Anna-Katharina<EMAIL_ADDRESS>
Catherine<EMAIL_ADDRESS>
Department Mathematik

Friedrich-Alexander-Universität Erlangen-Nürnberg

Cauerstraße 11, 91058 Erlangen, Germany

June 9, 2023

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to  invariants of oriented surfaces
defined in terms of ribbon graphs. For every ribbon graph this yields an object in the category, defined up to isomorphism, that depends only on the homeomorphism class of the associated surface. This object is
constructed via (co)equalisers and images and equipped with a mapping class group action. It  can be viewed as a categorical generalisation of the ground state of  Kitaev's quantum double model or of a representation variety for a surface. We apply the construction to group objects in cartesian monoidal categories, in particular to simplicial groups as group objects in $\SSet$ and to crossed modules as group objects in $\Cat$. The former yields a simplicial set consisting of representation varieties, the latter a groupoid whose sets of objects and morphisms are obtained from representation varieties.

§ INTRODUCTION

Constructions that assign algebraic or geometric objects  with mapping class group actions to oriented surfaces  are of interest in many contexts.
They arise in 3d topological quantum field theories (TQFTs) of  Turaev-Viro-Barrett-Westbury   or Reshetikhin-Turaev type [43, 5, 40]  and are encoded in the weaker notion of a modular functor, see  <cit.>. For a recent construction of modular functors from finite tensor categories, see Fuchs, Schweigert and Schaumann <cit.>, for a classification  via  factorisation homology,  Brochier and Woike [8].

Modular functors also arise  in Hamiltonian quantisation formalisms for  representation varieties. The associated mapping class group actions were first discovered in the combinatorial quantisation formalism by Alekseev, Grosse, Schomerus [1, 2, 3] and Buffenoir and Roche <cit.>, subsequently related to factorisation homology by Ben-Zvi, Brochier and Jordan <cit.> and studied by Faitg [17, 18].

Objects with mapping class group actions also arise from  correlators in conformal field theories, see  the work by Fuchs, Schweigert and Stigner  [20, 21]. They are also present
in models from condensed matter physics and topological quantum computing such as Levin-Wen models <cit.> and Kitaev's quantum double model [25].
Due to the work of Lyubashenko [29, 30, 31]  it is well-understood how to construct projective mapping class group actions  from Hopf algebras in abelian ribbon categories.

Many of these constructions are based on assignments of algebraic data to certain  graphs on surfaces, and they require linear categories   with duals,  often  also  abelian,  finite or semisimple. These restrictions are of course well-motivated from the context of TQFTs, in the quantisation of gauge theories or  in condensed matter physics. However, it  is also desirable to go beyond them.

In this article we show that any involutive Hopf monoid $H$ in  a complete and finitely cocomplete symmetric monoidal category $\mac$ yields an invariant of  oriented surfaces. We compute this invariant for  examples, such as simplicial groups as Hopf monoids in $\SSet$ and crossed modules as Hopf monoids in $\Cat$.

The construction is based on the choice of a ribbon graph. It  determines an  object in $\mac$, defined up to isomorphisms, that depends only  on the homeomorphism class of the    surface obtained by attaching discs to the faces of the graph. As Hopf monoids are categorical generalisations of Hopf algebras, it can be viewed as a categorical generalisation of Kitaev's quantum double model or  of representation varieties or moduli spaces of flat bundles on surfaces.

More precisely, we consider for  a ribbon graph the $\vert E \vert$-fold tensor product $H^{\oo E}$, where $E$ is the edge set of the graph. We  use the structure morphisms of the Hopf monoid to associate  $H$-module structures to its vertices and  $H$-comodule structures to its faces. This requires a choice of a marking for each vertex or face, and each  marking  defines a Yetter-Drinfeld module structure over $H$.
The  object assigned to the graph is  obtained  by equalising the $H$-comodule structures, by coequalising the $H$-module structures and by combining them  via a categorical image. It generalises  the protected space or ground state of Kitaev's quantum double model, and we therefore call it the protected object. We then show

(Theorem <ref>) The isomorphism class of the protected object for  a ribbon graph depends only on the homeomorphism class of the associated surface.

Essentially the same construction was  used by Meusburger and Voß  in  [35] to construct mapping class group actions from pivotal Hopf monoids in symmetric monoidal categories. These mapping class group actions are obtained from graphs with a single vertex and face  and act on the associated protected object. However, it was not established in [35] that this object is independent of the graph. By combining our results with the ones from [35] we obtain

(Theorem <ref>)  The protected object for a Hopf monoid $H$ and a surface $\Sigma$ of genus $g\geq 1$ is equipped with an action of the mapping class group $\mathrm{Map}(\Sigma)$ by automorphisms.

The remainder of this article is dedicated to the study of examples. For a finite-dimensional semisimple Hopf algebra $H$ as a Hopf monoid in  $\mathrm{Vect}_\C$ the protected object assigned to a surface coincides with the protected space of the associated quantum double model, as defined by Kitaev and by Buerschaper et a. in [25, 11]. However, our model is also defined in the non-semisimple case. For instance, for a group algebra $k[G]$  over a commutative ring $k$ as a Hopf monoid in $k$-Mod and a surface $\Sigma$ of genus $g\geq 1$, the protected object is the free $k$-module generated by the representation variety $\Hom(\pi_1(\Sigma), G)/G$.

A large class of examples of involutive  Hopf monoids  are group objects in cartesian monoidal categories, where the tensor product is a categorical product. In all our examples of this type, the protected object is given in terms of representation varieties. For a group $H$ as a Hopf monoid in $\Set$, the object assigned to a connected surface $\Sigma$ is the representation variety $\Hom(\pi_1(\Sigma), H)/H$.  For a simplicial group $H=(H_n)_{n\in\Z}$ as a Hopf monoid in $\SSet$, it is a simplicial set given by the representation varieties for the  groups $H_n$ and post-composition with the face maps and degeneracies. In this sense the construction can be viewed as a  generalisation of representation varieties from groups to group objects. The module structures at vertices correspond to the  group action and the comodule structures at faces of the graph to moment maps.

