^3     100.719     O_23^1     2     14.3305     O_33^2     3     7.0283

O_21^1     7     4.3772     O_13^1     1     23.0099
O_23^4     163.446     O_24^1     1     20.3588     O_43^1     3     8.0283

O_24^2     1     20.3588     O_43^1     3     8.0283

O_22^2     1     11.4055     O_23^1     2     14.3305

O_21^1     7     4.3772     O_13^3     1     37.3404
O_23^5     191.032     O_22^3     2     13.3305     O_23^1     2     14.3305

O_22^4     2     13.3305     O_23^1     2     14.3305

O_24^3     115.049     O_23^1     2     14.3305     O_34^1     3     8.0283
O_24^4     143.088     O_21^1     7     4.3772     O_14^1     1     32.6893

O_21^1     7     4.3772     O_14^2     1     32.6893
O_24^5     271.392     O_22^4     2     13.3305     O_24^1     1     20.3588

O_22^3     2     13.3305     O_24^2     1     20.3588

O_21^1     7     4.3772     O_14^3    1     62.0013
O_33^6     49.3969     O_33^2     3     7.0283     O_33^2     3     7.0283

O_33^3     3     7.0283     O_33^3     3     7.0283
O_33^8     255.411     O_33^5     1     36.3404     O_33^3     3     7.0283

O_33^4     1     36.3404     O_33^2     3     7.0283

O_33^3     3     7.0283     O_33^4     1     36.3404

O_33^2     3     7.0283     O_33^5     1     36.3404
O_34^2     56.4252     O_33^2     3     7.0283     O_34^1     3     8.0283

O_33^3     3     7.0283     O_34^1     3     8.0283
O_34^5     291.751     O_33^4     1     36.3404     O_34^1     3     8.0283

O_33^5     1     36.3404     O_34^1     3     8.0283

O_32^1     2     14.3305     O_24^1     1     20.3588

O_32^1     2     14.3305     O_24^2     1     20.3588

Let us briefly summarise the interesting subfactor lattices encoded in the above table.

There are four lines with $Z = O_{11}^6$, so there are four intermediate subfactors of the index $255.411\ldots$ subfactor corresponding to $O_{11}^6$.
Note that $O_{11}^6$ denotes the 6th smallest object in $\cE\cH_1$, which is $P'$, so this subfactor is the reduced subfactor corresponding to $P'$.
For two of these intermediates the lower inclusion is Extended Haagerup while the upper inclusions have index $58.3502\ldots$ and come from the reduced subfactor construction for $O_{12}^3$ or $O_{12}^4$ (these are the odd vertices near the ends of the Extended Haagerup principal graph).  For the other two, the lower and upper parts are switched.
We next want to see how these fit together into a lattice.  From the next two lines in the table we see that the index $58.3502\ldots$ subfactors themselves each have a single intermediate, with one inclusion being Extended Haagerup and the other being the $2$-supertransitive subfactor of index $13.3305\ldots$ with principal graph shown in Equation (<ref>).
It follows that the lattice is a hexagon, where the upper and lower edges are the Extended Haagerup subfactor and the middle edges are the index $13.3305\ldots$ subfactors.

Note that since none of the other entries in the $X$ or $Y$ columns also occurs in the $Z$ column, other than the hexagon every lattice will just be $M_n$, the lattice with one maximal element, one minimal element, and $n$ incomparable elements between them.  The number of such incomparable entries is simply the number of rows with that $Z$; for example, $O_{13}^6$ has intermediate subfactor lattice $M_6$.

In addition to the hexagon there are a few notable examples of quadrilaterals where all inclusions are at least $2$-supertransitive.

* A quadrilateral from $O_{33}^6 = U$ which follows from $X \otimes X \cong U \cong \overline{X} \otimes \overline{X}$. This quadrilateral is commuting and cocommuting.  This quadrilateral suggests a possible diagrammatic presentation for $\cE\cH_3$, described in Appendix <ref>.

* A quadrilateral from $O_{22}^8 = \jw{6}$ which follows from $A \otimes A \cong \jw{6} \cong \jw{6} \cong B \otimes B$ in $\cE\cH_2$.  This quadrilateral is commuting and cocommuting.  As for the previous quadrilateral, this may lead to a diagrammatic presentation for $\cE\cH_2$.

* A quadrilateral from $O_{23}^5$ whose dual is a non-commuting but cocommuting $(2,2)$-supertransitive quadrilateral where the subfactor is the index $14.3305\ldots$ one from Equation (<ref>) and the upper quadrilateral is the index $13.3305\ldots$ from Equation (<ref>).

* A quadrilateral from $O_{34}^2$ whose dual is a non-commuting but cocommuting $(3,3)$-supertransitive quadrilateral where the subfactor is the index $8.0283\ldots$ one from Equation (<ref>) and the upper quadrilateral is the index $7.0283\ldots$ from Equation (<ref>).

In <cit.> non-commuting cocommuting supertransitive quadrilaterals are divided into three cases (called Classes II, III, and IV, while Class I referred to non-cocommuting) based on whether the Galois group is trivial, cyclic of order $2$, or cyclic of order $3$.
The third and fourth examples above both fit into Class II.
We expect that the fourth example is the smallest quadrilateral in Class II with indices above $4$.
Note that the smallest example with indices above $4$ in Class III comes from the Asaeda–Haagerup subfactor, while the smallest example with indices above $4$ in Class IV comes from the Haagerup subfactor.
It is striking that Extended Haagerup appears to be the smallest example in the remaining case.

§ MODULE CATEGORIES AND GRAPH PLANAR ALGEBRA EMBEDDINGS

The graph planar algebra embedding theorem from <cit.> states that any subfactor planar algebra embeds in the graph planar algebra <cit.> of either of its principal graphs.  Peters observed in <cit.> that it is possible for a subfactor planar algebra to embed in the graph planar algebra of other graphs; in particular she
found that the Haagerup planar algebra embeds in the graph planar algebra of a third graph, called the `broom'.
In this section we strengthen the graph planar algebra embedding theorem, to obtain a classification of embeddings in graph planar algebras. In particular, we show that a subfactor planar algebra embeds into the graph planar algebra of a bipartite graph if and only if the graph is the fusion graph of a unitary module category with a compatible trace.

We begin with the simple observation that a module category $\cM$ for a tensor category $\scrC$
is exactly the data of a tensor functor $\scrC \rightarrow \End(\cM)$.
As we proceed through this section, we elaborate this observation in various directions, eventually obtaining our theorem.
This involves four adjustments:

* describing endofunctors in $\End(\cM)$ as graphs,

* adapting to the shaded setting required for subfactor planar algebras,

* working in the unitary setting, and

* understanding the additional data corresponding to pivotal structures.

Note that in order to be able to characterize maps out of the Extended Haagerup subfactor planar algebra (and hence characterize modules for the module categories),
we will rely on the unitary pivotal structure (see Remark <ref>). Thus even if the reader is only interested in the algebraic classification of modules over the Extended Haagerup fusion categories, they still need to understand the unitary pivotal version of the graph planar algebra embedding theorem!

We will assume that the reader is familiar with tensor categories following <cit.>,
but we will not assume previous familiarity with graph planar algebras.
We take this pedagogical approach for several reasons.
First, it was this algebraic perspective that allowed us to see that one should expect a GPA embedding theorem for modules.
Thus this approach unifies
(unitary) module category classification results like <cit.>
GPA embedding constructions like <cit.>, which will hopefully make GPA embeddings more accessible to algebraists.
Second, an independent purely subfactor theoretic proof of our classification of embeddings will appear soon in <cit.>, using towers of algebras.
Subfactor experts may prefer to read that paper as a replacement for this section.

Here is a more detailed breakdown of this section.  In <ref> we recall some background on module categories, Morita equivalences, and the endofunctor embedding theorem (that giving a module category $\cM$ over $\scrC$ is the same as giving a functor $\scrC \rightarrow \End(\cM)$).  In <ref> we introduce an “unbiased" definition of monoidal categories which we call monoidal algebras.  Monoidal algebras are an analogue of planar algebras without rotational symmetry.  In Section <ref> we introduce the graph monoidal algebra (which is an analogue of the graph planar algebra), explain its relationship to $\End(\cM)$, and see that the endomorphism embedding theorem yields a graph monoidal algebra embedding theorem for module categories.  This is the simplest non-technical version of our main result, and contains the major idea of this section.

The rest of the section is dedicated to adapting the graph monoidal algebra embedding theorem for module categories to the pivotal and unitary pivotal settings where it becomes the graph planar algebra embedding theorem for appropriate pivotal and unitary pivotal analogues of module categories.  These analogues of module categories  involve both structure on $\cM$ and compatibility of that structure with the module action.
In the semisimple pivotal setting Schaumann <cit.> showed that the appropriate structure is a choice of trace on $\cM$.
In <ref>, we recall the definitions of planar algebra, unitary dual functors, and unitary pivotal structure, and we explain the relationship between planar algebras and unitary pivotal fusion categories.
In <ref>,  we recall Schaumann's notion of trace and modify this notion to the unitary setting, we then define (unitary) pivotal modules, prove a (unitary) pivotal analogue of endofunctor embedding, and translate that into the desired graph planar algebra embedding theorem.

§.§ Module categories, Morita equivalences, and endofunctor embedding

Recall that a multitensor category over $\mathbb{C}$ is a $\mathbb{C}$-linear abelian category together with a tensor product $\otimes$, a unit object $1_\scrC$, and unitors and associators satisfying natural axioms, where every object $X$ has a dual $X^\vee$.
We call $\scrC$ a tensor category if $1_\scrC$ is simple.
A multifusion category is a semisimple multitensor category with finitely many isomorphism classes of simples, and a fusion category is a multifusion category with $1_\scrC$ simple.
For further details see the book <cit.>.

Tensor categories can be thought of as categorical analogues of ordinary algebras.
Many ordinary algebraic notions have analogues for tensor categories, and in particular the analogues of modules, bimodules, and Morita equivalences play a key role in studying tensor categories, as pioneered by Ocneanu, Müger, Ostrik, and others <cit.>.
Again see <cit.> for further details.

In particular, we have the following two important problems about the “representation theory” of fusion categories.

[Classification of Modules]

Classify all indecomposable semisimple module categories over a given fusion category $\scrC$.

[Morita Equivalence]

Classify all fusion categories $\cD$ (up to tensor equivalence) which are Morita equivalent to $\scrC$, and all the Morita equivalences between them (up to bimodule equivalence).
Furthermore, understand the Brauer-Picard groupoid, which describes the compositions of these Morita equivalences under balanced tensor product ${}_\scrC \cM \boxtimes_\cD \cN_\cE$.

From a higher categorical perspective it is somewhat unnatural to only study equivalence classes, and it is more natural to consider Etingof-Nikshych-Ostrik's Brauer-Picard $3$-groupoid <cit.> which consists of fusion categories Morita equivalent to $\scrC$, Morita equivalences between them, bimodule equivalences between these Morita equivalences, and bimodule natural isomorphisms.
The higher structure of this $3$-groupoid is essential for classifying $G$-graded extensions of fusion categories.
The Morita equivalence problem asks for the fundamental $1$-groupoid of this $3$-groupoid.
As it turns out, for the examples considered in this paper, the higher structure of the $3$-groupoid is trivial (see Corollary <ref>).

