Hamburger Beiträge zur Mathematik Nr. 953

December 2023

The evaluation of graphs on surfaces for state-sum models with defects

Julian Farnsteiner and Christoph Schweigert
Fachbereich Mathematik

Universität Hamburg

Bereich Algebra und Zahlentheorie

Bundesstraße 55

D – 20146 Hamburg

The evaluation of graphs on 2-spheres is a central
ingredient of the Turaev-Viro construction of
three-dimensional topological field theories.
In this article, we introduce a class of graphs,
called extruded graphs, that is relevant for
the Turaev-Viro construction with general defect configurations involving defects of various dimensions. We define
the evaluation of extruded graphs and show that it
is invariant under a set of moves. This ensures the
computability and uniqueness of our evaluation.

§ INTRODUCTION

Topological field theories have mediated many relations between topology and representation theory: they also have applications in
the theory of topological phases of matter, in quantum computing and for the
construction of correlators of 2-dimensional conformal field theories.
As a consequence, topological field theories, in particular in three dimensions,
have been intensely
studied in the last few decades. Various constructions of topological field
theories have been developed; the state-sum construction combines the virtues
of providing fully local theories, together explicit tools for concrete computations.

It is by now well-accepted that topological field theories should be studied
on stratified manifolds where strata of various codimensions can be endowed with representation-theoretic
decorations. Such strata, also called defects,
are central for applications to representation theory; they also
provide insight in
symmetries and dualities of the topological field theories; for an early
study in the case of 3-dimensional Dijkgraaf-Witten theories see e.g. <cit.>.

In the context of state-sum constructions, defects have recently been studied
intensely; a selection of papers includes <cit.>. The
stratifications discussed in these papers are, however, of limited generality.
We describe a more general case as follows:
consider, for simplicity, a closed oriented 3-manifold, together with a skeleton in the sense
of <cit.>. We label the 3-cells of the skeleton by spherical fusion categories
and the 2-cells by appropriate bimodule categories with trace. Figure (<ref>) shows two 3-cells
labeled by $\cal A$ and $\cal B$ as well as 2-cells labeled by $\cal K$, $\cal M$ and
$\cal N$.

\begin{equation}\label{pic_intro_1}
\pica{1}{pic_intro_1}
\end{equation}

In the simplest situation, which corresponds to the theory described in, e.g. <cit.>,
all 3-cells are labeled by the same spherical fusion category $\cal A$ and all
2-cells by the regular $\acat$-$\acat$-bimodule category, which is the category $\cal A$ with left and right action
given by the tensor product in $\cal A$.
We refer to this situation as the monochromatic labeling.
In this way, we recover the classical Turaev-Viro construction.

At a 1-cell (or edge) of a skeleton, finitely many  2-cells labeled by possibly different bimodule categories  meet.
For example, at the 1-cell in (<ref>)
that is labeled by $x$, four 2-cells meet.
The orientation of the 1-cells, together with the orientation of the ambient 3-manifold, provides a cyclic
order for the adjacent 2-cells.
The labeling is such that we can assign the cyclically balanced Deligne
product of the bimodule categories to the 1-cell.
We decorate each 1-cell with an object in this balanced Deligne product.
These objects are denoted by $v,w,x,y,z$ in (<ref>). We call such a labeled 1-cell a
generalized Wilson line.

In the monochromatic case, the balanced Deligne product is equivalent to
the Drinfeld center of $\cal A$. Indeed, ordinary bulk Wilson lines are labeled
by objects in the Drinfeld center.
Since the Drinfeld center is monoidal, there is a distinguished transparent label for a bulk Wilson line: the monoidal unit of the Drinfeld center.
(It should be appreciated that the treatment of bulk Wilson lines in <cit.>
is slightly different: they enter as additional geometric data and are not realized as 1-cells in the skeleton.)

Finally, we consider vertices where several 1-cells meet. An example
in (<ref>) is the vertex labeled by $\varphi$ in which 1-cells labeled
by $v,w,x,y$ and $z$ meet.
In the monochromatic case of ordinary Wilson lines labeled by
objects $x_1,\ldots x_n$ in a Drinfeld center $\mathcal{Z}(\mathcal{A})$,
the label for the vertex is an
invariant tensor $\mathrm{Hom}_{\mathcal{Z}(\mathcal{A})}(1,x_1\otimes \ldots \otimes x_n)$.
In fact, such a labeling requires auxiliary choices like a linear ordering of the objects
$x_i$. This can be avoided by considering a small sphere around the vertex;
this sphere has labeled marked points where
the bulk Wilson lines pierce the sphere. In this situation, we have the corresponding
block space for $\mathcal{Z}(\mathcal{A})$ on the sphere at our disposal, which does not depend on auxiliary
choices. We therefore assign an element in the block space to the vertex. To apply a similar strategy beyond the monochromatic case,
we use that block spaces, even in the presence of defects, have been
constructed in <cit.>.
While the block spaces in <cit.> have been constructed for framed surfaces, we here use
a variant for oriented surfaces. As explained in <Ref>, this is not a problem in the
situations we consider.

We call a 3-manifold with embedded labeled skeleton a labeled defect 3-manifold;
a topological field theory
has to assign a scalar to such a manifold.

The state-sum construction of such an invariant then proceeds as follows: one chooses as state-sum variables
objects in the bimodule categories that are assigned to the 2-cells of the
skeleton. (At a later stage in the state-sum construction, there will be a summation
over these variables.) For a fixed choice of state-sum variables,
one constructs a vector space $V_{(e,\iota)}$ for each half-edge $(e,\iota)$, i.e. for  each pair consisting of a 1-cell $e$
and a homotopy class $\iota$ of embeddings $[0,1) \to e$ that send 0 to an end point of $e$.
The reader is invited to think about this
space as a space of invariants. This, however, is only literally true in the transparent
case; we construct the relevant vector spaces in <Ref>.
Each edge gives rise to a pair of half-edges. The trace structure of the bimodule
categories incident to the edge gives rise to a non-degenerate bilinear
pairing on the two spaces of invariants.

To the labeled defect manifold, together with a choice of skeleton and a
choice of state-sum variables, we associate the tensor product $V:=\bigotimes_{(e,\iota)} V_{(e,\iota)}$
of these vector spaces over all half-edges.
There is a non-degenerate bilinear pairing
for the two vector spaces associated to the two half-edges $(e,\iota_+), (e,\iota_-)$ of an edge $e$.
Hence the vector space $V_{(e,\iota_+)}\otimes V_{(e,\iota_-)}$
and also $V$ as a tensor product of such spaces comes with a distinguished non-zero vector $\star$.

The vector space $V$ can also be organized as a tensor product over vertices,

$$ V=\bigotimes_{v} \bigotimes_{(e,\iota) \text{ incident to } v} V_{(e,\iota)} \,\,\, . $$

Around each vertex, we obtain the
following situation which is depicted in (<ref>) and which we call an extruded graph.

\begin{equation}\label{pic_intro_2}
\pica{1}{pic_intro_2}
\end{equation}

In the monochromatic situation, without bulk Wilson lines, we obtain a labeled graph
on a small sphere surrounding the vertex. Its edges are the intersection of the
2-cells with the sphere and hence labeled by the state-sum objects. Its vertices
are the intersection of 1-cells with the sphere and hence labeled with
invariant tensors. By the standard graphical calculus of spherical fusion categories
such a graph can be evaluated to a scalar, giving a linear form $\evaluate{-} : V \to \field$. Applying all these vertex evaluations to the
distinguished vector $\star\in V$ gives a scalar $\evaluate{\star}$; the weighted sum over these scalars for different choices state-sum variables is expected to be independent of the choice
of skeleton; it is a natural candidate for the Turaev-Viro invariant.

The vertex evaluation is thus a central ingredient of the Turaev-Viro
construction. A Turaev-Viro construction for a general defect 3-manifold
needs such an evaluation for extruded graphs.
The main result of this paper is such an evaluation, along with the proof that
it has all the properties that make it explicitly computable.

To define such an evaluation one could
envisage the following (standard) strategy: exhibit a series of moves that reduces the
extruded graph to a standard graph on the sphere. A good candidate for such a standard graph would
be a loop on the sphere with a marked point labeled by an endomorphism. The
standard graph could then
be evaluated by a trace. This strategy has a crucial problem: one has to show that
any combination of moves that simplifies a graph gives a trace with the same numerical value.
This requires a careful analysis of relations between different moves. In our situation, this
is a very difficult problem, since the combinatorics  of extruded graphs is involved and, to our knowledge, has not been developed.

Hence, in this paper, we use a rather different strategy: we define an
evaluation for all extruded graphs and then
prove theorems stating that a certain set of natural moves does not change the value
of the evaluation. The main idea of our definition is as follows: the block space $\blockspace_v$
in which the label $\varphi$ for a vertex $v$ is chosen comes by definition
<cit.> as a subspace
of a pre-block space $\preblock_v$. We show that the labels we have chosen
for the state-sum variables and the invariant tensors
combine to an element in the pre-block space $\preblock$. We then use the spherical structure on the fusion categories
to exhibit $\blockspace$ not only as a subspace, but as a retract of $\preblock$. This defines
the evaluation. We then introduce a set of moves and prove that
the moves leave the evaluation
invariant. The set of moves allows us to reduce the graph so that in the end,
we evaluate a trace. This ensures that our evaluation procedure
can be explicitly implemented  in practice.

For the practitioner, the content of this article can thus be reduced
to the following statements: there is a well-defined evaluation procedure for extruded graphs, and using the
moves described in <Ref>, the evaluation of any extruded graph can be reduced to the computation
of a trace on a bimodule category.

Concretely, this paper is organized as follows:
<Ref> fixes conventions and notation; and some algebraic notions, such as bimodule categories with trace and relative Deligne products, are recalled.
Moreover, diagrammatic notation used to manipulate morphisms is introduced in <Ref>, and in <Ref>, we define (split) equalizers of bi-balanced functors.

<Ref> starts with the <Ref> of extruded graphs.
We continue by describing the block space from <Ref>, which is associated to an extruded graph, as a split equalizer of a bi-balanced functor; this uses semisimplicity and the existence of spherical structures on the involved fusion categories.
In <Ref>, the evaluation of extruded graphs is introduced as a linear map associated to an extruded graph.
We then show in <Ref> that loop graphs are evaluated to traces.

<Ref> is dedicated to moves of invariance, see <Ref>.
We give a list of six selected moves in <Ref>, among which three are sufficient to make a uniqueness statement: our evaluation is unique in that a) it is left invariant by these moves and b) loop graphs are evaluated to traces; this is proved in <Ref>.

The two appendices are reserved for more technical proofs, in particular for the proof that the moves are really moves of invariance.

§ ALGEBRAIC PRELIMINARIES

We need to fix notation, and recall standard concepts, such as bimodule categories, categorical centers and the relative Deligne product.
Lesser-known structures which are reviewed include bimodule traces in <Ref>.

Throughout this paper, let us fix an algebraically closed field $\field$ of characteristic 0.
We will mainly work with $\field$-linear categories, using standard terminology that can be found in <cit.>.
When not stated otherwise, all linear categories are also assumed finite and semisimple.

The following notational conventions serve to shorten the presentation:

* Given any category $\ccat$ with objects $x,y \in \ccat$, we denote Hom-sets by angled brackets:

\begin{equation}
\homset{x}{y} := \homsetin{\ccat}{x}{y} := \mathrm{Hom}_\ccat (x,y).
\end{equation}

* Lowercase bold letters ($\si{x}, \si{y}, \dots$) are used for simple objects.
In a finite semisimple category $\ccat$, the notation

\begin{equation}
\sum_{\si{x}\in \ccat} \cdots
\end{equation}

stands for a sum over the set of isomorphism classes of simple objects of $\ccat$, with $\si{x}$ assuming the value of a representative for each class in the sum.

* We denote the opposed category $\opcat{\ccat}$ to any category $\ccat$ by an overline.
If $\begin{tikzcd}
x \arrow[r, "g"] & y \arrow[r, "f"] & z
\end{tikzcd}$ are objects and morphisms in $\ccat$, then we also denote the corresponding objects and morphism in $\opcat{\ccat}$ by an overline:
\opcat{z} \arrow[r, "\opcat{f}"] & \opcat{y} \arrow[r, "\opcat{g}"] & \opcat{x}.
\end{tikzcd}$
In other words,
\opcat{f \circ g} = \opcat{g} \circ \opcat{f}.

* We treat the Deligne product $\xcat \deligne \ycat$ of (as always, finite) linear categories $\xcat$ and $\ycat$ as strictly associative and use tacitly that the equivalence $\xcat \deligne \ycat \cong \ycat \deligne\xcat$ is symmetric.

* A morphism between finite direct sums $f : \bigoplus_{i\in I} X_i \to \bigoplus_{j\in J} Y_j$ is determined by the family of morphisms

\begin{equation}
\ete{f}{i}{j} :=  \projend{j} \circ f \circ \coincl{i} \; : X_i \to Y_j,
\end{equation}

which we call the matrix elements or components of $f$.

§.§ Calabi-Yau Categories

A linear category $\ccat$ equipped with a trace $\trace{\ccat}$, that is, a collection of maps $\trace{\ccat} : \homsetin{\ccat}{x}{x} \to \field$ for each object $x\in \ccat$ satisfying

* Symmetry: $\trace{\ccat}(f\circ g) = \trace{\ccat}(g\circ f)$ for morphisms $f:x \to y$, $g: y\to x$ in $\ccat$; and

* Non-Degeneracy: The assignment $f \mapsto \trace{\ccat}(f\circ -)$ defines an isomorphism $\homsetin{\ccat}{x}{y} \cong \homsetin{\ccat}{y}{x}^*$ for each $f : x \to y$ in $\ccat$,

is called a Calabi-Yau Category.
For any object $x\in \ccat$, the scalar $\pdim{x}:=\trace{\ccat}(\id_x)$ is called the dimension of $x$.

The trace of a (finite semisimple) Calabi-Yau category $\ccat$ is determined by its dimension vector $(\pdim{\si{x}})$, that is, the finite list of dimensions $\pdim{\si{x}}$ of (representatives of) simple objects $\si{x} \in \ccat$. The entries of the dimension vector are non-zero, $\pdim{\si{x}}\neq 0$.
Conversely, any list of non-zero scalars for each isomorphism class of simple objects defines a trace.

The "squared norm"

\begin{equation}
\catdim{\ccat} := \sum_\si{x} \pdim{\si{x}}^2
\end{equation}

of the dimension vector is called the dimension of the Calabi-Yau category $\ccat$.
If $\ccat$ and $\dcat$ are two Calabi-Yau categories, then the Deligne product $\ccat \deligne \dcat$ is a Calabi-Yau category with trace

\begin{equation}\label{eq_trace_deligne}
\trace{\ccat \deligne\dcat} (f \otimes g) := \trace{\ccat}(f) \, \trace{\dcat}(g)
\end{equation}

for endomorphisms $f$ in $\ccat$ and $g$ in $\dcat$.
The dimension of the product category is the product of the dimensions of the factors:

\begin{equation}\label{eq_dimension_deligne_product}
\catdim{\ccat \deligne\dcat} = \sum_{\si{z}\in \ccat\deligne\dcat} \pdim{\si{z}}^2 = \sum_{\substack{\si{x} \in \ccat \\ \si{y} \in \dcat}} \pdim{\si{x} \deligne \si{y}}^2 = \sum_{\substack{\si{x} \in \ccat \\ \si{y} \in \dcat}} \pdim{\si{x}}^2 \pdim{\si{y}}^2 = \catdim{\ccat} \catdim{\dcat}.
\end{equation}

§.§ Monoidal Categories

We treat all monoidal categories as strict.
For the monoidal product of two objects $a$ and $b$, we simply write $ab$, with the exception of the tensor product $\otimes$ of vector spaces.
Reversing the product of a monoidal category $\acat$ leads to another monoidal category, denoted $\revcat{\acat}$.

The monoidal categories we consider are pivotal.
Thanks to the coherence theorem <cit.>, we treat the pivotal structure as strict as well.
We will therefore not distinguish between left and right dualities, and denote by $a^*$ the dual of an object $a$ in a pivotal category.
We occasionally make use of the fact that a pivotal structure on a category $\acat$ gives rise to a distinguished monoidal equivalence

\begin{equation}
\opcat{\acat} \cong \revcat{\acat}.
\end{equation}

We denote the evaluation and coevaluation maps by

\begin{equation}\label{eq_ev_coev}
\ev{a} : a \; a^* \to \one \qquad\text{and} \qquad \coev{a} : \one \to  a^*\; a.
\end{equation}

Instead of distinguishing between left and right evaluation and coevaluation, we stick to the morphisms (<ref>) and use

\begin{equation}
\ev{a^*} : a^* \; a \to \one \qquad\text{and} \qquad \coev{a^*} : \one \to  a\; a^*
\end{equation}

for the other duality.

Both the left and the right pivotal trace endow a given pivotal fusion category $\acat$ with the structure of a Calabi-Yau category.
If the two traces agree, the category $\acat$ is called spherical, and it carries the structure of a Calabi-Yau category in a canonical way.

§.§ Diagrammatic Notations

We use the diagrammatic notation of string diagrams. These are always read bottom-to-top: for example, given morphisms $f:x\otimes y^* \to z$, $g: y \to w^*$ in a pivotal category $\acat$, we write

\begin{equation}\label{eq_strdiag_ex}
(z\otimes g) \circ (f \otimes y) \circ (x\otimes \coev{y}) = \pica{1}{pic_diag_example}.
\end{equation}

For generalities on string diagrams, we refer to <cit.>.

