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§ INTRODUCTION

The crucial algebraic structure underlying  rational two-dimensional conformal field theories (RCFT) is that of a modular tensor category [1, 2]. The approach to  RCFTs based on modular tensor categories and
Frobenius algebra objects was developed in [3, 4, 5, 6, 7, 8, 9, 10, 11].
Structure constants in this approach are calculated from ribbon graphs using a three-dimensional topological field theory (TFT).
We will refer to this approach as the TFT approach. A review  can be found in  [12], [13] and a historical overview of RCFT that puts the TFT approach into perspective can be found in  [14].

In this introductory section we will sketch the main structures
we will be working with leaving the precise definitions to the later sections.
At the chiral symmetry level an RCFT can be described in terms of a modular tensor category ${\mathcal C}$. One of the key data specifying  ${\mathcal C}$
is the associator morphisms spaces for the tensor products of objects. Choosing bases for the relevant morphisms these associators are represented by the
fusing matrices also called $\mathrm{F}$-symbols or 6j-symbols.  Such matrices satisfy a system of non-linear equations called the pentagon equations, which in general are very difficult to solve.  The defining observables of a  full RCFT are local bulk operators which combine a holomorphic and anti-holomorphic representation content of ${\mathcal C}$. It was shown in the papers cited above  that to construct such operators and their correlators it suffices to have a Frobenius algebra object $A$
in ${\mathcal C}$ with certain additional properties.   Such an algebra object can be constructed  from a conformal boundary condition. All conformal boundary conditions can then be associated with modules over $A$ and form a module category over $\mathcal{C}$. The multiplication structure of $A$, in addition to the fusing and braiding matrices of ${\mathcal C}$, provides further data satisfying non-trivial non-linear constraints.
One of the important  computational advantages of the TFT approach is
that   all structure constants of the CFT associated with the pair ${\mathcal C}, A$
can be found solving some linear problems. It was shown that the structure constants  will satisfy all the necessary non-linear constraints by virtue
of the non-linear equations satisfied by the data defining ${\mathcal C}$ and $ A$.

Besides the conformal boundary conditions, described as modules over $A$, another important ingredient of the categorical approach are the topological interfaces.   Given two RCFTs built on the same chiral symmetry category ${\mathcal C}$ from two Frobenius algebra objects: $A$ and $B$, a topological interface between them can be described as an $A$-$B$-bimodule object in ${\mathcal C}$. For the case $A=B$ such bimodules correspond to  topological defects. A new tensor product structure arises on the category of all $A$-$A$-bimodules giving rise to a fusion category. It has its own fusing matrices but no natural braiding. Furthermore, a number of non-trivial operations arise involving both conformal boundary conditions and topological defects (or more generally interfaces).
Firstly, we can define a tensor product (fusion) of a topological interface labelled by an $A$-$B$-bimodule  $X$ and a conformal boundary condition labelled by a $B$-module $M$. This tensor product   gives a conformal boundary condition described by the $A$-module $X\otimes_BM$, see Figure <ref>.

\begin{equation*}
\vcenter{\hbox{\begin{tikzpicture}[font=\scriptsize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) ;
\vertex [right=1cm of i1] (j1);
\vertex [above=0.5cm of i1] (i2) [label=left:\(X\)];
\vertex [above=0.5cm of j1] (j2) [label=left:\(M\)];
\vertex [above=3cm of i1] (i3);
\vertex [right =1cm of i3] (j3) ;
\vertex  [left=0.5cm of i3] (a1)  ;
\vertex [below= 0.4cm of a1] (a2);
\vertex [right=1cm of a2] (b1);
\vertex[right=0.5cm of j3] (top);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (top) rectangle (j1);
\diagram*{
(i1)--[thick,middlearrow={latex}] (i3),
(j1)--[thick] (j3)
\end{feynman}
\end{tikzpicture}}}
~ \;\xrightarrow{\text{fuse}}\;~
\vcenter{\hbox{\begin{tikzpicture}[font=\scriptsize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) ;
\vertex [above=0.5cm of i1] (i2) [label=left:\(X\otimes_B\!M\)];
\vertex [above=3cm of i1] (i3);
\vertex  [left=0.5cm of i3] (a1)  ;
\vertex [below= 0.4cm of a1] (a2);
\vertex[right=0.5cm of i3] (top);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (top) rectangle (i1);
\diagram*{
(i1)--[thick] (i3)
\end{feynman}
\end{tikzpicture}}}
\end{equation*}

Fusion of a topological interface with a conformal boundary condition.

Secondly, we can define a junction of a defect with a boundary.
More precisely,  for a given RCFT a topological defect can always end on a conformal boundary condition with the end point in general being a defect operator of non-trivial dimension. In particular cases, when this dimension is zero, the junction is topological. We will often refer to such topological junctions as open topological defects.
In correlation functions involving bulk and boundary operators an open defect  can be freely moved both in the bulk and on the boundary, as long as the defect does not pass through an insertion. We depict a topological junction  on Figure <ref> where we also show an insertion of a boundary operator.

\begin{equation*}
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.3);
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=left:\(N\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=left:\(L\)];
\vertex [below=1.6cm of j2] (j6);
\vertex   [above=2.3cm of j1] (j5);
\vertex [small,dot] [below=1.2cm of j5] (p1) [label=right:\(\Psi_i^\beta\)] {};
\vertex[above=0.6cm of p1] (p2) [label=left:\(M\)];
\vertex [left=0.15cm of j6] [label=left:\(\alpha\)];
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.25,2.9) [label=above:\(X\)];
\draw [thick]    (j7) to[out=90,in=-90] (x1);
\draw [fill=yellow] (1.06,2.4) rectangle (0.8, 2.3);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6),
(x1)--[thick,middlearrow={latex}] (i2)
\end{feynman}
\end{tikzpicture}}}
\end{equation*}

Open defect junction and an insertion of a boundary  operator.

A nice piece of terminology related to the two constructions was proposed in [15]. If a boundary condition is a simultaneous eigenstate
under fusion with a subring of topological defects we say that this boundary condition is strongly symmetric with respect to this subring.
If a  subset of topological defects each admits a  topological junction  with a boundary condition we say that that boundary condition is weakly symmetric with respect to  this subset. In the latter case one can fuse such junctions by putting them on top of each other. Such a fusion is governed by a new set of $\mathrm{F}$-matrices which we denote as $\mathrm{T}$ and  show schematically on  Figure <ref>.

\begin{equation*}
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0.2,3.5) ;
\vertex[above=0.5cm of x1] (x11);
\vertex[below left=0.001cm of x1] (x5) [label=below left:\(X_y\)];
\vertex[left=0.75cm of x5] (l1) [label=below left:\(X_x\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_c\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_a\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  [label=left:\(\alpha\)];
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.5,2.6) ;
\vertex[above=1.4cm of y] (y11);
\vertex [below left=0.001cm of y] (y1) ;
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) [label=left:\(\beta\)];
\draw [thick]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1.1) rectangle (0.8, 1);
\draw (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick]    (j) to[out=90,in=-90] (y);
\vertex[above=0.65 of j55] [label=right:\(M_b\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1),
(x1)--[thick,middlearrow={latex}] (x11),
(y)--[thick,middlearrow={latex}] (y11)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~= \mathlarger{\sum}_{v,\gamma,\delta} \mathrm{T}^{(x\,y\,a)c}_{\alpha b\beta,\gamma v\delta} ~
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [ empty dot] (x1) at (-0.5,2.3) [label=left:\(\gamma\)] {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_c\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=right:\(M_a\)];
\vertex   [above=1cm of j1] (j5);
\vertex   [above=1.1cm of j1] (j6);
\vertex [left=0.15cm of j5] [label=left:\(\delta\)];
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.3,1.4) [label=right:\(X_v\)];
\draw [thick] [middlearrow={latex}]   (j7) to[out=90,in=-90] (x1);
\draw  (1.06,1) rectangle (0.8, 1.1);
\vertex (k1) at (-1,2.8) ;
\vertex (k2) at (-1,3.4)  [label=left:\(X_x\)];
\vertex (k3) at (0,2.8) ;
\vertex (k4) at (0,3.4) [label=right:\(X_y\)];
\vertex[above=1.1cm of k2] (k5) ;
\vertex[above=0.6cm of k4] (k6);
\vertex[left=0.25cm of k5] (k55);
\vertex[right=0.25cm of j2] (j11);
\vertex[below=0.5cm of k5] (k7) ;
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k1) --(k7);
\draw [myptr={latex},thick,rounded corners=1mm]  (x1) -- (k3) --(k6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6)
\end{feynman}
\end{tikzpicture}}}
\end{equation*}

Fusion of open defects.

The rules for passing a topological defect through a bulk operator were formulated in [16] (see  [17] for a nice discussion in the context of the Ising model). In general this will result in a defect operator but in some special cases the passing through  just multiplies the bulk
operator by a constant. As discussed in [18] such a situation has non-trivial implications for bulk RG flows triggered by the bulk operator at hand.
If the constant is 1 the bulk operator commutes with some topological defects which have to survive the flow. If the constant is -1  the bulk operator anticommutes with a topological defect and such a defect must be present between the CFTs (possibly trivial) obtained as the end points with the opposite signs of the coupling. A slightly different take on the role of defects was presented in [15]. For a massive flow the infrared vacuum can be described by
a conformal boundary state [19], [20] specifying an RG boundary. In the presence of topological defects commuting with a perturbation
the RG boundary must be strongly symmetric. Recently non-invertible and categorical symmetries received a lot of attention, see e.g. [21], [22], [23] for a review. The constraints from topological defects on RG flows were discussed in this broader framework in
[24], [25], [26], [27].

Another arena of applications of topological defects is the boundary RG flows which are triggered by a boundary deformation switched on a conformal boundary condition. Both the fusion of Figure <ref> and the junctions on Figure <ref> proved to be useful. As shown in [28] any boundary RG flow
can be fused with a topological defect to give another RG flow triggered by an operator of the same weight. Constraints on the same flow arise in the situation when one has a topological junction that either commutes or anti-commutes with a relevant boundary field.  Similarly to  the bulk case discussed above moving a topological junction past a boundary insertion in general introduces  a new boundary condition between the insertion and the junction.
Such a move is depicted on Figure <ref>.

\begin{equation*}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=0.8 of j1] (j2);
\vertex[ above =0.9 of j1] (j3);
\vertex [small,dot] [ above=2.5cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1) ;
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_i^\beta\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8) [label=left:\(\alpha\)];
\vertex [above =4cm of j1] (j6);
\vertex [below=0.4cm of j6] (p1)  [label=right:\(M_a\)];
\draw  (0.06,0.8) rectangle (-0.2, 0.9);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_b\)];
\vertex[above=0.9cm of psi] (m2);
\vertex[left=1.2cm of psi] (x) [label=left:\(X_x\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_c\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.25cm of x] (x2);
\vertex[above=1.5cm of x] (xtop);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (xtop);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (psi),
(psi) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
~=\mathlarger{\sum}_{d,\gamma,\delta}{\mathrm{\dot{T}}}^{\,(x\,a\,i)c}_{\alpha b\beta,\gamma d\delta}~
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (j1) at (0,0);
\vertex [above=2.4 of j1] (j2);
\vertex[ above =2.5 of j1] (j3);
\vertex [small,dot] [ above=0.9cm of j1] (psi)  {};
\vertex[right=0.2cm of psi] (psi1);
\vertex[below=0.2cm of psi1] (psi2) [label=\(\,\,\,\Psi_i^\delta\)];
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7);
\vertex [left=0.15cm of j3] (j8) [label=left:\(\gamma\)];
\vertex [above =4cm of j1] (j6);
\vertex[below=0.4cm of j6] (p2) [label=right:\(M_a\)];
\draw (0.06,2.4) rectangle (-0.2, 2.5);
\vertex[below=0.1cm of j6] (m1) ;
\vertex[above=1.6cm of j1] (m3) [label=right:\(M_d\)];
\vertex[above=0.9cm of j2] (m2) ;
\vertex[above=0.5cm of j4] (xx);
\vertex[left=1.2cm of xx] (x);
\vertex[below=0.1cm of x] (x100)  [label=left:\(X_x\)];
\vertex[above=0.3 of j1] (jj) [label=right:\(M_c\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.15cm of x] (x2);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.6cm of j6] (j7);
\vertex[right=0.25cm of j7] (j77);
\vertex[left=0.25cm of x2] (x22);
\vertex[above=0.05cm of x] (x3);
\vertex[left=0.25cm of x3] (x33);
\vertex[above=0.1cm of x33] (x333);
\vertex[above=0.4cm of x2] (x4);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
\end{equation*}

Moving an open topological defect past a boundary field.

It is governed by a new set of fusing matrices which we denote[In this paper we introduced a new notation for three kinds of fusing matrices: $\mathrm{T}$, $\dot {\mathrm{T}}$ and the fusing matrices for junctions of topological defects which we denote as $\mathrm{Y}$. We decided to break with the tradition of keeping the letter $\mathrm{F}$ in the notation in favour of some mnemonic rule that would make  it easier to remember which situation the fusing matrix describes. Thus, the  triple junctions of topological defects resemble the letter Y while the junctions of defects with boundaries are depicted as letter-T-shaped junctions. The dot in the notation  $\dot {\mathrm{T}}$ is meant to remind of an insertion of a boundary operator.  ] by $\dot{\mathrm{T}}$. In the case when the boundary condition remains the same and the boundary field is left intact we say that the open defect commutes with the boundary operator. When
moving past the insertion amounts only to multiplication of the boundary field by $-1$ we say that it anti-commutes. Concrete examples of such situations were presented in [29] for the Virasoro minimal models. The commuting situation means that the topological junction survives into the infrared fixed point. When there are several such topological junctions their fusion algebra must also be an invariant under the RG flow. This means that the relevant
fusing matrices $\mathrm{T}$  defined on Figure <ref>  must be invariant (up to a change of basis).

Another fusion algebra is obtained when we look at defects with both ends attached to the UV boundary condition. On Figure <ref> this is depicted as defect lines stretched across a strip.

\begin{equation*}
\vcenter{\hbox{\begin{tikzpicture}[font=\scriptsize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) ;
\vertex[right=3cm of i1] (j1);
\vertex[above=0.5cm of i1] (i2);
\vertex[above=0.28cm of i1] (i22) ;
\vertex[right=0.2cm of i22] (i222) ;
\vertex[below=0.08cm of i222] (i2222) [label=right:\(M_a\)];
\path[pattern=north east lines,pattern color=ashgrey,very thin] (i2) rectangle (j1);
\vertex[below=2cm of i1] (i3);
\vertex[right=3cm of i3] (j2);
\vertex[below=0.5cm of i3] (i4);
\vertex[below=0.35cm of i3] (i33) ;
\vertex[right=0.2cm of i33] (i333);
\vertex[above=0.12cm of i333] (i3333)  [label=right:\(M_f\)];
\path[pattern=north east lines,pattern color=ashgrey,very thin] (i4) rectangle (j2);
\vertex[right=1cm of i1] (d1)[label=below right:\(\alpha\)];
\vertex[right=1cm of d1] (d2)[label=below right:\(\beta\)];
\vertex[below=2cm of d1] (d3)[label=above right:\(\delta\)];
\vertex[right=1cm of d3] (d4)[label=above right:\(\gamma\)];
\vertex[right=0.5cm of d1] (m1) [label=above:\(M_b\)];
\vertex[right=0.5cm of d2] (m1) [label=above:\(M_c\)];
\vertex[right=0.5cm of d3] (m1) [label=below:\(M_e\)];
\vertex[right=0.5cm of d4] (m1) [label=below:\(M_d\)];
\diagram*{
(i1)--[thick] (j1),
(j2)--[thick] (i3),
(d3)--[thick,middlearrow={latex},edge label=$X_x$] (d1),
(d4)--[thick,middlearrow={latex},edge label=$X_y$] (d2)
\end{feynman}
\end{tikzpicture}}}
~ \;\xrightarrow{\text{fuse}}\;\mathlarger{\sum}_u\mathlarger{\sum}_{\rho,\sigma,\varphi}\overline{\mathrm{T}}^{(x\,y\,c)a}_{\rho u\sigma,\alpha b\beta}\,\mathrm{T}^{(x\,y\,d)f}_{\delta e\gamma,\rho u\varphi}~
\vcenter{\hbox{\quad\begin{tikzpicture}[font=\scriptsize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) ;
\vertex[right=3cm of i1] (j1);
\vertex[above=0.5cm of i1] (i2);
\vertex[above=0.28cm of i1] (i22);
\vertex[right=0.2cm of i22] (i222)  ;
\vertex[below=0.08cm of i222] (i2222) [label=right:\(M_a\)];
\path[pattern=north east lines,pattern color=ashgrey,very thin] (i2) rectangle (j1);
\vertex[below=2cm of i1] (i3);
\vertex[right=3cm of i3] (j2);
\vertex[below=0.5cm of i3] (i4);
\vertex[below=0.35cm of i3] (i33) ;
\vertex[right=0.2cm of i33] (i333) ;
\vertex[above=0.12cm of i333] (i3333)  [label=right:\(M_f\)];
\path[pattern=north east lines,pattern color=ashgrey,very thin] (i4) rectangle (j2);
\vertex[right=1.5cm of i1] (d1)[label=below right:\(\sigma\)];
\vertex[below=2cm of d1] (d3)[label=above right:\(\varphi\)];
\vertex[right=0.5cm of d2] (m1) [label=above:\(M_c\)];
\vertex[right=0.5cm of d4] (m1) [label=below:\(M_d\)];
\diagram*{
(i1)--[thick] (j1),
(j2)--[thick] (i3),
(d3)--[thick,middlearrow={latex},edge label=$X_u$] (d1)
\end{feynman}
\end{tikzpicture}}}
\end{equation*}

Fusion of compact open defects.

It is not hard to show  that the fusion algebra of such compact open defects is associative. This was demonstrated in [30] for the case of charge-conjugation modular invariant (i.e. the trivial Frobenius algebra $A={\bf 1}$). We do this in the general case in section <ref>. When the boundary condition is the direct sum of all irreducible boundary conditions with multiplicity 1 the corresponding associative algebra  was considered in [31] under the name of the tube algebra. This notion   goes back to [32] and also emerges in bulk theories, see [18], [33],
[34], [35] for recent discussions of the tube algebras arising in bulk QFTs and lattice theories.
The structure constants of the  algebras
associated with compact open defects are expressed in terms of the matrices $\mathrm{T}$ and the dual matrices $\overline{ \mathrm{T}}$ as on
Figure <ref>.