The following theorem shows that the two problems above are closely related.
Recall that if $X$ is an invertible object, then conjugation by $X$ is an inner autoequivalence.

If $\scrC$ is a fusion category and $\cM$ is a semisimple $\scrC$-module category,
then $\cC-\cD$ bimodule category structures on $\cM$ which extend the $\cC$-module structure correspond exactly to functors $\cF: \cD \rightarrow \End_{\scrC}(\cM)$, and such a bimodule is a Morita equivalence if and only if $\cF$ is an equivalence of multitensor categories.
Two such bimodule categories are equivalent if and only if the functors differ by an inner autoequivalence.
Furthermore, $\End_{\scrC}(\cM)$ is a tensor category (with simple unit object) if and only if $\cM$ is indecomposable.

In particular, in order to solve the Problem <ref> about Morita equivalence, it is enough to solve Problem <ref> about modules, and further solve the following.

[Outer Automorphisms]
For each $\cD$ in the Morita equivalence class of $\scrC$, find the outer automorphism group of $\cD$.

None of the fusion categories we study in this article have outer automorphisms.
Thus classifying modules and Morita equivalences are essentially the same.
However, the reader should note that given $\scrC$ and $\cM$, actually calculating the structure of the dual category $\End_{\scrC}(\cM)$ may be quite difficult.
The dual categories $\End_{\scrC}(\cM)$ are essentially the same thing as the dual parts of GHJ subfactors <cit.>.
We refer the reader to <cit.> for a notable concrete example where understanding the detailed structure of the dual category is difficult.

All of the above problems are “external” problems, relating $\scrC$ to other tensor categories and module categories.
However, they are closely related by a theorem of Ostrik <cit.> to “internal” problems about algebra objects or Q-systems inside $\scrC$.
Two such algebras are internally Morita equivalent if there is an invertible bimodule object between them.

Given $A$, a connected semisimple algebra, $\Mod_\scrC(A)$ is an indecomposable module category. Moreover every indecomposable $\cC$-module category $\cM$ is equivalent to one of this form, by taking $A = \underline{\End}_\cC(m)$ for any simple $m \in \cM$.

The collection of connected semisimple algebras $\left \{B \mid \Mod_\scrC(B) \cong \Mod_\scrC(A) \right \}$ is exactly the internal Morita equivalence class of $A$.

The dual category $\End_{\scrC}(\Mod_\scrC(A))$ is canonically identified with the category of $A-A$ bimodules in $\cC$.

This theorem shows that the above problems are closely related to Ocneanu's “maximal atlas” <cit.>.

Let $\scrC$ be a fusion category.
A maximal atlas for $\scrC$ is a choice of a semisimple connected algebra $A$ in each internal Morita equivalence class.
From such a maximal atlas, one gets a collection of fusion categories $\Bim_\scrC(A,A)$ and Morita equivalences $\Bim_\scrC(A,B)$.[
The distinction between thinking of the maximal atlas as a collection of algebras and bimodules or as a collection of tensor categories and Morita equivalences is often elided.

In general, a maximal atlas will contain less information than the Brauer-Picard groupoid, because it does not remember the tensor equivalences between the fusion categories $\Bim_\scrC(A,A)$

For $\scrC = \Vec(\mathbb{Z}/3\mathbb{Z})$, a maximal atlas is given by $1$ and the group algebra $A = \mathbb{C}[\mathbb{Z}/3\mathbb{Z}]$ (with each group element in its own grade).
The category of bimodules $\Bim_\scrC(A,A)$ is $\Rep(\mathbb{Z}/3\mathbb{Z})$, which is (non-canonically!) equivalent to $\Vec(\mathbb{Z}/3\mathbb{Z})$.
The outer automorphism group of  $\Vec(\mathbb{Z}/3\mathbb{Z})$ is the group of units $(\mathbb{Z}/3\mathbb{Z})^\times$ acting by permuting simple objects, so we get two distinct equivalences $\Rep(\mathbb{Z}/3\mathbb{Z}) \cong\Vec(\mathbb{Z}/3\mathbb{Z})$.
Thus the aforementioned maximal atlas of $\scrC$ consists of two tensor categories (which happen to be tensor equivalent) and a single bimodule between the two, while the Brauer-Picard groupoid consists of one tensor category and four Morita autoequivalences.

One can then determine the group structure of this set of four autoequivalences.
By a result of Etingof-Nikshych-Ostrik <cit.>, this Brauer-Picard group must be the split orthogonal group $O_2(\mathbb{F}_3 \oplus \mathbb{F}_3^*)$, which is the Klein four group.
Note that in the maximal atlas formalism one can not even ask about the structure of this group.
In a sense the maximal atlas is a “universal cover” of the Brauer-Picard groupoid, and has lost all the interesting topological information about the latter (while still retaining the combinatorial information).
However, for all examples in this article, the Brauer-Picard group is trivial, and so these subtleties between the Brauer-Picard groupoid and the maximal atlas do not play an important role.
(In contrast, this distinction was critical in the study of the Asaeda–Haagerup subfactor <cit.>, which has Brauer-Picard group the Klein four group.)

Just as a module $M$ over an algebra $A$ is equivalent to a homomorphism $A \to \End(M)$,
module categories $\cM$ over $\scrC$ are equivalent to tensor functors $\scrC \rightarrow \End(\cM)$ <cit.>.  Thus the module classification problem is equivalent to the following.

[Endofunctor embedding]
Classify all semisimple categories $\cM$ and all tensor functors $\scrC \rightarrow \End(\cM)$, up to conjugation by an autoequivalence of $\cM$.

The following omnibus theorem summarizes much of the above.

Suppose that $\scrC$ is a fusion category.

* Module category structures on a semisimple category $\cM$ correspond exactly to tensor functors $\scrC \rightarrow \End(\cM)$.

* A fusion category $\cD$ is Morita equivalent to $\scrC$ if and only if there is an indecomposable semisimple $\cC$-module category $\cM$ such that $\cD$ is tensor equivalent to $\End_\scrC(\cM)$.
Furthermore the Morita equivalences ${}_\scrC\cN_\cD$ such that ${}_\scrC\cN$ is equivalent to ${}_\scrC \cM$ are a torsor for the group of outer automorphisms $\mathrm{Out}(\cD)$.

* Pairs $(\cM, m)$, where $\cM$ is a semisimple indecomposable $\scrC$-module category and $m \in \cM$ is a simple object, correspond exactly to connected semisimple algebras $A$ in $\scrC$, via $A\mapsto \Mod_{\scrC}(A)$ and $(\cM,m) \mapsto \underline{\End}_{\scrC}(m)$.
The dual category $\End_\cC(\cM)$ corresponding to $A$ is the category of $A$-$A$ bimodules in $\scrC$.

§.§.§ Modules for multifusion categories

Recall that a multifusion category $\scrC$ is like a fusion category, except $1_\scrC$ is no longer simple.
Since $\scrC$ is semisimple and $\scrC(1_\scrC \to 1_\scrC)$ is a commutative algebra, $1_\scrC$ breaks up as a sum of $r$ distinct simple objects $1_\scrC = \bigoplus_{i=1}^r 1_i$.
We call such a multifusion category $r$-shaded.
We denote by $\scrC_{ij}$ the summand $1_i\otimes \scrC \otimes 1_j$.

If $\scrC$ is an $r$-shaded multifusion category, then each $\scrC_{ii}$ is a fusion category. When $\scrC$ is indecomposable as a multifusion category each $\scrC_{ij}$ is a Morita equivalence between $\scrC_{ii}$ and $\scrC_{jj}$.  Furthermore, the tensor product map $\scrC_{ij} \boxtimes_{\scrC_{jj}} \scrC_{jk} \rightarrow \scrC_{ik}$ is an equivalence.

Conversely, given fusion categories $\cD_{11},\dots, \cD_{rr}$ and a Morita equivalence $\cD_{1j}$ between $\cD_{11}$ and $\cD_{jj}$ for each $1< j\leq r$, we define $\cD_{ik}:= \cD_{1i}^{-1}\boxtimes_{\cD_{11}}  \cD_{1k}$ for each $i,k\in \{1,\dots, r\}$ to get an indecomposable multifusion category $\cD = \bigoplus_{i,k=1}^r (\cD_{ik})$. These constructions are mutually inverse.

One can avoid the relative tensor product to obtain a multifusion category equivalent to $\cD$ as follows.
First, choose a simple object $d_i\in \cD_{1i}$ for all $i=1,\dots, r$, and consider the connected algebra objects $A_i = \underline{\End}_{\cD_{11}}(d_i)$.
Then $\cD$ is equivalent to the category of $A-A$ bimodules internal to $\cD_{11}$ where $A=\bigoplus_{i=1}^r A_i$.

The forward direction is <cit.> where instead of a grading group we have a grading by the groupoid of standard matrix units $E_{ij}$.  The proof for groupoids is parallel to the proof for groups.  (See also <cit.> which shows the first two parts.)

For the converse direction $\bigoplus_{i,k=1}^r  \cD_{1i}^{-1}\boxtimes_{\cD_{11}}  \cD_{1k}$ has a monoidal structure given by

\begin{align*}
\left(\cD_{1i}^{-1}\boxtimes_{\cD_{11}}  \cD_{1k}\right) \boxtimes \left(\cD_{1k}^{-1}\boxtimes_{\cD_{11}}  \cD_{1\ell}\right)
& \rightarrow \cD_{1i}^{-1}\boxtimes_{\cD_{11}}  \cD_{1k} \boxtimes_{\cD_{kk}} \cD_{1k}^{-1}\boxtimes_{\cD_{11}}  \cD_{1\ell} \displaybreak[1] \\
& \rightarrow \cD_{1i}^{-1}\boxtimes_{\cD_{11}}  \cD_{11} \boxtimes_{\cD_{11}}  \cD_{1\ell}   \displaybreak[1] \\
& \rightarrow \cD_{1i}^{-1}\boxtimes_{\cD_{11}} \cD_{1\ell}
\end{align*}

That this monoidal category is rigid follows from the identification of the inverse of a Morita equivalence $\cD_{ij}$ with the opposite category $\cD_{ij}^{\text{op}}$ from <cit.>, where the dual of an object in $\cD_{ij}$ is the same object thought of in $\cD_{ij}^{\text{op}} \cong \cD_{ji}$.

To each multifusion category $\scrC$ there is a corresponding rigid $2$-category whose objects are the indices, whose 1-morphisms are the objects in the $\scrC_{ij}$, and whose $2$-morphisms are the morphisms in $\scrC_{ij}$.  There is not an important difference between this $2$-category and the multifusion category, but in this paper, we use the multifusion language to align with the results of <cit.>.