In the string diagram (<ref>), all strands are labeled by objects and all coupons are labeled by appropriate morphisms.
We need to work with linear maps between Hom-spaces and therefore extend the graphical calculus to  diagrams in which we leave one coupon empty, drawn with a double outline.
Such a diagram represents a linear map from the Hom-space associated with the empty coupon to the Hom-space in which the fully labeled string diagram evaluates to a morphism.
The following example shows a linear map between Hom-spaces $\homset{x\otimes y^*}{z} \to \homset{x}{z \otimes w^*}$:

\begin{equation}
\pica{1}{pic_emptybox_ex} = \left(f \mapsto (z\otimes g) \circ (f \otimes y) \circ (x\otimes \coev{y}) \right).
\end{equation}

Since they describe maps of Hom-spaces, these diagrams with empty coupons are composed by inserting the pre-composed diagram into the empty box of the post-composed diagram.
This is illustrated below, in an example which describes a composition of linear maps

\begin{equation}
\homsetin{\vect}{\homset{x y^* y}{z w^*}}{\homset{x}{z w^*}} \otimes \homsetin{\vect}{\homset{x y^*}{z}}{\homset{x y^* y}{z w^*}} \xrightarrow{\circ} \homsetin{\vect}{\homset{x y^*}{z}}{\homset{x}{z w^*}}.
\end{equation}

\begin{equation}\label{eq_emptybox_comp}
\pica{1}{pic_emptybox_postcomp} \circ \pica{1}{pic_emptybox_precomp} = \pica{1}{pic_emptybox_ex}
\end{equation}

In <Ref>, we will see further generalizations of these diagrams.

§.§ Hom-Space Contractions

For a finite-dimensional vector space $V$, we follow <cit.> and denote the image of $\id_V$ under the identification $\homset{V}{V} \cong V \otimes V^*$ by $\tenhomid \in V \otimes V^*$.
It is useful to recall that an explicit form of $\tenhomid$ is given as follows: Pick a basis $(\varphi_\alpha)$ of $V$ and denote the dual basis of $V^*$ by $(\varphi^*_\alpha)$. Then $\tenhomid = \sum_\alpha \varphi_\alpha \otimes \varphi_\alpha^*$.
The following Sweedler-type notation is convenient:
$\tenhomid = \sweed{\tenhomid}{V} \otimes \sweed{\tenhomid}{V^*} \in V \otimes V^*$.

Let $V$ be a Hom-space $V = \homsetin{\ccat}{x}{y}$ of a Calabi-Yau category $\ccat$.
It then makes sense to consider the image of $\tenhomid$ under the isomorphism $\homsetin{\ccat}{x}{y} \otimes \homsetin{\ccat}{x}{y}^* \cong \homsetin{\ccat}{x}{y} \otimes \homsetin{\ccat}{y}{x}$.
We denote this image by $\tenhomid$ as well, and its Sweedler components by $\tenhomid = \basisel{x}{y} \otimes \basisel{y}{x}$.
When using graphical notation, we write

\begin{equation}
\tenhomid = \pica{1}{pic_homcontraction_1} \otimes \pica{1}{pic_homcontraction_2}.
\end{equation}

This allows us to express  resolutions of the identity of an object $x$ in a Calabi-Yau category $\ccat$ diagrammatically:

\begin{equation}\label{eq_res_of_id}
\sum_\si{x} \pdim{\si{x}} \basisel{\si{x}}{x} \circ \basisel{x}{\si{x}} = \sum_\si{x} \pdim{\si{x}} \pica{1}{pic_resofid} = \id_x.
\end{equation}

Recall that according to our conventions, this is a sum over isomorphism classes of simple objects $\si{x}$ of $\ccat$.
The following variation of (<ref>) will also be useful:

\begin{equation}\label{eq_parallel_res}
\sum_\si{x} \pdim{\si{x}} (\basisel{\si{x}}{y} \circ \basisel{x}{\si{x}}) \otimes (\basisel{\si{x}}{x} \circ \basisel{y}{\si{x}}) = \basisel{x}{y} \otimes \basisel{y}{x}.
\end{equation}

The notation can also be used to conveniently express the multiplicity $\dim \homset{x}{\si{x}}$ of a simple object $\si{x}$ in an object $x$ in a Calabi-Yau category $\ccat$:

\begin{equation}\label{eq_trace_multiplicity}
\dim \homset{x}{\si{x}} = \trace{\ccat} (\basisel{\si{x}}{x} \circ \basisel{x}{\si{x}}).
\end{equation}

§.§ Traced Bimodule Categories

Given a monoidal category $\acat$, we consider left $\acat$-module categories $\mcat$.
This is a category equipped with an action functor $\acat \times \mcat \to \mcat$, denoted $(a,m) \mapsto am$, together with coherence data and axioms.
For details, we refer to <cit.>.

We are interested only in finite and semisimple module categories over fusion categories.
In particular, all of our module categories are exact.

There are the obvious related notions of right module categories and bimodule categories.
A left $\acat$-module category can be equivalently described as a right $\revcat{\acat}$-module category.
An $\acat$-$\bcat$-bimodule category is the same as a left $\acat \deligne \revcat{\bcat}$-module category.
Lastly, every linear category is canonically equipped with the structure of a left or right $\vect$-module category. Hence, a left $\acat$-module category can also be seen as an $\acat$-$\vect$-bimodule category.

We will make tacit use of these identifications.
To keep in mind which type of module category a particular category is, we sometimes write, for instance $\bimod{\acat}{\mcat}{\bcat} = \mcat$ for an $\acat$-$\bcat$-bimodule category, and similar for left and right module categories.

For every fusion category $\acat$, we can consider the regular bimodule category $\bimod{\acat}{\acat}{\acat}$, whose action functors are given by the monoidal product.

Because we work over finite tensor categories, the bimodule categories we consider have internal Homs with respect to each action.
This means that given objects $m,n$ in a bimodule category $\bimod{\acat}{\mcat}{\bcat}$, there are internal Hom-objects $\ihom{\acat}{m}{n} \in \acat$ and $\ihom{\bcat}{m}{n} \in \bcat$, characterized by the existence of isomorphisms

\begin{equation}\label{eq_internal_homs}
\homsetin{\acat}{a}{\ihom{\acat}{m}{n}} \cong \homsetin{\mcat}{am}{n}, \qquad \homsetin{\bcat}{b}{\ihom{\acat}{m}{n}} \cong \homsetin{\mcat}{bm}{n}.
\end{equation}

Pivotal structures on monoidal categories lead to a unique module structure on the opposite of a module category:
For a bimodule category $\bimod{\acat}{\mcat}{\bcat}$ over pivotal categories $\acat$ and $\bcat$, the opposite category $\opcat{\mcat}$ is canonically a $\bcat$-$\acat$-bimodule category, with actions defined by $b \opcat{m} a := \opcat{a^* \, m \, b^*}$.

If a bimodule category $\bimod{\acat}{\mcat}{\bcat}$ over pivotal categories $\acat$ and $\bcat$ also carries the structure of a Calabi-Yau category, it is natural to impose the following consistency condition:
Let $a\in \acat, \, b\in \bcat$, and $m \in \mcat$ be objects, and let $f : amb \to amb$ be an endomorphism in $\mcat$.
The morphism

\begin{equation}\label{eq_ptr_def}
\mathrm{ptr}(f) := (\ev{a^*} \, m \, \ev{b}) \circ (a^* \, f \, b^*) \circ (\coev{a} \, m \, \coev{b^*}),
\end{equation}

which is an endomorphism of $m$, is called the partial trace of $f$ with respect to $a$ and $b$.
The consistency condition between the Calabi-Yau structure and the module actions on $\mcat$ mentioned above is

\begin{equation}\label{eq_module_trace_cond}
\trace{\mcat} (f) = \trace{\mcat} (\mathrm{ptr}(f)).
\end{equation}

Equation (<ref>) can be illustrated in a string diagram <cit.>:

\begin{equation}
\trace{\mcat} \left(\pica{1}{tracedef_a_1}\right) = \trace{\mcat} \underbrace{\left(\pica{1}{tracedef_b_1}\right)}_{\mathrm{ptr}(f)}.
\end{equation}

Following <cit.>, we call a Calabi-Yau structure on a bimodule category over pivotal fusion categories which satisfies the partial trace property (<ref>) a bimodule trace.
The corresponding notion for a one-sided module category is called a module trace.
We refer to a (bi-)module category equipped with a (bi-)module trace as a traced (bi-)module category.
The proof of the next lemma is left to the reader.

Let $\acat$ be a pivotal fusion category. The following statements are equivalent.

* $\acat$ is a spherical fusion category.

* The regular bimodule category $\bimod{\acat}{\acat}{\acat}$ admits a bimodule trace.

If either statement is true, then the bimodule trace on $\bimod{\acat}{\acat}{\acat}$ is unique up to a normalization factor.
If in addition, the bimodule trace is normalized such that the dimension of the monoidal unit is 1, then the bimodule trace is equal to the (left or right) pivotal trace.

Given two $\acat$-module categories $\mcat$ and $\ncat$, the product category $\mcat \times \ncat$ can be equipped with the structure of an $\acat$-module category by simultaneous action on the components, which is denoted $\mcat \boxplus \ncat$.
This module category is called the direct sum of $\mcat$ and $\ncat$.
A module category $\mcat$ is called indecomposable if the existence of an equivalence of module categories $\mcat \cong \ncat \boxplus \kcat$ implies that either $\ncat$ or $\kcat$ is equivalent to $\mcat$, with the other being trivial.
The structure of a bimodule trace on an indecomposable bimodule category is, if it exists, unique up to a scalar <cit.>.
The question whether or not a (bi-)module category admits a bimodule trace can be stated as an eigenvalue problem <cit.>.

A module trace also provides a relation between the (pivotal) dimension of an internal Hom object, and the dimensions of the objects in the module category:

Given a left $\acat$-module category $\mcat$ over a spherical fusion category $\acat$ with a module trace, the dimension of the internal Hom for simple  objects $\si{m},\si{n} \in \mcat$ can be computed as follows:

\begin{equation}
\pdim{\ihom{\acat}{\si{m}}{\si{n}}} = \frac{\catdim{\acat}}{\catdim{\mcat}} \; \pdim{\si{m}}\pdim{\si{n}}.
\end{equation}

Following <cit.>, we call the square matrix $Q$ with entries

\begin{equation}
Q_{\si{nm}} := \pdim{\ihom{\acat}{\si{m}}{\si{n}}}
\end{equation}

for (representatives of isoclasses of) simple objects $\si{m}, \si{n} \in \mcat$ the dimension matrix of $\mcat$.
From <cit.>, we know that $Q$ is of rank 1 with only non-zero entries.
This implies that $Q$ is of the form $Q_\si{nm} = \xi_\si{n}\zeta_\si{m}$ for non-zero scalars $\xi_\si{n}, \,\zeta_\si{m} \in \field$.

Moreover, the module trace on $\mcat$ causes the dual of the internal Hom to behave much like the dual of the Hom-space:

\begin{equation} \label{eq_dual_inthom}
\ihom{\acat}{m}{n}^* = \ihom{\acat}{n}{m}
\end{equation}

for all objects $m,n \in \mcat$.
This is easy to check using the Yoneda lemma; for $a \in \acat$, we have:

\begin{align}
\begin{aligned}
\homset{a}{\ihom{\acat}{n}{m}} &\cong \homset{an}{m} \cong \homset{m}{an}^* \cong \homset{a^*m}{n}^* \\
&\cong \homset{a^*}{\ihom{\acat}{m}{n}}^* \cong \homset{\ihom{\acat}{m}{n}}{a^*} \cong \homset{a}{\ihom{\acat}{m}{n}^*}.
\end{aligned}
\end{align}

Due to (<ref>), the dimension matrix $Q$ is symmetric, which means that $\xi_\si{m} = \zeta_\si{m}$ for all simple representatives $\si{m} \in \mcat$.

The dimension vector is an eigenvector of the dimension matrix $Q$ with eigenvalue $\catdim{\acat}$, see <cit.>:

\begin{equation}
\sum_\si{m} Q_\si{nm} \pdim{\si{m}} = \sum_\si{m} \xi_\si{n}\xi_\si{m} \pdim{\si{m}} = \catdim{\acat} \pdim{\si{n}}.
\end{equation}

As $\xi_\si{n}$ and $\catdim{\acat}$ are non-zero, this is equivalent to the following:

\begin{equation}
\frac{\sum_\si{m} \xi_\si{m} \pdim{\si{m}}}{\catdim{\acat}} = \frac{\pdim{\si{n}}}{\xi_\si{n}} =: \lambda,
\end{equation}

where we introduced a non-zero scalar $\lambda$ that does not depend on $\si{n}$.
Substituting $\xi_\si{m} = \frac{\pdim{\si{m}}}{\lambda}$, we find

\begin{equation}
\frac{\sum_\si{m} \pdim{\si{m}} \pdim{\si{m}}}{\lambda \, \catdim{\acat}} = \lambda, \qquad \text{and hence} \qquad \frac{\catdim{\mcat}}{\catdim{\acat}} = \lambda^2.
\end{equation}

Finally, we obtain the statement of the lemma:

\begin{align}
\begin{aligned}
\pdim{\ihom{\acat}{\si{m}}{\si{n}}} &= Q_{\si{nm}} = \xi_\si{n}\xi_\si{m} = \frac{\pdim{\si{n}}\pdim{\si{m}}}{\lambda^2} = \frac{\catdim{\acat}}{\catdim{\mcat}} \; \pdim{\si{m}}\pdim{\si{n}}.
\end{aligned}
\end{align}

The following lemma extends the $\star$-notation from <Ref> to string diagrams in traced module categories.
More, precisely, the following lemma holds.

Let  $\mcat$ be a left $\acat$-module category over a spherical fusion category $\acat$ with a module trace.
Let there be objects $m$, $n\in \mcat$ and $a \in \acat$. Denote by $(\varphi_\alpha)$ a basis of the Hom-space $\homsetin{\mcat}{m}{an}$, and the elements of the dual base by $\varphi^*_\alpha \in \homset{an}{m}$.

Then the $\alpha$-indexed families of morphisms

\begin{equation}
(\ev{a^*} n) \circ (a^* \varphi_\alpha) \qquad \text{and} \qquad (a^* \varphi_\alpha^*) \circ (\coev{a}\; n)
\end{equation}

form dual bases of the Hom-sets $\homset{a^* m}{n}$ and $\homset{n}{a^* m}$, respectively.

Using the $\star$-notation introduced in <Ref>, this statement can be expressed as the equality

\begin{equation}
\basisel{a^* m}{n} \otimes \basisel{n}{a^* m} = \left( (\ev{a^*} n) \circ (a^* \basisel{m}{an}) \right) \otimes \left((a^* \basisel{an}{m}) \circ (\coev{a}\; n)\right),
\end{equation}

or graphically as

\begin{equation}
\pica{1}{pic_modtracedualbase_1} \otimes \pica{1}{pic_modtracedualbase_2} = \pica{1}{pic_modtracedualbase_3} \otimes \pica{1}{pic_modtracedualbase_4}.
\end{equation}

We merely need to show that the duality relation

\begin{equation}\label{eq_moduleduality_pf_1}
\trace{\mcat} \left( (\ev{a^*} n) \circ (a^* \varphi_\alpha) \circ (a^* \varphi_{\tilde{\alpha}}^*) \circ (\coev{a}\; n) \right) = \delta_{\alpha, \tilde{\alpha}}
\end{equation}

Using the compatibility (<ref>) of the module trace with the partial trace, it is easy to see that the duality relation (<ref>) is inherited from the duality relation of the original dual bases $\varphi_\alpha$, $\varphi_\alpha^*$:

\begin{align}
\begin{aligned}
\trace{\mcat} &\left( (\ev{a^*} n) \circ (a^* \varphi_\alpha) \circ (a^* \varphi_{\tilde{\alpha}}^*) \circ (\coev{a}\; n) \right)
&\eqwithref{eq_ptr_def} \trace{\mcat} \left( \mathrm{ptr}( \varphi_\alpha \circ  \varphi_{\tilde{\alpha}}^*) \right)
\eqwithref{eq_module_trace_cond} \trace{\mcat} \left(  \varphi_\alpha \circ  \varphi_{\tilde{\alpha}}^*\right)
= \delta_{\alpha, \tilde{\alpha}}.
\end{aligned}
\end{align}

§.§ Centers of Bimodule Categories

Let $\bimod{\acat}{\mcat}{\acat}$ be an $\acat$-$\acat$-bimodule category.
Given an object $m\in \mcat$, a family of isomorphisms $\halfbraid{z}_a : am \to ma$ for each object $a\in \acat$, satisfying the conditions $\halfbraid{z}_{ab} = (\halfbraid{z}_a \, b) \circ (a \, \halfbraid{z}_b)$ and $\halfbraid{z}_\one = \id_m$ is called a balancing for $m$.
The linear category $\modulecent{\acat}{\mcat}$, whose objects are pairs $z = (m, \halfbraid{z})$, where $m\in \mcat$ and $\halfbraid{z}$ is a balancing for $m$, and whose morphisms $f: z \to z' = (m', \halfbraid{z'})$ are morphisms $f:m\to m'$ satisfying the condition that $\halfbraid{z'}_a \circ (a \, f) = (f \,a) \circ \halfbraid{z}_a$, is called the center of $\mcat$
We denote the forgetful functor $z\mapsto m$, which forgets the balancing, by $U : \modulecent{\acat}{\mcat} \to \mcat$.
If $\mcat$ is the regular bimodule category $\bimod{\acat}{\acat}{\acat}$, the center $\modulecent{\acat}{\acat}$ is the Drinfeld center of $\acat$.

Given objects $z = \in \modulecent{\acat}{\mcat}$ and $a\in \acat$, we define a particular pair of morphisms in $\mcat$:

\begin{equation}
\brev{z}{a} : a U(z) a^* \to U(z) \quad\text{and}\quad \cobrev{z}{a} : U(z) \to a^* U(z) a,
\end{equation}

to be called braided evaluation and braided coevaluation.
They can be defined in two equivalent ways, which are the left- and right-hand sides of the equations below.

\begin{align}
\brev{z}{a} &:= (U(z) \, \ev{a}) \circ (\halfbraid{z}_a \,a^*) = (\ev{a} \, U(z)) \circ (a\, (\halfbraid{z}_{a^*})^{-1}), \label{eq_def_brev} \\
\cobrev{z}{a} &:= ((\halfbraid{z}_{a^*})^{-1} \, a) \circ (U(z) \, \coev{a}) = (a^* \, \halfbraid{z}_a) \circ (\coev{a} \, U(z)). \label{eq_def_cobrev}
\end{align}

In diagrammatic notation, we represent the braided evaluation and coevaluation as

\begin{equation}\label{eq_brev_cobrev_pics}
\brev{z}{a} = \pica{1}{pic_brev} \qquad \text{and} \qquad \cobrev{z}{a} = \pica{1}{pic_cobrev},
\end{equation}

where we introduced the convention that objects of the center are drawn as red lines.
Pictures such as (<ref>) containing objects in the center $\modulecent{\acat}{\mcat}$ of a bimodule category $\mcat$ are always diagrams in $\mcat$, not $\modulecent{\acat}{\mcat}$.
When evaluating such diagrams, the forgetful functor has to be applied to all objects labeling red lines, and crossings between red and black lines need to be replaced by balancings. Unlike in the diagrammatic calculus for a braided monoidal category, there are no over- and under-crossings that need to be distinguished.