When there is a boundary operator that commutes with some subset of topological junctions the compact open defects from this  subset generate an associative algebra which is an invariant of the RG flow triggered by this operator.
Examples of such algebras and boundary RG flows were considered in  [29] in the context of $A$-series Virasoro minimal models.
When we have an operator that anti-commutes with some topological junctions similar conclusions apply giving defects linking the two infrared boundary conditions obtained for the opposite signs of the boundary coupling. More generally we can consider boundary RG flows in a multi-coupling space which are triggered by switching on several boundary operators. In such cases topological junctions arise between the boundary conditions on different RG trajectories.
The assignment of a junction to pairs of RG trajectories can be expressed in terms of the fusing matrix $\dot {\mathrm{T}}$.
We will consider concrete examples of such situations in section  <ref>.

We see that for the applications to RG flows one needs to know the matrices ${\mathrm T}$ and $\dot {\mathrm{T}}$. When $A=\bf {1}$ each irreducible module and bimodule is described by an irreducible object in $\mathcal{C}$. It follows from this that in an appropriate basis the matrices $\mathrm{T}$ and $\dot {\mathrm{T}}$ are given by the fusing matrices of  $\mathcal{C}$. For a non-trivial $A$ this is no longer the case and the new fusing matrices  need to be calculated. The main purpose of this paper is to lay out in detail the general structure associated with these matrices for the  case of $A\ne {\bf 1}$ and in the presence of multiplicities in the fusion rules. Furthermore we work out in detail how these matrices can be computed from a given representation theoretic data for the pair $(\mathcal{C}, A)$. In the TFT approach, as with all structure constants, this task reduces to solving some linear algebra problems. While conceptually we don't offer anything new to the computational part, we work out the details of the linear problems to the level at which one can use software. We illustrate the  method on particular examples of the rational free boson theories. An altogether different approach to calculating the fusing matrices associated with
fusion categories which are Morita equivalent to the one with the known fusing matrices is developed in [31]. Another recent calculation of fusing matrices arising in a CFT with a triality defect was performed in [36].

The main body of the paper is organised as follows. In section <ref> we introduce the tensor product structures related to boundaries and interfaces arising in RCFT. We formally introduce the associators and the fusing matrices for those tensor structures and discuss the related pentagon identities. In section
<ref> we introduce the compact open defects and their fusion algebras. In section <ref>  we work out the general formulae for the new fusing matrices in terms of the data in $(\mathcal{C}, A)$. In section <ref> we apply our general formulae to obtain the fusing matrices for
some rational free boson theories. In section <ref> we describe some concrete applications to boundary RG flows in rational free boson examples. The appendices contain a detailed discussion of associators with proofs of pentagon equations, as well as calculational details related to the free boson examples.

§ TENSOR PRODUCT STRUCTURES RELATED TO RCFTS

In this paper we are following the TFT approach to RCFT (see  e.g. [12], [13] for a review). The  relevant mathematical structures used in this approach are well exposed in the literature (see e.g. [37, 38]).   We will thus focus on conventions and notations providing only certain definitions that will be of particular importance for our calculations. Throughout this section we will assume that the modular tensor category $\mathcal{C}$ that encodes the chiral symmetry algebra of our RCFT is fixed.

§.§ Conventions for morphisms in C

We represent morphisms in  $\mathcal{C}$ as graphs in which lines stand for identity morphisms. For example an identity morphism $\operatorname{id}_U$ and $f\in \operatorname{Hom}(U,V)$ are depicted as

\begin{equation}
\operatorname{id}_U=~
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U\)];
\vertex [above=2cm of j1] (j2) [label=above:\(U\)];
\diagram*{
(j1)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U\)];
\vertex [above=2cm of j1] (j2) [label=above:\(U\)];
\vertex [above=1.2cm of j1] (j4);
\draw [fill=yellow] (-0.3,0.8) rectangle (0.3, 1.2);
\vertex [above=0.8cm of j1] (j3) [label=above:\scriptsize\(f\)];
\diagram*{
(j1)--[thick] (j3),
(j4)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

Here the convention is that such pictures are read from bottom to top. Since $\operatorname{id}_{\mathbf{1}}=1\in\mathbb{C}$, lines labelled by $\mathbf{1}$ will be omitted, where $\mathbf{1}$ is the tensor unit of $\mathcal{C}$ (see definition <ref>).  Composition of morphisms corresponds to concatenation of lines, while the tensor product to juxtaposition.

To extract information from the axioms of $\mathcal{C}$ we usually need to make an explicit  choice of basis in the three-point coupling spaces.
To this end we choose representatives $U_i,\,i\in\mathcal{I}$ of the isomorphisms classes of the simple objects in $\mathcal{C}$ where  $\mathcal{I}$ is a finite indexing set. For these simple objects we fix  bases $\{\lambda^\alpha_{(i,j)k}\}$ in the hom-spaces $\operatorname{Hom}(U_i\otimes U_j,U_k)$ as well as dual bases $\{\bar{\lambda}_{\alpha}^{(i,j)k}\}$ in $\operatorname{Hom}(U_k,U_i\otimes U_j)$. These bases will be depicted as

\begin{equation}\label{bases1}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U_i\)];
\vertex [above=1.7cm of j1] (j4);
\draw [fill=yellow] (-0.25,1.8) rectangle (0.75, 1.3);
\vertex [above=1.3cm of j1] (j3) [label=above:\( \hspace{5mm}\lambda^\alpha_{(i,j)k}\)];
\vertex [right=0.5cm of j1] (i1) [label=below:\(U_j\)];
\vertex (x) at (0.25,3) [label=above:\(U_k\)];
\vertex [below=1.2cm of x] (x2);
\vertex[above=1.7 cm of i1] (i4);
\vertex[ above=1.3 cm of i1] (i3);
\diagram*{
(j1)--[thick] (j3),
(i1)--[thick] (i3),
(x2)--[thick] (x)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(U_i\)];
\vertex [right=1cm of i1] (j1) [label=below:\(U_j\)];
\vertex [above=0.5cm of i1] (i2);
\vertex [above =0.5cm of j1] (j2);
\vertex  [right=0.5cm of i1] (k1) ;
\vertex [small,orange, dot][above =1.5cm of k1] (k2)[label=right:\(\alpha\)] {};
\vertex [above=1.5cm of k2] (k3) [label=above:\(U_k\)];
\draw [thick,rounded corners=1mm] (i1)--(i2)--(k2);
\draw [thick,rounded corners=1mm] (j1)--(j2)--(k2);
\diagram*{
(k2)--[thick] (k3)
\end{feynman}
\end{tikzpicture}}}
\hspace{20mm}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=above:\(U_i\)];
\vertex [below=1.3cm of j1] (j2);
\draw [fill=yellow] (-0.25,-1.8) rectangle (0.75, -1.3);
\vertex [below=1.8cm of j1] (j3) [label=\( \hspace{5mm}\bar{\lambda}_{\alpha}^{(i,j)k}\)];
\vertex [right=0.5cm of j1] (i1) ;
\vertex[below=0.04cm of i1] (xx) [label=above:\(U_j\)];
\vertex (x) at (0.25,-3) [label=below:\(U_k\)];
\vertex [above=1.2cm of x] (x2);
\vertex[below=1.3cm of i1] (i2);
\diagram*{
(j1)--[thick] (j2),
(i1)--[thick] (i2),
(x)--[thick] (x2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(U_k\)];
\vertex [small,orange, dot][above=1.5cm of i1] (i2) [label=right:\(\bar{\alpha}\)]{};
\vertex [above=3cm of i1] (k1);
\vertex [right =0.5cm of k1] (k2) ;
\vertex[below=0.04cm of k2] (xx) [label=above:\(U_j\)];
\vertex  [left=0.5cm of k1] (k3) [label=above:\(U_i\)] ;
\vertex [below=0.5cm of k3] (k4);
\vertex [below=0.5cm of k2] (k5);
\draw [thick,rounded corners=1mm] (k2)--(k5)--(i2);
\draw [thick,rounded corners=1mm] (k3)--(k4)--(i2);
\diagram*{
(i2)--[thick] (i1)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

where $\alpha\in \{ 1,\ldots,\tensor{N}{_{ij}}{^k}\}$ with $\tensor{N}{_{ij}}{^k}$ being the fusion coefficients defined as

\begin{equation}
\tensor{N}{_{ij}}{^k}:=\operatorname{dim}\operatorname{Hom}(U_i\otimes U_j,U_k) \, .
\end{equation}

Duality of these bases means that the following normalisation holds

\begin{equation} \label{e5}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  [small,orange, dot] (del) at (0,0) [label=below:\(\alpha\)] {} ;
\vertex  [small,orange, dot] (gam) at (0,-2) [label=above:\(\beta\)] {} ;
\vertex [below=1cm of gam] (j1) [label=below:\(U_j\)];
\vertex [above=1cm of del] (j2) [label=above:\(U_m\)];
\diagram*{
(j2)--[thick] (del),
(j1)--[thick] (gam),
(del) --[half left, thick, looseness=1.2,edge label=$U_k$] (gam),
(gam) --[half left, thick, looseness=1.2, edge label=$U_i$] (del)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{5mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (1) at (0,1) [label=above:\(U_j\)];
\vertex (2) at (0,-3) [label=below:\(U_j\)];
\diagram*{
(1)--[thick] (2)
\end{feynman}
\end{tikzpicture}}}\end{equation}

These bases also satisfy a completeness relation due to the semisimplicity of the category $\mathcal{C}$ [7]:

\begin{equation}\label{completelambda}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  (i1) at (0,0) [label=below:\(U_i\)]  ;
\vertex [above=3cm of i1] (i2) [label=above:\(U_i\)];
\vertex [right=1cm of i1] (j1) [label=below:\(U_j\)];
\vertex [right=1cm of i2] (j2) [label=above:\(U_j\)];
\diagram*{
(i1)--[thick] (i2),
(j1)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  [small,orange,dot] (up) at (0,0) [label=above:\(\gamma\)] {};
\vertex [small,orange,dot] [below=1cm of up] (down) [label=below:\(\gamma\)] {};
\vertex [above left=0.75cm of up] (i1);
\vertex [above=0.6cm of i1] (i2) [label=above:\(U_i\)];
\vertex [above right=0.75cm of up] (j1);
\vertex [above=0.6cm of j1] (j2) [label=above:\(U_j\)];
\draw [thick,rounded corners=1mm] (up) -- (i1) --(i2);
\draw [thick,rounded corners=1mm] (up) -- (j1) --(j2);
\vertex [below left=0.75cm of down] (i3) ;
\vertex [below=0.6cm of i3] (i4) [label=below:\(U_i\)];
\vertex [below right=0.75cm of down] (j3);
\vertex [below=0.6cm of j3] (j4) [label=below:\(U_j\)];
\draw [thick,rounded corners=1mm] (down) -- (j3) --(j4);
\draw [thick,rounded corners=1mm] (down) -- (i3) --(i4);
\diagram*{
(down)--[thick,edge label=$U_k$] (up)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

In the literature one finds other normalisation choices than (<ref>). In particular the choice made in  [30], for theories with  ${\tensor{N}{_{ij}}{^k}} \in\{0,1\}$, can be characterised by the equation

\begin{equation}\label{koj1}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[inner sep=2pt,font=\footnotesize]
\begin{feynman}
\vertex  [small,orange, dot] (l) at (0,0)  {} ;
\vertex [above right=1.25cm of l] [small,orange, dot] (alpha)  {};
\vertex [above=1cm of alpha] (i) [label=above:\(U_i\)];
\vertex [below right=1cm of alpha] (j1);
\vertex [below right=0.3cm of j1] (j2);
\vertex [below=0.6cm of j2] (j) [label=below:\(U_j\)];
\vertex [below=0.7cm of l] (1) [label=below:\(\mathbf{1}\)];
\vertex [above left=0.8cm of l] (k1) ;
\vertex [above left=0.3cm of k1] (k2);
\vertex [above=1.1cm of k2] (k) [label=above:\(U_k\)];
\draw [thick,rounded corners=1mm] (alpha) -- (j1)  -- (j2)--(j);
\draw [thick,rounded corners=1mm] (l) -- (k1)  -- (k2)--(k);
\diagram*{
(l) --[thick,edge label= $U_{\bar{k}}$] (alpha),
(alpha) --[thick] (i),
(l) --[scalar] (1)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(U_j\)];
\vertex [small,orange, dot][above=1.5cm of i1] (i2) {};
\vertex [above=3cm of i1] (k1);
\vertex [right =0.5cm of k1] (k2) ;
\vertex[below=0.04cm of k2] (xx) [label=above:\(U_i\)];
\vertex  [left=0.5cm of k1] (k3) [label=above:\(U_k\)] ;
\vertex [below=0.5cm of k3] (k4);
\vertex [below=0.5cm of k2] (k5);
\draw [thick,rounded corners=1mm] (k2)--(k5)--(i2);
\draw [thick,rounded corners=1mm] (k3)--(k4)--(i2);
\diagram*{
(i2)--[thick] (i1)
\end{feynman}
\end{tikzpicture}}}\end{equation}

where $\bar{k}$  is the label of the dual object of $U_k$.

§.§ Frobenius algebras and  modules over them

Let $A$ be a symmetric special Frobenius algebra in $\mathcal{C}$. We  introduce  the following graphical notation for the multiplication, unit, comultiplication and counit morphisms of  $A$

\begin{equation}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(A\)];
\vertex [right=1cm of i1] (j1) [label=below:\(A\)];
\vertex [above=0.5cm of i1] (i2);
\vertex [above =0.5cm of j1] (j2);
\vertex  [right=0.5cm of i1] (k1) ;
\vertex [small, scarlet, dot][above =1.5cm of k1] (k2) {};
\vertex [above=1.5cm of k2] (k3) [label=above:\(A\)];
\draw [thick,rounded corners=1mm] (i1)--(i2)--(k2);
\draw [thick,rounded corners=1mm] (j1)--(j2)--(k2);
\diagram*{
(k2)--[thick] (k3)
\end{feynman}
\end{tikzpicture}}}
\hspace{12mm}
\eta~=~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex(i1) at (0,0) [label=below:\(\mathbf{1}\)]{};
\draw[black,fill=baby] (0,0) circle (.5ex);
\vertex [above=1cm of i1] (i2) [label=above:\(A\)];
\diagram*{
(i1)--[thick] (i2)
\end{feynman}
\end{tikzpicture}}}
\hspace{12mm} \Delta ~=~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(A\)];
\vertex [small,blue, dot][above=1.5cm of i1] (i2){};
\vertex [above=3cm of i1] (k1);
\vertex [right =0.5cm of k1] (k2) [label=above:\(A\)];
\vertex  [left=0.5cm of k1] (k3) [label=above:\(A\)] ;
\vertex [below=0.5cm of k3] (k4);
\vertex [below=0.5cm of k2] (k5);
\draw [thick,rounded corners=1mm] (k2)--(k5)--(i2);
\draw [thick,rounded corners=1mm] (k3)--(k4)--(i2);
\diagram*{
(i2)--[thick] (i1)
\end{feynman}
\end{tikzpicture}}}
\hspace{12mm}
\varepsilon~=~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex(i1) at (0,0) [label=above:\(\mathbf{1}\)]{};
\draw[black,fill=aqua] (0,0) circle (.5ex);
\vertex [below=1cm of i1] (i2) [label=below:\(A\)];
\diagram*{
(i1)--[thick] (i2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

As $\mathcal{C}$ is semisimple,   $A$ is a finite direct sum of  $U_i$. We fix  all bases $\{\imath^A_{i,\alpha}\}$ in the embedding spaces $\operatorname{Hom}(U_i,A)$, as well as dual bases $\{\jmath_A^{i,\alpha}\}$ in $\operatorname{Hom}(A,U_i)$ which we draw as

\begin{equation}\label{embed0}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U_i\)];
\vertex [above=2.1cm of j1] (j2) [label=above:\(A\)];
\vertex [above=1.3cm of j1] (j4);
\draw [fill=yellow] (-0.3,0.8) rectangle (0.3, 1.3);
\vertex [above=0.8cm of j1] (j3) [label=above:\scriptsize\(\imath^A_{i,\alpha}\)];
\diagram*{
(j1)--[thick] (j3),
(j4)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U_i\)];
\vertex [above=2.1cm of j1] (j2) [label=above:\(A\)];
\vertex [above=0.9cm of j1] (j4);
\node[isosceles triangle,
minimum size =0.4cm] (T1)at (j4){};
\vertex [below=0.1cm of j4] (j5) [label=above:\tiny\(\alpha\)];
\diagram*{
(j1)--[thick] (j5),
(T1)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\qquad\quad \text{and}\qquad \quad
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(A\)];
\vertex [above=2.1cm of j1] (j2) [label=above:\(U_i\)];
\vertex [above=1.3cm of j1] (j4);
\draw [fill=yellow] (-0.3,0.8) rectangle (0.3, 1.3);
\vertex [above=0.8cm of j1] (j3) [label=above:\scriptsize\(\jmath_A^{i,\alpha}\)];
\diagram*{
(j1)--[thick] (j3),
(j4)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(A\)];
\vertex [above=2.1cm of j1] (j2) [label=above:\(U_i\)];
\vertex [above=1.1cm of j1] (j4);
\node[isosceles triangle,
minimum size =0.4cm] (T1)at (j4){};
\vertex [below=0.2cm of j4] (j5) [label=\tiny\(\alpha\)];
\diagram*{
(j1)--[thick] (T1),
(T1)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

They satisfy the following duality and completeness relations:

\begin{align}
&\jmath_A^{(j,\beta)}\circ \imath^A_{i,\alpha}=\delta_{i,j}\delta_{\alpha,\beta}\id_{U_i} \\
& \sum_{i, \alpha}  \imath^A_{i,\alpha} \circ  \jmath_A^{i,\alpha}  = \id_{M}\,.
\end{align}