The non-pivotal algebraic analogue of an irreducible finite depth subfactor $N \subset M$ is a pair $(\scrC, A)$ where $\scrC$ is a fusion category (which corresponds to the $N-N$ bimodules generated by $M$) and $A\in \scrC$ is a semisimple connected algebra object (which corresponds to $M$).
Given such an $A \in \scrC$, we get a Morita equivalence $\Mod_\scrC(A)= \Bim_\scrC(1_\scrC, A)$ between $\scrC$ and $\Bim_\scrC(A)$, and we get a 2-shaded indecomposable multifusion category as in Footnote <ref> by

\Bim_\scrC(1_\scrC \oplus A , 1_\scrC \oplus A)
\begin{pmatrix}
\Bim_\scrC(1_\scrC, 1_\scrC) & \Bim_\scrC(1_\scrC, A)
\\
\Bim_\scrC(A, 1_\scrC) & \Bim_\scrC(A,A)
\end{pmatrix}

with tensor product given by $\otimes_\scrC$, $\otimes_A$, or zero as appropriate.
Notice that $\Bim_\scrC(1_\scrC, 1_\scrC) = \scrC$.

There's an analogue of Prop. <ref> for module categories.

Suppose that $\cD$ is an $r$-shaded multifusion category with components $\cD_{ij}$.  Suppose that $\cM$ is an indecomposable module category over $\cD$.  Then $\cM = \bigoplus_{j=1}^r \cM_j$ where $\cM_i =1_i \vartriangleright \cM$.  Furthermore, the action maps $\cD_{ij} \boxtimes_{\cD_{jj}} \cM_j \rightarrow \cM_i$ are equivalences.

Conversely, given an indecomposable module category $\cM_1$ over $\cD_{11}$, we define
$$\cF(\cM_1) := \bigoplus_{i=1}^r \cD_{i1} \boxtimes_{\cD_{11}} \cM_1.$$
We can endow $\cF(\cM_1)$ with the structure of a $\cD$-module category via

$$\cD_{kj} \boxtimes \cF(\cM_1) \cong \bigoplus_{i=1}^r \cD_{kj} \boxtimes \cD_{i1} \boxtimes_{\cD_{11}} \cM_1 \rightarrow \cD_{kj} \boxtimes_{\cD_{jj}} \cD_{j1} \boxtimes_{\cD_{11}} \cM_1 \cong \cD_{k1} \boxtimes_{\cD_{11}} \cM_1 \subseteq \cF(\cM_1).$$

These constructions are mutually inverse with the isomorphism $\cF(\cM_1)\rightarrow \cM$ being the direct sum of the action maps $\cD_{i1} \boxtimes_{\cD_{11}} \cM_1 \cong \cM_i$.

The only nontrivial step is that $\cD_{ij} \boxtimes_{\cD_{jj}} \cM_j \rightarrow \cM_i$ is an equivalence.  This follows either by the techniques of <cit.> or of <cit.>.
Choose a simple object $m_j$ in $\cM_j$, and let $B_j = \underline{\End}_{\cD_{jj}}(m_j)$ be the internal endomorphisms of $m_j$ in $\cD_{jj}$.
Similarly, choose a simple object $x_{ij}$ in $\cD_{ij}$ and let $A_{ij}={}^\vee x_{ij} \otimes x_{ij}$ be its internal endomorphism algebra in $\cD_{jj}$.
We have an equivalence:

\cM \rightarrow \Bim_{\cD}(A_{ij},B_j)

via $m \mapsto {}^\vee x_{ij} \otimes \underline{\Hom}_{\cD}(m,m_j)$.
The restriction of this functor to $\cM_{i}$ then gives an inverse to the map  $\cD_{ij} \boxtimes_{\cD_{jj}} \cM_j \rightarrow \cM_i$.

Thus classifying modules for $\cD_{11}$ (answering Problem <ref>) is equivalent to classifying modules for $\cD$.
In particular, given an algebraic analogue of a subfactor $A \in \scrC$, we can instead solve the module problem over the corresponding indecomposable $2$-shaded multifusion category $\Bim_\scrC(1_\scrC \oplus A, 1_\scrC \oplus A)$ which is the purely algebraic, non-pivotal analogue of the subfactor planar algebra.
That is, we construct module categories for $\cE\cH_1$ by constructing module categories over the indecomposable $2$-shaded multifusion category which combines $\cE\cH_1$ and $\cE\cH_2$. This strategy is successful because the extended Haagerup subfactor planar algebra has a better skein theoretic description than either of the fusion categories $\cE\cH_1$ and $\cE\cH_2$ individually.

§.§.§ Module Cstar categories for unitary multitensor categories

In the nomenclature of <cit.>,
a unitary multitensor category $\scrC$ is a Cauchy complete rigid tensor $\Cstar$ category, which is semisimple by <cit.>.
We call $\scrC$ a unitary tensor category if $1_\scrC$ is simple.
Similar to the above characterization of module categories, given a (Cauchy complete) $\Cstar$ category $\cM$, endowing $\cM$ with the structure of a $\scrC$-module $\Cstar$ category is equivalent to supplying a dagger tensor functor $\scrC \to \End^\dag(\cM)$, the $\Cstar$ category of dagger endofunctors of $\cM$.[
In order for $\End^\dag(\cM)$ to be $\Cstar$, we only work with bounded natural transformations, i.e., those $\theta: F \Rightarrow G$ such that $\sup_{c\in \scrC} \|\theta_c\| <\infty$.
One then defines $\theta^\dag$ component-wise: $(\theta^\dag)_c := (\theta_c)^\dag$.
We provide the following lemma for those less familiar with $\Cstar$ categories, which also appears as <cit.>.

Suppose $\scrC$ is a unitary multitensor category and $\cM$ is a $\Cstar$ category.
Equipping $\cM$ with the structure of a $\scrC$-module $\Cstar$ category is equivalent to supplying a dagger tensor functor $(\Psi,\mu):\scrC \to \End^\dag(\cM)$.

We show how each structure induces the other, and we leave it to the reader to check these two processes are mutually inverse (up to dagger equivalence).

Suppose $\cM$ is a $\scrC$-module dagger category.
Note that $c\vartriangleright - $ is a dagger functor in $\End^\dag(\cM)$ for each $c\in \scrC$.
Moreover, if $f\in \scrC(a\to b)$, then $(f\vartriangleright - )^\dag = f^\dag \vartriangleright -$.
Hence $\Psi:\scrC \to \End^\dag(\cM)$ given by $\Psi(c) = c\vartriangleright -$ and $\Psi(f) = f\vartriangleright -$ defines a dagger functor.
Now defining

\mu^{a,b} : \Psi(a) \circ \Psi(b) = a\vartriangleright b\vartriangleright - \Rightarrow a\otimes b\vartriangleright - = \Psi(a\otimes b)

by $\mu^{a,b}_m := \alpha_{a,b,m} : a\vartriangleright b\vartriangleright m \to a\otimes b \vartriangleright m$ defines a unitary natural isomorphism, equipping $\Psi$ with the structure of a dagger tensor functor.

Conversely, suppose $(\Psi,\mu) : \scrC \to \End^\dag(\cM)$ is a dagger tensor functor.
For $c\in \scrC$ and $m\in \cM$, define $c\vartriangleright m := \Psi(c)(m)$.
For $c\in\scrC$ and $g\in \cM(m\to n)$, define $\id_c \vartriangleright g := \Psi(c)(f)$.
For $f\in \scrC(a\to b)$ and $m\in \cM$, define $f\vartriangleright \id_m := \Psi(f)_m$.
To show that $\vartriangleright : \scrC \times \cM \to \cM$ defines a bifunctor, it suffices to prove the exchange relation, which follows immediately from naturality.
That is, for $f\in \scrC(a\to b)$ and $g\in \cM(m\to n)$, the following diagrams commute:

\begin{equation*}
\begin{tikzcd}%[column sep]
\Psi(c)(m)\ar[r, "\Psi(f)_m"] \ar[d, "\Psi(c)(g)"]  & \Psi(d)(m)\ar[d, "\Psi(d)(g)"]
\\
\Psi(c)(n)\ar[r, "\Psi(f)_n"]  & \Psi(d)(n)
\end{tikzcd}
\qquad
\qquad
\begin{tikzcd}%[column sep]
c\vartriangleright m \ar[r, "f\vartriangleright \id_m"] \ar[d, "\id_c \vartriangleright g"]  &d \vartriangleright m\ar[d, "\id_d \vartriangleright g"]
\\
c \vartriangleright m\ar[r, "f\vartriangleright \id_n"]  &d \vartriangleright n
\end{tikzcd}
%(\id_c\vartriangleright g)\circ (f \vartriangleright \id_m)
%(f \vartriangleright \id_m) \circ(\id_c\vartriangleright g).
\end{equation*}

W define the natural unitary associator isomorphism $\alpha_{a,b,m} \in \cM( a\vartriangleright b \vartriangleright m \to a\otimes b \vartriangleright m)$ by $\alpha_{a,b,m} := \mu^{a.b}_m : [\Psi(a) \circ \Psi(b)](m) \to \Psi(a\otimes b)(m)$.
Notice that $\mu^{a,b} : \Psi(a)\circ \Psi(b) \Rightarrow \Psi(a\otimes b)$ is unitary if and only if $\mu^{a,b}_m$ is unitary for all $m\in \cM$.
Now one calculates
$(f \vartriangleright \id_m)^\dag = \Psi(f)_m^\dag :=(\Psi(f)^\dag)_m = \Psi(f^\dag)_m = f^\dag \vartriangleright \id_m$
$(\id_c \vartriangleright g)^\dag = \Psi(c)(g)^\dag = \Psi(c)(g^\dag) = \id_c \vartriangleright g^\dag$.
Thus $\cM$ is a $\scrC$-module dagger category.

We do not yet state
a $\Cstar$ version of Theorem <ref> as the above theorem implicitly uses rigidity for the statements on Morita equivalence and algebras.
When $\scrC$ is $\Cstar$, it is natural to impose compatibility conditions between the duality functor (implementing rigidity) and the dagger structure.
We will explain this in detail in <ref> below.

§.§ Monoidal algebras

Most algebraic structures have both a biased definition, like the usual definition of an algebra which emphasizes multiplying exactly two elements together, and an unbiased definition, like the definition of an algebra in which you can multiply arbitrary strings.[
See <cit.> for a delightful elementary discussion of the unbiased definition of an algebra.
The usual definition of monoidal category is biased as it emphasizes tensoring two objects and composing two morphisms.
In Definition <ref> below, we give an unbiased definition of monoidal category using the graphical calculus; we will see in <ref> below that planar algebras are the analogous unbiased definition of a pivotal monoidal category.

As a warm-up, we first recall the diagrammatic version of the unbiased definitions of an algebra and of a linear category.

The $E_1$-operad (or little intervals operad) consists of 1D Swiss cheese diagrams <cit.> consisting of a large interval, several removed subintervals called holes, all considered up to diffeomorphism.[In the literature, $E_1$ is an $\infty$-operad, which means that instead of being considered up to isotopy, we instead have the isotopies induce higher isomorphisms.
Since we only care about algebras over operads valued in the ordinary category of vector spaces, we can safely ignore this issue.]
These interval diagrams can be composed by plugging some new big intervals into the holes to get a new diagram.