Any linear category $\xcat$ can be seen as a $\vect$-$\vect$-bimodule category.
With respect to these $\vect$-actions, every object $x\in \xcat$ admits a balancing, and all balancings are isomorphic. Hence, there is an equivalence of categories $\xcat \cong \modulecent{\vect}{\xcat}$.

An alternative model for the center is given by the category of bimodule functors and bimodule natural transformations $\bimodfun{\acat}{\acat}{\acat}{\mcat}$: There is a canonical equivalence of categories

\begin{equation}
\bimodfun{\acat}{\acat}{\acat}{\mcat} \cong \modulecent{\acat}{\mcat}.
\end{equation}

§.§ Balanced Functors and their Diagrams

The center of a module category $\mcat$ over a monoidal category $\acat$
can also be characterized by a universal property:
A balanced functor consists of a functor $F: \mcat \to \xcat$, equipped with a family of isomorphisms
$\gamma_{a,m} : F(am) \to F(ma)$, natural in $a \in \acat$ and $m\in \mcat$, which moreover satisfies $\gamma_{ab,m} = \gamma_{b,ma} \circ \gamma_{a,bm}$.
Balanced functors form a category $\balfun{\mcat}{\xcat}$, whose morphisms are balanced natural transformations, i.e. those natural transformations $\eta: F \Rightarrow G$ which are compatible with the balancings $\gamma^F$ and $\gamma^G$ of $F$ and $G$ in that $\gamma_{a,m}^G \circ \eta_{am} = \eta_{ma} \circ \gamma_{a,m}^F$.
Up to equivalence, the center $\modulecent{\acat}{\mcat}$ is the unique category such that there is an equivalence of functor categories

\begin{equation}\label{eq_center_uniprop}
\balfun{\mcat}{\xcat} \cong \funcat{\modulecent{\acat}{\mcat}}{\xcat}.
\end{equation}

Let $\mcat$ be a (left) module category over any monoidal category $\acat$, and let $\ncat$ be a right $\acat$-module category.
The category $\ncat\deligne \mcat$ is an $\acat$-$\acat$-bimodule category. The balanced functors out of categories of this form are of particular interest to us.
We usually denote the balancing of a functor $F : \ncat\deligne \mcat \to \xcat$ into a category $\xcat$ as $\bal_{n,a,m} : F(na,m) \to F(n,am)$.

The prototypical example of such a balanced functor is the Hom-functor of a module category  $\mcat$ over a rigid monoidal category $\acat$:
Let $\ncat = \opcat{\mcat}$, which is a right $\acat$-module category with action $\opcat{m} a := \opcat{a^*m}$, where $a^*$ here denotes the right dual.
Then the evaluation morphism defines isomorphisms

\begin{equation}\label{eq_hom_balancing}
\homset{\opcat{m}a}{m'} = \homset{\opcat{a^*m}}{m'} \cong \homset{\opcat{m}}{am'},
\end{equation}

which assemble into a balancing of the Hom-functor.
The Hom-functor of a bimodule category $\bimod{\acat}{\mcat}{\bcat}$ has  even more structure, as it is balanced with respect to the $\acat\deligne\revcat{\bcat}$-action on $\mcat$.
In general, if $\ncat$ is a $\bcat$-$\acat$-bimodule category and $\mcat$ is an $\acat$-$\bcat$-bimodule category, then we say a functor

\begin{equation}\label{eq_bibal_conn}
F : \bimod{\bcat}{\ncat}{\acat} \deligne \bimod{\acat}{\mcat}{\bcat} \to \xcat
\end{equation}

is bi-balanced, if it is equipped with a balancing between the right $\acat\deligne\revcat{\bcat}$-action on $\ncat$ and the left $\acat\deligne\revcat{\bcat}$-action on $\mcat$.

We also call functors

\begin{equation}\label{eq_bibal_disconn}
G : \bimod{\acat}{\ncat}{\acat} \deligne \bimod{\bcat}{\mcat}{\bcat} \to \xcat,
\end{equation}

bi-balanced, if they are equipped with two balancings, one for $\mcat$ and one for $\ncat$, which are required to commute.
In the ambiguous case $\acat = \bcat$, a functor can be bi-balanced in either way:

\begin{align}\label{eq_bibal_ambiguous_disconn}
\wick{F : \c1 {_\acat} {\ncat} \c1 {_\acat} \deligne \c2 {_\acat}{\mcat}\c2 {_\acat} \to \xcat} && \text{"disconnected" balancings} \\
\wick{F : \c2 {_\acat} {\ncat} \c1 {_\acat} \deligne \c1 {_\acat}{\mcat}\c2 {_\acat} \to \xcat} && \text{"connected" balancings} \label{eq_bibal_ambiguous_conn}
\end{align}

We next develop a graphical calculus for bi-balanced functors of the "connected" type that is tailored to our purposes.
In particular, given a bi-balanced functor $S: \opcat{\ncat} \times \mcat \to \xcat$, we will be able to pictorially represent those morphisms in $\xcat$ which are built from

* morphisms of the form $S(f,g)$ where $f$ and $g$ are morphisms in $\ncat$ and $\mcat$, respectively; and

* the balancing isomorphisms of $S$, and their inverses.

Our diagrams specialize to the diagrams in <Ref> for the choice of the Hom-functor as bi-balanced functor $S$.

For morphisms $m_1 \xrightarrow{f} m_2 \xrightarrow{g} {m_3}$ in $\mcat$ and $n_2 \xrightarrow{h} n_1$ in $\ncat$, we write

\begin{equation}\label{eq_funcdiag_profun}
\pica{1}{pic_funcdiag_profun_2} := S\left(h, \pica{1}{pic_strdiag_gf} \right) = S(h, g\circ f),
\end{equation}

which represents a morphism $S(n_1, m_1) \to S(n_2, m_3)$ in $\xcat$.
Given objects $a \in \acat$, $m \in \mcat$, and $n \in \ncat$, we draw the following  diagram to represent the left balancing of $S$.

\begin{equation}\label{eq_funcdiag_bal}
\pica{1}{pic_funcdiag_bal} := \bal_{n,a,m} : S(a^*n,m) \to S(n,am).
\end{equation}

As usual, the juxtaposition of a strands labeled by $a\in \acat$ and $m \in \mcat$ stands for the object $am \in \mcat$.
For bi-balanced functors, bending around strands is allowed both on the left and on the right-hand side of diagrams.
To illustrate this, consider the following picture, which represents a morphism $S(a^*n,mb^*) \to S(nb,m')$ in $\xcat$.

\begin{equation}\label{eq_funcdiag_bibal_ex}
\pica{1}{pic_funcdiag_bibal_ex} = S(f, nb) \circ \bal_{n,b,am}^{\text{right}\;-1} \circ \bal_{n,a,mb}^{\text{left}}.
\end{equation}

If we consider the special case where $\ncat = \mcat = \bcat = \bimod{\acat}{\acat}{\acat}$ is the regular bimodule category, and $\xcat = \vect$, we can choose $S$ to be the Hom-functor of $\acat$, which is a bi-balanced functor.
Diagrams such as (<ref>) then specialize to the diagrams we encountered in <Ref>, with the modification that the previously empty double-box is now labeled with the name of a functor, in this case Hom.
The reason why these diagrams really specialize to the diagrams from <Ref> for the Hom-functor is that "bending around strands" as in (<ref>) encodes the Hom-functor's balancing, which is defined using a coevaluation; on the other hand, the coevaluation is also represented by bending around a strand in the diagrams from <Ref>, which are derived from string diagrams.
The Hom-functor of a module category $\mcat$ over $\acat$ also comes with a linear map $\homset{m}{m'} \to \homset{am}{am'}$ for each triple of objects $a\in \acat$, $m,m'\in \mcat$, which is part of the data of module action functor.
If $\acat$ is a rigid and pivotal, this linear map can be expressed using the balancing of the Hom-functor, together with either an evaluation or a coevaluation.
This is true of any balanced functor $S : \opcat{\ncat} \times \mcat \to \xcat$ over a pivotal category $\acat$:
The two expressions

\begin{equation}\label{eq_funcdiag_sidestrand}
\bal_{an,a,m} \circ S(\coev{a}\,n, m)= \pica{1}{pic_funcdiag_sidestrand} = \bal^{-1}_{n,a^*, am} \circ S(n, \ev{a^*}\, m).
\end{equation}

are equal.

Of course, diagrams for bi-balanced functors can be composed, and the composition operation is the same we already encountered in (<ref>): inserting the pre-composed diagram into the double-box of the post-composed diagram.
In this way, our diagrams are similar to other approaches to graphical calculi, such as the "corollas" appearing, for example, in <cit.>.

We leave it to the reader to convince themself that the evaluation of diagrams for bi-balanced functors is well-defined and that isotopic diagrams evaluate to equal morphisms.
In any case, the diagrams throughout this work are meant as illustrations that can, at any point, be translated into standard notation.

§.§ (Split) Equalizers of Bi-Balanced Functors

Let now $\bimod{\acat}{\ncat}{\acat}$ and $\bimod{\acat}{\mcat}{\acat}$ be $\acat$-$\acat$-bimodule categories, and let $S : \opcat{\ncat} \times \mcat \to \xcat$ be a bi-balanced functor (in the "connected" sense of (<ref>)).
From $S$, we will now construct another bilinear functor $\eq S : \modulecent{\acat}{\opcat{\ncat}} \times \modulecent{\acat}{\mcat} \to \xcat$, to be called the equalizer of $S$.

Given objects $x \in \modulecent{\acat}{\ncat}$ and $y \in \modulecent{\acat}{\mcat}$, the value $\eq S(x,y) \in \xcat$ is the essentially unique object such that the diagram

\begin{equation}\label{diag_balanced_limit}
\begin{tikzcd}
\eq S(x,y) \arrow[r,hook] & S(U(x),U(y)) \arrow[rrr, shift left=2mm, "f= \bigoplus_\si{a} {\pica{0.8}{pic_funcdiag_cross_top}}"{above}] \arrow[rrr, shift right=2mm, "g= \bigoplus_\si{a}{\pica{0.8}{pic_funcdiag_cross_bot}}"{below}] & & & \bigoplus_\si{a} S(\si{a}U(x), U(y)\si{a})
\end{tikzcd},
\end{equation}

is an equalizer.
It is straightforward to verify that a morphism $\mu: y \to y'$ in $\modulecent{\acat}{\mcat}$ defines a morphism of equalizer diagrams, making $\eq S$ functorial in the second (and in the same way, the first) argument.

If $\mcat = \ncat$ and $S: \opcat{\mcat}\times \mcat \to \vect$ is the Hom-functor, then the equalizer of $S$ on $x,y \in \modulecent{\acat}{\mcat}$ is the Hom-functor of the center:  $\eq \homsetin{\mcat}{-}{-} \cong \homsetin{\modulecent{\acat}{\mcat}}{-}{-}$.

The assignment $S \mapsto \eq S$ is functorial:
A bi-balanced natural transformation $\eta : S\Rightarrow S'$ from $S$ to another bi-balanced functor $S'$ defines a morphism of diagrams between the diagram (<ref>) and the diagram obtained from (<ref>) by replacing $S$ with $S'$.
Therefore, $\eta$ defines a morphism from the equalizer of $S$ to the equalizer of $S'$.

To proceed, let us recall the concept of a split equalizer.

Let $u,v,w$ be objects of any category $\xcat$.
A split equalizer is a diagram of the form

\begin{equation}\label{diag_split_equalizer}
\begin{tikzcd}
u \arrow[r, "e"{below}] & v \arrow[r, shift left=2mm, "f"{below}] \arrow[r, shift right=2mm, "g"{below}] \arrow[l, bend right, "r"{above}] & w \arrow[l, bend right, shift right=3mm, "t"{above}]
\end{tikzcd},
\end{equation}

where $t\circ f = \id_v$, $t\circ g = e \circ r$, $f\circ e = g\circ e$, and $r\circ e = \id_u$.
In particular, $r$ and $t$ are retracts of $e$ and $f$, respectively.

A contractible pair is a pair of parallel morphisms $f,g : v\to w$, together with a morphism $t: w \to v$, such that
an equalizer of $f$ and $g$ exists, $t\circ f = \id_v$, and $f \circ t \circ g = g\circ t \circ g$.

The following result is well-known, see <cit.>.

Split equalizers are in one-to-one correspondence to contractible pairs with a choice of equalizer.

The endomorphism

\begin{equation}\label{eq_idempotent_abstract}
h := t \circ g,
\end{equation}

of $v$, which is defined both for split equalizers and contractible pairs, plays a central role in the proof of <Ref>.
This is because $h$ is an idempotent, and its image is $u$.
The next statement is the reason why we are interested in split equalizers.

Let $\acat$ be a spherical fusion category, and let $\mcat$ and $\ncat$ be exact $\acat$-$\acat$-bimodule categories.
Let $S: \opcat{\ncat} \times \mcat \to \xcat$ be a bi-balanced functor into a linear category $\xcat$.
Then the spherical structure on $\acat$ determines a splitting of the equalizer (<ref>).
Hence, the spherical structure on $\acat$ exhibits the equalizer $\eq S$ of $S$ not only as a sub-object, but as a retract.

For objects $x \in \modulecent{\acat}{\ncat}$ and $y \in \modulecent{\acat}{\mcat}$, the idempotent

\begin{equation}
h \in \homsetin{\xcat}{S(U(x), U(y))}{S(U(x), U(y))}
\end{equation}

from (<ref>) is in this case given by

\begin{equation}\label{eq_idemp}
h= \frac{1}{\catdim{\acat}} \; \sum_\si{a} \pdim{\si{a}} \; \pica{1}{pic_funcdiag_idemp}.
\end{equation}

We call the top and bottom arrows of the equalizer diagram (<ref>) $f$ and $g$, respectively.
By <Ref>, we only need to build a morphism $t: \bigoplus_\si{a} S(\si{a} U(x), U(y)\si{a}) \to S(U(x),U(y))$ using the pivotal structure of $\acat$, such that $(f,g,t)$ form a contractible pair.
We  define $t$  by

\begin{equation}
t := \frac{1}{\catdim{\acat}} \; \sum_\si{a}\pdim{\si{a}} \pica{1}{pic_funcdiag_tmap} \quad : \; S(\si{a} U(x), U(y)\si{a}) \to S(U(x),U(y)).
\end{equation}

To prove that $(f,g,t)$ is a contractible pair, two conditions need to be checked.
The first is that $t$ is a retraction of $f$:

\begin{align}\label{eq_tmap_retract}
\begin{aligned}
t\circ f
&= \frac{1}{\catdim{\acat}} \; \sum_\si{a} \pdim{\si{a}} \; \pica{1}{pic_funcdiag_tmap_section} = \id_{S(U(x),U(y))}.
\end{aligned}
\end{align}

The second condition is that $f\circ t \circ g  = g \circ t \circ g$ holds.
To verify this, we abbreviate $h:= t\circ g$, which is an endomorphism of $S(U(x),U(y))$, and find by a calculation similar to (<ref>) that $h$ is indeed given by (<ref>).
Since $\frac{1}{\catdim{\acat}}$ is a global factor in the following computation in the Hom-space $\homsetin{\xcat}{S(U(x),U(y))}{\bigoplus_\si{b} S(\si{b} U(x), U(y)\si{b})}$, we omit it:

\begin{align}
\begin{aligned}
f\circ h &= \bigoplus_\si{b} \sum_\si{a}\pdim{\si{a}} \; \pica{1}{pic_funcdiag_eq_1} = \bigoplus_\si{b} \sum_{\si{a},\si{c}}\pdim{\si{a}}\pdim{\si{c}} \; \pica{1}{pic_funcdiag_eq_2} \\
&= \bigoplus_\si{b} \sum_{\si{a},\si{c}}\pdim{\si{a}}\pdim{\si{c}} \; \pica{1}{pic_funcdiag_eq_3} = \bigoplus_\si{b} \sum_{\si{c}}\pdim{\si{c}} \; \pica{1}{pic_funcdiag_eq_4} = g\circ h.
\end{aligned}
\end{align}

Hence, $f\circ h = g \circ h$, and the proof is complete.

The idempotent $h$ from (<ref>) is independent of the choice of spherical structure on $\acat$.
This follows from a direct calculation using the insight that the square of the dimension of a simple object $\si{a} \in \acat$ is equal for all spherical structures, which can be found in  <cit.>.

Given an $\acat$-$\acat$-bimodule category $\mcat$, the inclusion of the Hom-space

\begin{equation}\label{eq_homspace_inclusion}
\homsetin{\modulecent{\acat}{\mcat}}{x}{y} \hookrightarrow \homsetin{\mcat}{U(x)}{U(y)}
\end{equation}

has, for a given spherical structure on $\acat$, a canonical retraction $r : \homsetin{\mcat}{U(x)}{U(y)} \to \homsetin{\modulecent{\acat}{\mcat}}{x}{y}$, given by

\begin{align} \label{eq_homspace_projection}
\begin{aligned}
r (f) &= \frac{1}{\catdim{\acat}} \sum_\si{a} \pdim{\si{a}}\, \brev{y}{\si{a}} \circ ( \si{a}\, f \, \si{a}^* ) \circ \cobrev{x}{\si{a}^*} = \frac{1}{\catdim{\acat}} \sum_\si{a} \pdim{\si{a}}\,\pica{1}{pic_centerproj} \;\;\;.
\end{aligned}
\end{align}

This generalizes <cit.>, and is a direct corollary of <Ref>: The bi-balanced functor in this case is the Hom-functor in $\mcat$, as discussed in <Ref>.