Using these bases we can expand the comultiplication morphism in terms of $\bar{\lambda}^{(i,j)k}_{\bar{\alpha}}$ as

\begin{equation}\label{comultiplication}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0)  ;
\vertex[below=0.8cm of l] (w1);
\vertex[above=0.5cm of l] ;
\vertex[right=0.1cm of l] (phi) ;
\vertex  [above=0.8cm of l] (gam) {};
\vertex [small,blue,dot] [above left=1.25cm of gam] (del) {} ;
\vertex [above right=1cm of gam] (k1);
\vertex [above=1.4cm of k1] (k) ;
\vertex [above left=0.7cm of del] (i1);
\vertex [above=0.7cm of i1] (i) ;
\vertex [above right=0.7cm of del] (j1);
\vertex [above=0.7cm of j1] (j) ;
\draw [thick,rounded corners=1mm] (del) -- (j1) --(j);
\draw [thick,rounded corners=1mm] (del) -- (i1) --(i);
\vertex[above=0.8cm of i] (i3) [label=above:\(U_i\)];
\vertex[above=0.8cm of j] (j3) [label=above:\(U_j\)];
\vertex[above=0.8cm of k] (k3) ;
\vertex[above right=0.1cm of k] (tau);
\vertex[below=0.5cm of i] (i4) [label=left:\(A\)];
\vertex[below=0.5cm of j] (j4) [label=right:\(A\)];
\vertex[below=0.8cm of k] (k4);
\node[isosceles triangle,
minimum size =0.35cm] (T3)at (j){};
\node[isosceles triangle,
minimum size =0.35cm] (T2)at (i){};
\vertex[below=0.24cm of j] (b1) [label=:\tiny\(\beta\)];
\vertex[below=0.17cm of i] (a1) [label=:\tiny \(\alpha\)];
\vertex[below=1cm of del] (c1) ;
\vertex[above=0.4cm of c1] (c3) [label=above right:\(A\)];
\node[isosceles triangle,
minimum size =0.35cm] (T1)at (c1){};
\vertex[below=0.12cm of c1] (c2) [label=:\tiny\(\hspace{-0.5mm}\gamma\)];
\vertex[below=0.7cm of c1] (bot) [label=below:\(U_k\)];
\diagram*{
(T2)--[thick] (i3),
(T3)--[thick] (j3),
(T1)--[thick] (del),
(T1)--[thick] (bot)
\end{feynman}
\end{tikzpicture}}}
=\mathlarger{\sum}_{\delta=1}^{\tensor{N}{_{ij}}{^k}} \Delta_{k\gamma;\delta}^{i\alpha,j\beta} \vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(U_k\)];
\vertex [small,orange, dot][above=2cm of i1] (i2) [label=right:\(\bar{\delta}\)]{};
\vertex [above=3.8cm of i1] (k1);
\vertex [right =0.5cm of k1] (k2) ;
\vertex[below=0.04cm of k2] (xx) [label=above:\(U_j\)];
\vertex  [left=0.5cm of k1] (k3) [label=above:\(U_i\)] ;
\vertex [below=0.65cm of k3] (k4);
\vertex [below=0.65cm of k2] (k5);
\draw [thick,rounded corners=1mm] (k2)--(k5)--(i2);
\draw [thick,rounded corners=1mm] (k3)--(k4)--(i2);
\diagram*{
(i2)--[thick] (i1)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

The numbers $\Delta_{k\gamma;\delta}^{i\alpha,j\beta}\in{\mathbb C}$ are called the comultiplication numbers of $A$.
Similarly we can introduce the multiplication numbers by

\begin{equation}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (a1) at (0,0) [label=below:\(U_i\)];
\vertex [right=1cm of a1] (b1) [label=below:\(U_j\)];
\vertex[above=0.8cm of a1] (a2) ;
\vertex [above=0.8cm of b1] (b2) ;
\node[isosceles triangle,
minimum size =0.4cm] (T1)at (a2){};
\node[isosceles triangle,
minimum size =0.4cm] (T2)at (b2){};
\vertex[above=0.7cm of a1] (a2) [label=\tiny\(\alpha\)];
\vertex [right=0.5cm of a1] (x);
\vertex [small,scarlet, dot][above=1.8cm of x] (x1){};
\vertex [above=0.7cm of b1] (b2) [label=\tiny\(\beta\)];
\vertex[above=0.4cm of a2] (a3);
\vertex[above=0.4cm of b2] (b3);
\vertex[above=0.8cm of x1] (c1);
\node[isosceles triangle,
minimum size =0.4cm] (T3)at (c1){};
\vertex[above=0.6cm of x1] (c11) [label=above:\tiny\(\gamma\)];
\vertex [above=0.6cm of T3] (c) [label=above:\(U_k\)];
\diagram*{
(a1)--[thick] (T1),
(b1)--[thick] (T2),
(a3) --[quarter left, thick, looseness=0.9] (x1),
(b3) --[quarter right, thick, looseness=0.9] (x1),
(T3)--[thick] (c),
(x1)--[thick] (T3)
\end{feynman}
\end{tikzpicture}}}
~=\mathlarger{\sum}_{\delta=1}^{\tensor{N}{_{ij}}{^k} }m_{i\alpha,j\beta}^{k\gamma;\delta} ~
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(U_i\)];
\vertex [right=1cm of i1] (j1) [label=below:\(U_j\)];
\vertex [above=0.5cm of i1] (i2);
\vertex [above =0.5cm of j1] (j2);
\vertex  [right=0.5cm of i1] (k1) ;
\vertex [small,orange, dot][above =1.5cm of k1] (k2)[label=right:\(\delta\)] {};
\vertex [above=1.5cm of k2] (k3) [label=above:\(U_k\)];
\draw [thick,rounded corners=1mm] (i1)--(i2)--(k2);
\draw [thick,rounded corners=1mm] (j1)--(j2)--(k2);
\diagram*{
(k2)--[thick] (k3)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

As shown in  [7] in the case $\tensor{N}{_{ij}}{^k}\in\{0,1\}$ and $\dimh(U_k,A)\in\{0,1\}$ the following relation between these two sets of numbers holds:

\begin{equation}\label{comultnumbers}
\Delta_k^{i,j}=\frac{1}{\operatorname{dim}(A)} \cdot \frac{m_{\bar{i}, k}^j}{m_{\bar{i}, i}{ }^0} \cdot \frac{\mathrm{F}_{j, 0}^{(i\, \bar{i}\, k) k}}{\mathrm{F}_{0,0}^{(i\, \bar{i}\, i) i}} \, .
\end{equation}

Conformal boundary conditions for an RCFT are labelled by modules of the algebra $A$. We remind the reader the following two definitions  (see e.g. [7]) we will be using later.

Let $(A,m,\eta)$ be an algebra in a tensor category $\mathcal{C}$. A left $A$-module is a pair $N=(\dot{N},\rho^N)$ where $\dot{N}\in\operatorname{Obj}(\mathcal{C})$ and $\rho^N\in\Hom(A\otimes \dot{N},\dot{N})$ such that

\begin{equation}
\rho^N \circ\left(m \otimes \operatorname{id}_{\dot{N}}\right)=\rho^N \circ\left(\operatorname{id}_A \otimes \rho^N\right) \qquad \text { and } \qquad \rho^N \circ\left(\eta \otimes \operatorname{id}_{\dot{N}}\right)=\operatorname{id}_{\dot{N}}\,.
\end{equation}

\begin{equation}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(A\)];
\vertex [above=1.7cm of j1] (j4);
\draw [fill=yellow] (-0.25,1.8) rectangle (0.75, 1.3);
\vertex [above=1.3cm of j1] (j3) [label=above:\( \hspace{5mm}\rho^N\)];
\vertex [right=0.5cm of j1] (i1) [label=below:\(\dot{N}\)];
\vertex (x) at (0.25,3) [label=above:\(\dot{N}\)];
\vertex [below=1.2cm of x] (x2);
\vertex[above=1.7 cm of i1] (i4);
\vertex[ above=1.3 cm of i1] (i3);
\diagram*{
(j1)--[thick] (j3),
(i1)--[thick] (i3),
(x2)--[thick] (x)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (a1) at (0,0) [label=below:\(A\)];
\vertex [right=1cm of a1] (n1) [label=below:\(\dot{N}\)];
\vertex [above=1.4cm of n1] (r) ;
\vertex [above=1.5cm of r] (n2) [label=above:\(\dot{N}\)];
\vertex [above=0.5cm of a1] (a2);
\vertex [left=0.3cm of r] (r1);
\vertex [above=0.2cm of r] (r2);
\draw[fuc, thick] (r) -- (r1) -- (r2) -- cycle;
\draw [thick,rounded corners=1mm] (a1) -- (a2) --(r1);
\fill[fuc] (r) -- (r1) -- (r2) -- cycle;
\diagram*{
(n1)--[thick] (r),
(r)--[thick] (n2)
\end{feynman}
\end{tikzpicture}}}
\hspace{17mm}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (a1) at (0,0) [label=below:\(A\)];
\vertex [right=0.8cm of a1] (n1) [label=below:\(\dot{N}\)];
\vertex [above=1.3cm of n1] (r) ;
\vertex [above=3cm of n1] (n2) [label=above:\(\dot{N}\)];
\vertex [above=0.5cm of a1] (a2);
\vertex [left=0.3cm of r] (r1);
\vertex [above=0.2cm of r] (r2);
\draw[fuc, thick] (r) -- (r1) -- (r2) -- cycle;
\draw [thick,rounded corners=1mm] (a1) -- (a2) --(r1);
\fill[fuc] (r) -- (r1) -- (r2) -- cycle;
\vertex [above=0.6cm of r2] (s1) ;
\vertex [above=0.2cm of s1] (s2);
\vertex [left=0.3cm of s1] (s3);
\draw[fuc, thick] (s1) -- (s2) -- (s3) -- cycle;
\fill[fuc] (s1) -- (s2) -- (s3) -- cycle;
\vertex[left=0.7cm of a1] (aa1) [label=below:\(A\)];
\vertex [above =0.5cm of aa1] (aa2);
\draw [thick,rounded corners=1mm] (aa1) -- (aa2) --(s3);
\diagram*{
(n1)--[thick] (n2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (a1) at (0,0) [label=below:\(A\)];
\vertex [right=0.8cm of a1] (n1) [label=below:\(\dot{N}\)];
\vertex [above=2cm of n1] (r) ;
\vertex [above=3cm of n1] (n2) [label=above:\(\dot{N}\)];
\vertex [above=0.5cm of a1] (a2);
\vertex [left=0.3cm of r] (r1);
\vertex [above=0.2cm of r] (r2);
\vertex [left=0.8cm of a1] (aa1) [label=below:\(A\)];
\vertex[ left=0.4cm of a1] (b1);
\vertex[above=0.5cm of aa1] (aa2);
\vertex [small,red, dot][above=0.7cm of b1] (b2) {};
\vertex[above=0.5cm of b2] (b3);
\draw[fuc, thick] (r) -- (r1) -- (r2) -- cycle;
\draw [thick,rounded corners=1mm] (a1) -- (a2) --(b2);
\draw [thick,rounded corners=1mm] (aa1) -- (aa2) --(b2);
\draw [thick,rounded corners=1mm] (b2) -- (b3) --(r1);
\fill[fuc] (r) -- (r1) -- (r2) -- cycle;
\diagram*{
(n1)--[thick] (r),
(r)--[thick] (n2)
\end{feynman}
\end{tikzpicture}}}
\hspace{17mm}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (a1) at (0,0) ;
\vertex [above=0.4cm of a1] (aa1);
\vertex[above=0.5cm of aa1] (aa2);
\vertex [right=1cm of a1] (n1) [label=below:\(\dot{N}\)];
\vertex [above=1.4cm of n1] (r) ;
\vertex [above=1.5cm of r] (n2) [label=above:\(\dot{N}\)];
\vertex [above=0.5cm of a1] (a2);
\vertex [left=0.3cm of r] (r1);
\vertex [above=0.2cm of r] (r2);
\draw[fuc, thick] (r) -- (r1) -- (r2) -- cycle;
\draw [thick,rounded corners=1mm] (aa1) -- (aa2) --(r1);
\fill[fuc] (r) -- (r1) -- (r2) -- cycle;
\draw[black,fill=baby] (aa1) circle (.5ex);
\diagram*{
(n1)--[thick] (r),
(r)--[thick] (n2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (n1) at (0,0) [label=below:\(\dot{N}\)] ;
\vertex[above=3cm of n1] (n2) [label=above:\(\dot{N}\)];
\diagram*{
(n1)--[thick] (n2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

Let  $(\dot{N},\rho^N)$ and $(\dot{M},\rho^M)$ be left $A$-modules. An $A$-module intertwiner is a morphism $f\in\Hom(\dot{N},\dot{M})$ such that

\begin{equation}
f\circ \rho^N=\rho^M\circ \left(\operatorname{id}_A\otimes f\right)\,.
\end{equation}

\begin{equation}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (a1) at (0,0) [label=below:\(A\)];
\vertex [right=1cm of a1] (n1) [label=below:\(\dot{N}\)];
\vertex [above=1.4cm of n1] (r) ;
\vertex [above=3cm of n1] (n2) [label=above:\(\dot{N}\)];
\vertex [above=0.5cm of a1] (a2);
\vertex [left=0.3cm of r] (r1);
\vertex [above=0.2cm of r] (r2);
\vertex [above=0.5cm of r2] (f1) [label=above:\( f\)];
\vertex[above=0.5cm of f1] (f2);
\draw[fuc, thick] (r) -- (r1) -- (r2) -- cycle;
\draw [thick,rounded corners=1mm] (a1) -- (a2) --(r1);
\fill[fuc] (r) -- (r1) -- (r2) -- cycle;
\draw [fill=yellow] (0.75,2.1) rectangle (1.25, 2.6);
\vertex [above=0.5cm of r2] (f1) [label=above:\(f\)];
\diagram*{
(n1)--[thick] (r),
(r)--[thick] (f1),
(f2)--[thick] (n2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (a1) at (0,0) [label=below:\(A\)];
\vertex [right=1cm of a1] (n1) [label=below:\(\dot{N}\)];
\vertex [above=2.1cm of n1] (r) ;
\vertex [above=3cm of n1] (n2) [label=above:\(\dot{N}\)];
\vertex [above=0.6cm of a1] (a2);
\vertex [left=0.3cm of r] (r1);
\vertex [above=0.2cm of r] (r2);
\vertex [above=0.9cm of n1] (f1) [label=above:\(f\)];
\vertex[above=0.5cm of f1] (f2);
\draw[fuc, thick] (r) -- (r1) -- (r2) -- cycle;
\draw [thick,rounded corners=1mm] (a1) -- (a2) --(r1);
\fill[fuc] (r) -- (r1) -- (r2) -- cycle;
\draw [fill=yellow] (0.75,0.9) rectangle (1.25, 1.4);
\vertex [above=0.9cm of n1] (f1) [label=above:\(f\)];
\diagram*{
(n1)--[thick] (f1),
(f2)--[thick] (r),
(r)--[thick] (n2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

Given an algebra $A$ in $\mathcal{C}$ we can obtain the category of left $A$-modules $\mathcal{C}_A$ whose objects are the $A$-modules and whose morphisms are the $A$-module intertwiners, i.e., elements of the subspaces

\begin{equation}
\operatorname{Hom}_A(N, M):=\left\{f \in \operatorname{Hom}(\dot{N}, \dot{M}) \mid f \circ \rho^N=\rho^M \circ\left(\mathrm{id}_A \otimes f\right)\right\}\,.
\end{equation}

A simple object in  $\mathcal{C}_A$ is called a simple $A$-module. We will denote the representatives for the isomorphism classes of simple $A$-modules by $M_a$ and the corresponding indexing set by $\mathcal{J}_A$. Some of the properties of $\mathcal{C}$ are inherited by $\mathcal{C}_A$ . For instance in our case $\mathcal{C}_A$ is semisimple, see [5].

Recall that a module category $\mathcal{M}$ over a tensor category $\mathcal{C}$ is a category together with a bifunctor [The usual convention is to denote this functor by the same symbol as the tensor product bifunctor of $\mathcal{C}$.]
\otimes: \mathcal{M}\times\mathcal{C} \to
\mathcal{M}
that obeys generalised unit and associativity constraints. More specifically, $\mathcal{M}$ is equipped with a natural isomorphism with components

\begin{equation}
m_{M,U,V}: M\otimes (U\otimes V)\to (M\otimes U)\otimes V, \qquad U,V\in\operatorname{Obj}(\mathcal{C}),\, M\in\operatorname{Obj}(\mathcal{M})
\end{equation}

satisfying the module pentagon identity (<ref>).

The category $\mathcal{C}_A$ is a module category over $\mathcal{C}$  as for any $U\in\objc$ and $M\in\operatorname{Obj}(\mathcal{C}_A)$ the object $M\otimes U$ has a natural left $A$-module structure. In particular, choosing $M=A$ we obtain:

Let $U\in\objc$ and $A$ an algebra in $\mathcal{C}$. The induced left module $\operatorname{Ind}_A(U)$ is the left $A$-module whose underlying object is $A\otimes U$ and whose representation morphism is $m\otimes \operatorname{id}_U$, that is

\begin{equation}
\operatorname{Ind}_A(U):=\left( A\otimes U,\,m\otimes \operatorname{id}_U\right)\,.
\end{equation}

Since $M\otimes U_i$ is a left $A$-module and thus labels a conformal boundary condition, we can  define boundary fields as follows. Boundary fields which join a segment of boundary labelled by $M$ to a segment labelled by $N$ and that transform in the representation $U_i$ of $\mathfrak{V}$ are elements of $\Hom_{A}(M\otimes U_i,N)$.