Our main example  is the case where the underlying symmetric monoidal category is the category $\Cat$ of small categories and functors between them. Group objects in $\Cat$ are precisely crossed modules. They are given by a group homomorphism $\partial: A\to B$ and an action $\brhd: B\times A \to A$ by group automorphisms, subject to some consistency conditions. As Cat is complete and cocomplete, the associated protected objects exist, but are difficult to determine concretely.
We relate them to simplicial groups  via the nerve functor and its left adjoint, which yields an explicit description of the protected objects and their mapping class group actions.

(Theorem <ref>, Corollary <ref>) The protected object for a crossed module $(B, A,\brhd,\partial)$  as a group object in $\Cat$ and a  surface $\Sigma$ of genus $g\geq 1$ is a groupoid $\mathcal G$  with $\mathrm{Ob}\mathcal G=\Hom(\pi_1(\Sigma), B)/B$  and  with equivalence classes of group homomorphisms $\tau: \pi_1(\Sigma)\to A\rtimes B$ as morphisms. The action of the mapping class group $\mathrm{Map}(\Sigma)$ is induced by its action on $\Hom(\pi_1(\Sigma), A\rtimes B)/A\rtimes B$.

The equivalence classes of morphisms are given by the equivalence relation $\tau_1\circ \tau_2\sim \tau'_1\circ \tau'_2$ on the set of group homomorphisms $\tau: F_{2g}\to A\rtimes B$, whenever the composites exist and $\tau_1$, $\tau'_1$ and $\tau'_2$,$\tau_2$ are conjugate. We compute the protected object and the associated mapping class group action explicitly for some simple examples of crossed modules.

To our knowledge this construction is new and differs substantially from the  constructions with crossed modules in  higher gauge theory settings such as the work of Martins and Picken [33, 34] on higher holonomies, the work [13, 14] by Bullivant et. al. on higher lattices and topological phases, the work  [26] by Koppen, Martins and Martin on topological phases from crossed modules of Hopf algebras and  the recent work [42] by Sozer and Virelizier on 3d homotopy quantum field theory.
In those settings the structure maps of the crossed module often encode higher categorical structures in the topological data, such as homotopies between paths, or relate data  in triangulations or cell decompositions.
In our approach they enter the formalism as data - a specific example of a group object - but the crossed module structure is not required
to encode the topology or geometry.

The article is structured as follows. Section <ref> introduces the algebraic background for the article. In Section <ref> we summarise the background on  Hopf monoids in symmetric monoidal categories. In Section <ref> we discuss their (co)modules and the construction of their (co)invariants via (co)equalisers and images in complete and finitely cocomplete symmetric monoidal categories. Section <ref> contains the required background on ribbon graphs and surfaces.

In Section <ref> we formulate the categorical counterpart of Kitaev's quantum double model for an involutive Hopf monoid $H$ in a complete and finitely cocomplete symmetric monoidal category. This is a simple generalisation of the formulation in [11] for  finite-dimensional semisimple complex Hopf algebras, and an almost identical construction was used in [35].

For each ribbon graph we consider the tensor product $H^{\oo E}$, where $E$ is the edge set of the graph. We assign to each marked vertex a $H$-module structure and to each marked face an $H$-comodule structure on $H^{\oo E}$.  The protected object  is  constructed by (co)equalising these (co)module structures and taking an image.

In Section <ref> we show that the protected object defined by a ribbon graph depends only on the homeomorphism class of the associated oriented surface $\Sigma$.
We first demonstrate that moving the markings for the (co)module structures and edge reversals yield isomorphic protected objects. We then consider a number of graph transformations that are sufficient to reduce every connected ribbon graph to a standard graph and prove that these induce isomorphisms of the protected object.

These sections are necessarily rather technical.
The reader primarily interested in the results may skip to the main theorem in Section <ref>, where we also treat some examples. In particular, we show that the protected object for a group $H$ as a Hopf monoid in $\Set$ and a connected surface $\Sigma$ is the representation variety $\Hom(\pi_1(\Sigma), H)/H$. We then consider  group algebras $H=k[G]$ and their duals $k[G]^*$ for a commutative ring $k$ as Hopf monoids in $k\text{-Mod}$. The associated protected objects are  the free $k$-module $\langle \Hom(\pi_1(\Sigma), G)/G\rangle_k$ and the set of maps $\Hom(\pi_1(\Sigma), G)/G\to k$.

Section <ref> treats the example of a simplicial group $H=(H_n)_{n\in\N_0}$ as a Hopf monoid in $\SSet$. In this case, the protected object is a simplicial set given by the representation varieties $\Hom(\pi_1(\Sigma), H_n)/H_n$ and by post-composition with the face maps and degeneracies of $H$. This result is required for the construction of the protected object of a crossed module as a group object in $\Cat$.

The construction of the protected object for group objects in $\Cat$ is more involved and treated in Section <ref>.  We start by summarising the required background on crossed modules in Section <ref>. In Section <ref> we discuss equalisers and coequalisers in $\Cat$ and summarise how the latter can be constructed via the nerve functor $N:\Cat\to \SSet$ and its left adjoint. In Section <ref> we apply these results to (co)equalise the (co)module structures over Hopf monoids in $\Cat$.
In Section <ref> we apply this to the (co)module structures associated with a ribbon graph and determine the protected object for the associated surface. We describe it explicitly and treat a simple example.