<Ref> can be used to investigate the center of a traced bimodule category.
Concretely, we find that the restriction of a bimodule trace to the center is again a trace:

Let $\acat$ be a spherical fusion category, and let $\mcat$ be a traced $\acat$-$\acat$-bimodule category.
The center $\modulecent{\acat}{\mcat}$ can be equipped with the structure of a Calabi-Yau category, whose trace function is defined, for an endomorphism $f : x \to x$ in $\modulecent{\acat}{\mcat}$, by

\begin{equation}\label{eq_center_trace}
\trace{\modulecent{\acat}{\mcat}}(f) := \frac{1}{\catdim{\acat}}\trace{\mcat}(f).
\end{equation}

Of course, the normalization factor $\frac{1}{\catdim{\acat}}$ in (<ref>) is a convention.
The reason for this particular choice will be made clear by <Ref>.
Before we prove <Ref>, we need some intermediate results.
First, observe that the  idempotent $h : \homsetin{\mcat}{U(x)}{U(y)} \to \homsetin{\mcat}{U(x)}{U(y)}$ leaves the trace invariant:

Let $x \in \modulecent{\acat}{\mcat}$ and let $f$ be an endomorphism of $U(x)$ in $\mcat$.

\begin{equation}
\trace{\mcat}(f) = \trace{\mcat}(h(f)).
\end{equation}

The proof is a direct calculation:

\begin{align}
\begin{aligned}
\trace{\mcat}(h(f)) &= \frac{1}{\catdim{\acat}} \sum_\si{a} \pdim{\si{a}}\,\trace{\mcat}\Left({4}\pica{1}{pic_traceofret_1}\Right){4} = \frac{1}{\catdim{\acat}} \sum_\si{a} \pdim{\si{a}}\,\trace{\mcat}\Left({4}\pica{1}{pic_traceofret_2}\Right){4}\\
&= \frac{1}{\catdim{\acat}} \sum_\si{a} \pdim{\si{a}}\,\trace{\mcat}\Left({4}\pica{1}{pic_traceofret_3}\Right){4} = \trace{\mcat}(f).
\end{aligned}
\end{align}

The first equality is (<ref>), the second uses the symmetry of the trace, and the third uses the compatibility (<ref>) of the bimodule trace on $\mcat$ with the pivotal structure on $\acat$.

It is an even more straightforward observation that, for objects $x,y,z \in \modulecent{\acat}{\mcat}$, a morphism $f:y \to z$ in $\modulecent{\acat}{\mcat}$, and a morphism $g: U(x) \to U(y)$ in $\mcat$, the equality

\begin{equation}\label{eq_retraction_sesquimult}
f\circ r(g) = r(f\circ g)
\end{equation}

Now we are in a position to prove the lemma.

It is clear that the candidate $\trace{\modulecent{\acat}{\mcat}}$ for the trace function defined in (<ref>) inherits the symmetry property from the trace $\trace{\mcat}$ on $\mcat$.
It remains to check whether it is also non-degenerate.
To this end, let $f:y\to z$ be a morphism as before.
We need to check that if for every $g \in \homsetin{\modulecent{\acat}{\mcat}}{z}{y}$, we have $\trace{\modulecent{\acat}{\mcat}}(f \circ \tilde{g}) = 0$, then $f=0$.
As the normalization factor $\frac{1}{\catdim{\acat}}$ in the definition of $\trace{\modulecent{\acat}{\mcat}}$ is not relevant for the validity of the statement, the condition $\forall \tilde{g}\in \homset{z}{y}, \trace{\modulecent{\acat}{\mcat}}(f \circ \tilde{g}) = 0$ is equivalent to:

\begin{equation}\label{eq_centercalabiyau_calc_0}
\forall \tilde{g}\in \homset{z}{y}, \trace{\mcat}(f \circ \tilde{g}) = 0
\end{equation}

Since $r$ is surjective, there is some $g \in \homsetin{\mcat}{U(z)}{U(y)}$ such that $\tilde{g} = r(g)$.
Hence, (<ref>) is equivalent to $\forall g\in \homset{U(z)}{U(y)}, \trace{\mcat}(f \circ r(g)) = 0$.
Moreover, from (<ref>) and <Ref>, we know that

\begin{equation}
\trace{\mcat}(f \circ r(g)) = \trace{\mcat}(r(f \circ g)) = \trace{\mcat}(f \circ g).
\end{equation}

Thus, the statement we want to prove is that
$\forall g\in \homset{U(z)}{U(y)}, \trace{\mcat}(f \circ g) = 0$ implies $f=0$.
This is true by the non-degeneracy of the trace on $\mcat$.

§.§ Shifting Actions under (Co-)Ends

In this section, we are concerned with a structure which looks like, but is not quite, a balancing.
Let $\mcat$ be a left module category over a pivotal fusion category $\acat$, and let $S : \opcat{\mcat} \times \mcat \to \xcat$ be a bilinear functor.
We define isomorphisms

\begin{equation}\label{eq_coendbal_def}
\coendbal_a : \bigoplus_\si{m} S(\si{m},a\si{m}) \to \bigoplus_\si{m} S(a^*\si{m},\si{m})
\end{equation}

for $a \in  \acat$ in the following way.
Pick an arbitrary (auxiliary) Calabi-Yau structure on $\mcat$.
Then define $\coendbal_a$ in terms of their matrix elements

\begin{align}\label{eq_beta_components}
\begin{aligned}
\cte{(\coendbal_a)}{\si{n}}{\si{k}} &:= \pdim{\si{n}}\,S(\basisel{a^*\si{k}}{\si{n}}, (\ev{a} \si{k}) \circ (a \basisel{\si{n}}{a^*\si{k}})) \quad : \; S(\si{n}, a\si{n}) \to S(a^* \si{k} , \si{k})\\
&= \pdim{\si{n}} \pica{1}{pic_betabal}.
\end{aligned}
\end{align}

Although the definition (<ref>) seems ad hoc, the expression arises naturally when considering the term $\bigoplus_\si{m} S(\si{m},a\si{m})$ as an end, as is done in <cit.>.
It is then also obvious that $\coendbal_a$ does not depend on the auxiliary Calabi-Yau structure.

The morphisms $\coendbal_{a}$ are of interest to us for some particular choices of $S$, which we now discuss.

* For $S = - \deligne -$, the family of isomorphisms $\coendbal_{a}$ forms a balancing for the object $\bigoplus_\si{m} \opcat{\si{m}} \deligne \si{m}$ of $\opcat{\mcat} \deligne \mcat$.
Their components are given by

\begin{align}\label{eq_ewid_balancings_components}
\begin{aligned}
\cte{(\coendbal_a)}{\si{n}}{\si{k}} &= \pdim{\si{n}} \basisel{a^*\si{k}}{\si{n}} \deligne \left((\ev{a} \si{k}) \circ (a \, \basisel{\si{n}}{a^*\si{k}})\right).
\end{aligned}
\end{align}

* Let $\acat$ be a spherical fusion category, let $\bimod{\acat}{\ncat}{\acat}$ be an $\acat$-$\acat$-bimodule category, fix $m\in \ncat$ and set $S_m := - \, m \, (-)^*$ as a functor $\lmod{\acat}{\acat} \times \rmod{\opcat{\acat}}{\acat} \to \bimod{\acat}{\ncat}{\acat}$.
The morphisms $\coendbal_a$ associated with $S_m$ again assemble into a balancing.
We can thus define a functor $\leftind : \ncat \to \modulecent{\acat}{\ncat}$, called the induction functor via

\begin{equation}\label{eq_induction_functors}
\leftind(m) := (\bigoplus_\si{a} \si{a} m \si{a}^*, \coendbal) .
\end{equation}

The components of the balancings are given by

\begin{equation}
\cte{(\coendbal_a)}{\si{b}}{\si{c}} = \pdim{\si{b}}\basisel{a^*\si{c}}{\si{b}} m \left((\ev{a} \si{c}) \circ (a \, \basisel{\si{b}}{a^*\si{c}})\right).
\end{equation}

The induction functor is left and right adjoint to the forgetful functor $U : \modulecent{\acat}{\ncat} \to \ncat$:

\begin{equation} \label{eq_induction_adjunction}
\leftind \dashv U \dashv \rightind.
\end{equation}

The adjunctions (<ref>) are witnessed by isomorphisms

\begin{equation} \label{eq_induction_adjunction_r_iso}
\indadjr{m}{x} : \homsetin{\ncat}{m}{U(x)} \to \homsetin{\modulecent{\acat}{\ncat}}{\leftind(m)}{x}
\end{equation}

\begin{equation} \label{eq_induction_adjunction_l_iso}
\indadjl{m}{x} : \homsetin{\ncat}{U(x)}{m} \to \homsetin{\modulecent{\acat}{\ncat}}{x}{\rightind(m)},
\end{equation}

natural in $m \in \ncat$ and $x \in \modulecent{\acat}{\ncat}$.
For morphisms $f : m \to U(x)$, $g: U(x) \to m$, $f' : \leftind(m) \to x$ and $g' : x \to \rightind(m)$, these adjunction isomorphisms are given explicitly by the following expressions.

\begin{align}
\indadjr{m}{x} (f) = \bigoplus_\si{a} \pdim{\si{a}} \brev{x}{\si{a}} \circ (\si{a} \, f \, \si{a}^*)  &\quad \text{and} \quad
(\indadjr{m}{x})^{-1}(f') = f' \circ \coincl{\one}; \label{eq_def_indadjr}
\\
\indadjl{m}{x} (g) = \bigoplus_\si{a} (\si{a} \, g \, \si{a}^*) \circ \cobrev{x}{\si{a}^*} &\quad \text{and} \quad
(\indadjl{m}{x})^{-1}(g') = \projend{\one} \circ g'.\label{eq_def_indadjl}
\end{align}

§.§ Relative Deligne Products.

Let $\bcat$ be a spherical fusion category, and let $\rmod{\mcat}{\bcat}$ be a right and $\lmod{\bcat}{\ncat}$ be a left $\bcat$-module category.
We make use of the relative Deligne product of $\mcat$ and $\ncat$.
For us, it is convenient to realize it as the center of the $\bcat$-$\bcat$-bimodule category $\rmod{\mcat}{\bcat} \deligne \lmod{\bcat}{\ncat}$:

\begin{equation}\label{eq_def_reldel}
\mcat \reldelov{\bcat} \ncat := \modulecent{\bcat}{\rmod{\mcat}{\bcat} \deligne \lmod{\bcat}{\ncat}}.
\end{equation}

This makes it obvious that there is a forgetful functor into the ordinary Deligne product $U: \mcat \reldelov{\bcat} \ncat \to \mcat \deligne \ncat$, which has the left and right adjoint $\leftind$ from (<ref>).
In notation, we sometimes omit the spherical fusion category $\bcat$ when there is no ambiguity, simply writing

\begin{equation}
\mcat \reldel \ncat := \mcat \reldelov{\bcat} \ncat.
\end{equation}

By <Ref>, the relative Deligne product specializes to the ordinary Deligne product in the case that $\bcat = \vect$, i.e.

\begin{equation}
\mcat \reldelov{\vect} \ncat \cong \mcat \deligne \ncat.
\end{equation}

Let now $\acat$ and $\ccat$ be additional spherical fusion categories, and suppose that $\bimod{\acat}{\mcat}{\bcat}$ and $\bimod{\bcat}{\ncat}{\ccat}$ are bimodule categories.
The category $\mcat \reldel \ncat$ then still caries a left $\acat$-action inherited from $\mcat$ and a right $\ccat$-action inherited from $\ncat$, and is thus an $\acat$-$\ccat$-bimodule category.
If $\acat = \ccat$, there is a canonical equivalence

\begin{equation}
\modulecent{\acat}{\mcat \reldelov{\bcat} \ncat} \cong \modulecent{\bcat}{\ncat \reldelov{\acat}\mcat}.
\end{equation}

We also use "dangling product" notation for this category, as in

\begin{equation}\label{eq_dangling_product}
\mcat \reldel \ncat \reldel := \modulecent{\acat}{\mcat \reldelov{\bcat} \ncat}.
\end{equation}

As a shorthand for the induction functor $\leftind$ from (<ref>), we write for $m\in \mcat$ and $n \in \ncat$:

\begin{equation}\label{eq_induction_reldel_objects}
m \rexreldel n := \leftind(m\deligne n).
\end{equation}

The dangling product notation from (<ref>) can be used here as well: We may write $m \rexreldel n \rexreldel$ for an object in $\mcat \reldel \ncat \reldel$.

The relative Deligne product is associative (in the sense that it comes with an appropriately coherent set of equivalences witnessing associativity) and unital with respect to the regular bimodule category $\bimod{\acat}{\acat}{\acat}$.
We slightly abuse notation by treating the relative Deligne product as if it were strictly associative: we do not pay attention to bracketing and make no explicit use of associators.
Unitality requires an equivalence of $\acat$-$\bcat$-bimodule categories

\begin{equation}\label{eq_reldel_unitality}
\lmod{\acat}{\acat} \reldel \rmod{\mcat}{\bcat} \cong \bimod{\acat}{\mcat}{\bcat}
\end{equation}

for every $\acat$-$\bcat$-bimodule category $\mcat$.
This equivalence is given, on objects of the form $a \lexreldel m \in \acat \reldel \mcat$, by

\begin{equation}\label{eq_reldel_unitality_explicit}
a \lexreldel m \mapsto am \in \mcat, \qquad \text{with pseudo-inverse} \qquad n \mapsto \one \lexreldel n \in \acat \reldel \mcat,
\end{equation}

for $n \in \mcat$.

A construction of a trace on $\mcat \reldel \ncat$ is provided by <Ref>, but it is not immediately clear that this trace respects the $\acat$-$\ccat$-bimodule structure.
On the other hand, a bimodule trace for $\mcat \reldel \ncat$  is constructed in <cit.>.
These traces are indeed equal, making calculations involving the trace on the relative Deligne product easy to perform:

The bimodule trace on $\mcat \reldelov{\bcat} \ncat$ defined in <cit.> is the trace for the center of a traced bimodule category defined in <Ref>.
This implies the following property: For endomorphisms $f : m\to m$ in $\mcat$ and $g: n\to n$ in $\ncat$,

\begin{equation}\label{eq_reldel_trace_product}
\trace{\mcat\reldel\ncat} (f \rexreldel g)  = \trace{\mcat}(f) \trace{\ncat}(g).
\end{equation}

In particular,

\begin{equation}\label{eq_reldel_dimensions_product}
\pdim{m \rexreldel n} =  \pdim{m} \, \pdim{n}.
\end{equation}

The unitor equivalence

\begin{equation}
\acat \reldel \acat \cong \acat
\end{equation}

from (<ref>) for a spherical fusion category $\acat$ is an equivalences between two traced bimodule categories. <Ref> implies that this equivalence preserves the trace.
This compatibility is the reason for the normalization factor $\frac{1}{\catdim{\acat}}$ in the definition (<ref>) of the Calabi-Yau structure for the center of a traced bimodule category.

Due to its definition as a center (<ref>), the relative Deligne product $\mcat \reldelov{\bcat} \ncat$ comes with a forgetful functor $U : \mcat \reldelov{\bcat} \ncat \to \mcat \deligne \ncat$.
We will often use a form of Sweedler notation for this functor, writing

\begin{equation}\label{eq_reldel_forget_sweed}
U(x) = \sweed{x}{\mcat} \deligne \sweed{x}{\ncat}
\end{equation}

for $x \in \mcat \reldel \ncat$.

§.§ Silent Objects.

Let $\xcat$ be a linear category.
In accordance with <cit.>, we call the object

\begin{align}\label{eq_def_silentobjs}
\begin{aligned}
\ewid{\xcat} &:= \bigoplus_\si{x} \opcat{\si{x}} \deligne \si{x} \in \opcat{\xcat} \deligne \xcat
\end{aligned}
\end{align}

the silent object of the category $\opcat{\xcat} \deligne \xcat$.
We are interested in silent objects for two reasons.

The first is the natural isomorphism

\begin{equation}\label{eq_soc_def}
\soc : \homsetin{\xcat}{-}{-} \cong \homsetin{\opcat{\xcat}\deligne \xcat}{-\deligne-}{\ewcoid{\xcat}}
\end{equation}

between Hom-functors, which we will use frequently.
Explicitly, $\soc$ is given on a morphism $f:x\to x'$ and in components for simple $\si{y}\in \xcat$ by

\begin{equation}\label{eq_sil}
\tc{\soc(f)}{\si{y}} = \pdim{\si{y}} \; \basisel{\si{y}}{x'} \deligne (\basisel{x'}{\si{y}} \circ f)
\end{equation}

with inverse

\begin{equation}\label{eq_sil_inv}
\soc^{-1}(g) = \sum_{\si{y}} \sweed{(\tc{g}{\si{y}})}{\opcat{\xcat}} \circ \sweed{(\tc{g}{\si{y}})}{\xcat}
\end{equation}

with $g = \sweed{g}{\opcat{\xcat}} \otimes \sweed{g}{\xcat} \; : \; \opcat{x'} \deligne x \to \bigoplus_\si{y} \opcat{\si{y}} \deligne \si{y}$.

The second reason has to do with the Eilenberg-Watts equivalence

\begin{equation}\label{eq_ew}
\ew \, : \, \funcat{\xcat}{\ycat} \cong \opcat{\xcat} \deligne \ycat
\end{equation}

for linear categories $\xcat$ and $\ycat$, see <cit.>.
The equivalence is defined by

\begin{equation}\label{eq_ew_exp}
\ew (F) := \bigoplus_\si{x} \opcat{\si{x}} \deligne F(\si{x}),
\end{equation}

and comparison with (<ref>) reveals that $\ewid{\xcat} = \ew (\id_\xcat)$.