\left(
\begin{tikzpicture}[baseline=-.1cm, scale=.7]
\draw[very thick] (0,0) -- (1,0);
\draw[very thick] (2,0) -- (3,0);
\draw[very thick] (4,0) -- (5,0);
\filldraw[thick] (0,0) circle (.05cm);
\filldraw[thick, fill=white] (1,0) circle (.05cm);
\filldraw[thick, fill=white] (2,0) circle (.05cm);
\filldraw[thick, fill=white] (3,0) circle (.05cm);
\filldraw[thick, fill=white] (4,0) circle (.05cm);
\filldraw[thick] (5,0) circle (.05cm);
\node at (1.5,0) {\scriptsize{$1$}};
\node at (3.5,0) {\scriptsize{$2$}};
\end{tikzpicture}
\right)
\circ_2
\left(
\begin{tikzpicture}[baseline=-.1cm, scale=.7]
\draw[very thick] (0,0) -- (1,0);
\draw[very thick] (2,0) -- (3,0);
\draw[very thick] (4,0) -- (5,0);
\filldraw[thick] (0,0) circle (.05cm);
\filldraw[thick, fill=white] (1,0) circle (.05cm);
\filldraw[thick, fill=white] (2,0) circle (.05cm);
\filldraw[thick, fill=white] (3,0) circle (.05cm);
\filldraw[thick, fill=white] (4,0) circle (.05cm);
\filldraw[thick] (5,0) circle (.05cm);
\node at (1.5,0) {\scriptsize{$1$}};
\node at (3.5,0) {\scriptsize{$2$}};
\end{tikzpicture}
\right)
\left(
\begin{tikzpicture}[baseline=-.1cm, scale=.7]
\draw[very thick] (0,0) -- (1,0);
\draw[very thick] (2,0) -- (3,0);
\draw[very thick] (4,0) -- (5,0);
\draw[very thick] (6,0) -- (7,0);
\filldraw[thick] (0,0) circle (.05cm);
\filldraw[thick, fill=white] (1,0) circle (.05cm);
\filldraw[thick, fill=white] (2,0) circle (.05cm);
\filldraw[thick, fill=white] (3,0) circle (.05cm);
\filldraw[thick, fill=white] (4,0) circle (.05cm);
\filldraw[thick, fill=white] (5,0) circle (.05cm);
\filldraw[thick, fill=white] (6,0) circle (.05cm);
\filldraw[thick] (7,0) circle (.05cm);
\node at (1.5,0) {\scriptsize{$1$}};
\node at (3.5,0) {\scriptsize{$2$}};
\node at (5.5,0) {\scriptsize{$3$}};
\end{tikzpicture}
\right)

An $E_1$-algebra in vector spaces is an algebra for this operad, which means it consists of a vector space $A$ together with a linear map $A^{\otimes h} \rightarrow A$ attached to each linear Swiss cheese diagram with $h$ holes.
These maps must be compatible with the operad structure (i.e., plugging elements of $A$ into holes, and plugging diagrams into larger diagrams, associates).
Unpacking this definition, an $E_1$-algebra in vector spaces consists of multiplication maps $\mu_n: A^{\otimes n} \rightarrow A$ for every natural number $n$ ($n=0$ gives the unit) which satisfy the appropriate associativity relations.
This is exactly the unbiased definition of a unital associative algebra.

The colored operad of linear tangles with label set $S$ consists of a large interval with several holes removed, together with a labelling by an element of $S$ for each connected component of the diagram modulo diffeomorphism.
(To match this up with future examples, it helps to think of these components as “strings" connecting each hole to the next hold or to the outside interval.)

\begin{tikzpicture}[baseline=-.1cm, scale=.7]
\draw[very thick] (0,0) -- node[above] {\scriptsize{$w$}} (1,0);
\draw[very thick] (2,0) -- node[above] {\scriptsize{$x$}} (3,0);
\draw[very thick] (4,0) -- node[above] {\scriptsize{$y$}} (5,0);
\draw[very thick] (6,0) -- node[above] {\scriptsize{$z$}} (7,0);
\filldraw[thick] (0,0) circle (.05cm);
\filldraw[thick, fill=white] (1,0) circle (.05cm);
\filldraw[thick, fill=white] (2,0) circle (.05cm);
\filldraw[thick, fill=white] (3,0) circle (.05cm);
\filldraw[thick, fill=white] (4,0) circle (.05cm);
\filldraw[thick, fill=white] (5,0) circle (.05cm);
\filldraw[thick, fill=white] (6,0) circle (.05cm);
\filldraw[thick] (7,0) circle (.05cm);
\node at (1.5,0) {\scriptsize{$1$}};
\node at (3.5,0) {\scriptsize{$2$}};
\node at (5.5,0) {\scriptsize{$3$}};
\end{tikzpicture}

Again the operadic structure comes from gluing linear tangles into the holes, but since substitution only makes sense when the labels match, this is a colored operad.
An algebra ${}_\bullet V_\bullet$ for this operad consists of a family of vector spaces $\{{}_{x} V_y\}_{x,y\in S}$ together with an action of linear tangles with holes.
That is, to each linear tangle $T$ with components labelled by $x_1,\dots, x_n$, we get a linear map
$Z(T) : {}_{x_1}\! V_{x_2} \otimes \cdots  \otimes{}_{x_{n-1}}\!V_{x_n} \rightarrow {}_{x_1}\!V_{x_n}$ which is compatible with composition of linear tangles with holes.
It is not difficult to see that an algebra for the operad of linear tangles with label set $S$ gives an unbiased definition of a linear category whose set of objects is $S$, whose hom spaces $\Hom(x\to y)$ are the vector spaces ${}_{x} V_{y}$, and whose composition of morphisms is the action of linear tangles.  The identity morphisms come from the tangles with no holes.

We now give an unbiased definition of monoidal category.
This is quite similar to the above definition, but we now have two dimensional diagrams where the vertical direction represents composition using labelled strings as before, and the horizontal direction represents tensor product without strings.

A monoidal tangle with label set $S$ is a rectangle, with several smaller rectangles (with edges parallel to those of the big one) removed, and some non-crossing smooth strings labelled by elements of $S$ which are oriented upward, have no minima or maxima, and begin and end on the tops or bottoms of the rectangles.
We say a monoidal tangle $T$ has type $((s_0,t_0); (s_1,t_1),\dots, (s_k,t_k))$ where $s_0,\dots, s_k, t_0,\dots, t_k$ are finite words on $S$ if the tangle $T$ has $k$ input rectangles, and there are $|s_i|, |t_i|$ strings attached to the bottom and top respectively of the $i$-th rectangle (the zeroth rectangle is the output rectangle and $1\leq i\leq k$ corresponds to the $i$-th input rectangle), which are labelled by the characters in the words $s_i, t_i$ respectively.
Here is an example of a tangle with
$S = \{
\ColorDot{red}
\,,\,
\ColorDot{blue}
\,,\,
\ColorDot{DarkGreen}
\}$,
where we color the strings instead of labelling them:

\begin{tikzpicture}[baseline = -.1cm, scale=.6]
\draw[very thick, rounded corners = 5pt] (-2,-2) rectangle (2,2);
\draw[very thick, rounded corners = 5pt] (-1.5,.5) rectangle (-.5,1.5);
\draw[very thick, rounded corners = 5pt] (-1,-1.5) rectangle (.5,-.5);
\draw[very thick, rounded corners = 5pt] (.25,.5) rectangle (1.25,1.5);
\node at (-1,1) {\scriptsize{$1$}};
\node at (-.25,-1) {\scriptsize{$2$}};
\node at (.75,1) {\scriptsize{$3$}};
\draw[thick, red] (-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.75,-.5);
\draw[thick, blue] (-1.5,-2) .. controls ++(90:.5cm) and ++(270:.5cm) .. (-1.2,.5);
\draw[thick, blue] (.25,-.5) .. controls ++(90:.5cm) and ++(270:.5cm) .. (.5,.5);
\draw[thick, DarkGreen] (-1,1.5) -- (-1,2);
\draw[thick, red] (-.5,-2) -- (-.5,-1.5);
\draw[thick, red] (-.25,-.5) -- (-.25,2);
\draw[thick, DarkGreen] (0,-2) -- (0,-1.5);
\draw[thick, DarkGreen] (1,-2) -- (1,.5);
\draw[thick, blue] (1.5,-2) -- (1.5,2);
\end{tikzpicture}
\qquad
\text{has type}
\qquad
\underbrace{\ColorDot{blue}\,\ColorDot{red}\,\ColorDot{DarkGreen}\,\ColorDot{DarkGreen}\,\ColorDot{blue}}_{s_0}
\,,\,
\underbrace{\ColorDot{DarkGreen}\,\ColorDot{red}\,\ColorDot{blue}}_{t_0}
\underbrace{\ColorDot{blue}\,\ColorDot{red}}_{s_1}
\,,\,
\underbrace{\ColorDot{DarkGreen}}_{t_1}
\underbrace{\ColorDot{red}\,\ColorDot{DarkGreen}}_{s_2}
\,,\,
\underbrace{\ColorDot{red}\,\ColorDot{red}\,\ColorDot{blue}}_{t_2}
\underbrace{\ColorDot{blue}\,\ColorDot{DarkGreen}}_{s_3}
\,,\,
\underbrace{\emptyset}_{t_3}

Monoidal tangles are considered up to isotopy (through diagrams that again have no minima or maxima).
Monoidal tangles form a colored operad, because you can insert monoidal tangles into the rectangles of a large monoidal tangle to get a new monoidal tangle.