In Section <ref> we describe the mapping class group action on the protected object. By combining our results with the ones from [35],
we obtain that the mapping class group of an oriented surface $\Sigma$ acts on the associated protected objects. We show that  in the case of a simplicial group $H=(H_n)_{n\in\N_0}$ this action is the one induced by its  action on the representation varieties $\Hom(\pi_1(\Sigma), H_n)/H_n$.
For the case of a crossed module  $(B, A,\brhd,\partial)$ as a group object in $\Cat$  we obtain a mapping class group action by invertible endofunctors on the associated groupoid, which is
induced by the action on the representation variety $\Hom(\pi_1(\Sigma), A\rtimes B)/A\rtimes B$.

§ ALGEBRAIC BACKGROUND

§.§ Involutive Hopf monoids

Throughout the article $\mac$ is a symmetric monoidal category with unit object $e$ and  braidings $\tau_{X,Y}: X\oo Y\to Y\oo X$.
We also suppose that $\mac$ is complete and finitely cocomplete.
In  formulas, we suppress associators and unit constraints and  coherence data of monoidal functors.

* A Hopf monoid in $\mac$ is
an object $H$ in $\mac$ together with morphisms $m: H\oo H\to H$, $\eta:e\to H$, $\Delta: H\to H\oo H$, $\epsilon: H\to e$ and $S: H\to H$, the multiplication, unit, comultiplication, counit and antipode,  such that

* the (co)multiplication satisfies the (co)associativity and (co)unitality conditions

\begin{align}\label{eq:counit}
&m\circ (m\oo 1_H)=m\circ (1_H\oo m), & &m\circ (\eta\oo 1_H)=m\circ (1_H\oo \eta)=1_H,\\
&(\Delta\oo 1_H)\circ\Delta=(1_H\oo \Delta)\circ \Delta, & &(\epsilon\oo 1_H)\circ\Delta
=(1_H\oo \epsilon)\circ \Delta=1_H,\nonumber
\end{align}

* comultiplication and counit are monoid morphisms

\begin{align}
&\Delta\circ \eta=\eta\oo \eta, & &\Delta\circ m=(m\oo m)\circ (1_H\oo \tau_{H,H}\oo 1_H)\circ (\Delta\oo \Delta)\\
&\epsilon\circ\eta=1_e, & & \epsilon\circ m=\epsilon\oo \epsilon,\nonumber
\end{align}

* $S$ satisfies the antipode condition

\begin{align}
m\circ (S\oo 1_H)\circ\Delta=m\circ (1_H\oo S)\circ \Delta=\eta\circ\epsilon.
\end{align}

It is called involutive if $S\circ S=1_H$.

* A morphism of Hopf monoids in $\mac$ is a morphism $f: H\to H'$ in $\mac$ with

\begin{align}
&f\circ m=m'\circ (f\oo f), & &f\circ \eta=\eta', & &(f\oo f)\circ\Delta=\Delta'\circ f, & &\epsilon'\circ f=\epsilon.
\end{align}

We denote by $\mathrm{Hopf}(\mac)$ the category of Hopf monoids and morphisms of Hopf monoids in $\mac$.

The antipode of a Hopf monoid is unique, and it is an anti-monoid and anti-comonoid morphism

\begin{align}\label{eq:antialg}
&S\circ m=m^{op}\circ (S\oo S), & S\circ \eta=\eta, &  & (S\oo S)\circ \Delta=\Delta^{op}\circ S, & &\epsilon\circ S=\epsilon,
\end{align}

see for instance Porst <cit.>. If $H$ is involutive, the antipode satisfies the additional identities

\begin{align}\label{eq:EightWithAntipode}
m^{op}\circ (S\oo 1_H)\circ\Delta=m^{op}\circ (1_H\oo S)\circ\Delta=\eta\circ\epsilon.
\end{align}

Every morphism of Hopf monoids $f: H\to H'$ satisfies $f\circ S=S'\circ f$. This follows  as for Hopf algebras by considering the convolution monoid $\Hom_\mac(H,H)$ with the  product $f\star g=m\circ (f\oo g)\circ \Delta$.

In the following, we use generalised Sweedler notation for the coproduct in a Hopf monoid and write $\Delta(h)=\low h 1\oo \low h 2$, $(\Delta\oo 1_H)\circ\Delta(h)=(1_H\oo\Delta)\circ\Delta(h)=\low h 1\oo \low h 2\oo \low h 3$ etc. This is analogous to Sweedler notation for a Hopf algebra. It can  be viewed as a shorthand notation for a diagram that describes a morphism in a symmetric monoidal category, see [35] for examples. We also write $m^{(n)}: H^{\oo (n+1)}\to H$ and $\Delta^{(n)}: H\to H^{\oo (n+1)} $ for $n$-fold products and coproducts.

* For any commutative ring $k$ a  Hopf monoid in   $k$-Mod  is a Hopf algebra over $k$. In particular, for any field $\F$  a Hopf monoid in $\mathrm{Vect}_\F$ is a Hopf algebra over $\F$.

* For any finite group $G$ and commutative ring $k$, the group algebra $k[G]$ and its dual $k[G]^*$  are Hopf monoids in $k\mathrm{-Mod}$.

* The tensor product  of two Hopf monoids  in $\mathcal C$ has a Hopf monoid structure  given by  the tensor product of (co)units, (co)multiplications  and antipodes and the braiding morphisms. Any tensor product of Hopf monoid morphisms is a morphism of Hopf monoids.

* Every Hopf monoid $H=(H,m,\eta,\Delta,\epsilon, S)$ in a symmetric monoidal category $\mac$  defines a Hopf monoid $ H^*=(H,\Delta,\epsilon, m,\eta, S)$ in the symmetric monoidal category $\mac^{op}$. This generalises the dual Hopf algebra  in $\mathrm{Vect}_\F$.

The following example yields many  subexamples, which are a focus in this article.

Let $(\mac, \times)$ be a cartesian monoidal category with terminal object $\bullet$. Let  $\epsilon_X: X\to \bullet$ be the terminal morphism  and    $\Delta_X: X\to X\times X$ the diagonal morphism for an object $X$.