For computational purposes, it is useful to express the representation map $\rho^M$ introduced in (<ref>)  with respect to a  basis. To that end, for any given module $(\dot{M},\rho^M)$ we choose bases $b^M_{i,\alpha}$ for $\Hom(U_i,\dot{M})$ and dual bases $b_M^{j,\beta}$ for $\Hom(\dot{M},U_j)$
depicted as

\begin{equation}\label{emb1}
b^M_{i,\alpha}~=:~ \vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U_i\)];
\vertex [above=2.1cm of j1] (j2) [label=above:\(\dot{M}\)];
\vertex [above=0.9cm of j1] (j4);
\node[isosceles triangle,
minimum size =0.3cm] (T1)at (j4) {};
\vertex [below=0.1cm of j4] (j5) ;
\vertex [above right=0.12 of j5] (jj5) [label=above right:\scriptsize\(\alpha\)];
\diagram*{
(j1)--[thick] (j5),
(T1)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\hspace{25mm}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(\dot{M}\)];
\vertex [above=2.1cm of j1] (j2) [label=above:\(U_j\)];
\vertex [above=1.1cm of j1] (j4);
\node[isosceles triangle,
minimum size =0.3cm] (T1)at (j4){};
\vertex [below=0.25cm of j4] (j5) ;
\vertex [right=0.23cm of j5] (jj5) [label=\scriptsize\(\beta\)];
\diagram*{
(j1)--[thick] (T1),
(T1)--[thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

These morphisms are
normalised so  that

\begin{equation}
b_M^{j,\beta}\circ b^M_{i,\alpha}=\delta_{i,j}\delta_{\alpha,\beta}\operatorname{id}_{U_i}\,
\end{equation}

and satisfy the completeness condition
∑_i, α  b^M_i,α ∘b_M^i,α  = id_M   .
Using these bases we have the decomposition

\begin{equation}\label{rep2}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0) [label=below:\(U_i\)];
\vertex [above=2.8cm of j1] (j2) ;
\vertex [above=1.4cm of j1] (j4);
\vertex [above=2.1cm of j1] (j5) [label=right:\(\dot{M}\)];
\node[isosceles triangle,
minimum size =0.3cm] (T1)at (j4)  {};
\vertex [below=0.1cm of j4] (j5) ;
\vertex [above right=0.15cm of j5] (jj5) [label=right:\scriptsize\(\beta\)];
\node[isosceles triangle,
minimum size =0.3cm] (T2)at (j2){};
\vertex [below right=0.1cm of j2] (jj2) [label= right:\scriptsize\(\gamma\)];
\vertex [above=0.8cm of j2] (top) [label=above:\(U_j\)];
\vertex [left=1cm of j1] (a1) [label=below:\(U_a\)];
\vertex [above=0.6cm of a1] (a2) [label=above:\scriptsize\(\alpha\)];
\vertex [above=0.6cm of T1] (a3);
\vertex [left=0.3cm of a3] (a4);
\vertex [above=0.2cm of a3] (a5);
\draw[fuc, thick] (a3) -- (a4) -- (a5) -- cycle;
\fill[fuc] (a3) -- (a4) -- (a5) -- cycle;
\node[isosceles triangle,
minimum size =0.4cm] (T3)at (a2)  {};
\vertex[above=0.6cm of T3] (a5);
\draw [thick,rounded corners=1mm] (T3) -- (a5) --(a4);
\vertex [above=0.5cm of a1] (a2) [label=above:\scriptsize\(\alpha\)];
\diagram*{
(j1)--[thick] (j5),
(T1)--[thick] (T2),
(T2)--[thick] (top),
(a1)--[thick] (T3)
\end{feynman}
\end{tikzpicture}}}~\;=\mathlarger{\sum}_{\delta=1}^{\tensor{N}{_{ai}}{^j}} \,\mathlarger{\rho}^{M\;(j\gamma);\delta}_{(a\alpha)(i\beta)} \;
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=below:\(U_a\)];
\vertex [right=1cm of i1] (j1) [label=below:\(U_i\)];
\vertex [above=0.7cm of i1] (i2);
\vertex [above =0.7cm of j1] (j2);
\vertex  [right=0.5cm of i1] (k1) ;
\vertex [small,orange, dot][above =1.8cm of k1] (k2)[label=right:\(\delta\)] {};
\vertex [above=1.8cm of k2] (k3) [label=above:\(U_j\)];
\draw [thick,rounded corners=1mm] (i1)--(i2)--(k2);
\draw [thick,rounded corners=1mm] (j1)--(j2)--(k2);
\diagram*{
(k2)--[thick] (k3)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

where $\rho^{M\;(j\gamma);\delta}_{(a\alpha)(i\beta)}\in{\mathbb C}$ are the representation numbers of $M$.  Similarly we can define representation numbers in the case where we have a right module over $A$.

§.§ Bimodules over Frobenius algebras and their fusion

In this subsection we consider topological interfaces between two RCFTs sharing the same chiral symmetry algebra described by ${\mathcal C}$.

In the TFT approach such interfaces are described  by $A$-$B$-bimodules where the RCFTs on each side of the interface are the ones obtained from the Frobenius algebras $A$ and $B$ respectively [7, 16].

* Let $A,B$  be algebra objects in a tensor category $\mathcal{C}$. An $A$-$B$-bimodule $M$ is a triple $(\dot{M},\rho^M,\tilde{\rho}^M)$ where $\dot{M}\in\operatorname{Obj}(\mathcal{C})$, $\rho^M\in\Hom(A\otimes\dot{M},\dot{M})$ and $\tilde{\rho}^M\in\Hom(\dot{M}\otimes B,\dot{M})$ such that $(\dot{M},\rho^M)$ is a left $A$-module, $(\dot{M},\tilde{\rho}^M)$ is a right $B$-module and
two actions commute:

\begin{equation}
\rho^M\circ(\operatorname{id}_A\otimes \tilde{\rho}^M)=\tilde{\rho}^M\circ(\rho^M\otimes \operatorname{id}_B)\,.
\end{equation}

* Given two $A$-$B$-bimodules $X,Y$, the space of bimodule intertwiners is defined as the subspace

\begin{equation}
\Hom_{A|B}(X,Y):=\{f\in\Hom(\dot{X},\dot{Y})\,|\,f\circ \rho^X=\rho^Y\circ(\operatorname{id}_A\otimes f),\; f\circ \Tilde{\rho}_X=\tilde{\rho}^Y\circ (f\otimes \operatorname{id}_B)\}\,.
\end{equation}

Thus, we obtain the category $\prescript{}{A}{\mathcal{C}}_B$ whose objects are $A$-$B$-bimodules and whose morphisms are $A$-$B$-bimodule intertwiners. Certain properties of $\mathcal{C}$ are inherited by  $\prescript{}{A}{\mathcal{C}}_B$. For example, if $\mathcal{C}$ has a finite number of isomorphism classes of simple objects, then so has $\dc$. In the following, we shall choose particular representatives of the isomorphism classes of simple $A$-$B$-bimodules denoted $\{X_x\}$ for $x$ in an indexing set $\mathcal{K}_{AB}$. An object $X\in\operatorname{Obj}(\prescript{}{A}{\mathcal{C}}_B)$ will  be represented graphically as

\begin{equation}
\vcenter{\hbox{\begin{tikzpicture}[block/.style ={rectangle, draw=black, thick, text width=5em,align=center, minimum height=1em},font=\scriptsize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) ;
\vertex [right=1cm of i1] (j1);
\vertex [above=0.5cm of i1] (i2) [label=right:\(X\)];
\vertex [above=0.5cm of j1] (j2) ;
\vertex [above=3cm of i1] (i3);
\vertex [right =1cm of i3] (j3) ;
\vertex  [left=0.5cm of i3] (a1)  ;
\vertex [below= 0.4cm of a1] (a2);
\node at (a2) [rectangle,draw] (a3) {$A$};
\vertex [right=1cm of a2] (b1);
\node at (b1) [rectangle,draw] (b3) {$B$};
\vertex [right=1cm of b1] (c1);
\diagram*{
(i1)--[thick,middlearrow={latex}] (i3)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

While we will only work with the bimodules in $\prescript{}{A}{\mathcal{C}}_B$ with the arrow directed upwards, the opposite arrow represents the dual objects in $\prescript{}{B}{\mathcal{C}}_A$.

Unlike $\mathcal{C}_A$, $\prescript{}{A}{\mathcal{C}}_A$ has a structure of a tensor category. To see this, consider two simple defects  running parallel to each other. In the limit of vanishing distance they fuse to a single defect. Thus emerges the notion of a fusion algebra of defects [39]. In the TFT framework  this  fusion corresponds to the tensor product of bimodules over an algebra. Before we formally define this we need to introduce the tensor product of a right $A$-module $M$ and a left $A$-module $N$.

Let $(\dot{M},\rho^M)$ be a right $A$-module and $(\dot{N},\rho^N)$ a left $A$-module where $A$ is a Frobenius algebra in a modular tensor category $\mathcal{C}$. The tensor product over $A$ denoted $M\otimes_AN$ is the object

\begin{equation}\label{coker}
\end{equation}

The corresponding idempotent $P_{M,N}\in\operatorname{End}(M\otimes N)$ with $\operatorname{Im}(P_{M,N})=M\otimes_AN$ is found to be [Note that in the following, in order to lighten the notation in diagrams we will often identify the module with its underlying object $M\equiv \dot{M}$.]

\begin{equation}\label{proj}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (m1) at (0,0) [label=below:\(M\)];
\vertex [above=2.1cm of m1] (m2);
\vertex [right=0.3cm of m2] (m3);
\vertex [above=0.2cm of m2] (m4);
\draw[fuc, thick] (m2) -- (m3) -- (m4) -- cycle;
\fill[fuc] (m2) -- (m3) -- (m4) -- cycle;
\vertex[above=3cm of m1] (m5) [label=above:\(M\)];
\vertex [right=1.5cm of m1] (n1) [label=below:\(N\)];
\vertex [above=2.1cm of n1] (n2);
\vertex [left=0.3cm of n2] (n3);
\vertex [above=0.2cm of n2] (n4);
\draw[fuc, thick] (n2) -- (n3) -- (n4) -- cycle;
\fill[fuc] (n2) -- (n3) -- (n4) -- cycle;
\vertex [above=3cm of n1] (n5) [label=above:\(N\)];
\vertex[right=0.75cm of m1] (a1);
\vertex [above=0.6cm of a1] (a2);
\vertex  [small,blue, dot][above=0.5cm of a2] (a3) {};
\vertex [above left=0.5cm of a3] (a4);
\vertex[above right=0.5cm of a3] (a5);
\draw [thick,rounded corners=1mm,looseness=3] (a3) -- (a4) --(m3);
\draw [thick,rounded corners=1mm,looseness=3] (a3) -- (a5) --(n3);
\diagram*{
(m1)--[thick] (m5),
(n1) --[thick] (n5),
(a2)--[thick] (a3)
\draw[black,fill=baby] (a2) circle (.5ex);
\end{feynman}
\end{tikzpicture}}}
\end{equation}

The tensor product $M\otimes_AN$ is not in general an $A$-module. Interesting situations occur when the initial objects have bimodule structures. More specifically, consider an $A$-$B$-bimodule $X$ and a $B$-$C$-bimodule $Y$ where $A,B$ and $C$ are Frobenius algebras in $\mathcal{C}$. Then $X\otimes_ BY$ is an $A$-$C$-bimodule. Also, noticing that $X\otimes Y$ is an $A$-$C$-bimodule one can show that $P_{X,Y}$ is a bimodule intertwiner $P_{X,Y}\in\Hom_{A|C}(X\otimes Y,X\otimes Y)$. Since $\mathcal{C}$ is abelian, there exists an object $\operatorname{Im}(P_{X,Y})$ in $\prescript{}{A}{\mathcal{C}}_C$, a monomorphism $e_{X,Y}\in\Hom_{A|C}(\operatorname{Im}(P_{X,Y}),X\otimes Y)$ and an epimorphism $r_{X,Y}\in\Hom_{A|C}(X\otimes Y,\operatorname{Im}(P_{X,Y}))$ such that

\begin{equation}\label{proj11}
P_{X,Y}=e_{X,Y}\circ r_{X,Y}\,.
\end{equation}

Then we define

\begin{equation}
\end{equation}

Furthermore, defining the action on morphisms $f\in\Hom_{A|B}(X,X^\prime)$ and $g\in \Hom_{B|C}(Y,Y^\prime)$ as

\begin{equation}\label{newprod}
f \otimes_B g:=r_{X^{\prime}, Y^{\prime}} \circ(f \otimes g) \circ e_{X, Y}
\end{equation}

we obtain a bifunctor $\otimes_B:\mathcal{C}_{A|B}\times\mathcal{C}_{B|C}\to\mathcal{C}_{A|C}$.

We still have to find an appropriate associator for $\prescript{}{A}{\mathcal{C}}_A$ to define a tensor category structure. It will prove useful to start from the most general tensor structure that can appear between three bimodules. Let $X\in\operatorname{Obj}(\prescript{}{A}{\mathcal{C}}_B),\, Y\in\operatorname{Obj}(\prescript{}{B}{\mathcal{C}}_C),\,Z\in\operatorname{Obj}( \prescript{}{C}{\mathcal{C}}_D)$. We define  the associator maps

\begin{equation}
\check{\alpha}_{X,Y,Z}:X\otimes_B(Y\otimes_CZ)\to(X\otimes_BY)\otimes_CZ
\end{equation}

\begin{equation}\label{associator2}
\check{\alpha}_{X,Y,Z}=r_{X\otimes_BY,Z}\circ\left(r_{X,Y}\otimes \operatorname{id}_{Z}\right)\circ\left(\operatorname{id}_{X}\otimes e_{Y,Z}\right)\circ e_{X,Y\otimes_CZ}\,.
\end{equation}

Specialising to the case  $A=B=C=D$ we  introduce the associator

\begin{equation}\label{associator11}
\check{\alpha}_{X,Y,Z}:X\otimes_A(Y\otimes_AZ)\to(X\otimes_AY)\otimes_AZ
\end{equation}

of $\prescript{}{A}{\mathcal{C}}_A$ defined by

\begin{equation}\label{associator1}
\check{\alpha}_{X,Y,Z}=r_{X\otimes_AY,Z}\circ\left(r_{X,Y}\otimes \operatorname{id}_Z\right)\circ\left(\operatorname{id}_X\otimes e_{Y,Z}\right)\circ e_{X,Y\otimes_AZ}\,.
\end{equation}

It is shown in appendix  <ref> that this associator satisfies  the pentagon identity turning $\prescript{}{A}{\mathcal{C}}_A$ into a tensor category.

Similarly to the fusion of two interfaces described above, we can define a fusion of an interface and a conformal boundary condition.
Consider an $A$-$B$-interface described by a $A$-$B$-bimodule $X$ running close to a $B$-boundary described by a left $B$-module $M$.
The result of the fusion is the object $X\otimes_BM$ defined in (<ref>) which has a left $A$-module structure and thus represents a new boundary condition.

§.§ Junctions

In this subsection we introduce various types of junctions between interfaces and boundaries. Such junctions appear in a network of topological interfaces and conformal boundary conditions and have been studied in [16, 18, 30].

The first type of junctions we consider joins  three topological interfaces.  At this point we choose the conventions of [16] so that the morphisms in the diagrams which involve interfaces flow from top to bottom. Then the junctions labelled by morphisms $\alpha\in \operatorname{Hom}_{A|B}(X^\prime,X)$ and $\beta \in \Hom_{A|C}(Z,X\otimes_BY)$ are represented graphically as

\begin{equation}\label{jun1}
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=above right:\(X\)] ;
\vertex [dot,yellow] [above =1.5cm of i1] (i2) [label=right:\(\alpha\)] {};
\vertex[above=0.2cm of i2] (i4);
\vertex[above=0.4cm of i4] (i6) [label=right:\(X^\prime\)];
\vertex [above=3cm of i1] (top) ;
\vertex [left=0.5cm of top] (a1);
\vertex [below=0.3cm of a1] (a2);
\node at (a2) [rectangle,draw] (a3) {$A$};
\vertex [right=0.5cm of top] (b1);
\vertex [below=0.3cm of b1] (b2);
\node at (b2) [rectangle,draw] (b3) {$B$};
\diagram*{
(i1)--[thick,middlearrow={latex}] (i2),
(i2)--[thick,middlearrow={latex}] (top)
\end{feynman}
\end{tikzpicture}}}
~\qquad \text{and}\qquad~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) [label=above left:\(X\)] ;
\vertex[right=1.3cm of i1] (j1)[label=above right:\(Y\)];
\vertex[right=0.65cm of i1] (b1);
\node at (b1) [rectangle,draw] (b2) {$B$};
\vertex [dot,yellow] [above=1.5cm of b1] (b3) [label=right:\(\beta\)] {};
\vertex[left=0.2cm of b3] (b5);
\vertex[above=0.2cm of b5] (b6);
\vertex[above=3cm of b1] (top);
\vertex[above=0.2cm of b3] (b7);
\vertex[above=0.6cm of b7] (b8) [label=right:\(Z\)];
\vertex[left=0.7cm of top] (a1);
\vertex[below=0.4cm of a1] (a2);
\node at (a2) [rectangle,draw] (a3) {$A$};
\vertex[right=0.9cm of top] (c1);
\vertex[below=0.4cm of c1] (c2);
\node at (c2) [rectangle,draw] (c3) {$C$};
\diagram*{
(i1)--[thick,middlearrow={latex}] (b3),
(j1)--[thick,middlearrow={latex}] (b3),
(b3)--[thick,middlearrow={latex}] (top)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

For  calculations  we introduce  bases in  morphism spaces:

\begin{equation}\label{base2}
\Lambda^\alpha_{(x,y)z}\in\Hom_{A|C}(X_x\otimes_BX_y,X_z)\qquad \text{and}\qquad \bar{\Lambda}^\alpha_{(x,y)z}\in\Hom_{A|C}(X_z,X_x\otimes_BX_y)
\end{equation}

where $\alpha=1,\ldots,\operatorname{dim}\Hom_{A|C}(X_x\otimes_BX_y,X_z,)$ and $x\in\mathcal{K}_{AB},\;y\in\mathcal{K}_{BC},\;z\in\mathcal{K}_{AC}$. We fix their normalisation by

\begin{equation}\label{normLambda}
\Lambda^\alpha_{(x,y)z}\circ \bar{\Lambda}^\beta_{(x,y)w}=\delta_{z,w}\delta_{\alpha,\beta}\operatorname{id}_{X_z}
\end{equation}

or in graphical notation

\begin{equation} \label{defnorm}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  [small,yellow, dot] (del) at (0,0) [label=below:\(\beta\)] {} ;
\vertex  [small,yellow, dot] (gam) at (0,-2) [label=above:\(\alpha\)] {} ;
\vertex [below=1cm of gam] (j1) [label=below:\(X_z\)];
\vertex [above=1cm of del] (j2) [label=above:\(X_w\)];
\vertex (kek) at (0.8,-1) [label=right:\(X_y\)];
\diagram*{
(del)--[thick,middlearrow={latex}] (j2),
(j1)--[thick,middlearrow={latex}] (gam),
(gam) --[half right, thick, looseness=1.2,middlearrow={latex}] (del),
(gam) --[half left, thick, looseness=1.2, edge label=$X_x$,middlearrow={latex}] (del)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{5mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (1) at (0,1) [label=above:\(X_z\)];
\vertex (2) at (0,-3) [label=below:\(X_z\)];
\diagram*{
(2)--[thick,middlearrow={latex}] (1)
\end{feynman}
\end{tikzpicture}}}\end{equation}

Let us note that the basis morphisms (<ref>) satisfy a completeness relation analogous to (<ref>):

\begin{equation}\label{complete3def}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  (i1) at (0,0) [label=below:\(X_x\)]  ;
\vertex [above=3cm of i1] (i2) [label=above:\(X_x\)];
\vertex [right=1cm of i1] (j1) [label=below:\(X_y\)];
\vertex [right=1cm of i2] (j2) [label=above:\(X_y\)];
\diagram*{
(i1)--[middlearrow={latex},thick] (i2),
(j1)--[middlearrow={latex},thick] (j2)
\end{feynman}
\end{tikzpicture}}}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  [small,yellow,dot] (up) at (0,0) [label=above:\(\gamma\)] {};
\vertex [small,yellow,dot] [below=1cm of up] (down) [label=below:\(\gamma\)] {};
\vertex [above left=0.75cm of up] (i1);
\vertex [above=0.6cm of i1] (i2) [label=above:\(X_x\)];
\vertex [above right=0.75cm of up] (j1);
\vertex [above=0.6cm of j1] (j2) [label=above:\(X_y\)];
\draw [middlearrow={latex},thick] (up) -- (i1) --(i2);
\draw [middlearrow={latex},thick] (up) -- (j1) --(j2);
\vertex [below left=0.75cm of down] (i3) ;
\vertex [below=0.6cm of i3] (i4) [label=below:\(X_x\)];
\vertex [below right=0.75cm of down] (j3);
\vertex [below=0.6cm of j3] (j4) [label=below:\(X_y\)];
\draw [myptr={latex},thick] (j4) -- (j3) --(down);
\draw [myptr={latex},thick] (i4) -- (i3) --(down);
\diagram*{
(down)--[middlearrow={latex},thick,edge label=$X_z$] (up)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

due to the semisimplicity of the category $_A\mathcal{C}_C$ as established in [16].