A monoidal algebra with label set $S$ is an algebra for the operad of monoidal tangles with label set $S$.
Unpacking this definition, a monoidal algebra $\cP_{\bullet \to \bullet}$ consists of a family of finite dimensional vector spaces $\cP_{s\to t}$ where $s,t$ are finite words in $S$, together with an action of monoidal tangles.
To each monoidal tangle $T$ of type $((s_0,t_0); (s_1,t_1,\dots, (s_k,t_k)))$, we associate a multilinear map
$Z(T):\prod_{j=1}^k \cP_{s_j\to t_j}\to \cP_{s_0 \to t_0}$,
and composition of monoidal tangles corresponds to composition of multilinear maps.
Here is an an  example:

\begin{tikzpicture}[baseline = -.1cm, scale=.6]
\draw[very thick, rounded corners = 5pt] (-2,-2) rectangle (2,2);
\draw[very thick, rounded corners = 5pt] (-1.5,.5) rectangle (-.5,1.5);
\draw[very thick, rounded corners = 5pt] (-1,-1.5) rectangle (.5,-.5);
\draw[very thick, rounded corners = 5pt] (.25,.5) rectangle (1.25,1.5);
\node at (-1,1) {\scriptsize{$1$}};
\node at (-.25,-1) {\scriptsize{$2$}};
\node at (.75,1) {\scriptsize{$3$}};
\draw[thick, red] (-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.75,-.5);
\draw[thick, blue] (-1.5,-2) .. controls ++(90:.5cm) and ++(270:.5cm) .. (-1.2,.5);
\draw[thick, blue] (.25,-.5) .. controls ++(90:.5cm) and ++(270:.5cm) .. (.5,.5);
\draw[thick, DarkGreen] (-1,1.5) -- (-1,2);
\draw[thick, red] (-.5,-2) -- (-.5,-1.5);
\draw[thick, red] (-.25,-.5) -- (-.25,2);
\draw[thick, DarkGreen] (0,-2) -- (0,-1.5);
\draw[thick, DarkGreen] (1,-2) -- (1,.5);
\draw[thick, blue] (1.5,-2) -- (1.5,2);
\end{tikzpicture}
\,\,\right)
\cP_{
\ColorDot{blue}\,\ColorDot{red}
\,\to\,
\ColorDot{DarkGreen}
\times
\cP_{
\ColorDot{red}\,\ColorDot{DarkGreen}
\,\to\,
\ColorDot{red}\,\ColorDot{red}\,\ColorDot{blue}
\times
\cP_{
\ColorDot{blue}\,\ColorDot{DarkGreen}
\,\to\,
\emptyset
\to
\cP_{
\ColorDot{blue}\,\ColorDot{red}\,\ColorDot{DarkGreen}\,\ColorDot{DarkGreen}\,\ColorDot{blue}
\,\to\,
\ColorDot{DarkGreen}\,\ColorDot{red}\,\ColorDot{blue}
%\underbrace{\scrC(X^{\otimes 2}\to X)}_{\cP_{2\to 1}}
%\underbrace{\scrC(X^{\otimes 2} \to X^{\otimes 3})}_{\cP_{2\to 3}}
%\underbrace{\scrC(X^{\otimes 2} \to 1_{\scrC})}_{\cP_{2\to 0}}
%\underbrace{\scrC(X^{\otimes 5} \to X^{\otimes 3})}_{\cP_{5\to 3}}.

A monoidal algebra is called semisimple if for every pair of words $s,t$ on $S$, the $2\times2$ linking algebra

\begin{equation}
\label{eq:LinkingAlgebra}
\cL(s,t)
\begin{pmatrix}
\cP_{s\to s}
\cP_{t\to s}
\\
\cP_{s\to t}
\cP_{t\to t}
\end{pmatrix}
\end{equation}

whose multiplication given by matrix multiplication together with the appropriate `stacking' multiplication tangles is a finite dimensional semisimple algebra.

Suppose $\scrC$ is a linear monoidal category with a set of objects $\mathscr{S}:=\{X_s\}_{s\in S}$ which monoidally generates $\scrC$, i.e., every object in $\scrC$ is isomorphic to a tensor product of objects in $\mathscr{S}$.
We define a monoidal algebra $\cP(\scrC, \mathscr{S})_{\bullet \to \bullet}$ with label set $S$ as follows.
For $s_1,\dots, s_k,t_1,\dots, t_\ell\in S$, we define

\cP(\scrC, \mathscr{S})_{s_1\cdots s_k\to t_1\cdots t_\ell} := \scrC(X_{s_1}\otimes \cdots \otimes X_{s_k} \to X_{t_1}\otimes \cdots \otimes X_{t_\ell}).

We use the convention that if $\emptyset$ is the empty word on $S$, then the empty tensor product of objects is $1_{\scrC}$.
The action of tangles is just the graphical calculus for tensor categories.
See <cit.> for a summary of many versions of the graphical calculus; additional resources include <cit.> and <cit.>.

The monoidal algebra $\cP(\scrC, X)_{\bullet \to\bullet}$ is similar in spirit to the way the term `monoidal algebra' is used in the work of Wenzl on constructing and classifying subfactors and fusion categories from quantum groups <cit.> which is based on the original towers of algebras approach to subfactor theory <cit.>.

There is an equivalence of categories [
Pairs $(\scrC, \{X_{s}\}_{s\in S})$ form a $2$-category where between any two 1-morphisms, there is at most one 2-morphism, which is necessarily invertible when it exists <cit.>.
Hence this 2-category is equivalent to its truncation to a 1-category.

\[
\left\{\,
\parbox{7cm}{\rm Monoidal algebras $\cP_{\bullet\to \bullet}$ with label set $S$ and finite dimensional box spaces $\cP_{m\to n}$}\,\left\}
\,\,\,\,\cong\,\,
\left\{\,\parbox{7.8cm}{\rm Pairs $(\scrC, \{X_s\}_{s\in S})$ with $\scrC$ a linear monoidal category with generators $X_s\in \scrC$ for $s\in S$}\,\right\}.
\right.\right.
\]

The equivalence here is given by taking the idempotent category of a monoidal algebra; the details are standard, following e.g. <cit.>. We leave as an exercise to the reader to check that the definition of semisimplicity given above corresponds to the usual definition for the Cauchy completion of the idempotent category.

In the case we care most about, $\scrC$ is a semisimple monoidal category, and it is hopeless to expect to have a finite monoidal generating set.
Instead, one typically has a collection of objects $\mathscr{S}:=\{X_s\}_{s\in S}$ labelled by $S$ which Cauchy tensor generates $\scrC$ in the sense that every object in $\scrC$ is a direct summand of a direct sum of tensor products of objects in $\mathscr{S}$.
This is not a big problem because of the following well-known theorem:

Suppose $\scrC$ is a semisimple monoidal category and $\mathscr{S}:=\{X_s\}_{s\in S}$ is a set of objects that Cauchy tensor generates $\scrC$.

* Let $\scrC_{\mathscr{S}}$ be the full monoidal subcategory of $\scrC$ whose objects are tensor products of objects in $\mathscr{S}$.  Then $\scrC_{\mathscr{S}}$ is a monoidal category which is monoidally generated by $\mathscr{S}$.

* The monoidal category $\scrC$ is monoidally equivalent to the idempotent completion of the additive envelope (also known as the Cauchy completion, pseudo-abelian envelope, or Karoubi envelope) of $\scrC_{\mathscr{S}}$.

Combining Theorems <ref> and <ref>, given a semisimple linear monoidal category $\scrC$ and a set $\mathscr{S}:=\{X_s\}_{s\in S}$ of objects that Cauchy tensor generate $\scrC$, we get a semisimple monoidal algebra from $\scrC_{\mathscr{S}}$.
Conversely, given a semisimple monoidal algebra $\cP_{\bullet \to \bullet}$ with label set $S$, we can take the idempotent completion of the corresponding monoidal category to recover the semisimple monoidal category $\scrC$ and a set of Cauchy tensor generating objects $\mathscr{S}:=\{X_s\}_{s\in S}$ corresponding to the strands labelled by $s\in S$.

There is an equivalence of categories
(see Footnote <ref>)

\[
\left\{\,
\parbox{5.5cm}{\rm Semisimple monoidal algebras $\cP_{\bullet\to \bullet}$ with label set $S$}\,\left\}
\,\,\,\,\cong\,\,
\left\{\,\parbox{7cm}{\rm Pairs $(\scrC, \{X_s\}_{s\in S})$ with $\scrC$ a semisimple linear monoidal category with Cauchy tensor generators $X_s\in \scrC$ for $s\in S$}\,\right\}.
\right.\right.
\]

Expanding on Example <ref>, given a semisimple linear monoidal category $\scrC$ and an object $X\in \scrC$ which Cauchy tensor generates $\scrC$, we get a semisimple monoidal algebra $\cP(\scrC, X)_{\bullet \to \bullet}$ with label set $\bbN_{\geq 0}$ by defining $\cP(\scrC, X)_{m\to n} := \scrC(X^{\otimes m} \to X^{\otimes n})$.

§.§.§ Shaded semisimple monoidal algebras and semisimple monoidal categories

We next extend the discussion in <ref> on $r$-shaded multifusion categories to the case of $r$-shaded semisimple monoidal categories.
Suppose $\scrC$ is a semisimple linear monoidal category, and $1_{\scrC} = \bigoplus_{i=1}^r 1_i$ is a decomposition of $1_\scrC$ into simples.
We call such a $\scrC$ an $r$-shaded semisimple monoidal category.
We write $\scrC_{ij} = 1_i \otimes \scrC \otimes 1_j$, and we note that $\scrC = \bigoplus_{i,j=1}^r \scrC_{ij}$.
We also have distinguished idempotents $p_i \in \scrC(1_\scrC \to 1_\scrC)$ corresponding to the summand $1_i$ for $1\leq i\leq r$.
In the graphical calculus, we represent these projections, which freely float about in their regions, as a single shading.
For example, we could denote

\tikz[baseline=.1cm]{\draw[fill=\ShadeOne, rounded corners=5, very thin, baseline=1cm] (0,0) rectangle (.5,.5);}=p_i
\qquad
\tikz[baseline=.1cm]{\draw[fill=\ShadeTwo, rounded corners=5, very thin, baseline=1cm] (0,0) rectangle (.5,.5);}=p_j

Then for objects $a,b\in \scrC_{ij}$, we would denote a morphism $f\in \scrC(a\to b)$ by

\begin{tikzpicture}
\fill[\ShadeOne] (-.5,-.7) rectangle (0,.7);
\fill[\ShadeTwo] (.5,-.7) rectangle (0,.7);
\draw (0,-.7) -- (0,.7);
\roundNbox{fill=white}{(0,0)}{.3}{0}{0}{$f$}
\end{tikzpicture}

This motivates the following definition.