A Hopf monoid in $\mac$ is a group object in $\mac$:   an object $H$ together with morphisms $m: H\times H\to H$, $\eta :\bullet\to H$ and $I:H\to H$
such that the following diagrams commute

\begin{align}\label{eq:groupobjdiags}
H\times H\times H \ar[r]^{\quad 1_H\times m} \ar[d]_{m\times 1_H}& H\times H\ar[d]^m\\
H\times H\ar[r]_m & H
} &
&\xymatrix{H\cong \bullet\times H \ar[r]^{\;\eta\times 1_H}\ar[rd]_{1_H} & H\times H\ar[d]^m & \ar[l]_{1_H\times \eta} H\times \bullet\cong H \ar[ld]^{1_H}\\
& H
&\xymatrix{ H\times H \ar[rr]^{I\times 1_H} & & H\times H \ar[d]^m\\
H \ar[u]^{\Delta_H} \ar[r]^{\epsilon_H} & \bullet \ar[r]^{\eta}  & H
H\ar[d]_{\Delta_H} \ar[r]^{\epsilon_H} & \bullet \ar[r]^{\eta}  & H\\
H\times H \ar[rr]^{1_H\times I} & & H\times H. \ar[u]_m
\end{align}

A morphism of Hopf monoids  is a morphism of group objects: a morphism $F: H\to H'$ with

\begin{align}\label{eq:groupmorphism}
F\circ m=m'\circ (F\times F).
\end{align}

Note that this implies  $F\circ \eta=\eta'$ and $I'\circ F=F\circ I$.

* A group object in the cartesian monoidal category
$(\Set,\times)$ is a group.

* A group object in the cartesian monoidal category
$(\Top,\times)$ is a topological group.

* A group object in the cartesian monoidal category $(\Cat,\times)$ of small categories and functors between them  is a crossed module (cf. Definition <ref>).

* Let $G$ be a group  and $G\mathrm{-Set}=\Set^{\mathrm{B} G}$ the cartesian monoidal category of $G$-sets and $G$-equivariant maps. A
group object in  $G\mathrm{-Set}$   is a group   with a $G$-action by  automorphisms.

* A group object in the cartesian monoidal category $\SSet=\Set^{\Delta^{op}}$ of simplicial sets and simplicial maps  is a simplicial group (cf. Definition <ref>).

The last two examples in Example <ref> have counterparts for any  functor category $\mac^\mad$, where $\mad$ is small and $\mac$ symmetric monoidal. In this case the functor category $\mac^\mad$ inherits a symmetric monoidal structure from $\mac$, and we have

For any symmetric monoidal category $\mathcal C$ and a small category $\mathcal D$ the monoidal categories $\mathrm{Hopf}(\mac^\mad)$ and $\mathrm{Hopf}(\mac)^\mad$
are  symmetric monoidally equivalent.

The equivalence is given by
the functor $R: \mathrm{Hopf}(\mac^\mad)\to \mathrm{Hopf}(\mathcal C)^\mad$ that sends a Hopf monoid $(H, m,  \eta, \Delta, \epsilon, S)$ to the functor $K: \mathcal D\to \mathrm{Hopf}(\mathcal C)$ with $K(D)=H(D)$ and the component morphisms $m_D$, $\eta_D$, $\Delta_D$, $\epsilon_D$, $S_D$ for $D\in \text{Ob}(\mathcal D)$ and with $K(f)=H(f)$ for a morphism $f$ in $\mathcal D$. Hopf monoid morphisms  in $\mac^\mad$ are sent to themselves. The functor $R$ has an obvious inverse, and both functors  are symmetric monoidal.

Further examples are obtained by taking the images of Hopf monoids under symmetric monoidal functors. If both of the categories  are cartesian monoidal,  it is sufficient that
the functor preserves finite products, which holds in particular for any right adjoint functor.

* Let $F: \mac\to\mac'$ be a symmetric monoidal functor. Then for every Hopf monoid $H$ in $\mac$ the image  $F(H)$ has a canonical Hopf monoid structure.

* If $\mac,\mac'$ are  cartesian monoidal categories and $F:\mac\to\mac'$ a functor that preserves finite products, then  $F$ is symmetric monoidal, and for every group object $H$ in $\mac$ the image $F(H)$ is a group object in $\mac'$.

§.§ (Co)modules and their (co)invariants

As their definitions involve only structure maps, (co)modules  over Hopf monoids in symmetric monoidal categories are defined analogously to (co)modules  over Hopf algebras. The only difference is that linear maps are replaced by morphisms.

Let $H$ be a Hopf monoid in  $\mac$.

* An $H$-module in $\mac$ is an object $M$ in $\mac$ with a morphism $\rhd: H\oo M\to M$ satisfying

\begin{align}
\rhd\circ (m\oo 1_M)=\rhd\circ (1_H\oo\rhd), \qquad \rhd \circ (\eta\oo 1_M)=1_M.
\end{align}

A morphism of $H$-modules is a morphism $f: M\to M'$ in $\mac$ with
$\rhd'\circ (1_H \oo f)=f\circ \rhd$.

* An $H$-comodule in $\mac$ is an object $M$ in $\mac$ with a morphism $\delta: M\to  H\oo M$ satisfying

\begin{align}\label{eq:comod}
(\Delta\oo 1_M)\circ \delta=(1_H\oo \delta)\circ \delta, \qquad (\epsilon\oo 1_M)\circ\delta=1_M.
\end{align}

A morphism of $H$-comodules is a morphism $f: M\to M'$ in $\mac$ with
$(1_H \oo f)\circ \delta=\delta'\circ f$.

There are analogous notions of right (co)modules and  bi(co)modules  and morphisms between them.
Just as in the case of a Hopf algebra, there are also various compatibility conditions that can be imposed between modules and comodule structures.
The most important one in the following is the one for Yetter-Drinfeld modules.