The silent object has more structure if the category $\xcat$ is instead a module category $\lmod{\acat}{\mcat}$.
In this case, we saw in (<ref>) that the object $\ewid{\mcat}$ of the $\acat$-$\acat$-bimodule category $\opcat{\mcat} \deligne\mcat$ has a balancing $\beta$.
In a slight abuse of terminology,

\begin{equation}\label{eq_def_zewid}
\zewid{\mcat} := (\ewid{\mcat}, \beta) \in \modulecent{\acat}{\opcat{\mcat} \deligne\mcat} = \opcat{\mcat} \reldel \mcat
\end{equation}

will also be called a silent object.
A direct consequence of (<ref>) is the relation $U(\zewid{\mcat}) = \ewid{\mcat}$ between the two notions of silent objects.

There is also a module Eilenberg-Watts equivalence

\begin{equation}\label{eq_mew}
\mew \,:\, \lmodfun{\acat}{\mcat}{\ncat} \cong \opcat{\mcat} \reldel \ncat
\end{equation}

for left $\acat$-module categories $\mcat$ and $\ncat$.
The equivalence (<ref>) is constructed from (<ref>) by noticing that the module constraint of a module functor corresponds to a balancing of the object in the Deligne product.

We conclude this section with a remark on the dimensions of silent objects.

Let $\xcat$ be a Calabi-Yau category.
It is easy to see that

\begin{equation}\label{eq_dim_silent}
\pdim{\ewid{\xcat}} = \catdim{\xcat}.
\end{equation}

If $\mcat$ is a traced left module category over a spherical fusion category $\acat$, then by <Ref>,

\begin{equation}\label{eq_dim_silent_mod}
\pdim{\zewid{\mcat}} = \frac{\catdim{\mcat}}{\catdim{\acat}}.
\end{equation}

If $\mcat$ is irreducible, then $\mathrm{Fun}_\acat(\mcat,\mcat)$ is a spherical fusion category with respect to a canonical pivotal structure that does not depend on the choice of trace on $\mcat$ <cit.>.
Therefore, the monoidal unit $\id_\mcat$ has pivotal dimension 1.
This shows that in general, the module Eilenberg-Watts equivalences (<ref>) do not preserve traces.
At several points, categorical dimension factors will be present in our equations, whose appearance is rooted in this fact.

§ EXTRUDED GRAPHS AND THEIR EVALUATION

§.§ Defect Manifolds and Extruded Graphs

The evaluation procedure to be constructed in <Ref> is defined for surfaces endowed with a specific type of labeled graphs which we call extruded graphs.
By a surface, we always mean a smooth, oriented, compact 2-manifold $\sigsurf$, possibly with boundary.

We now introduce additional structure on $\sigsurf$: in a first step, we allow for a finite number of embedded disks in its interior $\mathrm{Int}(\sigsurf)$ and a finite number of embedded intervals on its boundary $\partial \sigsurf$.  The embedded disks and intervals are called nodes. The surface (<ref>) pictured below has three nodes: two disks, and one interval.

\begin{equation}\label{pic_nodesurf}
\pica{1}{prelims_surface_insertions}
\end{equation}

An unlabeled defect surface has even more structure: it is endowed with finitely many embedded disjoint oriented intervals, called defect lines, that must meet the following criteria:

* The end points of each defect line must lie on the boundary of a disk-shaped node or in the interior of an interval node.

* The interior of the defect lines must be disjoint from the disk and interval nodes.

* Each disk node must contain at least one end-point of a defect line.

* The boundary $\partial \sigsurf$ of $\sigsurf$ is covered entirely by interval nodes and defect lines.

The connected components of the complement of defect lines and disk and interval nodes in $\sigma$ are called domains.
In addition to these requirements, we only allow fine unlabeled defect surfaces, which means that all domains are disks.

\begin{equation}\label{pic_unl_defsurf}
\pica{1}{prelims_defsurf}
\end{equation}

We introduce the following labeling of defect surfaces by three labels of algebraic data:

Level 1 To each of the (two-dimensional) domains, we assign a spherical fusion category $\acat, \bcat, \dots$

Level 2 To each defect line adjacent to a domain labeled by $\acat$ on the left and a domain labeled by $\bcat$ on the right, we assign a traced $\acat$-$\bcat$-bimodule category $\mcat$.
In case the defect line lies on the boundary of the surface, there is only one adjacent domain, and the algebraic label is a one-sided (left or right) module category, depending on the orientation of the defect line.

We illustrate this labeling in the example (<ref>), which contains only one domain, labeled by $\acat$.
In this case, all bimodule categories are thus $\acat$-$\acat$-bimodule categories, with the exception of $\mcat_1$, which labels a defect line on the boundary and is thus merely a traced right $\acat$-module category.

\begin{equation}\label{pic_defsurf}
\pica{1}{prelims_2labeleddefsurf}
\end{equation}

The complete labeling of a defect surface as in (<ref>) requires one more layer of algebraic data.

\begin{equation}\label{pic_extgraph}
\pica{1}{prelims_4labeledthickdefsurf}
\end{equation}

Level 3 To each defect line labeled by a bimodule category $\mcat$, we assign in addition an object $m \in \mcat$.

For each node $L$, we describe a category $\raycat(L)$, and then assign an object $x \in \raycat(L)$ to the node.
The category $\raycat(L)$ is obtained as follows.
A node of a defect surface with algebraic labels introduced in the levels 1 and 2 inherits the labels illustrated in the following figure:

\begin{equation}\label{pic_defonemf_labels}
\pica{1}{prelims_labeled_def1mf}
\end{equation}

Here, the orientation of the nodes' boundaries is defined using the orientation of the underlying surface, and the signs indicate whether an adjacent defect line is oriented towards or away from the node.
The orientation endows the set of end points of defect lines on a node with a (cyclic or linear) order, and we thus obtain a (cyclically or linearly) composable string $(\mcat_1 , \mcat_2, \dots)$ of bimodule categories which label the adjacent defect lines.
If the sign of the end point is negative, we use the opposite bimodule category $\opcat{\mcat_i}$ instead.
The ray category of $L$ is then the relative Deligne product

\begin{equation}\label{eq_def_raycat}
\raycat (L) := \mcat_1 \reldel \mcat_2 \reldel \cdots.
\end{equation}

A defect surface with all three layers of labels is called an extruded graph.

We briefly discuss some differences to <cit.>.
Here, all manifolds are oriented as opposed to 2-framed.
To compensate this loss of topological structure, we need additional algebraic structure:
The tensor categories considered in	<cit.> are only rigid, whereas we here require them to be equipped with a spherical structure.
We also use a spherical structure on the bimodule categories.
Furthermore, all categories we consider are semisimple: by <cit.>,
we cannot expect to construct a 3-dimensional state-sum model with defects from non-semisimple data, which is our motivation for considering extruded graphs.
We do not discuss the question where a similar evaluation can be defined for extruded graphs
with non-semisimple labels.

Another minor difference is that we do not admit circular defect lines: all defect lines must start and end at a node.
Taking these points into account, our labeled defect surfaces corresponds to labeled defect
surfaces in the terminology of <cit.>.

The terminology extruded graph needs justification.
In <Ref>, they have been introduced as surfaces with decorations and algebraic labels.
An alternative view is three-dimensional, by considering the cylinder over the surface.
One component of the boundary of the cylinder then contains defect lines and nodes.
From the nodes, perpendicular circular rays emanate and connect the node to the other boundary component.
These rays carry the ray label.
From this perspective, nodes are no longer labeled.
Note that a special case of ray	categories are Drinfeld centers which are braided and thus categorically inherently three-dimensional objects.
Further discussion can be found in <cit.>.
Mathematically, both perspectives are completely equivalent; they are simply different ways of thinking about the same data.

§.§ The State-Sum Modular Functor

The evaluation of extruded graphs takes values in vector spaces provided by the state-sum modular functor described in <cit.>.
We briefly review an alteration of this modular functor, namely the oriented/spherical version as opposed to the framed/rigid version.
Being a symmetric monoidal functor between bicategories, the modular functor consists of several pieces of data.
One of these pieces of data is the block space $\blockspace$, which is a vector space assigned to an extruded graph $\sigsurf$.

We do not need to explicitly construct an oriented/spherical version of the modular functor
from <cit.>, since we only need block spaces.
Moreover, we only consider fine defect surfaces, for which block spaces are easier to define.
In fact, moving to an oriented/spherical setting simplifies the construction in <cit.> where the role
of the framing was to control powers of the double dual (which in our situation is trivialized by
the pivotal structure).
Nevertheless, it is useful to think of the block space as a component of a larger structure,
the modular functor, cf. <cit.>.

We now fix an extruded graph $\sigsurf$. Let $\raycat(\sigsurf)$ denote the Deligne product over the ray categories for all nodes $L$ (intervals and disks) in $\sigsurf$:

\begin{equation}
\raycat(\sigsurf) := \bigdeligne_L \raycat(L).
\end{equation}

Let $x \btimes y \btimes \cdots \in \raycat(\sigsurf)$ be the node labels of $\sigsurf$. We proceed to recall the definition of the associated block space $\blockspace = \blockspace(x \btimes y \btimes \cdots)$.

The choices of objects that label defect lines and nodes of $\sigsurf$ permit us to associate a vector space to each node $L$:
Let $x \in \raycat(L)$ denote the node label, and let $(m_1 \in \mcat_1, m_2 \in \mcat_2, \dots)$ be the list of objects that label the defect lines adjacent to $L$; they define an object in the node category of $L$, which is the ordinary Deligne product

\begin{equation}
\nodecat (L) := \mcat_1 \deligne \mcat_2 \deligne \cdots
\end{equation}

of the bimodule categories that label the defect lines adjacent to $L$.
Notice that there is a forgetful functor $U: \raycat(L) \to \nodecat(L)$ from the ray category to the node category, which forgets the balancings that come with the relative Deligne product.
We use $U$ to define the vector space

\begin{equation}\label{eq_nodespace}
\nodespace(m_1,m_2,\dots) := \homsetin{\nodecat(L)}{m_1 \deligne m_2 \deligne \cdots}{U(x)},
\end{equation}

to be called the node space for the node $L$.

Varying the objects $m_1, m_2, \dots$ in the expression (<ref>) gives rise to the node space functor for a node $L$ of $\sigsurf$:

\begin{equation}\label{eq_nodespacefun}
\nodespace(L) := \homsetin{\nodecat(L)}{-}{U(x)} : \opcat{\nodecat(L)} \to \vect.
\end{equation}

Tensoring over all nodes $L$ of $\sigsurf$ gives the (total) node space functor associated to the surface $\sigsurf$:

\begin{equation}\label{eq_total_nodespace}
\nodespace := \bigotimes_L \nodespace(L) : \bigdeligne_L \opcat{\nodecat(L)} = \opcat{\nodecat} \to \vect.
\end{equation}

Here, we call the Deligne product $\bigdeligne_L \opcat{\nodecat(L)} = \opcat{\nodecat}$ denotes the (total) node category for the surface $\sigsurf$. As each defect line starts and ends at a node, the  bimodule category $\mcat$ associated to a defect line appears precisely twice in the total node category, once as $\mcat$ and once as $\opcat\mcat$.
Thus the terms of the node category $\nodecat$ can be rearranged as follows:

\begin{equation}
\bigdeligne_L \nodecat(L) = \nodecat = \bigdeligne_\mcat \mcat \deligne \opcat\mcat,
\end{equation}

where now $\mcat$ runs over defect lines of $\sigsurf$.
Recall from (<ref>) that there is a silent object

\begin{equation}
\ewid{\mcat} = \bigoplus_\si{m} \si{m}\deligne \opcat{\si{m}} \in \mcat \deligne\opcat{\mcat}.
\end{equation}

The Deligne product over all of these objects $\ewid{\mcat}$ provides us with a silent object in the total node category:

\begin{equation}
\bigdeligne_\mcat \ewid{\mcat} \cong \ewid{(\deligne_\mcat \mcat)} \in \nodecat.
\end{equation}

Following <cit.>, we define the pre-block space as the value of the total node functor on this object (using the Sweedler-like notation from (<ref>) for the forgetful functor $U$):

\begin{equation}\label{eq_def_preblock}
\preblock := \nodespace(\ewid{(\deligne_\mcat \mcat)}) = \homsetin{\nodecat}{\bigoplus_{\si{m},\si{n}} \si{m} \deligne \opcat{\si{m}} \deligne \si{n} \deligne\opcat{\si{n}} \deligne\cdots}{\sweed{x}{\mcat} \deligne \opcat{\sweed{y}{\mcat}} \deligne \sweed{z}{\ncat} \deligne \opcat{\sweed{x}{\ncat}} \deligne\cdots}.
\end{equation}

By (<ref>), the pre-block space is isomorphic to the tensor product of hom-spaces of bimodule categories

\begin{equation}\label{eq_preblock_shortform}
\preblock \cong \homset{\sweed{y}{\mcat}}{\sweed{x}{\mcat}} \otimes \homset{\sweed{x}{\ncat}}{\sweed{z}{\ncat}} \otimes \cdots.
\end{equation}

The block space $\blockspace$ from <cit.>, which we work towards defining, is obtained as a subspace of the pre-block space $\preblock$.
It consists of those vectors satisfying a condition that we will now describe. It can be thought of as a flatness condition, see <cit.>.

We pick a domain labeled by a spherical fusion category $\acat$.
Assume that one of the defect lines adjacent to $\acat$ is labeled by a spherical bimodule category $\mcat$.
Moreover, we for now restrict to the case where the two domains adjacent to the defect line are distinct.
Without loss of generality, take $\mcat$ to be a left $\acat$-module.
Denote the labels of the other defect lines adjacent to $\acat$ by $\ncat$, $\kcat$, $\lcat$, …, and nodes by $x,y,z,\dots$.

\begin{equation}\label{pic_clean_domain}
\pica{1}{holonomy_situation}
\end{equation}

Given the functor $S_\mcat: \acat \deligne \opcat{\acat} \to \vect$ defined by

\begin{equation}\label{eq_def_sm}
S_\mcat(a, \opcat{a'}) := \nodespace_{x,y,\dots}(a \ewid{\mcat} \opcat{a'} \deligne \ewid{\ncat} \deligne \ewid{\kcat} \deligne \ewid{\lcat} \deligne, \cdots).
\end{equation}

a comparison of (<ref>) and (<ref>) reveals that the pre-block space $\preblock$ is obtained from $S_\mcat$ as

\begin{equation}\label{eq_preblock_from_bibal_func}
\preblock \cong S_\mcat (\one, \opcat{\one}).
\end{equation}

We now equip $S_\mcat$ with the structure of a bi-balanced functor, using the explicit form of the pre-block space given in (<ref>).
A particular component of the balancing is given by

\begin{equation}\label{eq_def_holbal}
\begin{tikzcd}[row sep=4mm]
S_\mcat(a, \opcat{\one}) = \bigoplus_{\si{mnkl}} \homset{\si{k} \deligne \opcat{a\si{m}} \deligne \cdots}{U(x)} \otimes \homset{\si{m} \deligne \opcat{\si{n}} \deligne \cdots}{U(y)} \otimes \homset{\si{n} \deligne \opcat{\si{l}} \deligne \cdots}{U(z)} \otimes \cdots \arrow[d, "{\text{Hom-balancing}}"]
\arrow[gray, dashed, rounded corners, to path={
([xshift=11.5em,yshift=.8em]\tikztostart.south west)
-- ([xshift=11.5em,yshift=.4em]\tikztostart.south west)
-- ([xshift=12.3em,yshift=0.1em]\tikztotarget.north west)
-- ([xshift=12.3em,yshift=-0.5em]\tikztotarget.north west)}]{d} \\
\bigoplus_{\si{mnkl}} \homset{\si{k} \deligne \opcat{\si{m}} \deligne \cdots}{U(x)\,a} \otimes \homset{\si{m} \deligne \opcat{\si{n}} \deligne \cdots}{U(y)} \otimes \homset{\si{n} \deligne \opcat{\si{l}} \deligne \cdots}{U(z)} \otimes \cdots \arrow[d, "{\text{balancing of $x$}}"]
\arrow[gray, dashed, rounded corners, to path={
([xshift=12.3em,yshift=.8em]\tikztostart.south west)
-- ([xshift=12.3em,yshift=.4em]\tikztostart.south west)
-- ([xshift=10.1em,yshift=0.1em]\tikztotarget.north west)
-- ([xshift=10.1em,yshift=-0.5em]\tikztotarget.north west)}]{d} \\
\bigoplus_{\si{mnkl}} \homset{\si{k} \deligne \opcat{\si{m}} \deligne \cdots}{a\,U(x)} \otimes \homset{\si{m} \deligne \opcat{\si{n}} \deligne \cdots}{U(y)} \otimes \homset{\si{n} \deligne \opcat{\si{l}} \deligne \cdots}{U(z)} \otimes \cdots \arrow[d, "{\text{Hom-balancing}}"]
\arrow[gray, dashed, rounded corners, to path={
([xshift=10.1em,yshift=.8em]\tikztostart.south west)
-- ([xshift=10.1em,yshift=.4em]\tikztostart.south west)
-- ([xshift=4.4em,yshift=0.1em]\tikztotarget.north west)
-- ([xshift=4.4em,yshift=-0.5em]\tikztotarget.north west)}]{d} \\
\bigoplus_{\si{mnkl}} \homset{a^*\,\si{k} \deligne \opcat{\si{m}} \deligne \cdots}{U(x)} \otimes \homset{\si{m} \deligne \opcat{\si{n}} \deligne \cdots}{U(y)} \otimes \homset{\si{n} \deligne \opcat{\si{l}} \deligne \cdots}{U(z)} \otimes \cdots \arrow[d, dotted, "\text{several balancings}"{left}]
\arrow[gray, dashed, rounded corners, no head, to path={
([xshift=4.4em,yshift=.8em]\tikztostart.south west)
-- ([xshift=4.4em,yshift=.4em]\tikztostart.south west)
-- ([xshift=9em,yshift=-0.1em]\tikztostart.south west)}]
\arrow[gray, dashed, rounded corners, to path={
([xshift=19em,yshift=-0.4em]\tikztostart.south west)
-- ([xshift=26.2em,yshift=0.1em]\tikztotarget.north west)
-- ([xshift=26.2em,yshift=-0.5em]\tikztotarget.north west)}]{d}
\\
\bigoplus_{\si{mnkl}} \homset{\si{k} \deligne \opcat{\si{m}} \deligne \cdots}{U(x)} \otimes \homset{\si{m} \deligne \opcat{\si{n}} \deligne \cdots}{U(y)} \otimes \homset{\si{n} \deligne \opcat{a\si{l}} \deligne \cdots}{U(z)} \otimes \cdots \arrow[d, dotted, "\text{several balancings}"{left}]
\arrow[gray, dashed, start anchor={[xshift=26.2em,yshift=.7em]south west}, end anchor = {[xshift=31.9em,yshift=.4em]south west},bend right=40] %-2.4
\arrow[gray, dashed, start anchor={[xshift=31.8em,yshift=.7em]south west}, end anchor = {[xshift=29.9em,yshift=.5em]south west},bend left=30]
\arrow[gray, dashed, rounded corners, crossing over, to path={
([xshift=29.6em,yshift=.8em]\tikztostart.south west)
-- ([xshift=29.6em,yshift=.4em]\tikztostart.south west)
-- ([xshift=24.7em,yshift=0.1em]\tikztotarget.north west)
-- ([xshift=24.7em,yshift=-0.5em]\tikztotarget.north west)}]{d}\\
\bigoplus_{\si{mnkl}} \homset{\si{k} \deligne \opcat{\si{m}} \deligne \cdots}{U(x)} \otimes \homset{\si{m} \deligne \opcat{\si{n}} \deligne \cdots}{U(y)} \otimes \homset{a^*\si{n} \deligne \opcat{\si{l}} \deligne \cdots}{U(z)} \otimes \cdots
\arrow[dashed, gray, start anchor={[xshift=24.7em,yshift=.7em]south west}, end anchor = {[xshift=16.2em,yshift=.4em]south west},bend left=40, looseness=0.8]
\arrow[gray, dashed, rounded corners, crossing over, to path={
([xshift=22.0em,yshift=.8em]\tikztostart.south west)
-- ([xshift=22.0em,yshift=.4em]\tikztostart.south west)
-- ([xshift=20.2em,yshift=0.1em]\tikztotarget.north west)
-- ([xshift=20.2em,yshift=-0.5em]\tikztotarget.north west)}]{d}
\arrow[gray, dashed, start anchor={[xshift=16.4em,yshift=.8em]south west}, end anchor = {[xshift=21.7em,yshift=.5em]south west},bend right=30, looseness=0.6]
\arrow[d, dotted, crossing over, "\text{several balancings}\qquad"{left}]
\\
\bigoplus_{\si{mnkl}} \homset{\si{k} \deligne \opcat{\si{m}} \deligne \cdots}{U(x)} \otimes \homset{\si{m} \deligne \opcat{\si{n}} \deligne \cdots}{aU(y)} \otimes \homset{\si{n} \deligne \opcat{\si{l}} \deligne \cdots}{U(z)} \otimes \cdots
\arrow[gray, dashed, rounded corners, to path={
([xshift=20.2em,yshift=.8em]\tikztostart.south west)
-- ([xshift=20.2em,yshift=.4em]\tikztostart.south west)
-- ([xshift=14.5em,yshift=0.1em]\tikztotarget.north west)
-- ([xshift=14.5em,yshift=-0.5em]\tikztotarget.north west)}]{d}
\arrow[d, "{\text{Hom-balancing}}", crossing over]\\
\bigoplus_{\si{mnkl}} \homset{\si{k} \deligne \opcat{\si{m}} \deligne \cdots}{U(x)} \otimes \homset{a^* \si{m} \deligne \opcat{\si{n}} \deligne \cdots}{U(y)} \otimes \homset{\si{n} \deligne \opcat{\si{l}} \deligne \cdots}{U(z)} \otimes \cdots = S_\mcat(\one,\opcat{a}).
\end{tikzcd}
\end{equation}