An $R$-shaded monoidal tangle with label set $S$ is
a monoidal tangle with label set $S$ whose regions are shaded by the elements of $R$
such that
each element $x\in S$ has a left source shading $s_x\in R$ and a right target shading $t_y\in R$.
For example, for the shading set
$R = \{
\ColorDot{\ShadeOne}
\,,\,
\ColorDot{\ShadeTwo}
\,,\,
\ColorDot{\ShadeThree}
\}$,
and the label set
\{
\begin{tikzpicture}[baseline=-.1cm]
\fill[\ShadeOne] (-.2,-.2) rectangle (0,.2);
\fill[\ShadeTwo] (.2,-.2) rectangle (0,.2);
\draw (0,-.2) -- (0,.2);
\end{tikzpicture}
\,,\,
\begin{tikzpicture}[baseline=-.1cm, xscale =-1]
\fill[\ShadeOne] (-.2,-.2) rectangle (0,.2);
\fill[\ShadeTwo] (.2,-.2) rectangle (0,.2);
\draw (0,-.2) -- (0,.2);
\end{tikzpicture}
\,,\,
\begin{tikzpicture}[baseline=-.1cm]
\fill[\ShadeTwo] (-.2,-.2) rectangle (0,.2);
\fill[\ShadeThree] (.2,-.2) rectangle (0,.2);
\draw (0,-.2) -- (0,.2);
\end{tikzpicture}
\,,\,
\begin{tikzpicture}[baseline=-.1cm, xscale = -1]
\fill[\ShadeTwo] (-.2,-.2) rectangle (0,.2);
\fill[\ShadeThree] (.2,-.2) rectangle (0,.2);
\draw (0,-.2) -- (0,.2);
\end{tikzpicture}
\}
we have the following $R$-shaded monoidal tangle with label set $S$:

\begin{tikzpicture}[baseline = -.1cm, scale=.6]
\fill[\ShadeOne, rounded corners = 5pt] (-2,-2) rectangle (1,2);
\fill[\ShadeTwo, rounded corners = 5pt] (-.5,-2) rectangle (2,2);
\fill[\ShadeTwo] (-.8,1.5) rectangle (-1.2,2);
\fill[\ShadeTwo] (-1.5,-2) .. controls ++(90:.5cm) and ++(270:.5cm) .. (-1.2,.5) --
(-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.5,-.5) -- (-.5,-1.5) -- (-.5,-2) -- (-1.5,-2);
\fill[\ShadeThree] (0,-.5) .. controls ++(90:.5cm) and ++(270:.5cm) .. (.5,.5) --
(1,.5) -- (1,-2) -- (1.5,-2) -- (1.5,2) --(-.8,2) -- (-.8,1.5) --
(-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.5,-.5) -- (0,-.5);
\fill[\ShadeOne] (-.5,-2) rectangle (0,-1.5);
\draw[very thick, rounded corners = 5pt] (-2,-2) rectangle (2,2);
\draw[fill=white, very thick, rounded corners = 5pt] (-1.5,.5) rectangle (-.5,1.5);
\draw[fill=white, very thick, rounded corners = 5pt] (-1,-1.5) rectangle (.5,-.5);
\draw[fill=white, very thick, rounded corners = 5pt] (.25,.5) rectangle (1.25,1.5);
\node at (-1,1) {\scriptsize{$1$}};
\node at (-.25,-1) {\scriptsize{$2$}};
\node at (.75,1) {\scriptsize{$3$}};
\draw (-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.5,-.5);
\draw (-1.5,-2) .. controls ++(90:.5cm) and ++(270:.5cm) .. (-1.2,.5);
\draw (0,-.5) .. controls ++(90:.5cm) and ++(270:.5cm) .. (.5,.5);
\draw (-1.2,1.5) -- (-1.2,2);
\draw (-.8,1.5) -- (-.8,2);
\draw (-.5,-2) -- (-.5,-1.5);
\draw (0,-2) -- (0,-1.5);
\draw (1,-2) -- (1,.5);
\draw (1.5,-2) -- (1.5,2);
\end{tikzpicture}

An $R$-shaded monoidal algebra with label set $S$ is an algebra over the operad of $R$-shaded monoidal tangles with label set $S$.
Notice this means that the spaces $\cP_{x \to y}$ are only well-defined when
consecutive characters in the words $x$ and $y$ have compatible target and source shadings, and the source and target shadings of the words $x$ and $y$ agree.

We have the following shaded version Theorem <ref>.

There is an equivalence of categories
(see Footnote <ref>)

\[
\left\{\,
\parbox{5cm}{\rm Semisimple $\{1,\dots, r\}$-shaded monoidal algebras $\cP_{\bullet\to \bullet}$ with label set $S$}\,\left\}
\,\,\,\,\cong\,\,
\left\{\,\parbox{9cm}{\rm Pairs $(\scrC, \{X_y\}_{y\in S})$ with $\scrC$ an $r$-shaded semisimple monoidal category with
decomposition $1 = \bigoplus_{i=1}^r 1_i$ with Cauchy tensor generators $X_y\in \scrC_{s_y, t_y}$ for $y\in S$
\right.\right.
\]

§.§.§ Unitary monoidal algebras

A dagger monoidal algebra with label set $S$ is a monoidal algebra $\cP_{\bullet \to \bullet}$ with label set $S$ equipped with antilinear maps $\dag: \cP_{s \to t} \to \cP_{t\to s}$ for all words $s,t$ on $S$ such that

* $\dag\circ \dag = \id$ and

* for every monoidal tangle $T$, $T^\dag(x_1^\dag,\dots, x_k^\dag) = T(x_1,\dots, x_k)^\dag$ where $T^\dag$ denotes the vertical reflection of $T$ about the $x$-axis.

A dagger monoidal algebra is called a $\Cstar$ monoidal algebra if in addition

* Every $\dag$-algebra $\cP_{s\to s}$ with the stacking multiplication is a $\Cstar$ algebra,[

Being a $\Cstar$ algebra is a property of a complex $*$-algebra and not extra structure.
Indeed, every $\Cstar$ algebra has a unique $\Cstar$ norm, which can be recovered from the spectral radius, which is defined purely algebraically.
] and

* for all $f\in \cP_{s\to t}$, there is a $g\in \cP_{s\to s}$ such that $f^\dag \circ f = g^\dag \circ g$.

Finally, a unitary monoidal algebra is a semisimple $\Cstar$ monoidal algebra.

When $\scrC$ is a $\Cstar$ monoidal category, the unitary Cauchy completion (a.k.a unitary Karoubi envelope) is the orthogonal projection completion of the orthogonally additive envelope, i.e., we add formal orthogonal direct sums of objects, and then we take the category of orthogonal projections.
Since finite dimensional $\Cstar$ algebras are semisimple, we see that if $\scrC$ is a $\Cstar$ monoidal category whose endomorphisms spaces are finite dimensional, then the unitary Cauchy completion is semisimple.
In this case, we say a set of objects $\mathscr{S} = \{X_s\}_{s\in S}$ Cauchy tensor generates $\scrC$ if every object of $\scrC$ is unitarily isomorphic to an orthogonal direct summand of an orthogonal direct sum of tensor products of objects in $\mathscr{S}$.

We have the following unitary version Theorem <ref>.

There is an equivalence of categories
(see Footnote <ref>)

\[
\left\{\,
\parbox{6.3cm}{\rm (Semisimple) $\Cstar$ monoidal algebras $\cP_{\bullet\to \bullet}$ with label set $S$}\,\left\}
\,\,\,\,\cong\,\,
\left\{\,\parbox{7cm}{\rm Pairs $(\scrC, \{X_s\}_{s\in S})$ with $\scrC$ a (semisimple) $\Cstar$ monoidal category with Cauchy tensor generators $X_s\in \scrC$ for $s\in S$}\,\right\}.
\right.\right.
\]

§.§ Graph monoidal algebra embedding

In this section we relate endofunctor embeddings $\scrC \rightarrow \End(\cM)$ to embeddings of monoidal algebras into
graph monoidal algebras, which is the non-pivotal analog of embedding planar algebras into graph planar algebras.
We give a $2$-shaded multifusion version which applies to an algebraic analog of a finite depth subfactor standard invariant.

Let $J$ be a finite set.
The tensor category $\Vec(J\times J)$ of bi-$J$-graded vector spaces has objects finite dimensional vector spaces which decompose as direct sums $V = \bigoplus_{i,j\in J} V_{ij}$, morphisms linear maps which preserve the bi-grading, i.e., $f: V\to W$ is a sum $f=\sum_{ij} f_{ij}: V_{ij} \to W_{ij}$, and composition the composition of linear maps.
The tensor product of two bi-graded vector spaces is given by convolution

(V\otimes W)_{ik} := \bigoplus_{j\in J} V_{ij} \otimes W_{jk},

as is the tensor product of morphisms, i.e., if $f: V^1 \to V^2$ and $g: W^1 \to W^2$, then

(f\otimes g)_{ik}
\bigoplus_{j \in J}
f_{ij}\otimes g_{jk}
\bigoplus_{j \in J}
V^1_{ij}\otimes W^1_{jk}
\longrightarrow
\bigoplus_{j\in J}
V^2_{ij} \otimes W^2_{jk}.

It is straightforward to see that $\Vec(J\times J)$ is a finitely semisimple rigid tensor category.
A set of representatives of the simple objects is given by $\{E_{ij}\}_{i,j\in J}$, where $E_{ij}$ has a copy of $\bbC$ in the $ij$-graded component and the zero vector space everywhere else.
The dual of $V$ is given by $(V^\vee)_{ij} := (V_{ji})^\vee$, the space of linear functionals $V_{ji} \to \bbC$, with obvious evaluation and coevaluation maps.
Indeed, it is straightforward to verify that $\Vec(J\times J)$ is monoidally equivalent to the
tensor category $\Vec[\cG_r]$ of $\cG_r$-graded vector spaces, where $\cG_r$ is the groupoid with $r:=|J|$ objects and a unique isomorphism between any two object.
In turn, $\Vec[\cG_r]$ is easily seen to be monoidally equivalent to $\End(\cM)$ where $\cM$ is a finitely semisimple category such that a set of representatives of the simple objects $\Irr(\cM)$ is in bijection with $J$.

Given a bi-graded vector space $V\in \Vec(J\times J)$, we may think of it as a $\Vec$-enriched graph $\vec{\Gamma}=(J,V)$, whose vertices are the set $J$, and whose edges are the finite dimensional vector spaces $V_{ij}$.
We call a bi-graded vector space connected if given any two vertices $i,k\in J$, there is a sequence of vertices
$(i=j_0, j_1, \dots, j_n=k)$ such that $V_{j_{\ell-1}j_{\ell}}\neq (0)$ for all $\ell=1,\dots, n$.
Observe that if $\Gamma$ is connected, then $\Gamma$ Cauchy tensor generates $\Vec(J\times J)$.

Given a (connected) $\Vec$-graph $\vec{\Gamma} = (J, V)$,
we get an honest (connected) graph $\Gamma$ with vertex set $J$ and whose edges from $i$ to $j$ is are some choice of basis for the space $V_{ij}$.
Clearly picking different bases yields isomorphic graphs.

This approach is very similar to that in the classification of Temperley-Lieb module categories using weighted graphs from <cit.>.
In <cit.>, the authors classify unitary Temperley-Lieb module categories using bi-graded Hilbert spaces, which we discuss briefly (with a warning) in <ref> below.

Suppose $\Gamma$ is a connected directed graph with vertex set $J$ and edge set $V_{ij}$ from $i$ to $j$.
For an edge $\varepsilon \in V_{ij}$, we write $s(\varepsilon) = i$ and $t(\varepsilon) = j$, the source and target of $\varepsilon$.

We define the graph monoidal algebra $\cG\cM\cA(\Gamma)_{\bullet \to \bullet}$ as follows.
For $m,n\geq 0$, we define
$\cG\cM\cA(\Gamma)_{m \to n}$
to be the $\bbC$-vector space with distinguished basis the set of pairs $(p,q)$ where
$p,q$ are paths on $\Gamma$ of length $m,n$ respectively whose sources and targets agree.