Let $H$ be a Hopf monoid in  $\mac$.

* A
Yetter-Drinfeld module over $H$ is a triple $(M,\rhd,\delta)$ such that $(M,\rhd)$ is an $H$-module, $(M,\delta)$ is an $H$-comodule and

\begin{align*}
\delta\circ \rhd=(m^{(2)}\oo \rhd)\circ (1_{H^{\oo 2}}\circ \tau_{H,H}\oo 1_M)\circ (1_{H^{\oo 3}}\oo S\oo 1_M)\circ (1_H\oo \tau_{H^{\oo 2},H}\oo 1_M)\circ (\Delta^{(2)}\oo \delta).
\end{align*}

* A morphism of Yetter-Drinfeld modules is a morphism $f: M\to M'$ that is a module and a comodule morphism.

In Sweedler notation with  the conventions $\delta(m)=\low m 0\oo \low m 1$ and $\Delta(h)=\low h 1\oo \low h 2$ the Yetter-Drinfeld module condition in Definition <ref>  reads

\begin{align}
(h\rhd m)_{(0)}\oo (h\rhd m)_{(1)}=\low h 1 \low m 0 S(\low h 3)\oo (\low h 2\rhd \low m 1).
\end{align}

Yetter-Drinfeld modules over group objects in cartesian monoidal categories are especially simple to describe. In this case, composing the coaction morphism $\delta: M\to H\times  M$ with the projection morphism $\pi_1: H\times M\to H$ yields a morphism $F=\pi_1\circ \delta: M\to H$ reminiscent of a moment map. The Yetter-Drinfeld module condition  states that this morphism intertwines the $H$-module structure on $M$ and the conjugation action of $H$ on itself.

Let $H$ be a group object in a cartesian monoidal category, $(M, \rhd)$  a module and  $(M, \delta)$  a comodule over $H$.
Then $(M, \rhd, \delta)$ is a Yetter-Drinfeld module  over $H$
iff the morphism
$F:=\pi_1\circ \delta: M \to H$ satisfies

\begin{align}\label{eq:yddefgrp}
F\circ \rhd=m^{(2)}\circ (1_H\times\tau_{H,F(M)})\circ (1_H\times I\times 1_{F(M)})\circ(\Delta_H\times F).
\end{align}

If the objects of $\mac$ are sets,  condition (<ref>) reads $F(h\rhd m)=hF(m) h^\inv$ for all $h\in H$, $m\in M$. By an abuse of notation, we  sometimes write such formulas  for the general case to keep notation simple.

By Example <ref> the images of Hopf monoids  under symmetric monoidal functors are Hopf monoids. Analogous statements hold for their (co)modules.

* If $F: \mac\to\mac'$  is a symmetric monoidal functor and $M$ a (co)module over a Hopf monoid $H$ in $\mac$, then $F(M)$ is a (co)module over the Hopf monoid $F(H)$.

* Let $\mac,\mac'$ be cartesian monoidal categories and $F: \mac\to\mac'$ a functor that preserves finite products. Then for every (co)module $M$ over a group object $H$ in $\mac$ the image $F(M)$ is a (co)module over the group object $F(H)$.

(Co)invariants of (co)modules  cannot be generalised directly from Hopf algebras over fields to Hopf monoids in symmetric monoidal categories. To obtain  generalised notions of (co)invariants, we require that the symmetric monoidal category $\mac$ has all equalisers and coequalisers.

<cit.> Let $\mac$ be a symmetric monoidal category that has all equalisers and coequalisers, $H$ a Hopf monoid in $\mac$.

* The invariants of an $H$-module $(M,\rhd)$ are the coequaliser $(M^{H},\pi)$ of $\rhd$ and $\epsilon\oo 1_{M}$:

\begin{align}
\nonumber
\xymatrix{  H\oo M<EMAIL_ADDRESS>1_M}<EMAIL_ADDRESS>& M \ar[r]^{\pi} & M^H.
\end{align}

* The coinvariants of an $H$-comodule $(M,\delta)$ are the equaliser $(M^{coH},\iota)$ of $\delta$ and $\eta \oo 1_{M}$:

\begin{align}
\nonumber
\xymatrix{ M^{coH} \ar[r]^{\iota} & M<EMAIL_ADDRESS>1_M\quad}<EMAIL_ADDRESS>& H\oo M.
\end{align}

As expected, $H$-(co)module  morphisms induce morphisms between the (co)invariants. This follows directly from the universal properties of the (co)equalisers.

Suppose that $\mathcal C$ has all equalisers and coequalisers and $H$ is a Hopf monoid in $\mac$. Then for
every $H$-module morphism $f: (M, \triangleright) \to (M', \triangleright')$ there is a unique morphism $f^H: M^H \to M'^H $ with $f^H \circ \pi = \pi' \circ f$. Likewise, for every $H$-comodule morphism $f: (M, \delta) \to (M', \delta')$ there is a unique morphism $f^{coH}: M^{coH} \to M'^{coH}$ with $\iota' \circ f^{coH}= f\circ \iota$.

Note that all definitions in this section are symmetric with respect to a Hopf monoid $H$ in $\mac$ and the dual Hopf monoid $ H^*$ in $\mac^{op}$ from Example <ref>. Modules and comodules over $H$ in $\mac$ correspond to comodules and modules over $ H^*$ in $\mac^{op}$, respectively, and the same holds for their (co)invariants.
It is also directly apparent from the formula in Definition <ref> that Yetter-Drinfeld modules over $H$ correspond to Yetter-Drinfeld modules over $ H^*$.

For objects in a symmetric monoidal category $\mac$ that are both, modules and comodules over certain Hopf monoids in $\mac$, we combine the notion of invariants and coinvariants and impose both conditions. This requires that the category $\mac$ is equipped with images.  We work with a   general non-abelian notion of image, see Mitchell <cit.> and Pareigis <cit.>. There is an analogous notion of a coimage, which is  the image of the corresponding morphism   in  $\mac^{op}$, see <cit.>.