The left balancing for $S_\mcat$ defined in (<ref>) is called "holonomy" in <cit.>.
It can be pictured as "walking around the boundary" of the domain $\acat$ with an object $a \in \acat$:

\begin{equation}
\pica{1}{holonomy}.
\end{equation}

The right balancing of $S_\mcat$ is directly inherited from the balancing of the silent object $\ewid{\mcat}$.
Together, this turns $S_\mcat$ into a bi-balanced functor.

Balanced functors $S_\mcat$, $S_\ncat$, and similar can be constructed for each defect line adjacent to $\acat$, as long as those defect lines separate $\acat$ from a different domain, or lie on the boundary of the defect surface.
It may be the case, however, that a defect line has the domain $\acat$ as its left- and right-adjacent domain, as in the example (<ref>).

\begin{equation}\label{pic_internal_defline}
\pica{1}{pic_internal_defline}
\end{equation}

$\mcat$ is then an $\acat$-$\acat$-bimodule category.
In this case, $S_\mcat$ is a functor with two balancings as well, but in the "disconnected" sense as in (<ref>).

We use the equalizer $\eq {S_\mcat}$ of $S_\mcat$ defined in <Ref> to define the block spaces.
For this to be consistent, we need the result <cit.>, which in our language assumes the following form:

There are canonical isomorphisms

\begin{equation}
\baleqat{S_\mcat}{\one}{\one} \cong \baleqat{S_\ncat}{\one}{\one}
\end{equation}

between equalizers for each pair of defect lines $(\mcat,\ncat)$ adjacent to $\acat$.

In order to define block spaces, we need to consider all domains simultaneously.
For each domain $\acat$, $\bcat$, $\ccat$, …, choose a defect line $\mcat_\acat$ adjacent to $\acat$, $\mcat_\bcat$ adjacent to $\bcat$, $\mcat_\ccat$ adjacent to $\ccat$, and so on.

For an extruded graph $\sigsurf$, the associated block space is the vector space $\blockspace$ given by the intersection

\begin{equation}\label{eq_def_blockspace}
\blockspace := \bigcap_{\text{domains }\acat} \baleqat{S_{\mcat_\acat}}{\one}{\one}
\end{equation}

of the equalizers $\baleqat{S_{\mcat_\acat}}{\one}{\one}$, where for each domain $\acat$, an adjacent defect line $\mcat_\acat$ has been chosen.
<Ref> ensures that this is well-defined up to a canonical isomorphism.

When unraveling <Ref>, one can see that it agrees with the original definition of the block functor <cit.>.
Note that the block space $\blockspace$ is by definition a sub-vector space of the pre-block space $\preblock$.

This example is a special case of <cit.>.
Let $\acat$ and $\bcat$ be spherical fusion categories, and let $\mcat$ be an $\acat$-$\bcat$-bimodule category and $\ncat$ be  a $\bcat$-$\acat$-bimodule category.
Consider as a labeled defect surface the sphere with two nodes and two defect lines connecting the nodes, such that each node has an incoming and an outgoing defect line attached.
We label the domains by $\acat$ and $\bcat$, and the defect lines by $\mcat$ and $\ncat$, accordingly.
We pick node labels $x \in \opcat{\mcat} \reldel \ncat \reldel$ and $y \in \opcat{\ncat} \reldel \mcat \reldel$ in the ray categories associated to the nodes.
The pre-block space for this defect surface is the space of natural transformations

\begin{equation}
\preblock \cong \homsetin{\opcat{\ncat} \deligne \mcat}{U(\opcat{x})}{U(y)} \cong  \homsetin{\funcat{\mcat}{\ncat}}{\ew(U\opcat{x})}{\ew(U(y))} = \mathrm{Nat}(\ew(U(\opcat{x})),\ew(U(y)).
\end{equation}

Furthermore, the block space is given by the subspace of bimodule natural transformations

\begin{equation}
\blockspace \cong \homsetin{\opcat{\ncat} \reldel \mcat \reldel}{\opcat{x}}{y} \cong  \homsetin{\bimodfun{\acat}{\bcat}{\mcat}{\ncat}}{\mew(\opcat{x})}{\mew(y)} = \mathrm{Nat}(\mew(U(\opcat{x})),\mew(U(y))).
\end{equation}

The block space of a framed defect surface that is not fine has been expressed <cit.>, in the case of
the framed modular functor, as a limit over all framed fine refinements.
A fine refinement of a non-fine framed defect surface $\sigsurf$ is a fine defect surface, whose defect structure (nodes and defect lines) is a subdivision of the framed defect structure of $\sigsurf$.
A similar construction can be expected to exist for the oriented state-sum modular functor on
surfaces that are not fine, but has not been worked out to our knowledge, and is not needed for our purposes.

§.§ Definition of the Evaluation

To each extruded graph $\sigsurf$ we associate a linear map $\evaluate{\sigsurf} : \nodespace \to \blockspace$ from its node space to its block space, called its evaluation.
This section is dedicated to defining this evaluation, which factors through the pre-block space $\preblock$ and is thus a composition of two linear maps $\nodespace \xrightarrow{\delta} \preblock \xrightarrow{\holsurj} \blockspace$.
We start by explaining the second map, $\holsurj$.

In <Ref> of the block spaces, the equalizers of bi-balanced functors $\baleqat{S_{\mcat_\acat}}{\one}{\one}$ appear, which by <Ref> come with a retract structure.
In particular, there is an idempotent $\holidem_\acat$ on ${S_{\mcat_\acat}}({\one},{\one}) \eqwithref{eq_preblock_from_bibal_func} \preblock$ whose image is the sub-vector space $\baleqat{S_{\mcat_\acat}}{\one}{\one}$.
The idempotents $\holidem_\acat$ for different domains $\acat$ commute, allowing us to define:

The holonomy idempotent of an extruded graph $\sigsurf$ is given by the composition

\begin{equation}
\holidem := \holidem_\acat \circ \holidem_\bcat \circ \cdots
\end{equation}

of holonomy idempotents for each domain of $\sigsurf$.

It follows immediately from <Ref> that the image of $\holidem$ is the block space

\begin{equation}
\mathrm{im}(\holidem) = \blockspace \subset \preblock.
\end{equation}

The corestriction of the holonomy idempotent $\holidem$ onto its image is the surjection $\holsurj$, which was announced in the beginning of the section:

\begin{equation}\label{eq_def_holproj}
\begin{tikzcd}
\preblock \arrow[r, "\holidem"] \arrow[dr, "\holsurj", two heads] & \preblock \\
& \blockspace \arrow[u, hook]
\end{tikzcd}.
\end{equation}

Next, we need to define the map $\delta: \nodespace \to \preblock$.
It is obtained from the morphisms

\begin{equation}
\tau_m : \ewid{\mcat} = \bigoplus_\si{n} \si{n} \deligne \opcat{\si{n}} \to m \deligne\opcat{m},
\end{equation}

for $m \in \mcat$, defined by

\begin{equation}
\tau_m := \bigoplus_\si{n} \pdim{\si{n}} \; (\basisel{m}{\si{n}} \deligne \basisel{\si{n}}{m}).
\end{equation}

The inclined reader will notice that the $\tau_m$ are the structure morphisms of the end of the functor $-\deligne-$.
We then define $\delta$ as the linear map

\begin{equation}\label{eq_eval_delta}
\nodespace = \nodespace(m_1 \deligne \opcat{m_1} \deligne m_2 \deligne \opcat{m_2} \deligne \cdots) \xrightarrow{\nodespace(\tau_{m_1 \deligne m_2 \deligne \cdots})} \nodespace(\ewid{\mcat_1 \deligne \mcat_2 \deligne \cdots}) = \preblock.
\end{equation}

Let $\sigsurf$ be an extruded graph.
The evaluation $\evaluate{\sigsurf} : \nodespace \to \blockspace$ of $\sigsurf$ is the linear map

\begin{equation}\label{eq_def_eval_shortform}
\evaluate{\sigsurf} := \holsurj \circ \delta.
\end{equation}

We often would like to obtain a scalar from an extruded graph $\sigsurf$.
This is only possible with additional data: a vector $f \in \nodespace$ in the node space and a linear form $\varphi \in \blockspace^*$ on the block space.
We say that

\begin{equation}\label{eq_total_eval}
\totaleval{\varphi}{\sigsurf}{f} = \varphi \left( \evaluate{\sigsurf} f \right) \qquad \in \field
\end{equation}

is the scalar evaluation of the triple $(\sigsurf, f, \varphi)$.
The left-hand side of (<ref>) uses the pairing $(-,-) : \blockspace^* \otimes \blockspace \to \field$.

Our construction supposes that all (bi-)module categories are equipped with a trace.
However, the trace structure is never used in the definition of the evaluation procedure.
We will see in <Ref> that traces provide distinguished choices for linear forms $\varphi \in \blockspace^*$.

[Generalized 6j symbols]
A homeomorphism between the surface of a tetrahedron and the sphere $\mathbb{S}^2$ defines a graph with 4 vertices and 6 edges on the sphere.
Let us consider an extruded graph $\sigsurf$ on $\mathbb{S}^2$, whose nodes and defect lines are obtained from the vertices and edges of the tetrahedron (after picking an orientation for each edge).
The domains of $\sigsurf$ are labeled by arbitrary spherical fusion categories, and the defect lines by arbitrary traced bimodule categories.
The object labels for the defect lines, as well as the node labels, are all simple objects in the respective categories.
The evaluation of such an extruded graph, composed with some $\varphi \in \blockspace^*$ defines a linear form $\varphi \circ \evaluate{\sigsurf} \in \nodespace^*$ on the total node space of $\sigsurf$.
This linear form further generalizes the generalized 6j-symbol from <cit.>.

A state-sum modular functor defined on non-fine defect surfaces as mentioned in <Ref> would allow to evaluate also
non-fine extruded graphs. Such a construction is beyond the scope of this work.

§.§ Loop Graphs and Lasso Graphs

We now explicitly compute the invariant for two of the simplest possible types of extruded graphs: First, a lasso graph $Q$, whose underlying surface is the sphere $\mathbb{S}^2$. We will later see (<Ref>) that any extruded graph on the sphere can be transformed – using the moves from <Ref> – into this lasso graph.
A lasso graph is an extruded graph $Q$: It is a sphere with a single node and a single circular defect line starting and ending at the node.

\begin{equation} \label{pic_lassograph}
Q = \pica{1}{lassograph}.
\end{equation}

There are three levels of algebraic labels to be specified:

* The two hemispheres forming the connected components of the complement of the defect line are labeled by spherical fusion categories $\acat$ and $\bcat$.

* The single defect line is labeled by a traced bimodule category $\bimod{\acat}{\mcat}{\bcat}$.
This implies that the ray category associated to the single node is the balanced category $\raycat = \mcat \reldelov{\bcat} \opcat{\mcat} \reldelov{\acat}$. It will be important that this category contains a silent object $\zewid{\mcat}$, as described in <Ref>.
In contrast, the node category is given by the (unbalanced) category $\nodecat = \mcat \deligne \opcat{\mcat}$.

* The defect line carries an additional subordinate label: A choice of an object $m \in \mcat$.
Moreover, we label the node by an object $x \in \raycat$.
These choices determine the node space to be the Hom-space $\nodespace(m\btimes \opcat{m}) = \homsetin{\nodecat}{m\btimes \opcat{m}}{U(x)}$.

This completes the three levels of labels that determine the extruded graph $Q$.

The extruded graphs of the second type, loop graphs, form a subset of lasso graphs: they are characterized the fact that the node label $x = \zewid{\mcat}$ is silent.
Loop graphs are special because their evaluation against a particular "transparent" linear form $\theta \in \blockspace^*$ is a trace.
This is the main result of this section:

Let $O$ be a loop graph, that is, an extruded graph as in (<ref>) whose underlying surface is a sphere with a single node and a single defect line, labeled by some object $m \in \bimod{\acat}{\mcat}{\bcat}$ starting and ending at the node, whose node is labeled by the silent object $\zewid{\mcat}$.
Then the block space $T$ of $O$ is the endomorphism space $T= \homsetin{\mcat \reldel \opcat{\mcat} \reldel}{\zewid{\mcat}}{\zewid{\mcat}}$.
Moreover, the linear form

\begin{equation}\label{eq_dualoftheta}
\theta := \catdim{\acat} \catdim{\bcat} \, \trace{\mcat \reldel \opcat{\mcat} \reldel}(-) \qquad \in \blockspace^*,
\end{equation}

relates the evaluation $\evaluate{O}$ from <Ref> to the module trace of the bimodule category $\mcat$:

\begin{equation}\label{eq_elementary_graph}
\theta \circ \evaluate{O} = \trace{\mcat}\circ \soc^{-1} \qquad \in \nodespace^*.
\end{equation}

The isomorphism $\soc : \homset{m}{m} \to \homset{\ewid{\mcat}}{m \deligne \opcat{m}} = \nodespace$ appearing in (<ref>) was introduced in <Ref>.

If the module category $\mcat$ is indecomposable, the block space $\homset{\zewid{\mcat}}{\zewid{\mcat}}$ is one-dimensional.

<Ref> can be reformulated in the following useful fashion.

Let $O$ be a loop graph as in <Ref>, and let $f: m \to m$ be an endomorphism of $m \in \mcat$.
Then the scalar evaluation of $(O, \soc(f), \theta)$, see (<ref>), is the scalar

\begin{equation}
\left(\theta, \evaluate{O} \soc(f) \right) = \trace{\mcat}(f) \qquad \in \field.
\end{equation}

The proof of <Ref> relies on an intermediate result for general lasso graphs.
To simplify the presentation, we specialize in the following to $\bcat = \vect$.
In essence, $\mcat$ is now just a left $\acat$-module category.
On the topological level, $\vect$-labeled domains can be removed entirely, so we are left with a disk with a single boundary node as our defect surface.
The more general case where $\bcat$ is not necessarily $\vect$ does not provide additional insight.

The node category $\nodecat = \mcat \deligne \opcat{\mcat}$ is an $\acat$-$\acat$-bimodule category, whose center $\modulecent{\acat}{\nodecat} = \raycat$ is the ray category.
Because both the silent object $\zewid{\mcat}$ and the node label $x$ are objects of $\raycat$, the pre-block space

\begin{equation}
\preblock = \homsetin{\nodecat}{\ewid{\mcat}}{U(x)}
\end{equation}

has a subspace $\homsetin{\raycat}{\zewid{\mcat}}{x}$;
in <Ref>, we saw that there is a distinguished retraction $r: \homsetin{\nodecat}{\ewid{\mcat}}{U(x)} \to \homsetin{\raycat}{\zewid{\mcat}}{x}$ to this inclusion of hom-spaces.