The hom-spaces introduced in (<ref>) can be described as subspaces of larger hom-spaces – the ones defined using the tensor product in $\mathcal{C}$. Namely, the following two isomorphisms can be defined

\begin{align}\label{iso1}
\phi_{X,Y}^Z:{\Hom_{A|C}}_{(P_{X,Y})}(X\otimes Y,Z) &\stackrel{\cong}{\longrightarrow} \Hom_{A|C}(X\otimes_BY,Z) \quad \text{and}\nonumber\\
\bar{\phi}_{X,Y}^Z:\Hom_{A|C}^{\;\;\;\;\;(P_{X,Y})}(Z,X\otimes Y) &\stackrel{\cong}{\longrightarrow} \operatorname{Hom}_{A \mid C}\left(Z, X \otimes_B Y\right)
\end{align}

\begin{align}\label{spaces}
{\Hom_{A|C}}_{(P_{X,Y})}(X\otimes Y,Z)    &:=\{f \in  \Hom_{A|C}(X\otimes Y,Z) \mid f \circ P_{X,Y}=f\} \, ,  \nonumber\\ \Hom_{A|C}^{\;\;\;\;\;(P_{X,Y})}(Z,X\otimes Y) &:=\{g \in \Hom_{A|C}(Z,X\otimes Y) \mid P_{X,Y} \circ g=g\}
\end{align}

and the isomorphisms in (<ref>) act as

\begin{equation} \label{phi_action}
\phi_{X,Y}^Z (f) = f\circ e_{X,Y} \, , \qquad \bar{\phi}_{X,Y}^Z(g)=r_{X,Y}\circ g \, .
\end{equation}

The images of  the bases (<ref>) under the isomorphisms (<ref>) are

\begin{equation} \label{base3}
\Tilde{\Lambda}^\alpha_{(x,y)z}\in{\Hom_{A|C}}_{(P_{X_x,X_y})}(X_x\otimes X_y,X_z) \quad \text{and} \quad \bar{\Tilde{\Lambda}}^\alpha_{(x,y)z}\in\Hom_{A|C}^{\;\;\;\;\;(P_{X_x,X_y})}(X_z,X_x\otimes X_y)
\end{equation}

defined by

\begin{equation}
\Tilde{\Lambda}^\alpha_{(x,y)z}=  \Lambda^\alpha_{(x,y)z}\circ r_{X_x,X_y} \qquad \text{and} \qquad \bar{ \Tilde{\Lambda}^\alpha_{(x,y)z}} =e_{X_x,X_y}\circ \bar{\Lambda}^\alpha_{(x,y)z}\,.
\end{equation}

The inverse formula reads

\begin{equation}\label{baserel1}
\Lambda^\alpha_{(x,y)z}=\Tilde{\Lambda}^\alpha_{(x,y)z}\circ e_{X_x,X_y} \qquad \text{and} \qquad \bar{\Lambda}^\alpha_{(x,y)z}=r_{X_x,X_y}\circ \bar{\Tilde{\Lambda}}^\alpha_{(x,y)z} \, .
\end{equation}

On the diagrams we will depict the $ \Tilde{\Lambda}$ morphisms and their duals as hollow circles.

Next we consider junctions of topological interfaces with conformal boundaries   where an $A$-$B$-interface $X$ can start or end on the boundary [16], which we call open defects. These will correspond respectively to morphisms $\alpha\in\Hom_A(M,X\otimes_BN)$ and $\beta\in\Hom_A(X\otimes_BN,M)$ where $M$ is an $A$-module while $N$ is a $B$-module. Graphically

\begin{equation}\label{jun2}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (-0.25,0.9);
\vertex  (x2) at (0.75, 0.65);
\vertex[below=0.9cm of x1] (xbot);
\vertex [above=3cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=left:\(N\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=left:\(M\)];
\vertex [below=1.1cm of j2] (j6) ;
\vertex[left=0.15cm of j6] [label=left:\(\alpha\)];
\vertex   [above=1.8cm of j1] (j5) {};
\vertex [left=0.1cm of j5] (j7);
\vertex (x) at (-0.5,0.6) [label=above:\(X\)];
\draw [thick]    (x1) to[out=90,in=-90] (j7);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,3) rectangle (1.6,0);
\draw [fill=yellow] (1.06,1.8) rectangle (0.8, 1.9);
\draw [thick] (-0.1,1.4) rectangle (-0.5, 1.8);
\vertex at (-0.3,1.4) [label=\(A\)];
\draw [thick] (0.1,0.4) rectangle (0.5, 0);
\vertex at (0.3,0) [label=\(B\)];
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6),
(xbot)--[thick,middlearrow={latex}] (x1)
\end{feynman}
\end{tikzpicture}}}
~\hspace{14mm} \text{and} \hspace{14mm}~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (-0.25,2.2);
\vertex  (x2) at (0.75, 0.65);
\vertex [above=3cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=left:\(M\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.4cm of j2] (j4) [label=left:\(N\)];
\vertex [below=1.7cm of j2] (j6);
\vertex   [above=1.2cm of j1] (j5);
\vertex [left=0.15cm of j5] [label=left:\(\beta\)];
\vertex [left=0.1cm of j6] (j7);
\vertex (x) at (-0.5,2.1) [label=above:\(X\)];
\draw [thick]    (j7) to[out=90,in=-90] (x1);
\vertex[above=0.8cm of x1] (xtop);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,3) rectangle (1.6,0);
\draw [fill=yellow] (1.06,1.2) rectangle (0.8, 1.3);
\draw [thick] (0.1,2.6) rectangle (0.5, 3);
\vertex at (0.3,2.6) [label=\(B\)];
\draw [thick] (-0.1,1.2) rectangle (-0.5, 1.6);
\vertex at (-0.3,1.2) [label=\(A\)];
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6),
(x1)--[thick,middlearrow={latex}] (xtop)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

We choose bases for these junction spaces

\begin{equation}\label{base4}
\Omega^\beta_{(x,a)b}\in\Hom_A(X_x\otimes_BM_a,M_b)\qquad \text{and}\qquad \bar{\Omega}^\alpha_{(x,a)b}\in\Hom_A(M_b,X_x\otimes_BM_a)
\end{equation}

where $X_x$, $M_{a}$, $M_{b}$ are a simple bimodule and two simple modules respectively.
The basis elements satisfy the normalisation condition

\begin{equation} \label{junctions_norm}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (j1) at (0,0);
\vertex [above=0.5 of j1] (j2);
\vertex[ above =0.6 of j1] (j3);
\vertex  [ above=2cm of j1] (psi) {};
\vertex [above=3.4cm of j1] (j4) ;
\vertex[above=3.5cm of j1] (j5)  ;
\vertex[ left=0.15cm of  j4] (j7) [label=left:\(\alpha\)];
\vertex [left=0.15cm of j3] (j8) [label=left:\(\beta\)];
\vertex [above =4cm of j1] (j6);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\draw [fill=yellow] (0.06,0.5) rectangle (-0.2, 0.6);
\draw [fill=yellow] (0.06,3.4) rectangle (-0.2, 3.5);
\vertex[below=0.1cm of j6] (m1) [label=left:\(M_a\)];
\vertex[ above=0.1cm of j1] (m4) [label=left:\(M_c\)];
\vertex[above=1.3cm of j1] (m3) ;
\vertex[above=0.01cm of psi] (m2) [label=right:\(M_b\)];
\vertex[left=1.2cm of psi] (x) [label=left:\(X_x\)];
\vertex [below=0.25cm of x] (x1);
\vertex [above=0.25cm of x] (x2);
\draw [thick]    (j8) to[out=90,in=-90] (x1);
\draw [thick]    (x2) to[out=90,in=-90] (j7);
\draw [thick] [middlearrow={latex}]   (x1) to (x2);
\draw [thick] (-1.3,2.8) rectangle (-1.7, 3.2);
\vertex at (-1.5,2.8) [label=\(A\)];
\draw [thick] (-0.3,1.8) rectangle (-0.7, 2.2);
\vertex at (-0.5,1.8) [label=\(B\)];
\diagram*{
(j1) --[thick] (j2),
(j3) --[thick] (j4),
(j5)--[thick] (j6)
\end{feynman}
\end{tikzpicture}}}
~= \delta_{a,c}\delta_{\alpha,\beta} ~
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (1) at (0,0) [label=above left:\(M_a\)];
\vertex (2) at (0,4);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\draw [thick] (-0.4,1.8) rectangle (-0.8, 2.2);
\vertex at (-0.6,1.8) [label=\(A\)];
\diagram*{
(1)--[thick] (2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

The basis morphisms $\Omega$ also satisfy the following completeness relation due to the semisimplicity of the category $\mathcal{C}_A$ [5]:

\begin{equation}
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (i1) at (0,0) ;
\vertex [right=1cm of i1] (j1);
\vertex [above=0.5cm of i1] (i2) [label=left:\(X_x\)];
\vertex [above=0.5cm of j1] (j2) [label=left:\(M_a\)];
\vertex [above=4cm of i1] (i3);
\vertex [right =1cm of i3] (j3) ;
\vertex  [left=0.5cm of i3] (a1)  ;
\vertex [below= 0.4cm of a1] (a2);
\vertex [right=1cm of a2] (b1);
\vertex[right=0.5cm of j3] (top);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (top) rectangle (j1);
\diagram*{
(i1)--[thick,middlearrow={latex}] (i3),
(j1)--[thick] (j3)
\end{feynman}
\end{tikzpicture}}} = \mathlarger{\sum}_{b\in\mathcal{J}_A}\mathlarger{\sum}_\gamma   \vcenter{\hbox{\hspace{5mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.5);
\vertex  (x2) at (0, 0.6);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.1cm of j1] (j3) [label=left:\(M_a\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.1cm of j2] (j4) [label=left:\(M_a\)];
\vertex [below=1.4cm of j2] (j6) ;
\vertex[left=0.2cm of j6] (jjj) ;
\vertex[below=0.1cm of jjj] (jjjj) [label=left:\(\gamma\)];
\vertex   [above=1.5cm of j1] (j5);
\vertex [left=0.2cm of j5] [label=left:\(\gamma\)];
\vertex [left=0.1cm of j6] (j7);
\vertex[left=0.1cm of j5] (j0);
\vertex (x) at (-0.25,2.9) [label=above:\(X_x\)];
\vertex (yy) at (-0.25,0.7) [label=above:\(X_x\)];
\vertex[below=0.1cm of j6] (g1);
\vertex[above=0.1cm of j5] (g2);
\vertex[above=0.5cm of g2] (g3) [label=right:\(M_b\)];
\draw [thick]    (j7) to[out=90,in=-90] (x1);
\draw [thick]    (x2) to[out=90,in=-90] (j0);
\draw [fill=yellow] (1.06,2.5) rectangle (0.8, 2.6);
\draw [fill=yellow] (1.06,1.5) rectangle (0.8, 1.6);
\diagram*{
(j1)--[ thick] (j5),
(j2)--[ thick] (j6),
(g1)--[thick] (g2),
(x1)--[thick,middlearrow={latex}] (i2),
(i1)--[thick,myptr={latex}] (x2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

Similarly to the morphisms $\Tilde{\Lambda}$ introduced above we introduce  morphisms $\tilde \Omega$ using the isomorphism  (<ref>).
In more detail, consider an isomorphism

\begin{equation}\label{iso5}
\operatorname{Hom}_{A|\mathbf{1}}(X_x\otimes_BM_b,M_c)\cong  {\Hom_{A|\mathbf{1}}}_{(P_{X_x,M_b})}(X_x\otimes M_b,M_c)
\end{equation}

given by

ϕ_x,b^c: _A|1_(P_X_x,M_b)(X_x⊗M_b,M_c) →Hom_A|1(X_x⊗_BM_b,M_c)

as defined in (<ref>).
Here we have used $\mathbf{1}$ which is the tensor unit of $\mathcal{C}$  thought of as the trivial algebra.
This isomorphism introduces a basis  in the hom-space on the left of (<ref>):

\begin{equation}\label{omegatilde}
\tilde{\Omega}^\alpha_{(x,b)c} = \Omega^\alpha_{(x,b)c} \circ r_{x,b}
\end{equation}

and similarly the dual basis $ \bar{\Tilde{\Omega}}^\alpha_{(x,b)c} $. On the diagrams we will depict the morphisms $ \tilde{\Omega}$ and their
duals as hollow rectangles.

The final type of junction that we consider in this paper is a boundary field. As mentioned earlier, these morphisms in general are elements of $\operatorname{Hom}_A(M\otimes U,N)$, where $M,N$ are left $A$-modules and $U\in\objc$. We introduce the bases

\begin{equation}\label{bcco}
\Psi^\alpha_{(a,i)b}\in\operatorname{Hom}_A(M_a\otimes U_i,M_b) \qquad \text{and}\qquad \bar{\Psi}^\alpha _{(a,i)b}\in \operatorname{Hom}_A(M_b,M_a\otimes U_i)
\end{equation}

where  all the objects involved are simple.

We represent these bases of  boundary  fields  by black blobs on the boundary, as on the following picture

\begin{equation}\label{bccopic}
\Psi^\alpha_{(a,i)b}= \vcenter{\hbox{\hspace{5mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (a) at (0,0);
\vertex [above=4cm of a] (b);
\vertex [dot] [above=2cm of a] (psi) {};
\vertex[right=0.2cm of psi] (i) ;
\vertex[below=0.2cm of i] (ii) [label=:\(\;\;\,\Psi_i^\alpha\)];
\vertex [above=1.4cm of psi] (b1) [label=left:\(M_a\)];
\vertex [below=1.4cm of psi] (b2) [label=left:\(M_b\)];
\draw [thick] (-1.1,2.2) rectangle (-1.5, 1.8);
\vertex at (-1.3,1.8) [label=\(A\)];
\diagram*{
(a)--[thick] (psi),
(psi)--[thick] (b)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

We further  choose the normalisation condition

\begin{equation}
\Psi^\alpha_{(a,i)b} \circ \bar{\Psi}^\beta _{(a,i)c}=\delta_{b,c}\delta_{\alpha,\beta}\operatorname{id}_{M_b}
\end{equation}

that  graphically looks like

\begin{equation}
\vcenter{\hbox{\hspace{5mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\vertex (a) at (0,0);
\vertex [above=4cm of a] (b);
\vertex [dot] [above=1.4cm of a] (psi) {};
\vertex[above=0.6cm of psi] (ok) [label=left:\(M_a\)];
\vertex [dot] [above=2.6cm of a] (psi1){};
\vertex[right=0.2cm of psi] (i) ;
\vertex[below=0.2cm of i] (ii) [label=:\(\;\;\,\Psi_i^\alpha\)];
\vertex[right=0.2cm of psi1] (d1) ;
\vertex[below=0.2cm of d1] (d2) [label=:\(\;\;\,\bar{\Psi}_i^{\beta}\)];
\vertex [above=3.8cm of a] (b1) [label=left:\(M_c\)];
\vertex [above=0.2cm of a] (b2) [label=left:\(M_b\)];
\draw [thick] (-1.1,2.2) rectangle (-1.5, 1.8);
\vertex at (-1.3,1.8) [label=\(A\)];
\diagram*{
(a)--[thick] (psi),
(psi)--[thick] (b)
\end{feynman}
\end{tikzpicture}}}
= \delta_{b,c}\delta_{\alpha,\beta}\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (1) at (0,0) [label=above left:\(M_b\)];
\vertex (2) at (0,4);
\path[pattern=north east lines,pattern color=ashgrey,very thin] (0.01,4) rectangle (0.6,0);
\draw [thick] (-0.4,1.8) rectangle (-0.8, 2.2);
\vertex at (-0.6,1.8) [label=\(A\)];
\diagram*{
(1)--[thick] (2)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

The basis morphisms $\Psi$ satisfy the following completeness relation due to the semisimplicity of $\mathcal{C}_A$:

\begin{equation}
\id_{M_a\otimes U_i} =\sum_{b\in\mathcal{J}_A}\sum_\gamma \bar{\Psi}^\gamma_{(a,i)b}\circ \Psi^\gamma_{(a,i)b}
\end{equation}

§.§ Fusing matrices

In this section we will introduce various fusing matrices related to the tensor structures  introduced in the previous sections.

§.§.§ Fusing matrix of C

Recall that we assume $\mathcal{C}$ to be a strict category with trivial associator morphisms.
The fusing matrices of  $\mathcal{C}$ arise from two different representations of $\operatorname{Hom}(U_i\otimes U_j\otimes U_k,U_l)$:

$$\bigoplus_q \operatorname{Hom}(U_i\otimes U_j,U_q)\otimes \operatorname{Hom}(U_q\otimes U_k,U_l)$$
and as

$$\bigoplus_p \operatorname{Hom}(U_j\otimes U_k,U_p)\otimes \operatorname{Hom}(U_i\otimes U_p,U_l)\, .$$

Each representation plus the choice of basis (<ref>) gives $\operatorname{Hom}(U_i\otimes U_j\otimes U_k,U_l)$
a distinguished basis.