An  image of a morphism $f: C\to C'$ in $\mac$ is an object $\mathrm{im}(f)$ together with a pair $(P,I)$ of a monomorphism $I: \mathrm{im}(f)\to C'$ and a morphism $P: C\to \mathrm{im}(f)$ with $I\circ P=f$ and the following universal property:   for any pair $(Q,J)$ of a monomorphism $J: X\to C'$ and a morphism $Q: C\to X$ with $J\circ Q=f$ there is a unique morphism $v:\mathrm{im}(f)\to X$ with $I=J\circ v$.

Images are unique up to unique isomorphism. If $\mathcal C$ has all equalisers, then $P: C\to \mathrm{im}(f)$ is an epimorphism <cit.>. In an abelian category $\mathcal C$ this notion of  image coincides with the usual definition of an image as the kernel of the cokernel <cit.>. If $\mathcal C$ is complete, then all images exist, as any complete category has intersections <cit.>. This implies the existence of all images <cit.>.

<cit.>[Def. 2.8 in [35]  considers only the case $H=K$, as that is the only one required there.] Let $\mac$ be a complete and finitely cocomplete symmetric monoidal category and $H, K$  Hopf monoids in $\mac$.
The biinvariants of an   $H$-module and  $K$-comodule $M$ are the image  of the morphism $\pi\circ\iota: M^{coK}\to M^H$

\begin{align}\label{eq:imagedef}
\xymatrix{ M^{coK} \ar[rd]_P\ar[r]^\iota & M \ar[r]^\pi & M^H\\
& M_{inv}:=\mathrm{im}(\pi\circ \iota). \ar[ru]_{I}
\end{align}

Requiring $\mathcal C$ to be complete and finitely cocomplete ensures the existence of invariants, coinvariants and biinvariants. Examples of such categories are $\mathrm{Set}$, $\mathrm{Top}$, $\mathrm{Grp}$, $\mathrm{Vect}_{\F}$, $\mathrm{Cat}$, $k\mathrm{-Mod}$ and the category $\mathrm{Ch}_{k\mathrm{-Mod}}$ of chain complexes of $k$-modules.
For a small category $\mad$  and a complete and finitely cocomplete category $\mathcal C$ the category $\mac^\mad$ is also complete and finitely cocomplete, see for instance Pareigis <cit.>. Hence,  $G\mathrm{-Set}$ and $\mathrm{SSet}$ also satisfy the requirement.

As discussed in <cit.> one could also consider the coimage of the morphism $\pi\circ \iota$ instead of its image.
This amounts to passing from  modules and comodules over the Hopf monoids $H,K$ in $\mac$ to comodules and modules over the Hopf monoids $H^*$, $K^*$  in $\mac^{op}$  from Example <ref>.

We illustrate (co)invariants and biinvariants with a few simple examples. (Co)invariants of (co)modules over Hopf monoids in $\SSet$ and $\Cat$ and the associated biinvariants for Yetter-Drinfeld modules are treated in Sections <ref> and <ref>, respectively.

* A Hopf monoid $H$ in $\mac=\Set$ (in $\mac=\Top$) is a (topological) group $H$ and

* an $H$-module is a (continuous) $H$-Set $\rhd: H\times M\to M$,

* $M^H=\{H\rhd m\mid m\in M\}$  with  $\pi: M\to M^H$, $m\mapsto H\rhd m$ (and the  quotient topology),

* an $H$-comodule is given by a (continuous) map $F:M\to H$,

* $M^{coH}=F^\inv(1)$  with the inclusion $\iota: F^\inv(1)\to M$ (and  the subspace topology),

* $
M_{inv}=\pi(F^\inv(1))=\{H\rhd m\mid F(m)=1\}
(with the final topology induced by $\pi$).

An $H$-module and $H$-comodule $(M,\rhd, F)$ is a  Yetter-Drinfeld module iff $F(h\rhd m)=hF(m)h^\inv$ for all $m\in M$, $h\in H$.

* Let $G$ be a group and $H$ a group with a $G$-action by automorphisms, viewed  as a Hopf monoid in $G-\Set= \mathrm{Set}^{\mathrm{B}G}$. Then $H$-modules are $H\rtimes G$-sets, $H$-comodules are $G$-sets $M$ with $G$-equivariant maps $F: M\to H$ and

* $M^H=\{H\rhd m\mid m\in M\}$ is the orbit space for $H$ with the induced $G$-action and  $G$-equivariant canonical surjection $\pi: M\to M^H$,

* $M^{coH}=F^\inv(1)$ with the induced $G$-action and  $G$-equivariant inclusion $\iota: F^\inv(1)\to M$,

* $M_{inv}=\pi(F^\inv(1))$ with the induced $G$-action.

* For a Hopf algebra $H$ over a commutative ring $k$ as a Hopf monoid in $k$-Mod, $H$-(co)modules and Yetter-Drinfeld modules are  (co)modules and Yetter-Drinfeld modules over $H$ in the usual sense. Their (co)invariants and biinvariants are

* $M^H=M/\langle \{h\rhd m-\epsilon(h) m\mid h\in H, m\in M\}\rangle$,

* $M^{coH}=\{m\in M\mid \delta(m)=1\oo m\}$,

* $M_{inv}=\pi(M^{coH})$.

While the coinvariants in Example <ref>, 3. coincide with the usual  coinvariants for comodules over a Hopf algebra,
the invariants form a quotient rather than a subset.
This distinction is irrelevant in the case of semisimple Hopf algebras, but not in general. As our definition is symmetric with respect to Hopf monoids in a symmetric monoidal category $\mac$ and the dual Hopf monoids in $\mac^{op}$, it is more natural in our setting.
The following example illustrates this.