The action of tangles is given by a state-sum model similar to a graph planar algebra:

\begin{equation}
\label{eq:StateSumForGMA}
T((p_1,q_1), \dots, (p_k, q_k))
\sum_{
\text{states $\sigma$ on $T$}
\prod_{1\leq i\leq k}
\delta_{\sigma|_i =  (p_i,q_i)}
\sigma|_0
\end{equation}

Here, $T$ is a monoidal tangle with $k$ input disks, and the $(p_i,q_i)$ are basis elements (pairs of paths) in $\cG\cM\cA(\Gamma)_{m_i \to n_i}$.
A state $\sigma$ on a monoidal tangle $T$ is an assignment of vertices and edges of $\Gamma$ to the regions and strings of $T$ respectively such that if a string labelled by $\varepsilon$ separates the left region $R_\ell$ from the right region $R_r$, then $R_\ell$ is labelled by $s(\varepsilon)$ and $R_r$ is labelled by $t(\varepsilon)$.
Now $\sigma|_i$ denotes the pair of paths in $\cG\cM\cA(\Gamma)_{m_i \to n_i}$ obtained from reading the bottom and top boundaries of the $i$-th input disk from left to right respectively.
In other words, we only sum over states which are `compatible' with the paths we input.

Consider the following directed graph:

\begin{tikzpicture}[baseline = -.1cm, scale=.6]
\node[draw, inner sep=.5mm, circle, thick] (a) at (0,0) [below] {\scriptsize{$a$}};
\node[draw, inner sep=.5mm, circle, thick] (b) at (2,0) [below] {\scriptsize{$b$}};
\node[draw, inner sep=.5mm, circle, thick] (c) at (4,0) [below] {\scriptsize{$c$}};
\node[draw, inner sep=.5mm, circle, thick] (d) at (6,0) [below] {\scriptsize{$d$}};
\draw[->] (a) to[out=30, in=150] node [above] {\scriptsize{$\varepsilon$}} (b);
\draw[->] (b) to[out=-150, in=-30] node [below] {\scriptsize{$\varepsilon^*$}} (a);
\draw[->] (b) to[out=30, in=150] node [above] {\scriptsize{$\xi$}} (c);
\draw[->] (c) to[out=-150, in=-30] node [below] {\scriptsize{$\xi^*$}} (b);
\draw[->] (c) to[out=30, in=150] node [above] {\scriptsize{$\kappa$}} (d);
\draw[->] (d) to[out=-150, in=-30] node [below] {\scriptsize{$\kappa^*$}} (c);
\end{tikzpicture}

For the monoidal tangle displayed below on the left, there are exactly two compatible states for the input
$(x_1=(\varepsilon\xi, \varepsilon\xi), x_2=(\xi\xi^*, \varepsilon^*\varepsilon), x_3=(\emptyset, \xi^*\xi))$, which are displayed below on the right.

\begin{tikzpicture}[baseline = -.1cm, scale=.6]
\draw[very thick, rounded corners = 5pt] (-2,-2) rectangle (2,2);
\draw[fill=white, very thick, rounded corners = 5pt] (-1.5,.5) rectangle (-.5,1.5);
\draw[fill=white, very thick, rounded corners = 5pt] (-1,-1.5) rectangle (.5,-.5);
\draw[fill=white, very thick, rounded corners = 5pt] (.25,.5) rectangle (1.25,1.5);
\node at (-1,1) {\scriptsize{$1$}};
\node at (-.25,-1) {\scriptsize{$2$}};
\node at (.75,1) {\scriptsize{$3$}};
\draw (-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.5,-.5);
\draw (-1.5,-2) .. controls ++(90:.5cm) and ++(270:.5cm) .. (-1.2,.5);
\draw (0,-.5) .. controls ++(90:.5cm) and ++(270:.5cm) .. (.5,.5);
\draw (-1.2,1.5) -- (-1.2,2);
\draw (-.8,1.5) -- (-.8,2);
\draw (-.5,-2) -- (-.5,-1.5);
\draw (0,-2) -- (0,-1.5);
\draw (1,-2) -- (1,.5);
\draw (1.5,-2) -- (1.5,2);
\end{tikzpicture}
\qquad
\leadsto
\qquad
\begin{tikzpicture}[baseline = -.1cm, scale=.7]
\draw[very thick, rounded corners = 5pt] (-2,-2) rectangle (2,2);
\draw[fill=white, very thick, rounded corners = 5pt] (-1.5,.5) rectangle (-.5,1.5);
\draw[fill=white, very thick, rounded corners = 5pt] (-1,-1.5) rectangle (.5,-.5);
\draw[fill=white, very thick, rounded corners = 5pt] (.25,.5) rectangle (1.25,1.5);
\node at (-1,1) {\scriptsize{$x_1$}};
\node at (-.25,-1) {\scriptsize{$x_2$}};
\node at (.75,1) {\scriptsize{$x_3$}};
\draw (-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.5,-.5) ;
\draw (-1.5,-2) .. controls ++(90:.5cm) and ++(270:.5cm) .. (-1.2,.5);
\draw (0,-.5) .. controls ++(90:.5cm) and ++(270:.5cm) .. (.5,.5);
\draw (-1.2,1.5) -- (-1.2,2);
\draw (-.8,1.5) -- (-.8,2);
\draw (-.5,-2) -- (-.5,-1.5);
\draw (0,-2) -- (0,-1.5);
\draw (1,-2) -- (1,.5);
\draw (1.5,-2) -- (1.5,2);
\node at (-1.35,1.75) {\scriptsize{$\varepsilon$}};
\node at (-.65,1.75) {\scriptsize{$\xi$}};
\node at (1.75,0) {\scriptsize{$\xi^*$}};
\node at (-1.4,0) {\scriptsize{$\varepsilon$}};
\node at (-.9,0) {\scriptsize{$\xi$}};
\node at (1.15,0) {\scriptsize{$\xi$}};
\node at (0,0) {\scriptsize{$\xi^*$}};
\node at (-.75,-1.75) {\scriptsize{$\varepsilon^*$}};
\node at (.15,-1.75) {\scriptsize{$\varepsilon$}};
\end{tikzpicture}
\,,\,
\begin{tikzpicture}[baseline = -.1cm, scale=.7]
\draw[very thick, rounded corners = 5pt] (-2,-2) rectangle (2,2);
\draw[fill=white, very thick, rounded corners = 5pt] (-1.5,.5) rectangle (-.5,1.5);
\draw[fill=white, very thick, rounded corners = 5pt] (-1,-1.5) rectangle (.5,-.5);
\draw[fill=white, very thick, rounded corners = 5pt] (.25,.5) rectangle (1.25,1.5);
\node at (-1,1) {\scriptsize{$x_1$}};
\node at (-.25,-1) {\scriptsize{$x_2$}};
\node at (.75,1) {\scriptsize{$x_3$}};
\draw (-.8,.5) .. controls ++(270:.5cm) and ++(90:.5cm) .. (-.5,-.5) ;
\draw (-1.5,-2) .. controls ++(90:.5cm) and ++(270:.5cm) .. (-1.2,.5);
\draw (0,-.5) .. controls ++(90:.5cm) and ++(270:.5cm) .. (.5,.5);
\draw (-1.2,1.5) -- (-1.2,2);
\draw (-.8,1.5) -- (-.8,2);
\draw (-.5,-2) -- (-.5,-1.5);
\draw (0,-2) -- (0,-1.5);
\draw (1,-2) -- (1,.5);
\draw (1.5,-2) -- (1.5,2);
\node at (-1.35,1.75) {\scriptsize{$\varepsilon$}};
\node at (-.65,1.75) {\scriptsize{$\xi$}};
\node at (1.65,0) {\scriptsize{$\kappa$}};
\node at (-1.4,0) {\scriptsize{$\varepsilon$}};
\node at (-.9,0) {\scriptsize{$\xi$}};
\node at (1.15,0) {\scriptsize{$\xi$}};
\node at (0,0) {\scriptsize{$\xi^*$}};
\node at (-.75,-1.75) {\scriptsize{$\varepsilon^*$}};
\node at (.15,-1.75) {\scriptsize{$\varepsilon$}};
\end{tikzpicture}

Hence the output of the tangle on the left applied to the input $(x_1,x_2,x_3)$ is
$(\varepsilon \xi\xi^*,\varepsilon\varepsilon^*\varepsilon\xi\xi^*) + (\varepsilon \xi\kappa,\varepsilon\varepsilon^*\varepsilon\xi\kappa)$.

The graph monoidal algebra of $\Gamma$ is really the non-pivotal analog of the graph planar algebra of $\Gamma$.
The reader is encouraged to compare the above definition with that of the graph planar algebra of a bipartite graph in Definition <ref> below.

Given a connected $\Vec$-graph $\vec{\Gamma} = (J, V) \in \Vec(J\times J)$, the semisimple monoidal algebra $\cP(\Vec(J\times J),\vec{\Gamma})_{\bullet \to\bullet}$ from Example <ref> is isomorphic to the graph monoidal algebra $\cG\cM\cA(\Gamma)_{\bullet \to \bullet}$.

Denoting $\vec{\Gamma}^{\otimes n} = (J, V^{\otimes n})$, we have a canonical isomorphism

V^{\otimes n}_{ik}
\cong
\bigoplus_{j_1,\dots, j_{n-1} \in J} V_{ij_1} \otimes V_{j_1, j_2} \otimes \cdots \otimes V_{j_{n-1} k}.

Observe that an $f\in \Hom_{\Vec(J\times J)}(\vec{\Gamma}^{\otimes m} \to \vec{\Gamma}^{\otimes n})$ is completely determined by its component maps

\left\{
\bigoplus_{j_1,\dots, j_{m-1} \in J} V_{ij_1} \otimes V_{j_1 j_2} \otimes \cdots \otimes V_{j_{m-1} \ell}
\to
\bigoplus_{k_1,\dots, k_{n-1} \in J} V_{ik_1} \otimes V_{k_1 k_2} \otimes \cdots \otimes V_{k_{n-1} \ell}.
\right\}_{i,\ell\in J}.

Now fix a basis $\{\varepsilon_{i\ell}^k\}$ for each $V_{i\ell}$, and for each pair of paths on $\Gamma$ from $i$ to $\ell$

p = \varepsilon_{ij_1}^{p_1} \otimes \cdots \otimes \varepsilon_{j_{m-1}\ell}^{p_m}
\qquad
\qquad
q = \varepsilon_{ik_1}^{q_1} \otimes \cdots \otimes \varepsilon_{k_{n-1}\ell}^{q_n},

of lengths $m$ and $n$ respectively, we let $F^{i\ell}_{p\to q}\in \Hom_{\Vec(J\times J)}(\vec{\Gamma}^{\otimes m} \to \vec{\Gamma}^{\otimes n})$ be the unique $i\ell$-component map sending $p$ to $q$ and all other paths $p'$ from $i$ to $\ell$ of length $m$ to zero.
We see then that

\begin{equation}
\label{eq:BasisOfVecGMA}
\Vec(J\times J, \vec{\Gamma})_{m \to n}
\Hom_{\Vec(J\times J)}(\vec{\Gamma}^{\otimes m} \to \vec{\Gamma}^{\otimes n})
\bigoplus_{i,\ell\in J} \operatorname{span}_\bbC
\left\{
F^{i\ell}_{p\to q}
\right\}_{\text{$p,q$ paths $i$ to $\ell$}}.
\end{equation}

Now it is straightforward to verify that the linear extension

\Phi_{m\to n} : \Vec(J\times J, \vec{\Gamma})_{m \to n} \to \cG\cM\cA(\Gamma)_{m\to n}

of $F^{i\ell}_{p\to q} \mapsto (p,q)$ is a linear isomorphism for all $m,n\geq 0$.