The matrices that perform the transformation between the two bases are the fusion matrices of $\mathcal{C}$. Their entries are defined by

\begin{equation} \label{fmove1}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(U_l\)] ;
\vertex [small,orange, dot] (alpha) at (0,-1) [label=left:\footnotesize\(\alpha\)] {};
\vertex  [small,orange, dot] (beta) at (1,-2) [label=right:\footnotesize\(\beta\)] {};
\vertex  (i) at (-1,-3) [label=below:\(U_i\)];
\vertex  (left) at (-0.7,-2);
\vertex [above=0.5cm of i] (i2);
\vertex [below left=0.042cm of left] (i3);
\vertex (j) at (0.4,-3) [label=below:\(U_j\)];
\vertex (k) at (1.6,-3) [label=below:\(U_k\)];
\vertex  (j2) at (0.45,-2.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-2.6);
\vertex [above left=0.3cm of k2] (k3);
\draw [thick,rounded corners=1mm] (alpha) -- (left) -- (i2) -- (i);
\draw [thick,rounded corners=1mm] (beta)  -- (j2) -- (j);
\draw [thick,rounded corners=1mm] (beta)  -- (k2) -- (k);
\diagram*{
(l) --[thick]  (alpha)   ,
(alpha) -- [thick,edge label=$U_p$] (beta)
\end{feynman}
\end{tikzpicture}}}
~\hspace{5mm}=\mathlarger{\sum}_{q\in\mathcal{I}}\mathlarger{\sum}_{\gamma,\delta}\,\mathrm{F}_{\alpha p \beta,\gamma q\delta}^{(i\,j\,k)l}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(U_l\)] ;
\vertex [small,orange, dot] (del) at (0,-1) [label=left:\footnotesize\(\delta\)] {};
\vertex  [small, orange, dot] (gamma) at (-1,-2) [label=right:\footnotesize\(\gamma\)] {};
\vertex  (i) at (-1.6,-3) [label=below:\(U_i\)];
\vertex  (i2) at (-1.55,-2.6);
\vertex [below left=0.03cm of i2] (i3);
\vertex (j) at (-0.4,-3) [label=below:\(U_j\)];
\vertex (k) at (1,-3) [label=below:\(U_k\)];
\vertex (j2) at (-0.45,-2.6) ;
\vertex [below right=0.03cm of j2] (j3) ;
\vertex [above=0.5cm of k] (k2);
\vertex [above left=0.005cm of k2] (k3);
\vertex (right) at (0.7,-2);
\draw [thick,rounded corners=1mm] (del) -- (right) -- (k2) -- (k);
\draw [thick,rounded corners=1mm] (gamma) -- (i2)  -- (i);
\draw [thick,rounded corners=1mm] (gamma) -- (j2) -- (j);
\diagram*{
(l) --[thick]  (del)   ,
(del) -- [thick, edge label=$U_q$] (gamma)
\end{feynman}
\end{tikzpicture}}}\end{equation}

The same matrix appears in  an analogous relation involving dual morphisms

\begin{equation}
\vcenter{\hbox{\hspace{2mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=below:\(U_l\)] ;
\vertex [small,orange,dot] [above=1cm of l] (gam) [label=above:\footnotesize\(\delta\)] {};
\vertex [small,orange,dot] [above left=1.25cm of gam] (del) [label=above:\footnotesize\(\gamma\)] {} ;
\vertex [above right=1cm of gam] (k1);
\vertex [above=1.4cm of k1] (k) [label=above:\(U_k\)];
\vertex [above left=0.7cm of del] (i1);
\vertex [above=0.7cm of i1] (i) [label=above: \(U_i\)];
\vertex [above right=0.7cm of del] (j1);
\vertex [above=0.7cm of j1] (j) [label=above: \(U_j\)];
\draw [thick,rounded corners=1mm] (gam) -- (k1) --(k);
\draw [thick,rounded corners=1mm] (del) -- (j1) --(j);
\draw [thick,rounded corners=1mm] (del) -- (i1) --(i);
\diagram*{
(l)--[thick] (gam),
(gam) --[thick,edge label= $U_q$] (del)
\end{feynman}
\end{tikzpicture}}}
~=\;\mathlarger{\sum}_{p\in\mathcal{I}}\mathlarger{\sum}_{\alpha,\beta}\mathrm{F}^{(i\,j\,k)l}_{\alpha p\beta,\gamma q\delta}~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=below:\(U_l\)] ;
\vertex [small,orange,dot] [above=1cm of l] (beta) [label=above:\footnotesize\(\alpha\)] {};
\vertex [above left=1cm of beta] (i1);
\vertex [above=1.4cm of i1] (i) [label=above:\(U_i\)];
\vertex [small,orange,dot] [above right=1.25cm of beta] (alpha) [label=right:\footnotesize\(\,\beta\)] {};
\vertex[above left=0.7cm of alpha] (j1);
\vertex [above=0.7cm of j1] (j) [label=above:\(U_j\)];
\vertex [above right=0.7cm of alpha] (k1);
\vertex [above=0.7cm of k1] (k) [label=above:\(U_k\)];
\draw [thick,rounded corners=1mm] (alpha) -- (j1) --(j);
\draw [thick,rounded corners=1mm] (alpha) -- (k1) --(k);
\draw [thick,rounded corners=1mm] (beta) -- (i1) --(i);
\diagram*{
(alpha) --[thick,edge label= $U_p$] (beta),
(l)--[thick] (beta)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

The inverse of $\mathrm{F}$  is defined as

\begin{equation}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(U_l\)] ;
\vertex [small,orange, dot] (del) at (0,-1) [label=left:\footnotesize\(\beta\)] {};
\vertex  [small, orange, dot] (gamma) at (-1,-2) [label=right:\footnotesize\(\alpha\)] {};
\vertex  (i) at (-1.6,-3) [label=below:\(U_i\)];
\vertex  (i2) at (-1.55,-2.6);
\vertex [below left=0.03cm of i2] (i3);
\vertex (j) at (-0.4,-3) [label=below:\(U_j\)];
\vertex (k) at (1,-3) [label=below:\(U_k\)];
\vertex (j2) at (-0.45,-2.6) ;
\vertex [below right=0.03cm of j2] (j3) ;
\vertex [above=0.5cm of k] (k2);
\vertex [above left=0.005cm of k2] (k3);
\vertex (right) at (0.7,-2);
\draw [thick,rounded corners=1mm] (del) -- (right) -- (k2) -- (k);
\draw [thick,rounded corners=1mm] (gamma) -- (i2)  -- (i);
\draw [thick,rounded corners=1mm] (gamma) -- (j2) -- (j);
\diagram*{
(l) --[thick]  (del)   ,
(del) -- [thick, edge label=$U_p$] (gamma)
\end{feynman}
\end{tikzpicture}}}
~=\; \mathlarger{\sum}_{q,\gamma,\delta} \mathrm{G}^{(i\,j\,k)l}_{\alpha p\beta,\gamma q\delta}~
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(U_l\)] ;
\vertex [small,orange, dot] (alpha) at (0,-1) [label=left:\footnotesize\(\gamma\)] {};
\vertex  [small,orange, dot] (beta) at (1,-2) [label=right:\footnotesize\(\delta\)] {};
\vertex  (i) at (-1,-3) [label=below:\(U_i\)];
\vertex  (left) at (-0.7,-2);
\vertex [above=0.5cm of i] (i2);
\vertex [below left=0.042cm of left] (i3);
\vertex (j) at (0.4,-3) [label=below:\(U_j\)];
\vertex (k) at (1.6,-3) [label=below:\(U_k\)];
\vertex  (j2) at (0.45,-2.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-2.6);
\vertex [above left=0.3cm of k2] (k3);
\draw [thick,rounded corners=1mm] (alpha) -- (left) -- (i2) -- (i);
\draw [thick,rounded corners=1mm] (beta)  -- (j2) -- (j);
\draw [thick,rounded corners=1mm] (beta)  -- (k2) -- (k);
\diagram*{
(l) --[thick]  (alpha)   ,
(alpha) -- [thick,edge label=$U_q$] (beta)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

§.§.§ Fusing matrix Y for bimodules

Next we   define  fusing matrices for the case when all the objects involved are bimodules. More specifically, consider equation (<ref>) with all simple objects $U_i$ of $\mathcal{C}$ replaced with simple objects $X_x$ of  bimodule categories. Taking into account the construction of the fusion of bimodules, the most general definition  includes three initial simple bimodules: $X_x\in\operatorname{Obj}(\prescript{}{A}{\mathcal{C}}_B),\, X_y\in\operatorname{Obj}(\prescript{}{B}{\mathcal{C}}_C),\,X_z\in\operatorname{Obj}( \prescript{}{C}{\mathcal{C}}_D)$ while the  simple bimodule at the top of the diagram must be $X_w\in \operatorname{Obj}(\prescript{}{A}{\mathcal{C}}_D)$. The final ingredient needed for the resulting equation to make sense  is the associator

\begin{equation}
\check{\alpha}_{x,y,z}:X_x\otimes_B(X_y\otimes_CX_z)\to(X_x\otimes_BX_y)\otimes_CX_z
\end{equation}

already introduced in (<ref>). In the general case of distinct algebras $A$, $B$, $C$, $D$ it is defined as

\begin{equation}\label{associator22}
\check{\alpha}_{x,y,z}=r_{X_x\otimes_BX_y,X_z}\circ\left(r_{X_x,X_y}\otimes \operatorname{id}_{X_z}\right)\circ\left(\operatorname{id}_{X_x}\otimes e_{X_y,X_z}\right)\circ e_{X_x,X_y\otimes_CX_z}\,.
\end{equation}

We can now obtain a matrix $\mathrm{Y}$ that furnishes the vector space isomorphism

\begin{equation}\label{Ymatrixiso}
\Hom_{A|D}\left(X_x\otimes_B(X_y\otimes_C X_z),X_w\right)
\cong \Hom_{A|D}\left((X_x\otimes_BX_y)\otimes_C X_z,X_w\right)
\end{equation}

induced by the associator (<ref>). The entries of this matrix are defined by the equation

\begin{equation} \label{cross1}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [yellow, dot] (alpha) at (0,-1) [label=left:\(\alpha\)] {};
\vertex  [yellow, dot] (beta) at (1,-2) [label=right:\(\beta\)] {};
\vertex  (i) at (-1,-4.5) [label=below:\(X_x\)];
\vertex  (left) at (-0.7,-3);
\vertex [above=1cm of i] (i2);
\vertex [below left=0.042cm of left] (i3);
\vertex (j) at (0.4,-4.5) [label=below:\(X_y\)];
\vertex (k) at (1.6,-4.5) [label=below:\(X_z\)];
\vertex [above=0.8cm of j] (j11);
\vertex[above=0.8cm of k] (k11);
\vertex  (j2) at (0.45,-3.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-3.6);
\vertex [above left=0.3cm of k2] (k3);
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i2) -- (alpha);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j)  -- (j11) -- (beta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (k)  -- (k11) -- (beta);
\diagram*{
(alpha) --[middlearrow={latex},thick]  (l)   ,
(beta) -- [middlearrow={latex},thick,edge label=$X_u$] (alpha)
\end{feynman}
\end{tikzpicture}}}
~\hspace{5mm}=\mathlarger{\sum}_{v\in\mathcal{K}_{AC}}\mathlarger{\sum}_{\gamma,\delta}\,\mathrm{Y}_{\alpha u\beta,\gamma v\delta}^{(x\,y\,z)w}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [yellow, dot] (del) at (0,-1) [label=left:\(\delta\)] {};
\vertex  [yellow, dot] (gamma) at (-1,-2) [label=right:\(\gamma\)] {};
\vertex  (i) at (-1.6,-3.5);
\vertex  (i2) at (-1.55,-2.6);
\vertex [below left=0.03cm of i2] (i3);
\vertex (j) at (-0.4,-3.5) ;
\vertex (k) at (1,-3.5) ;
\vertex (j2) at (-0.45,-2.6) ;
\vertex [below right=0.03cm of j2] (j3) ;
\vertex [above=0.5cm of k] (k2);
\vertex [above left=0.005cm of k2] (k3);
\vertex (right) at (0.7,-2);
\vertex[above=0.7cm of i] (i11);
\vertex[above=0.7cm of j] (j11);
\vertex[above=1cm of k] (k11);
\vertex[below=0.4cm of i] (a1);
\vertex[left=0.25cm of a1] (a2);
\vertex[right=0.25cm of k] (c1);
\draw [fill=yellow] (a2) rectangle (c1);
\vertex[below=0.4cm of i] (a1) ;
\vertex[below=0.46cm of i] (iii) [label=above right:\(\quad\quad\check{\alpha}_{x,y,z}\)];
\vertex[below=0.6cm of a1] (a3) [label=below:\(X_x\)];
\vertex[below=1cm of j] (b1) [label=below:\(X_y\)];
\vertex[below=1cm of k] (c2) [label=below:\(X_z\)];
\vertex[below=0.4cm of j] (b2);
\vertex[below=0.4cm of k] (c3);
\draw [middlearrow={latex},thick,rounded corners=1mm] (a3) -- (a1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (b1) -- (b2)  ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (c2) -- (c3) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (k) -- (k11)  -- (del);
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i11)  -- (gamma);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j) -- (j11) -- (gamma);
\diagram*{
(del) --[middlearrow={latex},thick]  (l)   ,
(gamma) -- [middlearrow={latex},thick, edge label=$X_v$] (del)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

or in algebraic  form

\begin{equation}
\Lambda^\alpha_{(x,u)w}\circ\left(\operatorname{id}_{X_x}\otimes_B \Lambda^\beta_{(y,z)u}\right)=\sum_{v\in\mathcal{K}_{AC}}\sum_{\gamma,\delta}\,\mathrm{Y}_{\alpha u \beta,\gamma v\delta}^{(x\,y\,z)w} \,\Lambda^\delta_{(v,z)w}\circ\left(\Lambda^\gamma_{(x,y)v}\otimes_C\operatorname{id}_{X_z}\right)\circ \check{\alpha}_{x,y,z}\,.
\end{equation}

Notice that (<ref>) uses the convention of the diagrams flowing from bottom to top. Furthermore, to simplify notation in some places we will just write the labels $x,y,z$ instead of the corresponding bimodules $X_x,X_y,X_z$. To obtain a simpler expression for the matrix $\mathrm{Y}$ we compose (<ref>) with the epimorphism $r_{x,y\otimes_Cz}\circ\left(\id_x\otimes r_{y,z}\right)$ which leads to the equivalent equation

\begin{equation} \label{cross11}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [yellow, dot] (alpha) at (0,-1) [label=left:\(\alpha\)] {};
\vertex  [yellow, dot] (beta) at (1,-2) [label=right:\(\beta\)] {};
\vertex  (i) at (-1,-3) [label=below:\(X_x\)];
\vertex  (left) at (-0.7,-3);
\vertex [above=1cm of i] (i2);
\vertex [below left=0.042cm of left] (i3);
\vertex (j) at (0.4,-3) [label=below:\(X_y\)];
\vertex (k) at (1.6,-3) [label=below:\(X_z\)];
\vertex [above=0.5cm of j] (j11);
\vertex[above=0.5cm of k] (k11);
\vertex  (j2) at (0.45,-3.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-3.6);
\vertex [above left=0.3cm of k2] (k3);
\vertex[below=0.4cm of i] (a1);
\vertex[below=0.4cm of j] (b1);
\vertex[below=0.4cm of k] (c1);
\vertex[left=0.25cm of i] (a11);
\vertex[right=0.25cm of c1] (c11);
\draw [fill=yellow] (c11) rectangle (a11);
\vertex[left=0.25cm of i] (a11) [label=below right:\(\qquad \;r_{x,y\otimes_Cz}\)];
\vertex[below=1.6cm of a1] (a2) [label=below:\(X_x\)];
\vertex[below=0.5cm of b1] (b2);
\vertex[below=0.5cm of c1] (c2);
\vertex[left=0.25cm of b2] (b22);
\vertex[below=0.4cm of c2] (c3);
\vertex[right=0.25cm of c3] (c33);
\draw [fill=yellow] (c33) rectangle (b22);
\vertex[below=0.4cm of b2] (b3);
\vertex[below=0.7cm of c3] (c4) [label=below:\(X_z\)];
\vertex[below=0.7cm of b3] (b4) [label=below:\(X_y\)];
\draw [middlearrow={latex},thick,rounded corners=1mm] (b4) -- (b3) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (c4) -- (c3) ;
\vertex[left=0.25cm of b2] (b22) [label=below right:\(\quad r_{y,z}\)];
\draw [middlearrow={latex},thick,rounded corners=1mm] (a2) -- (a1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (b2) -- (b1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (c2) -- (c1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i2) -- (alpha);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j)  -- (j11) -- (beta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (k)  -- (k11) -- (beta);
\diagram*{
(alpha) --[middlearrow={latex},thick]  (l)   ,
(beta) -- [middlearrow={latex},thick,edge label=$X_u$] (alpha)
\end{feynman}
\end{tikzpicture}}}
~\hspace{5mm}=\mathlarger{\sum}_{v\in\mathcal{K}_{AC}}\mathlarger{\sum}_{\gamma,\delta}\,\mathrm{Y}_{\alpha u\beta,\gamma v\delta}^{(x\,y\,z)w}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [yellow, dot] (del) at (0,-1) [label=left:\(\delta\)] {};
\vertex  [yellow, dot] (gamma) at (-1,-2) [label=right:\(\gamma\)] {};
\vertex  (i) at (-1.6,-3);
\vertex  (i2) at (-1.55,-2.6);
\vertex [below left=0.03cm of i2] (i3);
\vertex (j) at (-0.4,-3) ;
\vertex (k) at (1,-3) ;
\vertex (j2) at (-0.45,-2.6) ;
\vertex [below right=0.03cm of j2] (j3) ;
\vertex [above=0.5cm of k] (k2);
\vertex [above left=0.005cm of k2] (k3);
\vertex (right) at (0.7,-2);
\vertex[above=0.7cm of i] (i11);
\vertex[above=0.7cm of j] (j11);
\vertex[above=1cm of k] (k11);
\vertex[below=0.4cm of i] (a1);
\vertex[left=0.25cm of a1] (a2);
\vertex[right=0.25cm of k] (c1);
\draw [fill=yellow] (a2) rectangle (c1);
\vertex[below=0.4cm of i] (a1) ;
\vertex[below=0.46cm of i] (iii) [label=above right:\(\quad\quad\check{\alpha}_{x,y,z}\)];
\vertex[below=0.6cm of a1] (a3) [label=below:\(X_x\)]; %top of 2nd box
\vertex[below=1cm of j] (b1) [label=below:\(X_y\)]; %top of 2nd box
\vertex[below=1cm of k] (c2) [label=below:\(X_z\)];%top of 2nd box
\vertex[below=0.4cm of j] (b2);
\vertex[below=0.4cm of k] (c3);
\vertex[below=0.4cm of c2] (z1); %bottom of 2nd box
\vertex[left=0.25cm of a3] (x11);
\vertex[right=0.25cm of z1] (z11);
\draw [fill=yellow] (x11) rectangle (z11);
\vertex[left=0.25cm of a3] (x11) [label=below right:\(\qquad\;r_{x,y\otimes_Cz}\)];
\vertex[below=0.4cm of a3] (a4); %bottom of 2nd
\vertex[below=0.4cm of b1] (b3); %bottom of 2nd
\vertex[below=1.3cm of a4] (a5) [label=below:\(X_x\)];
\vertex[below=0.5cm of b3] (b4); %top of 3rd
\vertex[below=0.5cm of z1] (z2); %top of 3rd
\vertex[below=0.4cm of z2] (z3); %bot of 3rd
\vertex[left=0.25cm of b4] (b44);
\vertex[right=0.25cm of z3] (z33);
\draw [fill=yellow] (b44) rectangle (z33);
\vertex[left=0.25cm of b4] (b44) [label=below right:\(\qquad r_{y,z}\)];
\vertex[below=0.4cm of b4] (b5);
\vertex[below=0.4cm of b5] (b6) [label=below:\(X_y\)];
\vertex[below=0.4cm of z3] (z4) [label=below:\(X_z\)];
\draw [middlearrow={latex},thick,rounded corners=1mm] (a5) -- (a4) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (b6) -- (b5) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (z4) -- (z3) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (a3) -- (a1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (b4) -- (b3) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (z2) -- (z1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (b1) -- (b2)  ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (c2) -- (c3) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (k) -- (k11)  -- (del);
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i11)  -- (gamma);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j) -- (j11) -- (gamma);
\diagram*{
(del) --[middlearrow={latex},thick]  (l)   ,
(gamma) -- [middlearrow={latex},thick, edge label=$X_v$] (del)
\end{feynman}
\end{tikzpicture}}}\end{equation}