For a finite group $G$ and a commutative ring $k$  the group algebra $k[G]$ and  its dual $k[G]^*$ are Hopf monoids in $k$-Mod.

For the group algebra $H=k[G]$

* the invariants of a $H$-module $(M,\rhd)$ are $M^{H}=M/\langle \{g\rhd m-m\mid m\in M, g\in G\}\rangle$,

* comodules are $G$-graded $k$-modules $M=\oplus_{g\in G} M_g$ with $\delta(m)=g\oo m$ for all $m\in M_g$,

* their coinvariants are $M^{coH}=M_1$.

A $k[G]$-module and comodule  $(M,\rhd, \delta)$ is a Yetter-Drinfeld module iff
$g\rhd M_h= M_{ghg^\inv}$ for all $g,h\in G$, and in this case
$M_{inv}\cong H_0(G, M_1)$.

For the dual Hopf monoid $H=k[G]^*$

* modules are $G$-graded $k$-modules $M=\oplus_{g\in G} M_g$ with $\delta_g\rhd m=\delta_g(h) m$ for $m\in M_h$,

* their invariants are $M^H=M/(\oplus_{g\in G,g\neq 1} M_g)\cong M_1$,

* comodules are $k[G]$-right modules $(M,\lhd)$ with $\delta(m)=\sum_{g\in G} \delta_g\oo (m\lhd g)$,

* their coinvariants are $M^{coH}=\{m\in M\mid m\lhd g=m\,\forall g\in G\}$.

A $k[G]^*$-module and comodule $(M,\rhd,\delta)$ is a Yetter-Drinfeld module iff $M_h\lhd g= M_{ghg^\inv}$ for all $g,h\in G$, and in this case $M_{inv}\cong H^0(G, M_1)$.

By Lemma <ref> morphisms of (co)modules over a Hopf monoid $H$ induce morphisms between their (co)invariants. The question if morphisms of both, modules and comodules, induce morphisms between the associated biinvariants is more subtle in general. It  is shown in <cit.> that this always holds for isomorphisms. As a direct generalisation  we have  in the notation of  (<ref>)

Let $\mathcal C$ be complete and finitely cocomplete, $H$, $K$ Hopf monoids in $\mathcal C$ and $\Phi: M \to M'$  an isomorphism of $H$-modules and $K$-comodules. There is a unique morphism $\Phi_{inv}: M_{inv}\to M'_{inv}$ with $\pi'\circ \Phi\circ \iota= I'\circ \Phi_{inv}\circ P$, and $\Phi_{inv}$ is an isomorphism.

§ RIBBON GRAPHS AND SURFACES

In this section we summarise the background on ribbon graphs, also called fat graphs or embedded graphs, for more details we refer to the textbooks of Lando et. al. [32] and Ellis-Monaghan and Moffatt [16].
Throughout this article, all graphs are  directed  graphs with a finite number of vertices and edges.  In contrast to [35] we do not require that the graphs are   connected and allow  isolated vertices with no incident edges.

A ribbon graph is a graph with a cyclic ordering of the  edge ends at each vertex.

The  cyclic ordering of edge ends  at the vertices of a ribbon graph allows one to thicken its edges  to strips or ribbons and defines the faces of
the ribbon graph.
One says that a path in a ribbon graph turns maximally left at a vertex if it
enters the vertex along an edge end and leaves it along an edge end that
comes directly before it with respect to the cyclic ordering. A face of a ribbon graph  is  defined as a cyclic equivalence class of closed paths that turn maximally left at each vertex and traverse each edge at most once in each direction. Each isolated vertex is also viewed as a face, and such a face is called an isolated face.

In the following we denote by $V,E,F$ the sets of vertices, edges and faces of a ribbon graph and by $s(\alpha), t(\alpha)$ the starting and target vertex of an edge $\alpha$. We  say that two edge ends incident at a  vertex $v\in V$ are neighbours or neighbouring if one of them comes directly before or after the other with respect to the cyclic ordering at $v$.
An edge $\alpha$ with $s(\alpha)=t(\alpha)$ is called a loop. A loop at $v$ whose starting and target end are neighbours  is called an isolated loop.
When drawing a  ribbon graph we take the cyclic  ordering of  edge ends at  vertices as the one in  the  drawing.

Ribbon graphs are directly related to embedded graphs on oriented surfaces. Every graph $\Gamma$ embedded into an oriented surface $\Sigma$  inherits a cyclic ordering of the edge ends at each vertex and hence a ribbon graph structure.
Attaching discs to the faces of the ribbon graph $\Gamma$ yields an oriented surface $\Sigma_\Gamma$ such that the connected components of $\Sigma_\Gamma\setminus \Gamma$ are discs and in bijection with faces of $\Gamma$, see Figure <ref>. If $\Gamma$ is embedded into an oriented surface $\Sigma$, the surface $\Sigma_\Gamma$  is homeomorphic to $\Sigma$ iff each connected component of $\Sigma\setminus\Gamma$ is a disc.  In this case, we call $\Gamma$ properly embedded in $\Sigma$.  Note that this implies a bijection between connected components of $\Gamma$ and of $\Sigma$, and    connected components of $\Sigma$ containing an isolated vertex are spheres. The genus $g$ of a connected component of $\Sigma$ is then determined by the Euler characteristic  $2-2g=|V|-|E|+|F|$, where $|V|,|E|,|F|$ are the number of vertices, edges and faces of the associated connected component of $\Gamma$.

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Attaching a disc to the face $f$ yields a torus.

Note that each ribbon graph or embedded graph has a Poincaré dual obtained by replacing each vertex  (face) with a face (vertex) and each edge with a dual edge.
This transforms the paths that characterise faces into paths that go counterclockwise around a vertex and vice versa. Edge ends  correspond to edge sides of the dual graph  and their
cyclic ordering  at a vertex  to the cyclic ordering of the edge sides  in the dual face.