Let $Q$ be a lasso graph with node label $x \in \raycat$.
The block space of $Q$ is the hom-space in the ray category:

\begin{equation}\label{eq_lasso_block}
\blockspace = \homsetin{\raycat}{\zewid{\mcat}}{x}.
\end{equation}

Moreover, the projection $\holsurj : \preblock \to \blockspace$ from (<ref>) is equal to the retraction into the hom-space of the center from <Ref>.

Both claims of the lemma follow immediately if we can show that the holonomy idempotent $h : \preblock \to \preblock$ as defined in <Ref> is equal to the idempotent from (<ref>).
Concretely, we must prove

\begin{equation}\label{eq_loop_holidem_result}
h = \sum_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \brev{x}{\si{a}^*} \circ (\si{a}^* \;  - \;  \si{a}) \circ \cobrev{\zewid{\mcat}}{\si{a}} = \sum_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \pica{1}{pic_loopcalc_7}.
\end{equation}

Starting with an element $g \in \preblock = \homsetin{\nodecat}{\ewid{\mcat}}{U(x)}$, we unravel <Ref> and calculate its image $\holidem(g)$ under the holonomy idempotent $\holidem$.
Since $\holidem$ is a composition of several maps, the calculation is broken down into intermediate steps.
The morphisms composing to $\holidem$ are given vertically in the left column of (<ref>). The right column contains the images of $g$ under the partial compositions of the morphisms in graphical notation. In particular, the bottom-most entry is equal to $\holidem(g)$.

\begin{equation}\label{calc_lasso_longcalc}
\begin{tikzcd}[row sep=4mm, column sep=20mm]
\homset{\ewid{\mcat}}{U(x)} \arrow[d, "{\frac{1}{\catdim{\acat}}} \; \text{evaluation}"]&
g \arrow[d, mapsto] \\
\bigoplus_\si{a} \homset{\si{a} \si{a}^* \ewid{\mcat}}{U(x)} \arrow[d, "{\text{balancing of } \zewid{\mcat}}"]&
\bigoplus_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \pica{0.8}{pic_loopcalc_1}  \arrow[d, mapsto]
\\
\bigoplus_\si{a} \homset{ \si{a}  \ewid{\mcat} \si{a}^*}{U(x)} \arrow[d, "{\text{Hom-balancing}}"]&
\bigoplus_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \pica{0.8}{pic_loopcalc_2} \arrow[d, mapsto]
\\
\bigoplus_\si{a} \homset{\si{a} \ewid{\mcat}}{U(x)\si{a}} \arrow[d, "{\text{balancing of }x}"]&
\bigoplus_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \pica{0.8}{pic_loopcalc_3} \arrow[d, mapsto]
\\
\bigoplus_\si{a} \homset{\si{a}  \ewid{\mcat}}{\si{a}U(x)} \arrow[d, "{\text{Hom-balancing}}"]&
\bigoplus_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \pica{0.8}{pic_loopcalc_4} \arrow[d, mapsto]
\\
\bigoplus_\si{a} \homset{\si{a}^* \si{a} \ewid{\mcat}}{U(x)} \arrow[d, "{\text{coevaluation}}"]&
\bigoplus_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \pica{0.8}{pic_loopcalc_5} \arrow[d, mapsto]
\\
\homset{\ewid{\mcat}}{U(x)} &
\sum_\si{a} \frac{\pdim{\si{a}}}{\catdim{\acat}}\; \pica{0.8}{pic_loopcalc_6}.
\end{tikzcd}
\end{equation}

From the bottom-most row, we can read off the expression for $h$, and recognize that we have obtained (<ref>).

The expression (<ref>) for the block space of a lasso graph is in agreement with <cit.>.

The contents of <Ref> generalize to the case where $\bcat$ is not necessarily $\vect$.
In particular, the general formula for the holonomy idempotent $\holidem$ is given by

\begin{equation}
\holidem(g) = \sum_{\si{a},\si{b}} \frac{\pdim{\si{a}}\pdim{\si{b}}}{\catdim{\acat}\catdim{\bcat}}\; \brev{x}{\si{a}^* \deligne \si{b}^*} \circ ((\si{a}^* \deligne \si{b}^*) g  (\si{a} \deligne \si{b})) \circ \cobrev{\zewid{\mcat}}{\si{a}\deligne \si{b}}.
\end{equation}

<Ref> is a step towards proving <Ref>.
The remainder of the proof is technical, and can be found in <Ref>.
We here continue with an application of <Ref>, which is a formula for the evaluation of lasso graphs.

Let $Q$ be a lasso graph with labels as in (<ref>).
Let moreover

* $f \in \nodespace = \homset{m \deligne\opcat{m}}{U(x)}$ be a node label,

* $\tilde{\varphi} \in \homsetin{\raycat}{x}{\zewid{\mcat}}$ be a morphism, and

* $\varphi \in \blockspace^*$ be the corresponding linear form on the block space defined by $\varphi := \trace{\raycat}(\tilde{\varphi} \circ -) \in \blockspace^* = \homset{\zewid{\mcat}}{x}^*$.

\begin{equation}
\left(\varphi, \evaluate{Q} f \right) = \trace{\mcat}\left(\soc^{-1}(\tilde{\varphi} \circ f)\right) \qquad \in \field.
\end{equation}

The proof is a direct calculation. In the second-to-last step, the loop graph $O$ from <Ref> appears.

\begin{align}
\begin{aligned}
\left(\varphi, \evaluate{Q} f\right) &= (\varphi \circ \holsurj \circ \delta)(f) = \trace{\raycat}\left(\tilde{\varphi} \circ \holsurj (\delta (f))\right) \eqwithref{eq_retraction_sesquimult} \trace{\raycat}\left(\holsurj (\tilde{\varphi} \circ \delta (f))\right) \\&\eqwithref{eq_eval_delta} \trace{\raycat}\left(\holsurj (\tilde{\varphi} \circ f \circ \tau_m)\right) = \trace{\raycat}\left(\holsurj (\delta (\tilde{\varphi} \circ f))\right) \eqwithref{eq_dualoftheta} \frac{1}{\catdim{\acat}\catdim{\bcat}}\theta \circ \holsurj \circ \delta (\tilde{\varphi} \circ f) \\&= \frac{1}{\catdim{\acat}\catdim{\bcat}}\left(\theta,\evaluate{O} (\tilde{\varphi} \circ f)\right) \hspace{7mm}\eqdesc{\text{\Cref{cor_loop_eval_scalar}}}\hspace{7mm} \trace{\mcat}\left(\soc^{-1}(\tilde{\varphi} \circ f)\right).
\end{aligned}
\end{align}

§ MOVES OF INVARIANCE

Moves are certain transformations of an extruded graph, changing both its topological and algebraic data.
They make the abstractly defined evaluation procedure easily computable.

§.§ Overview of Moves

A move $\mathbf{M}$ can be performed on a disk-shaped subset $\sigsurf$ of an extruded graph, and results in another extruded graph, which is obtained by replacing the region $\sigsurf$ with another disc region $\postmove{\sigsurf}$.
We require that the boundaries of the discs intersect
the respective extruded graphs in a generic position, pictured e.g. in (<ref>), (<ref>).
Each move $\mathbf{M}$ can only be performed on disk regions $\sigsurf$ which have a geometry specific to $\mathbf{M}$.
The data of a move are then given by:

* An assignment $\sigsurf \mapsto \postmove{\sigsurf}$ describing the local change in geometry.

* A prescription to determine the algebraic labels of $\postmove{\sigsurf}$ from the algebraic labels of $\sigsurf$.

* A linear map $\movemap : \nodespace \to \postmove{\nodespace}$ from the node space $\nodespace$ of $\sigsurf$ to the node space $\postmove{\nodespace}$ of $\postmove{\sigsurf}$.

While the regions $\sigsurf$ and $\postmove{\sigsurf}$ have well-defined node spaces in the sense of (<ref>), they do not have block- or pre-block spaces.
When in the following, we make a statement about "the pre-block space of $\sigsurf$", or similar, we really mean: Consider an extruded graph $\sigsurf_T$ which contains the region $\sigsurf$. Then the statement holds for the pre-block space of $\sigsurf_T$, for any choice of $\sigsurf_T$.

We proceed to define two more linear maps

\begin{align}
\Pmovemap : \preblock \to \postmove{\preblock}, \qquad &\Pmovemap := \bigoplus_\si{m} \postmove{\delta}_\si{m} \circ \movemap_\si{m}, \quad\text{ and} \label{eq_Pmovemap} \\
\Movemap : \blockspace \to \postmove{\blockspace}, \qquad &\Movemap := \holsurj \circ \Pmovemap \circ \iota \label{eq_Movemap}
\end{align}

between pre-block spaces and block spaces.
The direct sum in (<ref>) runs over simple object labels $\si{m}$ for the defect lines of $\sigsurf$.
For each such labeling, the move $\mathbf{M}$ provides a linear map between the appropriate node spaces, which is called $\movemap_\si{m} : \nodespace_\si{m} \to \postmove{\nodespace}_\si{m}$ in (<ref>).
The linear map $\postmove{\delta_\si{m}} : \postmove{\nodespace}_\si{m} \to \postmove{\preblock}$ is the one that appears in the definition of the evaluation (<ref>).
Equation (<ref>) involves the morphisms of the retraction $\begin{tikzcd}
\blockspace \arrow[r, "\iota"{below}] & \preblock \arrow[l, "\holsurj"{above}, bend right=20]
\end{tikzcd}$.
The map $\Movemap$ is needed for the following definition.

We call a move $\mathbf{M}$ between regions $\sigsurf$ and $\postmove{\sigsurf}$ with linear map $\movemap$ on the node spaces an invariance or a move of invariance iff  the diagram

\begin{equation}\label{diag_movediag}
\begin{tikzcd}
\nodespace \arrow[r, "\evaluate{\sigsurf_{T}}"{above}] \arrow[d, "\movemap"{left}] & \blockspace \arrow[d, "\Movemap"] \\
\postmove{\nodespace} \arrow[r, "\evaluate{\postmove{\sigsurf_{T}}}"{below}] & \postmove{\blockspace}
\end{tikzcd}
\end{equation}

commutes for every extruded graph $\sigsurf_{T}$ which contains the region $\sigsurf$. Here, $\postmove{\sigsurf_{T}}$ denotes the extruded graph obtained from $\sigsurf_{T}$ by replacing the region $\sigsurf$ with $\postmove{\sigsurf}$.

We now define a set of moves, wherein we distinguish between elementary moves and composite moves.
For the elementary moves, we show the invariance property using the definition of the evaluation procedure $\evaluate{-}$. The composite moves are combinations of elementary moves.

For all the moves in our list, the linear map $\Movemap$ is an isomorphism.

We consider the following elementary moves:

§.§.§ Elementary Moves

$\moveOR$ – Orientation Reversal
One of the most basic changes one can make to an extruded graph is the flipping of a defect line's orientation.

\begin{equation}\label{pic_moveOR}
\pica{1}{moveOR_1} \xrightarrow{\moveOR} \pica{1}{moveOR_2}
\end{equation}

In the $\moveOR$-move, a defect line labeled by an object $m$ in an $\acat$-$\bcat$-bimodule category $\mcat$ is replaced by a defect line with opposite orientation.
This new defect line needs a $\bcat$-$\acat$-bimodule category for a label.
For this we choose $\opcat{\mcat}$, and as the post-move object label we take $\opcat{m} \in \opcat{\mcat}$.
Note that $\opcat{\mcat}$ is unambiguously defined only because $\acat$ and $\bcat$ are equipped with pivotal structures.

We denote the node labels of the nodes adjacent to the $\mcat$-labeled defect line by $x$ and $y$.
In the definition of ray categories, a distinction is made between incoming and outgoing defect lines at a certain node.
This determines whether the bimodule category that labels the defect line is used, or its opposite.
Due to this, the ray categories (and the node categories) before and after the move are identical – up to the identification $\opcat{\opcat{\mcat}} \cong \mcat$, which becomes an equality for strict pivotal structures on $\acat$ and $\bcat$.
Hence, there is a canonical identification of the node spaces before and after the move, defining the map

\begin{equation*}
\movemap : \nodespace \to \postmove{\nodespace}.
\end{equation*}

The proof that the move $\moveOR$ is a move of invariance is the content of <Ref>.

$\moveC$ – Contraction
The contraction move allows us to transform two nodes, connected by a defect line, into a single node, thereby dissolving the defect line.

\begin{equation}\label{pic_moveC}
\pica{1}{moveC_1} \xrightarrow{\moveC} \pica{1}{moveC_2}
\end{equation}

The move $\moveC$ takes place inside a disk-shaped region $\sigsurf$ with two nodes and three defect lines, labeled by objects $k$, $m$ and $l$ in traced $\acat$-$\bcat$-bimodule categories $\kcat$, $\mcat$ and $\lcat$.
The nodes are labeled by $x \in \kcat\reldel\opcat{\mcat}\reldel$ and $y\in \mcat \reldel \opcat{\lcat}\reldel$.

After the move, $\sigsurf$ is replaced by the similar region $\postmove{\sigsurf}$ which only has one node, and from which the defect line labeled by $m$ has disappeared.
To obtain the node label of $\postmove{\sigsurf}$ from the node labels $x$ and $y$ of $\sigsurf$,
recall that the module Eilenberg-Watts equivalence (<ref>)
allows us to express the ray categories of $\sigsurf$ and $\postmove{\sigsurf}$ as categories of bimodule functors:

\begin{equation}
\kcat\reldel\opcat{\mcat}\reldel \cong \bimodfun{\acat}{\bcat}{\mcat}{\kcat}, \quad \mcat\reldel\opcat{\lcat}\reldel \cong \bimodfun{\acat}{\bcat}{\lcat}{\mcat}, \quad \kcat\reldel\opcat{\lcat}\reldel \cong \bimodfun{\acat}{\bcat}{\lcat}{\kcat}.
\end{equation}

The composition of functors defines a composition operation for these particular ray categories, here denoted by the node label $x \mitosis{\mcat} y \in \kcat\reldel\opcat{\lcat}\reldel$ for $\postmove{\sigsurf}$.
Explicitly, we have

\begin{equation}\label{eq_def_gencomp_U}
U(x \mitosis{\mcat} y) = \homsetin{\mcat}{\sweed{x}{\mcat}}{\sweed{y}{\mcat}} \otimes \; \sweed{x}{\kcat} \deligne \sweed{y}{\opcat{\lcat}} \qquad \in \kcat \deligne \opcat{\lcat}.
\end{equation}

The last datum needed to specify the move is the linear map

\begin{equation}
\movemap : \nodespace = \homset{k \deligne \opcat{m}}{U(x)} \otimes \homset{m \deligne \opcat{l}}{U(y)} \to \homset{k \deligne\opcat{l}}{U(x\mitosis{\mcat} y)} = \postmove{\nodespace}
\end{equation}

between node spaces, appearing in <Ref>.
For this move, $\movemap$ is given by the following composition of morphisms:

\begin{align}\label{eq_gencomp_def}
\begin{aligned}
&\qquad \homset{k \deligne \opcat{m}}{U(x)} \otimes \homset{m \deligne \opcat{l}}{U(y)}\\
&\cong \homset{k}{\sweed{x}{\kcat}} \otimes \homset{\opcat{m}}{\sweed{x}{\opcat{\mcat}}} \otimes \homset{m}{\sweed{y}{\mcat}} \otimes \homset{\opcat{l}}{\sweed{y}{\opcat{\lcat}}} \\
&= \homset{k}{\sweed{x}{\kcat}} \otimes \homset{\sweed{x}{\mcat}}{m} \otimes \homset{m}{\sweed{y}{\mcat}} \otimes \homset{\opcat{l}}{\sweed{y}{\opcat{\lcat}}} \\
&\xrightarrow{\text{compose}} \homset{k}{\sweed{x}{\kcat}} \otimes \homset{\sweed{x}{\mcat}}{\sweed{y}{\mcat}} \otimes \homset{\opcat{l}}{\sweed{y}{\opcat{\lcat}}} \\
&\cong \homset{k\deligne \opcat{l}}{\homsetin{\mcat}{\sweed{x}{\mcat}}{\sweed{y}{\mcat}} \otimes \; \sweed{x}{\kcat} \deligne \sweed{y}{\opcat{\lcat}}}\\
&= \homset{k\deligne \opcat{l}}{U(x \mitosis{\mcat} y)}.
\end{aligned}
\end{align}

There is also a "one-sided" version of the $\moveC$-move, where the $\lcat$-labeled defect line (or alternatively, the $\kcat$-labeled defect line) is not present:

\begin{equation}\label{pic_moveC_oneside}
\pica{1}{moveC_oneside_1} \xrightarrow{\moveC} \pica{1}{moveC_oneside_2}
\end{equation}

The composition operation for node labels is well-defined even in this case.
Note however that a move without $\kcat$ and $\lcat$ does not exist, as this would result in a node without any defect lines, which is not allowed.

The proof that the move $\moveC$ is a move of invariance is the content of <Ref>.

A special case we are interested in is that where $\mcat = \lcat = \kcat$ and $x = y = \zewcoid{\mcat}$.
In this case, the node spaces are

\begin{equation}
\nodespace \cong \homset{k}{m} \otimes \homset{m}{l} \qquad \text{and} \qquad \postmove{\nodespace} \cong \homset{k}{l},
\end{equation}

and the linear map $\psi$ specializes to the composition of morphisms in $\mcat$.