Using (<ref>) in the top part of (<ref>) as well as the explicit form of $\check{\alpha}_{x,y,z}$ leads to the following equation

\begin{equation}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [empty dot] (alpha) at (0,-1) [label=left:\(\alpha\)] {};
\vertex  [empty dot] (beta) at (1,-2) [label=right:\(\beta\)] {};
\vertex  (i) at (-1,-3) [label=below:\(X_x\)];
\vertex  (left) at (-0.7,-3);
\vertex [above=1cm of i] (i2);
\vertex [below left=0.042cm of left] (i3);
\vertex (j) at (0.4,-3) [label=below:\(X_y\)];
\vertex (k) at (1.6,-3) [label=below:\(X_z\)];
\vertex [above=0.5cm of j] (j11);
\vertex[above=0.5cm of k] (k11);
\vertex  (j2) at (0.45,-3.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-3.6);
\vertex [above left=0.3cm of k2] (k3);
\vertex[below=0.4cm of i] (a1);
\vertex[below=0.4cm of j] (b1);
\vertex[below=0.4cm of k] (c1);
\vertex[left=0.25cm of i] (a11);
\vertex[right=0.25cm of c1] (c11);
\draw [fill=yellow] (c11) rectangle (a11);
\vertex[left=0.25cm of i] (a11) [label=below right:\(\qquad \;\;P_{x,y\otimes z}\)];
\vertex[below=0.9cm of a1] (a2) [label=below:\(X_x\)];
\vertex[below=0.5cm of b1] (b2);
\vertex[below=0.5cm of c1] (c2);
\vertex[left=0.25cm of b2] (b22);
\vertex[below=0.4cm of c2] (c3);
\vertex[right=0.25cm of c3] (c33);
\vertex[below=0.4cm of b2] (b3);
\vertex[below=0.00001cm of c3] (c4) [label=below:\(X_z\)];
\vertex[below=0.00001cm of b3] (b4) [label=below:\(X_y\)];
\draw [middlearrow={latex},thick,rounded corners=1mm] (b4) -- (b1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (c4) -- (c1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (a2) -- (a1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i2) -- (alpha);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j)  -- (j11) -- (beta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (k)  -- (k11) -- (beta);
% \draw node (kek1) at (1, 2) [circ, label=left:{$\alpha$}];
\diagram*{
(alpha) --[middlearrow={latex},thick]  (l)   ,
(beta) -- [middlearrow={latex},thick,edge label=$X_u$] (alpha)
\end{feynman}
\end{tikzpicture}}}
~\hspace{5mm}=\mathlarger{\sum}_{v\in\mathcal{K}_{AC}}\mathlarger{\sum}_{\gamma,\delta}\,\mathrm{Y}_{\alpha u\beta,\gamma v\delta}^{(x\,y\,z)w}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [empty dot] (del) at (0,-1) [label=left:\(\delta\)] {};
\vertex  [empty dot] (gamma) at (-1,-2) [label=right:\(\gamma\)] {};
\vertex  (i) at (-1.6,-3.2);
\vertex  (i2) at (-1.55,-2.6);
\vertex [below left=0.03cm of i2] (i3);
\vertex (j) at (-0.4,-3.2) ;
\vertex (k) at (1,-3.2) ;
\vertex (j2) at (-0.45,-2.6) ;
\vertex [below right=0.03cm of j2] (j3) ;
\vertex [above=0.5cm of k] (k2);
\vertex [above left=0.005cm of k2] (k3);
\vertex (right) at (0.7,-2);
\vertex[above=0.7cm of i] (i11);
\vertex[above=0.7cm of j] (j11);
\vertex[above=1cm of k] (k11);
\vertex[below=0.4cm of i] (a1);
\vertex[left=0.25cm of a1] (a2);
\vertex[right=0.25cm of k] (c1);
\draw [fill=yellow] (a2) rectangle (c1);
\vertex[below=0.4cm of i] (a1) ;
\vertex[below=0.8cm of a1] (a3) [label=below:\(X_x\)];
\vertex[below=1.2cm of j] (b1) [label=below:\(X_y\)];
\vertex[below=1.2cm of k] (c2) [label=below:\(X_z\)];
\vertex[below=0.4cm of j] (b2);
\vertex[below=0.4cm of k] (c3);
\vertex (j) at (-0.4,-3.2) [label=below:\(P_{x\otimes y,z}\)];
\draw [middlearrow={latex},thick,rounded corners=1mm] (a3) -- (a1) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (b1) -- (b2)  ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (c2) -- (c3) ;
\draw [middlearrow={latex},thick,rounded corners=1mm] (k) -- (k11)  -- (del);
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i11)  -- (gamma);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j) -- (j11) -- (gamma);
\diagram*{
(del) --[middlearrow={latex},thick]  (l)   ,
(gamma) -- [middlearrow={latex},thick, edge label=$X_v$] (del)
\end{feynman}
\end{tikzpicture}}}\end{equation}

where the  junctions indicated by hollow black blobs represent morphisms of the form $\tilde{\Lambda}$ as in the first equation in (<ref>). For the left-hand side we notice that the morphisms in the top part satisfy

\begin{equation}\tilde{\Lambda}^\alpha_{(x,u)w}\circ\left(\id_x\otimes\tilde{\Lambda}^\beta_{(y,z)u}\right)\in {\Hom_{A|D}}_{(P_{X_x,X_y\otimes X_z})}(X_x\otimes( X_y\otimes X_z),X_w)\end{equation}
while for the right-hand side we see that
\begin{equation}\tilde{\Lambda}^\delta_{(v,z)w}\circ\left(\tilde{\Lambda}^\gamma_{(x,y)v}\otimes\id_z\right)\in {\Hom_{A|D}}_{(P_{X_x\otimes X_y,X_z})}((X_x\otimes X_y)\otimes X_z),X_w)\,.\end{equation}

\[
\vcenter{\hbox{\hspace{-10mm}{\scalebox{0.8}{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,2) [label=above:\(X_s\)] ;
\vertex [yellow, dot] (alpha) at (0,1) [label=left:\(\alpha\)] {};
\vertex  [yellow, dot] (beta) at (1,0) [label=right:\(\beta\)] {};
\vertex  (i) at (-1,-2) [label=below:\(X_x\)];
\vertex  (left) at (-0.7,-1);
\vertex [above=1cm of i] (i2);
\vertex [below left=0.042cm of left] (i3);
\vertex (j) at (0.4,-2) [label=below:\(X_y\)];
\vertex (k) at (1.8,-1.5) ;
\vertex (z1) at (1.4,-2) [label=below:\(X_z\)];
\vertex (w1) at (2.2,-2) [label=below:\(X_w\)];
\vertex[above=0.5cm of z1] (z2);
\vertex[above=0.5cm of w1] (w2);
\vertex [above=0.8cm of j] (j11);
\vertex [above=0.8cm of k][yellow, dot] [label=right:\(\gamma\)]  (k11){};
\vertex  (j2) at (0.45,-1.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-1.6);
\vertex [above left=0.3cm of k2] (k3);
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i2) -- (alpha);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j)  -- (j11) -- (beta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (z1) -- (z2) -- (k11);
\draw [middlearrow={latex},thick,rounded corners=1mm] (w1) -- (w2) -- (k11);
\draw[->, line width=0.5mm] (1.2,0.5) -- (2.5,1.5);
\node at (2,0.8) (eq1) {$\mathlarger{\mathrm{Y}}$};
\vertex (s1) at (4,6.5)  ;  %top of 2nd graph
\vertex[below=1cm of s1] [yellow, dot](alpha2){};
\vertex[below=1cm of alpha2] (eps1);
\vertex[right=1.3cm of eps1][yellow, dot] (eps){};
\vertex[below=1cm of eps] (del1);
\vertex[left=0.8cm of del1] (del)[yellow, dot] {};
\vertex[right=1.6cm of del] (ww1);
\vertex[below=1.2cm of ww1] (ww2);
\vertex[below=0.5cm of del] (del2);
\vertex[left=0.5cm of del2] (yy1);
\vertex[right=0.5cm of del2] (zz1);
\vertex[below=0.7cm of yy1] (yy2);
\vertex[below=0.7cm of zz1] (zz2);
\vertex[below=0.4cm of yy2] (yy3);
\vertex[left=0.25cm of yy3] (yy4);
\vertex[right=0.25cm of ww2] (ww3);
\draw [fill=yellow] (ww3) rectangle (yy4);
\vertex[below=0.075cm of yy3] [label=above right:\(\quad\check{\alpha}_{y,z,w}\)] (sad){};
\draw [middlearrow={latex},thick,rounded corners=1mm] (ww2) -- (ww1) -- (eps);
\draw [middlearrow={latex},thick,rounded corners=1mm] (zz2) -- (zz1) -- (del);
\draw [middlearrow={latex},thick,rounded corners=1mm] (yy2) -- (yy1) -- (del);
\vertex[left=1cm of alpha2] (xx1);
\vertex[below=1.5cm of xx1] (xx2);
\vertex[below=3cm of xx2] (xx3);
\vertex[below=0.9cm of yy3] (yy5);
\draw [middlearrow={latex},thick,rounded corners=1mm] (xx3) -- (xx2) -- (alpha2);
\vertex[below=0.4cm of zz2] (zz3);
\vertex[below=0.9 of zz3] (zz4);
\vertex[below=0.4cm of ww2] (ww4);
\vertex[below=0.9cm of ww4] (ww5);
\draw[->, line width=0.5mm] (6.5,2.5) -- (8.5,2.5);
\node at (7.5,2.2) (eq2) {$\mathlarger{\mathrm{Y}}$};
\vertex (sss1) at (11,6.5)  ; %top of 3rd graph
\vertex[below=1cm of sss1]  (tau1)[yellow, dot]{};
\vertex[right=1cm of tau1] (www1);
\vertex[below=1cm of www1] (www2);
\vertex[below=1cm of tau1] (sigma1);
\vertex[left=1.2cm of sigma1] (sigma2)[yellow, dot]{};
\vertex[below=0.5cm of sigma2] (xxx1);
\vertex[left=0.6cm of xxx1] (xxx2);
\vertex[right=0.7cm of xxx1] (qqq1);
\vertex[below=0.7cm of xxx2] (xxx3);
\vertex[below=0.7cm of qqq1] (qqq2)[yellow, dot]{};
\draw [middlearrow={latex},thick,rounded corners=1mm] (qqq2) -- (qqq1) -- (sigma2);
\vertex[below=0.5cm of qqq2] (qqq3);
\vertex[left=0.5cm of qqq3] (yyy1);
\vertex[right=0.5cm of qqq3] (zzz1);
\vertex[below=0.6cm of yyy1] (yyy2);
\vertex[below=0.6cm of zzz1] (zzz2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (zzz2) -- (zzz1) -- (qqq2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (yyy2) -- (yyy1) -- (qqq2);
\vertex[below=1.2cm of xxx3] (xxx4);
\draw [middlearrow={latex},thick,rounded corners=1mm] (xxx4) -- (xxx2) -- (sigma2);
\vertex[below=2.3cm of www2] (www3);
\draw [middlearrow={latex},thick,rounded corners=1mm] (www3) -- (www2) -- (tau1);
\vertex[right=0.25cm of www3] (www4);
\vertex[below=0.3cm of xxx4] (xxx5);
\vertex[left=0.25cm of xxx5] (xxx6);
\draw [fill=yellow] (xxx6) rectangle (www4);
\vertex[below=0.08cm of xxx5] (sssssss) [label=above right:\(\qquad\check{\alpha}_{x,y\otimes_Cz,w}\)];
\vertex[below=0.3cm of yyy2] (yyy3);
\vertex[below=0.3cm of zzz2] (zzz3);
\vertex[below=0.4cm of www3] (www5);
\vertex[below=1.4cm of xxx5] (xxx7);
\vertex[below=0.5cm of yyy3] (yyy4);
\vertex[below=0.5cm of zzz3] (zzz4);
\vertex[below=0.5cm of www5] (www6);
\vertex[below=0.4cm of yyy4] (yyy5);
\vertex[left=0.25cm of yyy5] (yyy6);
\vertex[right=0.25cm of www6] (www7);
\draw [fill=yellow] (www7) rectangle (yyy6);
\vertex[below=0.08cm of yyy5] (kek2) [label=above right:\(\quad\;\check{\alpha}_{y,z,w}\)];
\vertex[below=0.4cm of zzz4] (zzz5);
\vertex[below=0.4cm of www6] (www8);
\vertex[below=0.5cm of yyy5] (yyy7);
\vertex[below=0.5cm of zzz5] (zzz6);
\vertex[below=0.5cm of www8] (www9);
\draw[->, line width=0.5mm] (10,0) -- (11,-2);
\node at (10.2,-1) (eq1) {$\mathlarger{\mathrm{Y}}$};
\vertex (ssss1) at (12,-1.5) ;%top of 4th grap
\vertex[below=1cm of ssss1] (t1) [yellow, dot]{};
\vertex[left=1.2cm of t1] (lam1) ;
\vertex[below=1cm of lam1] (lam) [yellow, dot]{};
\vertex[below=1cm of t1] (wwww1);
\vertex[right=1cm of wwww1] (wwww2);
\vertex[below=3cm of wwww2] (wwww3);
\draw [middlearrow={latex},thick,rounded corners=1mm] (wwww3) -- (wwww2) -- (t1);
\vertex[below=1cm of lam] (theta1);
\vertex[left=1cm of theta1] (theta)[yellow, dot]{};
\vertex[right=1cm of theta1] (zzzz1);
\vertex[below=1cm of zzzz1] (zzzz2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (zzzz2) -- (zzzz1) -- (lam);
\vertex[below=0.5cm of theta] (theta2);
\vertex[left=0.5cm of theta2] (xxxx1);
\vertex[right=0.5cm of theta2] (yyyy1);
\vertex[below=0.5cm of xxxx1] (xxxx2);
\vertex[below=0.5cm of yyyy1] (yyyy2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (xxxx2) -- (xxxx1) -- (theta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (yyyy2) -- (yyyy1) -- (theta);
\vertex[below=0.4cm of xxxx2] (xxxx3);
\vertex[left=0.25cm of xxxx3] (xxxx4);
\vertex[right=0.25cm of zzzz2] (zzzz3);
\draw [fill=yellow] (zzzz3) rectangle (xxxx4);
\vertex[below=0.08cm of xxxx3] (kek4) [label=above right:\(\qquad\check{\alpha}_{x,y,z}\)];
\vertex[below=0.4cm of yyyy2] (yyyy3);
\vertex[below=0.4cm of zzzz2] (zzzz4);
\vertex[below=0.5cm of xxxx3] (xxxx5);
\vertex[below=0.5cm of yyyy3] (yyyy4);
\vertex[below=0.5cm of zzzz4] (zzzz5);
\vertex[right=0.25cm of wwww3] (wwww4);
\vertex[below=0.4cm of xxxx5] (xxxx6);
\vertex[left=0.25cm of xxxx6] (xxxx7);
\draw [fill=yellow] (xxxx7) rectangle (wwww4);
\vertex[below=0.08cm of xxxx6] (kek5) [label=above right:\(\qquad\check{\alpha}_{x,y\otimes_Cz,w}\)];
\vertex[below=0.4cm of yyyy4] (yyyy5);
\vertex[below=0.4cm of zzzz5] (zzzz6);
\vertex[below=0.4cm of wwww3] (wwww5);
\vertex[below=1.4cm of xxxx6] (xxxx8);
\vertex[below=0.5cm of yyyy5] (yyyy6);
\vertex[below=0.5cm of zzzz6] (zzzz7);
\vertex[below=0.5cm of wwww5] (wwww6);
\vertex[below=0.4cm of yyyy6] (yyyy7);
\vertex[left=0.25cm of yyyy7] (yyyy8);
\vertex[right=0.25cm of wwww6] (wwww7);
\draw [fill=yellow] (wwww7) rectangle (yyyy8);
\vertex[below=0.08cm of yyyy7] (kek7) [label=above right:\(\qquad\check{\alpha}_{y,z,w}\)];
\vertex[below=0.4cm of zzzz7] (zzzz8);
\vertex[below=0.4cm of wwww6] (wwww8);
\vertex[below=0.5cm of yyyy7] (yyyy9);
\vertex[below=0.5cm of zzzz8] (zzzz9);
\vertex[below=0.5cm of wwww8] (wwww9);
\draw[->, line width=0.5mm] (0.5,-2.75) -- (0.5,-4);
\node at (0.3,-3.3) (eq1) {$\mathlarger{\mathrm{Y}}$};
\vertex (stop) at (0.3,-4.4)  ; %top of last graph
\vertex[below=1cm of stop] (eta)  [yellow, dot]{};
\vertex[below=1cm of eta] (eta1) ;
\vertex[left=1 of eta1] (stheta) [yellow, dot]{};
\vertex[right=1cm of eta1] (sgamma) [yellow, dot]{};
\vertex[below=0.5cm of stheta] (stheta1);
\vertex[left=0.5cm of stheta1] (sx1);
\vertex[right=0.5cm of stheta1] (sy1);
\vertex[below=0.6cm of sx1] (sx2);
\vertex[below=0.6cm of sy1] (sy2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (sy2) -- (sy1) -- (stheta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (sx2) -- (sx1) -- (stheta);
\vertex[below=0.5cm of sgamma] (sgamma1);
\vertex[left=0.5cm of sgamma1] (sz1);
\vertex[right=0.5cm of sgamma1] (sw1);
\vertex[below=0.6cm of sz1] (sz2);
\vertex[below=0.6cm of sw1] (sw2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (sw2) -- (sw1) -- (sgamma);
\draw [middlearrow={latex},thick,rounded corners=1mm] (sz2) -- (sz1) -- (sgamma);
\vertex[below=0.4cm of sx2] (sx3);
\vertex[left=0.25cm of sx3] (sx4);
\vertex[right=0.25cm of sw2] (sw3);
\draw [fill=yellow] (sw3) rectangle (sx4);
\vertex[below=0.08cm of sx3] (kekw1) [label=above right:\(\qquad \check{\alpha}_{x,y,z\otimes_Dw}\)];
\vertex[below=0.4cm of sy2] (sy3);
\vertex[below=0.4cm of sz2] (sz3);
\vertex[below=0.4cm of sw2] (sw4);
\vertex[below=0.6cm of sx3] (sx4);
\vertex[below=0.6cm of sy3] (sy4);
\vertex[below=0.6cm of sz3] (sz4);
\vertex[below=0.6cm of sw4] (sw5);
\draw[->, line width=0.5mm] (2.5,-8) -- (4,-9);
\node at (3.3,-8.2) (eq1) {$\mathlarger{\mathrm{Y}}$};
\vertex (gtop) at (6,-7)  ; %top of 5th graph
\vertex[below=0.8cm of gtop] (gtau)[yellow, dot]{};
\vertex[below=1cm of gtau] (gtau1);
\vertex[right=1cm of gtau1] (gw1);
\vertex[below=1.8cm of gw1] (gw2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (gw2) -- (gw1) -- (gtau);
\vertex[left=1cm of gtau1] (glam)[yellow, dot]{};
\vertex[below=0.8cm of glam] (glam1);
\vertex[left=0.8cm of glam1] (gtheta)[yellow, dot]{};
\vertex[right=0.8cm of glam1] (gz1);
\vertex[below=1cm of gz1] (gz2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (gz2) -- (gz1) -- (glam);
\vertex[below=0.5cm of gtheta] (gtheta1);
\vertex[left=0.5cm of gtheta1] (gx1);
\vertex[right=0.5cm of gtheta1] (gy1);
\vertex[below=0.5cm of gx1] (gx2);
\vertex[below=0.5cm of gy1] (gy2);
\draw [middlearrow={latex},thick,rounded corners=1mm] (gy2) -- (gy1) -- (gtheta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (gx2) -- (gx1) -- (gtheta);
\vertex[below=0.4cm of gx2] (gx3);
\vertex[left=0.25cm of gx3] (gx4);
\vertex[right=0.25cm of gw2] (gw3);
\draw [fill=yellow] (gw3) rectangle (gx4);
\vertex[below=0.08cm of gx3] [label=above right:\(\qquad\check{\alpha}_{x\otimes_By,z,w}\)](kekz5);
\vertex[below=0.4cm of gy2] (gy3);
\vertex[below=0.4cm of gz2] (gz3);
\vertex[below=0.4cm of gw2] (gw4);
\vertex[below=0.5cm of gx3] (gx5);
\vertex[below=0.5cm of gy3] (gy4);
\vertex[below=0.5cm of gz3] (gz4);
\vertex[below=0.5cm of gw4] (gw5);
\vertex[below=0.4cm of gx5] (gx6);
\vertex[left=0.25cm of gx6] (gx7);
\vertex[right=0.25cm of gw5] (gw6);
\draw [fill=yellow] (gw6) rectangle (gx7);
\vertex[below=0.08cm of gx6] (kekz2) [label=above right:\(\qquad \check{\alpha}_{x,y,z\otimes_Dw}\)];
\vertex[below=0.4cm of gy4] (gy5);
\vertex[below=0.4cm of gz4] (gz5);
\vertex[below=0.4cm of gw5] (gw7);
\vertex[below=0.6cm of gx6] (gx7);
\vertex[below=0.6cm of gy5] (gy6);
\vertex[below=0.6cm of gz5] (gz6);
\vertex[below=0.6cm of gw7] (gw8);
\draw[->, line width=0.5mm] (6.8,-8) -- (8.5,-6.3);
\node at (7.2,-7) (eq6) {\eqref{genpent1}} ;
\diagram*{
(alpha) --[middlearrow={latex},thick]  (l)   ,
(beta) -- [middlearrow={latex},thick,edge label=$X_u$] (alpha),
(k11) -- [middlearrow={latex},thick,edge label=$X_r$] (beta),
(alpha2) --[middlearrow={latex},thick]  (s1),
(eps) -- [middlearrow={latex},thick] (alpha2),
(del) -- [middlearrow={latex},thick] (eps),
(yy5) -- [middlearrow={latex},thick] (yy3),
(zz4) -- [middlearrow={latex},thick] (zz3),
(ww5) -- [middlearrow={latex},thick] (ww4),
(tau1) --[middlearrow={latex},thick] (sss1),
(sigma2)--[middlearrow={latex},thick] (tau1),
(www6)--[middlearrow={latex},thick] (www5),
(zzz4)--[middlearrow={latex},thick] (zzz3),
(yyy4)--[middlearrow={latex},thick] (yyy3),
(xxx7)--[middlearrow={latex},thick] (xxx5),
(www9)--[middlearrow={latex},thick] (www8),
(zzz6)--[middlearrow={latex},thick] (zzz5),
(yyy7)--[middlearrow={latex},thick] (yyy5),
(t1)--[middlearrow={latex},thick] (ssss1),
(lam)--[middlearrow={latex},thick] (t1),
(theta)--[middlearrow={latex},thick] (lam),
(zzzz5)--[middlearrow={latex},thick] (zzzz4),
(yyyy4)--[middlearrow={latex},thick] (yyyy3),
(xxxx5)--[middlearrow={latex},thick] (xxxx3) ,
(wwww6)--[middlearrow={latex},thick] (wwww5),
(zzzz7)--[middlearrow={latex},thick] (zzzz6),
(yyyy6)--[middlearrow={latex},thick] (yyyy5),
(xxxx8)--[middlearrow={latex},thick] (xxxx6),
(wwww9)--[middlearrow={latex},thick] (wwww8),
(zzzz9)--[middlearrow={latex},thick] (zzzz8),
(yyyy9)--[middlearrow={latex},thick] (yyyy7),
(stheta)--[middlearrow={latex},thick] (eta),
(sgamma)--[middlearrow={latex},thick] (eta),
(eta)--[middlearrow={latex},thick] (stop),
(sw5)--[middlearrow={latex},thick] (sw4),
(sz4)--[middlearrow={latex},thick] (sz3),
(sy4)--[middlearrow={latex},thick] (sy3),
(sx4)--[middlearrow={latex},thick] (sx3),
(gtau)--[middlearrow={latex},thick] (gtop),
(glam)--[middlearrow={latex},thick] (gtau),
(gtheta)--[middlearrow={latex},thick] (glam),
(gw5)--[middlearrow={latex},thick] (gw4),
(gz4)--[middlearrow={latex},thick] (gz3),
(gy4)--[middlearrow={latex},thick] (gy3),
(gx5)--[middlearrow={latex},thick] (gx3),
(gw8)--[middlearrow={latex},thick] (gw7),
(gz6)--[middlearrow={latex},thick] (gz5),
(gy6)--[middlearrow={latex},thick] (gy5),
(gx7)--[middlearrow={latex},thick] (gx6)
\end{feynman}
\end{tikzpicture}}}}}
\]
Pentagon identity for the $\mathrm{Y}$-matrices.