It remains to see that this isomorphism is compatible with the action of monoidal tangles.
It suffices to show that $\Phi$ intertwines the actions of a single vertical strand with no input disk, vertical stacking tangles, and horizontal concatenation tangles, as these tangles generate the monoidal operad.
The vertical strand in $\cP(\Vec(J\times J), \vec{\Gamma})_{1\to 1}$ is given by

\bigoplus_{i,j\in J} \id_{V_{ij}}
\bigoplus_{i,j\in J}
\sum_{k} F^{ij}_{\varepsilon_{ij}^k\to \varepsilon_{ij}^k}
\overset{\Phi_{1\to 1}}{\longmapsto}
\bigoplus_{i,j\in J}
\sum_{k} (\varepsilon_{ij}^k , \varepsilon_{ij}^k)
\id_{\cG\cM\cA(\Gamma)_{1\to 1}}
\begin{tikzpicture}[baseline = -.1cm]
\draw[very thick, fill=white, rounded corners = 5pt] (-.3,-.3) rectangle (.3,.3);
\draw (0,-.3) -- (0,.3);
\end{tikzpicture}\,.

Hence $\Phi_{1\to 1}$ preserves the strand.

To see that $\Phi_{\bullet\to \bullet}$ preserves composition, we check on our basis (<ref>).
Suppressing subscripts on edges for simplicity, suppose

p = \varepsilon^{p_1} \otimes \cdots \otimes \varepsilon^{p_k}
\qquad
\qquad
q = \varepsilon^{q_1} \otimes \cdots \otimes \varepsilon^{q_\ell}
\qquad
\qquad
r = \varepsilon^{r_1} \otimes \cdots \otimes \varepsilon^{r_m}
\qquad
\qquad
s = \varepsilon^{s_1} \otimes \cdots \otimes \varepsilon^{s_n}

are paths on $\Gamma$ from $i$ to $j$.

\Phi_{k \to \ell}(F^{ij}_{r\to s})
\circ
\Phi_{m \to n}(F^{ij}_{p\to q})
\begin{tikzpicture}[baseline = -.1cm]
\draw (-.7,-1.2) -- (-.7,1.2);
\draw (-.3,-1.2) -- (-.3,1.2);
\draw (.7,-1.2) -- (.7,1.2);
\draw[very thick, fill=white, rounded corners = 5pt] (-1,-1) rectangle (1,-.4);
\draw[very thick, fill=white, rounded corners = 5pt] (-1,.4) rectangle (1,1);
\node at (0,.7) {$(r,s)$};
\node at (0,-.7) {$(p,q)$};
\node at (-.9,.17) {\scriptsize{$\varepsilon^{r_1}$}};
\node at (-.9,-.23) {\scriptsize{$\varepsilon^{q_1}$}};
\node at (-.5,.17) {\scriptsize{$\varepsilon^{r_2}$}};
\node at (-.5,-.23) {\scriptsize{$\varepsilon^{q_2}$}};
\node at (.05,0) {\scriptsize{$\cdots$}};
\node at (.5,.17) {\scriptsize{$\varepsilon^{r_m}$}};
\node at (.5,-.23) {\scriptsize{$\varepsilon^{q_\ell}$}};
\node at (-.7,1.4) {\scriptsize{$\varepsilon^{s_1}$}};
\node at (-.3,1.4) {\scriptsize{$\varepsilon^{s_2}$}};
\node at (.2,1.4) {\scriptsize{$\cdots$}};
\node at (.7,1.4) {\scriptsize{$\varepsilon^{s_n}$}};
\node at (-.7,-1.4) {\scriptsize{$\varepsilon^{p_1}$}};
\node at (-.3,-1.4) {\scriptsize{$\varepsilon^{p_2}$}};
\node at (.2,-1.4) {\scriptsize{$\cdots$}};
\node at (.7,-1.4) {\scriptsize{$\varepsilon^{p_k}$}};
\end{tikzpicture}
\delta_{\ell=m}
\delta_{q=r}
\delta_{\ell=m}
\delta_{q=r}
\Phi_{k \to n}(F^{ij}_{p\to s})
\Phi_{k \to n}(F^{ij}_{r\to s}\circ F^{ij}_{p\to q}).

As composition is multi-linear, the general case follows by taking linear combinations.

Finally, to show $\Phi_{\bullet\to \bullet}$ preserves tensor product, we again work with our basis (<ref>).
Again suppressing subscripts for simplicity, suppose

p^{gh} = \varepsilon^{p_1} \otimes \cdots \otimes \varepsilon^{p_k}
\qquad
\qquad
q^{gh} = \varepsilon^{q_1} \otimes \cdots \otimes \varepsilon^{q_\ell}
\qquad
\qquad
r^{ij} = \varepsilon^{r_1} \otimes \cdots \otimes \varepsilon^{r_m}
\qquad
\qquad
s^{ij} = \varepsilon^{s_1} \otimes \cdots \otimes \varepsilon^{s_n}

are paths on $\Gamma$,
where $p^{gh}, q^{gh}$ go from $g$ to $h$, and $r^{ij}, s^{ij}$ go from $i$ to $j$.
We calculate

\begin{align*}
\Phi_{k \to \ell}(F^{gh}_{p\to q})
\otimes
\Phi_{m \to n}(F^{ij}_{r\to s})
\begin{tikzpicture}[baseline = -.1cm]
\draw (-.5,-.5) -- (-.5,.5);
\draw (-.2,-.5) -- (-.2,.5);
\draw (.5,-.5) -- (.5,.5);
\draw (1.1,-.5) -- (1.1,.5);
\draw (1.4,-.5) -- (1.4,.5);
\draw (2.1,-.5) -- (2.1,.5);
\draw[very thick, fill=white, rounded corners = 5pt] (-.7,-.3) rectangle (.7,.3);
\draw[very thick, fill=white, rounded corners = 5pt] (.9,-.3) rectangle (2.3,.3);
\node at (0,0) {$(p,q)$};
\node at (-.5,.7) {\scriptsize{$\varepsilon^{q_1}$}};
\node at (-.2,.7) {\scriptsize{$\varepsilon^{q_2}$}};
\node at (.15,.7) {\scriptsize{$\cdots$}};
\node at (.5,.7) {\scriptsize{$\varepsilon^{q_\ell}$}};
\node at (-.5,-.7) {\scriptsize{$\varepsilon^{p_1}$}};
\node at (-.2,-.7) {\scriptsize{$\varepsilon^{p_2}$}};
\node at (.15,-.7) {\scriptsize{$\cdots$}};
\node at (.5,-.7) {\scriptsize{$\varepsilon^{p_k}$}};
\node at (1.6,0) {$(r,s)$};
\node at (1.1,.7) {\scriptsize{$\varepsilon^{s_1}$}};
\node at (1.4,.7) {\scriptsize{$\varepsilon^{s_2}$}};
\node at (1.75,.7) {\scriptsize{$\cdots$}};
\node at (2.1,.7) {\scriptsize{$\varepsilon^{s_n}$}};
\node at (1.1,-.7) {\scriptsize{$\varepsilon^{r_1}$}};
\node at (1.4,-.7) {\scriptsize{$\varepsilon^{r_2}$}};
\node at (1.75,-.7) {\scriptsize{$\cdots$}};
\node at (2.1,-.7) {\scriptsize{$\varepsilon^{r_m}$}};
\end{tikzpicture}
\delta_{h=i}
\begin{tikzpicture}[baseline = -.1cm]
\draw (-.5,-.5) -- (-.5,.5);
\draw (-.2,-.5) -- (-.2,.5);
\draw (.5,-.5) -- (.5,.5);
\draw (1.1,-.5) -- (1.1,.5);
\draw (1.4,-.5) -- (1.4,.5);
\draw (2.1,-.5) -- (2.1,.5);
\draw[very thick, fill=white, rounded corners = 5pt] (-.7,-.3) rectangle (2.3,.3);
\node at (.8,0) {$(pr,qs)$};
\node at (-.5,.7) {\scriptsize{$\varepsilon^{q_1}$}};
\node at (-.2,.7) {\scriptsize{$\varepsilon^{q_2}$}};
\node at (.15,.7) {\scriptsize{$\cdots$}};
\node at (.5,.7) {\scriptsize{$\varepsilon^{q_\ell}$}};
\node at (-.5,-.7) {\scriptsize{$\varepsilon^{p_1}$}};
\node at (-.2,-.7) {\scriptsize{$\varepsilon^{p_2}$}};
\node at (.15,-.7) {\scriptsize{$\cdots$}};
\node at (.5,-.7) {\scriptsize{$\varepsilon^{p_k}$}};
\node at (1.1,.7) {\scriptsize{$\varepsilon^{s_1}$}};
\node at (1.4,.7) {\scriptsize{$\varepsilon^{s_2}$}};
\node at (1.75,.7) {\scriptsize{$\cdots$}};
\node at (2.1,.7) {\scriptsize{$\varepsilon^{s_n}$}};
\node at (1.1,-.7) {\scriptsize{$\varepsilon^{r_1}$}};
\node at (1.4,-.7) {\scriptsize{$\varepsilon^{r_2}$}};
\node at (1.75,-.7) {\scriptsize{$\cdots$}};
\node at (2.1,-.7) {\scriptsize{$\varepsilon^{r_m}$}};
\end{tikzpicture}
\\&=
\delta_{h=i}
\Phi_{k+m \to \ell+n}(F^{gj}_{pr\to qs})
\Phi_{k+m \to \ell+n}(F^{gh}_{p\to q}\otimes F^{ij}_{r\to s}).
\end{align*}

Again, the general case follows by taking linear combinations.

Since the actions of the generating tangles agree, we are finished.

Suppose $\scrC$ is a semisimple monoidal category Cauchy tensor generated by $X$, and $\cM$ is a finitely semisimple module category.  Let $\Irr(\cM)=\{m_1,\dots,m_r\}$ be a set of representatives of simple objects of $\cM$, and define $J:= \{1,\dots, r\}$.  The fusion $\Vec$-graph $\vec{\Gamma}$ of $\cM$ with respect to $X$ is the $\Vec$-graph whose vertices are $J$ and whose edge spaces are given by

\cM(X \vartriangleright m_i \to m_j).

Suppose $\scrC$ is a semisimple monoidal category Cauchy tensor generated by $X$,
$\cM$ is a finitely semisimple  category with isomorphism classes of simple objects indexed by $J$, and
$\vec{\Gamma}$ is a $\Vec$-graph whose vertices are $J$.
Equipping $\cM$ with the structure of an indecomposable left $\scrC$-module category
whose connected fusion $\Vec$-graph with respect to $X$ is $\vec{\Gamma}$
is equivalent to embedding the monoidal algebra $\cP(\scrC,X)_{\bullet \to \bullet}$ into $\cG\cM\cA(\Gamma)_{\bullet \to \bullet}$.