$\moveEF$ – Edge Fusion
The edge fusion move allows us to merge parallel defect lines which share start and end nodes.

\begin{equation}
\label{pic_moveEF}
\pica{1}{moveEF_1} \xrightarrow{\moveEF} \pica{1}{moveEF_2}
\end{equation}

The region $\sigsurf$ is disk-shaped and contains two defect lines, running in parallel between the only two nodes that lie partially in $\sigsurf$.
In contrast, the post-move region $\postmove{\sigsurf}$ features just a single defect line connecting the node segments.
The three domains of $\sigsurf$ are labeled by spherical fusion categories $\acat$, $\bcat$ and $\ccat$, and the defect lines are labeled by objects in bimodule categories $m \in \bimod{\acat}{\mcat}{\bcat}$ and $n \in \bimod{\bcat}{\ncat}{\ccat}$, respectively.
The nodes are appropriately labeled by $x \in \cdots \reldel \opcat{\ncat} \reldel \opcat{\mcat} \reldel \cdots$ and $y \in \cdots \reldel \mcat \reldel \ncat \reldel \cdots$.
The single defect line in the region $\postmove{\sigsurf}$ is labeled by the $\acat$-$\ccat$-bimodule category $\mcat \reldel \ncat$.
The object label $m\lexreldel n$ for the defect line involves the induction functor, see (<ref>).
The ray categories $\raycat$ and $\postmove{\raycat}$ associated with $\sigsurf$ and $\postmove{\sigsurf}$ are canonically equivalent to those of the pre-move surface $\sigsurf$, allowing us to use the same objects $x$ and $y$ for the rays of $\postmove{\sigsurf}$.

However, the node categories $\nodecat$ and $\postmove{\nodecat}$ are different; they are related by the forgetful functor $U_\bcat: \postmove{\nodecat} \to \nodecat$, which forgets the $\bcat$-balancing.
Together with the forgetful functors $\postmove{U}: \postmove{\raycat} \to \postmove{\nodecat}$ and $U : \raycat \to \nodecat$, we have $U = U_\bcat \circ \postmove{U}$.

The linear map $\movemap : \nodespace \to \postmove{\nodespace}$ between node spaces is given by the composition

\begin{align}\label{eq_moveEF_movemap_def}
\begin{aligned}
\tilde{\movemap} \; : \; \nodespace &= \homset{\opcat{n}\deligne\opcat{m}\deligne \cdots}{U(x)} \otimes \homset{m\deligne n\deligne \cdots}{U(y)} \\
& \cong \homset{\opcat{m \lexreldel n} \deligne \cdots}{\postmove{U}(x)} \otimes \homset{m \rexreldel n \deligne \cdots}{\postmove{U}(y)} = \postmove{\nodespace},
\end{aligned}
\end{align}

scaled by the factor $\catdim{\bcat}$, meaning $\psi := \frac{\tilde{\psi}}{\catdim{\bcat}}$.
In (<ref>) the ambidextrous adjunction $U_\bcat \dashv \leftind$ from (<ref>) was used.

The proof that the move $\moveEF$ is a move of invariance is the content of <Ref>.

§.§.§ Composite Moves

In addition to these elementary moves, we will define the following composite moves $\moveDV$, $\moveL$, and $\moveDE$.
The loop move $\moveL$ removes a defect line that starts and ends at the same node, and the other two moves $\moveDV$ and $\moveL$ allow us to insert nodes and defect lines which are, in an appropriate sense, transparent.

${\moveDV}$ – Silent Node
The $\moveDV$-move can be performed on a region $\sigsurf$ containing only a defect line, labeled by some object $m \in \mcat$.
In the post-move region $\postmove{\sigsurf}$, the defect line is split in two by a new node, which is labeled by the silent object $\zewid{\mcat} \in \opcat{\mcat} \reldel \mcat \reldel$.

\begin{equation}\label{pic_moveDV}
\pica{1}{moveDV_1} \xrightarrow{\moveDV} \pica{1}{moveDV_2}
\end{equation}

Since the region has no node, the node space of $\sigsurf$ is $\nodespace = \field$, and the linear map $\psi : \nodespace \to \postmove{\nodespace}$ is specified by the vector
$\psi (1) := \soc (\id_m) \in \homset{\opcat{m} \deligne m}{\ewid{\mcat}} = \postmove{\nodespace}$.

The proof that the move $\moveDV$ is a move of invariance is the content of <Ref>.

${\moveL}$ – Loop Move
A loop attached to a node can be removed. Note that this is only possible when at least one other defect line is attached to the node, as otherwise, the move would result in a node without defect lines, which is not allowed.

\begin{equation}\label{pic_moveL}
\pica{1}{moveL_1} \xrightarrow{\moveL} \pica{1}{moveL_2}
\end{equation}

The node label $x'$ is given by the contraction $x' = \zewid{\mcat} \mitosis{\opcat{\mcat}\reldel \mcat} x$.
Beware that in general, $x$ and $x'$ live in non-equivalent ray categories – the silent object $\zewid{\mcat}$ is a unit with respect to the contraction $\mitosis{\mcat}$, not $\mitosis{\opcat{\mcat}\reldel \mcat}$.
Concretely, if $U(x) = \sweed{x}{\opcat{\mcat} \reldel \mcat} \deligne \sweed{x}{\rest}$, where $\sweed{x}{\rest}$ is a rest term, we have

\begin{align}\label{eq_lmove_postmove_raylabel}
\begin{aligned}
U(x') &=  \homsetin{\opcat{\mcat}\reldel \mcat}{\sweed{x}{\opcat{\mcat}\reldel \mcat}}{\zewcoid{\mcat}}\otimes \sweed{x}{\rest} .
\end{aligned}
\end{align}

Thus, we can specify the linear map $\movemap : \nodespace \to \postmove{\nodespace}$ on $f \in \nodespace$ as  $\movemap(f) : n \to \homsetin{\opcat{\mcat}\reldel \mcat}{\sweed{x}{\opcat{\mcat}\reldel \mcat}}{\zewcoid{\mcat}} \otimes \sweed{x}{\rest}$, where $n$ stands for the labels of the other defect lines attached to the node.
Let us write $f = \sweed{f}{\opcat{\mcat} \deligne \mcat} \otimes \sweed{f}{\rest}$, where

\begin{equation}
\sweed{f}{\opcat{\mcat} \deligne \mcat} : \opcat{m} \deligne m \to \sweed{x}{\opcat{\mcat} \deligne \mcat}, \quad\text{and} \quad \sweed{f}{\rest} : n \to \sweed{x}{\rest}.
\end{equation}

Then $\movemap(f)$ is given as

\begin{equation}\label{eq_lmove_nodelabel}
\movemap(f) := r(\soc(\id_m) \circ \sweed{f}{\opcat{\mcat} \deligne \mcat}) \otimes \sweed{f}{\rest},
\end{equation}

where the isomorphism $\soc : \homset{m}{m} \cong \homset{\opcat{m}\deligne m}{\ewid{\mcat}}$ was defined in (<ref>), and
$r: \homsetin{\opcat{\mcat}\deligne \mcat}{\sweed{x}{\opcat{\mcat}\deligne \mcat}}{\ewid{\mcat}} \to \homsetin{\opcat{\mcat}\reldel \mcat}{\sweed{x}{\opcat{\mcat}\reldel \mcat}}{\zewcoid{\mcat}}$ is the retraction from <Ref>.

The proof that the move $\moveL$ is a move of invariance is the content of <Ref>.

${\moveDE}$ – Transparent Edge
An edge which is transparent – that is, labeled by the regular bimodule category $\acat$, and moreover labeled by the monoidal unit $\one \in \acat$, can be inserted between two nodes as shown in (<ref>).

\begin{equation}\label{pic_moveDE}
\pica{1}{moveDE_1} \xrightarrow{\moveDE}\pica{1}{moveDE_2}
\end{equation}

More precisely, let $x \in \modulecent{\acat}{\mcat}$ and $y \in \modulecent{\acat}{\opcat\ncat}$ be objects in the centers of the $\acat$-$\acat$-bimodule categories $\mcat$ and $\opcat\ncat$.
As such, $x$ and $y$ are appropriate node labels for the nodes of the region $\sigsurf$.
After the move, the rays of the region $\postmove{\sigsurf}$ must be labeled with objects in the ray categories $\opcat{\acat} \reldel \mcat \reldel = \modulecent{\acat}{\opcat{\acat} \reldel \mcat}$ and $\opcat{\ncat} \reldel \acat \reldel = \modulecent{\acat}{\opcat{\ncat}\reldel \acat}$.
We denote the (partial) forgetful functors  $\modulecent{\acat}{\opcat{\acat} \reldel \mcat} \to \opcat{\acat} \reldel \mcat$ and $\modulecent{\acat}{\opcat{\ncat}\reldel \acat} \to \opcat{\ncat}\reldel \acat$ by $\tilde{U}$.
The equivalence of bimodule categories $\opcat{\ncat} \cong \opcat{\ncat} \reldel \acat$ from (<ref>) extends to an equivalence of linear categories $\modulecent{\acat}{\opcat\ncat} \cong \modulecent{\acat}{\opcat{\ncat}\reldel \acat}$ because $\modulecent{\acat}{-}$ is a 2-functor from bimodule categories to linear categories <cit.>.
The node label $y$ in the region $\postmove{\sigsurf}$ is the image of $y$ under this equivalence, and $\postmove{x}$ is defined similarly.
The precise form (<ref>) of the equivalence shows that

\begin{equation}
\tilde{U}(\postmove{x}) = \opcat{\one} \lexreldel U(x) \quad\text{and}\quad \tilde{U}(\postmove{y}) = U(y) \lexreldel \one,
\end{equation}

and applying the complete forgetful functors, we find using (<ref>) that

\begin{equation}\label{eq_demove_premove_rays}
U(\postmove{x}) = \bigoplus_\si{a} \opcat{\si{a}} \deligne \si{a}U(x) \quad\text{and}\quad U(\postmove{y}) = \bigoplus_\si{a} U(y)\si{a}^* \deligne \si{a}.
\end{equation}

Finally, we need to specify a linear map $\movemap : \nodespace \to \postmove{\nodespace}$ between node spaces.
We pic a vector $f \otimes g \in \nodespace$, where $f$ and $g$ are morphisms

\begin{equation}\label{eq_moveDE_postmove_nodes}
f : m \to  U(x) \quad\text{and}\quad g : \opcat{n} \to  U(y).
\end{equation}

The value of $\movemap$ on $f \otimes g$ is $\movemap(f\otimes g) = \postmove{f} \otimes \postmove{g}$, which is a tensor product of morphisms $\postmove{f}: \opcat{\one} \deligne m \to \bigoplus_\si{a} \opcat{\si{a}} \deligne \si{a}U(x)$ and $\postmove{g} : \opcat{n} \deligne\one \to \bigoplus_\si{a} U(y)\si{a}^* \deligne \si{a}$ of $\postmove{\sigsurf}$.
Due to Schur's lemma, all simple components of $\postmove{f}$ and $\postmove{g}$ except for $\tc{\postmove{f}}{\one}$ and $\tc{\postmove{g}}{\one}$ are zero.
Thus setting

\begin{equation}
\tc{\postmove{f}}{\one} := \id_{\opcat{\one}} \deligne f \quad \text{and}\quad \tc{\postmove{g}}{\one} := g \deligne \id_\one
\end{equation}

defines $\postmove{f}$ and $\postmove{g}$, and thus the linear map $\movemap$.

The proof that the move $\moveDE$ is a move of invariance can be found in <cit.>.

With the list of elementary and composite moves complete, we are ready to formulate the next theorem, which states that the moves leave the evaluation invariant.

The moves $\moveOR$, $\moveC$, $\moveEF$, $\moveDV$, and $\moveL$ defined above are moves of invariance in the sense of <Ref> for the evaluation $\evaluate{-}$ of extruded graphs from <Ref>.

Before we get to the proof of <Ref>, which is the content of <Ref>,
we will see that <Ref> implies the uniqueness result discussed in the next section.

§.§ Uniqueness of the Evaluation.

To state the uniqueness property of the evaluation procedure $\evaluate{-}$ defined in <Ref>, we introduce some terminology.

A pre-evaluation $\alteval{-}$ is a choice of linear map $\alteval{\sigsurf} : N \to T$ for every extruded graph $\sigsurf$ on $\mathbb{S}^2$ with node space $N$ and block space $T$.

Consider the set $E$ of triples

\begin{equation}
E := \class{(\sigsurf, f, \varphi)}{\begin{array}{l}
\text{$\sigsurf$ is an extruded graph on $\mathbb{S}^2$ with node space $\nodespace$ } \\
\text{and block space $\blockspace$, $f\in \nodespace$ and $\varphi \in \blockspace^*$.}
\end{array}}.
\end{equation}

A scalar pre-evaluation is a function $s: E \to \field$, which is linear in the variables $f$ and $\varphi$.

There is an obvious bijection between pre-evaluations and scalar pre-evaluations.
For a fixed extruded graph $\sigsurf$, a scalar pre-evaluation is a linear map $s_\sigsurf : \nodespace \otimes \blockspace^* \to \field$.
Likewise, a pre-evaluation is, for each extruded graph $\sigsurf$, a linear map $\alteval{\sigsurf} : \nodespace \to \blockspace$.
Since $T$ is finite-dimensional, these spaces of linear maps coincide.

Clearly, our evaluation from <Ref>, restricted to extruded graphs on the sphere, is an pre-evaluation in this sense.
A pre-evaluation $\alteval{-}$ is said to be compatible with a move $\mathbf{M}$, if the diagram

\begin{equation}\label{diag_preev_compatible}
\begin{tikzcd}
\nodespace \arrow[r, "\alteval{\sigsurf_{T}}"{above}] \arrow[d, "\movemap"{left}] & \blockspace \arrow[d, "\Movemap"] \\
\postmove{\nodespace} \arrow[r, "\alteval{\postmove{\sigsurf_{T}}}"{below}] & \postmove{\blockspace}
\end{tikzcd}
\end{equation}

commutes in a generalization of (<ref>).
We say that a scalar pre-evaluation $s$ is compatible with $\mathbf{M}$ if $s(\sigsurf_T, f, \varphi) = s(\postmove{\sigsurf_T}, \movemap(f), \varphi \circ \Movemap^{-1})$.

The evaluation $\evaluate{-}$ defined in <Ref> is the unique pre-evaluation such that lasso graphs are evaluated as in <Ref>, and which is compatible with the moves $\moveOR$, $\moveC$, and $\moveEF$.

The scalar evaluation defined in (<ref>) is the unique scalar pre-evaluation such that lasso graphs are evaluated as in <Ref>, and which is compatible with the moves $\moveOR$, $\moveC$, and $\moveEF$.

We consider any pre-evaluation $\alteval{-}$ which evaluates lasso graphs as in <Ref> and for which the moves $\moveOR$, $\moveC$, and $\moveEF$ are moves of invariance.
We then show that $\alteval{\sigsurf} = \evaluate{\sigsurf}$  for any extruded graph $\sigsurf$ on $\mathbb{S}^2$.

The network of $n$ defect lines and $k$ nodes in $\sigsurf$ form a connected graph with $n$ edges and $k$ vertices.
Any connected graph has a spanning tree, that is, a sub-graph without cycles which contains all vertices of the original graph.
We pick such a spanning tree, and in it an edge.
Note that this edge is not a loop.
We may thus apply the contraction move $\moveC$ to the corresponding defect line in $\sigsurf$, obtaining an extruded graph $\postmove{\sigsurf}$ with $(n-1)$ defect lines and $(k-1)$ nodes.
Since $\moveC$ is by assumption compatible with the pre-evaluation $\alteval{-}$, we find $\Movemap_\mathbf{C} \circ \alteval{\sigsurf} = \alteval{\postmove{\sigsurf}} \circ \movemap_\mathbf{C}$,
where $\movemap_\mathbf{C} : \nodespace \to \postmove{\nodespace}$ and $\Movemap_\mathbf{C} : \blockspace \to \postmove{\blockspace}$ denote the linear maps defined in <Ref>.
This step is repeated until we arrive at an extruded graph $\sigsurf_1$ with only a single node.
Viewing all individual $\moveC$-moves as one composite move, it is clear that this move is compatible with $\alteval{-}$: $\Movemap_1 \circ \alteval{\sigsurf} = \alteval{\sigsurf_1} \circ \movemap_1$.
As depicted in (<ref>) below, the defect lines of $\sigsurf_1$, all start and end at the unique node.

\begin{equation} \label{pic_bouquet_of_circles}
\sigsurf_1 = \pica{1}{bouquetofcircles}
\end{equation}

The sphere $\sigsurf_1$ with the interior of the node removed is a closed disk with embedded defect lines that start and end on the boundary.
Since the lines do not intersect, there is at least one defect line whose start and end points are neighbors on the boundary of the disk.
To a defect line with this property, the loop move $\moveL$ can be applied.
As we will see below (<Ref>), the $\moveL$-move can be realized as a composition of $\moveC$, $\moveOR$, and $\moveEF$-moves.
Thus, the evaluation $\alteval{-}$ is compatible with $\moveL$.
This step of applying the move $\moveL$ is repeated until only one defect line is left.

The resulting extruded graph is a lasso graph $Q$.
Having constructed a move between $\sigsurf$ and $Q$ compatible with $\alteval{-}$, we denote the associated linear maps between node spaces and block spaces by $\movemap : \nodespace \to \nodespace_Q$ and $\Movemap : \blockspace \to \blockspace_Q$.
By assumption, <Ref> holds for the pre-evaluation $\evaluate{-}$.
As a consequence, $\evaluate{-}$ and $\alteval{-}$ are equal on lasso graphs such as $Q$, and the following diagram commutes:

\begin{equation}
\begin{tikzcd}
\nodespace \arrow[d, "\alteval{\sigsurf}"{left}] \arrow[r, "\movemap"] & \nodespace_Q \arrow[d, "\alteval{Q}"{left}, "\evaluate{Q}"{right}] & \nodespace \arrow[d, "\evaluate{\sigsurf}"] \arrow[l, "\movemap"{above}] \\
\blockspace \arrow[r, "\Movemap"{below}] & \blockspace_Q & \blockspace \arrow[l, "\Movemap"]
\end{tikzcd}
\end{equation}

Since $\Movemap$ is an isomorphism, we can deduce that $\alteval{\sigsurf} = \evaluate{\sigsurf}$.

The statement for the scalar evaluation follows immediately from the first part of the theorem and <Ref>.

We are grateful to Catherine Meusburger and Alexis Virelizier for detailed
discussions and to Alexis Virelizier for insisting that extruded graphs can
be dealt with as two-dimensional objects.

We thank David Jaklitsch for helpful discussions and Katrin Suchowski for creating the pictures.
The authors are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under SCHW1162/6-1; CS is partially supported under Germany's Excellence Strategy - EXC 2121
“Quantum Universe” - 390833306.

§ DETAILS ON MOVES