Pentagon identity for the $\mathrm{Y}$-matrices.

Using the definition (<ref>) of these spaces we find  that the projection morphisms can be removed yielding an equation in which
all tensor products are those of the objects in  $\mathcal{C}$:

\begin{equation}\label{crossf1}
\vcenter{\hbox{\hspace{-10mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [empty dot] (alpha) at (0,-1) [label=left:\(\alpha\)] {};
\vertex  [empty dot] (beta) at (1,-2) [label=right:\(\beta\)] {};
\vertex  (i) at (-1,-4) [label=below:\(X_x\)];
\vertex  (left) at (-0.7,-3);
\vertex [above=1cm of i] (i2);
\vertex [below left=0.042cm of left] (i3);
\vertex (j) at (0.4,-4) [label=below:\(X_y\)];
\vertex (k) at (1.6,-4) [label=below:\(X_z\)];
\vertex [above=0.8cm of j] (j11);
\vertex[above=0.8cm of k] (k11);
\vertex  (j2) at (0.45,-3.6);
\vertex [above right=0.15cm of j2] (j3);
\vertex [above right=0.15cm of j3] (j4);
\vertex  (k2) at (1.55,-3.6);
\vertex [above left=0.3cm of k2] (k3);
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i2) -- (alpha);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j)  -- (j11) -- (beta);
\draw [middlearrow={latex},thick,rounded corners=1mm] (k)  -- (k11) -- (beta);
\diagram*{
(alpha) --[middlearrow={latex},thick]  (l)   ,
(beta) -- [middlearrow={latex},thick,edge label=$X_u$] (alpha)
\end{feynman}
\end{tikzpicture}}}
~\hspace{5mm}=\mathlarger{\sum}_{v\in\mathcal{K}_{AC}}\mathlarger{\sum}_{\gamma,\delta}\,\mathrm{Y}_{\alpha u\beta,\gamma v\delta}^{(x\,y\,z)w}
\vcenter{\hbox{\hspace{1mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\vertex (l) at (0,0) [label=above:\(X_w\)] ;
\vertex [empty dot] (del) at (0,-1) [label=left:\(\delta\)] {};
\vertex  [empty dot] (gamma) at (-1,-2) [label=right:\(\gamma\)] {};
\vertex  (i) at (-1.6,-4) [label=below:\(X_x\)];
\vertex  (i2) at (-1.55,-2.6);
\vertex [below left=0.03cm of i2] (i3);
\vertex (j) at (-0.4,-4) [label=below:\(X_y\)];
\vertex (k) at (1,-4) [label=below:\(X_z\)];
\vertex (j2) at (-0.45,-2.6) ;
\vertex [below right=0.03cm of j2] (j3) ;
\vertex [above=0.5cm of k] (k2);
\vertex [above left=0.005cm of k2] (k3);
\vertex (right) at (0.7,-2);
\vertex[above=0.7cm of i] (i11);
\vertex[above=0.7cm of j] (j11);
\vertex[above=1cm of k] (k11);
\draw [middlearrow={latex},thick,rounded corners=1mm] (k) -- (k11)  -- (del);
\draw [middlearrow={latex},thick,rounded corners=1mm] (i) -- (i11)  -- (gamma);
\draw [middlearrow={latex},thick,rounded corners=1mm] (j) -- (j11) -- (gamma);
\diagram*{
(del) --[middlearrow={latex},thick]  (l)   ,
(gamma) -- [middlearrow={latex},thick, edge label=$X_v$] (del)
\end{feynman}
\end{tikzpicture}}}
\end{equation}

The last equation is equivalent to the definition (<ref>) of the $\mathrm{Y}$ but it is in a form more suitable for calculations which we will later use to construct an explicit expression for $\mathrm{Y}$.

The associator (<ref>) obeys the pentagon identity

\begin{equation}\label{genpent1} \check{\alpha}_{x\otimes_By,z,w}\circ\check{\alpha}_{x,y,z\otimes_Dw}=\left(\check{\alpha}_{x,y,z}\otimes_D\operatorname{id}_{w}\right)\circ \check{\alpha}_{x,y\otimes_Cz,w}\circ\left(\id_{x}\otimes_B\check{\alpha}_{y,z,w}\right)
\end{equation}

which is a generalised version of (<ref>). Using this pentagon identity as well as the definition of the $\mathrm{Y}$-matrices (<ref>)
we   obtain a new pentagon identity that the $\mathrm{Y}$-matrices must obey. Graphically the sequence of moves for the new pentagon
is represented on figure <ref>. On that figure between the bottom graph and the one to its right   we have used the pentagon equation for the associator $\check{\alpha}$ (<ref>). Demanding that these two ways lead to the same result gives the following pentagon identity for the $\mathrm{Y}$-matrices

\begin{equation}\label{pentaY}
\sum_{q}\sum_{\delta,\varepsilon,\sigma} \mathrm{Y}^{(y\, z\,w)u}_{\beta r\gamma,\delta q\varepsilon}\,\mathrm{Y}^{(x\,q\,w)s}_{\alpha u\varepsilon,\sigma v\tau}\,\mathrm{Y}^{(x\,y\,z)v}_{\sigma q\delta,\theta p\lambda}=\sum_{\eta}\mathrm{Y}^{(x\,y\,r)s}_{\alpha u\beta,\theta p\eta}\,\mathrm{Y}^{(p\,z\,w)s}_{\eta r\gamma,\lambda v\tau}\,.
\end{equation}

§.§.§ Fusing matrix T for open defects

Next, we would like to define the fusing matrix for open topological interfaces.
To this end we consider the  associator

$\tilde{\alpha}:\prescript{}{A}{\mathcal{C}}_B\times\prescript{}{B}{\mathcal{C}}_C\times\mathcal{C}_C\to \prescript{}{A}{\mathcal{C}}_B\times\prescript{}{B}{\mathcal{C}}_C\times\mathcal{C}_C$  with components

\begin{align}\label{associator3}
& \tilde{\alpha}_{X,Y,M}:X\otimes_B(Y\otimes_CM)\to (X\otimes_BY)\otimes_CM, \nonumber \\
& \tilde{\alpha}_{X,Y,M}= r_{X\otimes_BY,M}\circ\left(r_{X,Y}\otimes\id_M\right)\circ\left(\id_X\otimes e_{Y,M}\right)\circ e_{X,Y\otimes_CM}\,.
\end{align}

This associator is required to obey a certain mixed pentagon identity the proof of which is presented in appendix <ref>. Notice that for the case $A=B=C$  this associator is exactly the module associativity constraint for the left module category $\mathcal{C}_A$ over the tensor category $\prescript{}{A}{\mathcal{C}}_A$ (see  <ref> for the definition).

Taking $X,Y$ and $M$ to be simple objects in their corresponding categories as well as fixing the bases for the relevant junction spaces we obtain the matrix $\mathrm{T}$ that furnishes the vector space isomorphism

\begin{equation}
\Hom_A\left(X_x\otimes_B(X_y\otimes_C M_a),M_c\right)\cong\Hom_A\left((X_x\otimes_BX_y)\otimes_C M_a,M_c\right)
\end{equation}

induced by the associator (<ref>). The entries of this matrix are defined by

\begin{equation}\label{cross4}
\vcenter{\hbox{\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex (x1) at (0,3.5) ;
\vertex[above=0.5cm of x1] (top1);
\vertex[below right=0.1cm of x1] (x5) [label=below left:\(X_y\)];
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);
\vertex[ above=0.4cm of j1] (j3) [label=right:\(M_c\)];
\vertex [right=1cm of i2] (j2);
\vertex[ below=0.2cm of j2] (j4) [label=right:\(M_a\)];
\vertex [below=1.4cm of j2] (j6);
\vertex   [above=1cm of j1] (j5) ;
\vertex[ above =1.1cm of j1] (j55);
\vertex [left=0.15cm of j5]  [label=left:\(\alpha\)];
\vertex [left=0.1cm of j55]  (j);
\vertex [left=0.1cm of j6] (j7);
\vertex (y) at (-0.7,2.6) ;
\vertex[above=1.4cm of y] (top2);
\vertex [below right=0.1cm of y] (y1) ;
\vertex[left=0.8cm of x5] (xx) [label=below left:\(X_x\)];
\vertex [below=1.4cm of j2] (k1);
\vertex [below=1.5cm of j2] (k2);
\vertex [left=0.15cm of k1] (k3);
\vertex [left=0.15cm of k2] (k4) [label=left:\(\beta\)];
\draw [thick]   (j7) to[out=90,in=-90] (x1);
\draw [fill=yellow] (1.06,1.1) rectangle (0.8, 1);
\draw [fill=yellow] (1.06,2.6) rectangle (0.8, 2.5);
\draw [thick]  (j) to[out=90,in=-90] (y);
\vertex[above=0.65 of j55] [label=right:\(M_b\)];
\draw [thick] (0,0.8) rectangle (0.4, 1.2);
\vertex at (0.2,0.8) [label=\(A\)];
\draw [thick] (0,2.3) rectangle (0.4, 2.7);
\vertex at (0.2,2.3) [label=\(B\)];
\draw [thick] (0.3,3.6) rectangle (0.7, 4);
\vertex at (0.5,3.6) [label=\(C\)];
\diagram*{
(j2)--[thick] (k1),
(k2)--[thick] (j55),
(j5)--[thick] (j1),
(x1)--[thick,myptr1={latex}] (top1),
(y)--[thick,myptr1={latex}] (top2)
\end{feynman}
\end{tikzpicture}}} \hspace{1mm}~= \mathlarger{\sum}_{v\in\mathcal{K}_{AC}}\mathlarger{\sum}_{\gamma=1}^{\tensor{\check{N}}{_{yx}}{^v}}\mathlarger{\sum}_{\delta=1}^{\tensor{B}{_{va}}{^c}} \mathrm{T}^{(x\,y\,a)c}_{\alpha b\beta,\gamma v\delta} ~
\vcenter{\hbox{\hspace{5mm}\begin{tikzpicture}[font=\footnotesize,inner sep=2pt]
\begin{feynman}
\path[pattern=north east lines,pattern color=ashgrey,very thin] (1.01,4) rectangle (1.6,0);
\vertex  (i1) at (0,0)  ;
\vertex [dot,yellow] (x1) at (-0.5,2.3) [label=left:\(\gamma\)] {};
\vertex  (x2) at (0.75, 0.65);
\vertex [above=4cm of i1] (i2);
\vertex [right=1cm of i1] (j1);