In the following we sometimes require a linear ordering of the edge ends at a vertex or of the edge sides in a face. This is achieved by inserting a marking, the cilium,   that separates the edge ends or edge sides of minimal and maximal order, see for instance Figure <ref>, Definition <ref> or Example <ref>. For faces this corresponds to the choice of a starting vertex for the associated cyclic equivalence class of paths.

* A ciliated vertex in a ribbon graph is a vertex with a choice of linear ordering of the incident edge ends that is compatible with their cyclic ordering.

* A  ciliated face in a ribbon graph is a closed path that turns maximally left at each vertex,  including the starting vertex, and traverses  each edge at most once in each direction.

A ciliated ribbon graph is a ribbon graph in which each face and vertex is assigned a cilium. Isolated vertices and faces are trivially ciliated.

For a closed surface $\Sigma$ of genus $g\geq 0$ we often work with  a ciliated  ribbon graph with a single vertex and  a single face that is given by a set of generators of the fundamental group

\begin{align}\label{eq:pi1gensurface}
\pi_1(\Sigma)=\langle \alpha_1,\beta_1,\ldots,\alpha_g,\beta_g\mid [\beta_g^\inv,\alpha_g]\cdots[\beta_1^\inv,\alpha_1]=1\rangle.
\end{align}

The standard graph of an oriented surface $\Sigma$ of genus $g\geq 1$ is the  graph

\begin{align}\label{eq:standardgraph}
\\[-7ex]
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\draw [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ]   (74,64.75) .. controls (3.21,51.88) and (41.22,123.79) .. (73.52,72.35) ;
\draw [shift={(74.5,70.75)}, rotate = 120.81] [color={rgb, 255:red, 208; green, 2; blue, 27 }  ,draw opacity=1 ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
%Shape: Circle [id:dp7701346359159704]
\draw  [color={rgb, 255:red, 74; green, 74; blue, 74 }  ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 74; blue, 74 }  ,fill opacity=1 ] (45.67,38.92) .. controls (45.67,38.27) and (45.14,37.75) .. (44.5,37.75) .. controls (43.86,37.75) and (43.33,38.27) .. (43.33,38.92) .. controls (43.33,39.56) and (43.86,40.08) .. (44.5,40.08) .. controls (45.14,40.08) and (45.67,39.56) .. (45.67,38.92) -- cycle ;
%Shape: Circle [id:dp8261969467192853]
\draw  [color={rgb, 255:red, 74; green, 74; blue, 74 }  ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 74; blue, 74 }  ,fill opacity=1 ] (39,45.58) .. controls (39,44.94) and (38.48,44.42) .. (37.83,44.42) .. controls (37.19,44.42) and (36.67,44.94) .. (36.67,45.58) .. controls (36.67,46.23) and (37.19,46.75) .. (37.83,46.75) .. controls (38.48,46.75) and (39,46.23) .. (39,45.58) -- cycle ;
%Shape: Circle [id:dp05595027998658675]
\draw  [color={rgb, 255:red, 74; green, 74; blue, 74 }  ,draw opacity=1 ][fill={rgb, 255:red, 74; green, 74; blue, 74 }  ,fill opacity=1 ] (38.33,56.25) .. controls (38.33,55.61) and (37.81,55.08) .. (37.17,55.08) .. controls (36.52,55.08) and (36,55.61) .. (36,56.25) .. controls (36,56.89) and (36.52,57.42) .. (37.17,57.42) .. controls (37.81,57.42) and (38.33,56.89) .. (38.33,56.25) -- cycle ;
% Text Node
\draw (130.5,89.9) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize]  {$\textcolor[rgb]{0.82,0.01,0.11}{\alpha _{1}}$};
% Text Node
\draw (108,13.4) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize]  {$\textcolor[rgb]{0.82,0.01,0.11}{\alpha }\textcolor[rgb]{0.82,0.01,0.11}{_{2}}$};
% Text Node
\draw (17,75.9) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize]  {$\textcolor[rgb]{0.82,0.01,0.11}{\alpha }\textcolor[rgb]{0.82,0.01,0.11}{_{g}}$};
% Text Node
\draw (129.5,34.9) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize,color={rgb, 255:red, 0; green, 0; blue, 0 }  ,opacity=1 ]  {$\textcolor[rgb]{0.01,0.11,0.82}{\beta _{1}}$};
% Text Node
\draw (36,15.9) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize,color={rgb, 255:red, 0; green, 0; blue, 0 }  ,opacity=1 ]  {$\textcolor[rgb]{0.01,0.11,0.82}{\beta }\textcolor[rgb]{0.01,0.11,0.82}{_{2}}$};
% Text Node
\draw (64.5,102.4) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize,color={rgb, 255:red, 0; green, 0; blue, 0 }  ,opacity=1 ]  {$\textcolor[rgb]{0.01,0.11,0.82}{\beta }\textcolor[rgb]{0.01,0.11,0.82}{_{g}}$};
% Text Node
\draw (92,101.23) node [anchor=north west][inner sep=0.75pt]  [font=\scriptsize]  {$v$};
\end{tikzpicture}\nonumber\\[-10ex]
\nonumber
\end{align}

with the face $f=[\beta_g^\inv, \alpha_g]\cdots[\beta_1^\inv,\alpha_1]$ and the ordering of edge ends at $v$ given by
$s(\alpha_1)<s(\beta_1)<t(\alpha_1)<t(\beta_1)<\ldots< s(\alpha_g)<s(\beta_g)<t(\alpha_g)<t(\beta_g)$. In particular, the standard graph for $S^2$ consists of a single isolated vertex and the associated isolated face.

In the following we use certain graph transformations  to relate properly embedded ribbon graphs in a connected surface $\Sigma$ to its standard graph.

Let $\Gamma$ be a ribbon graph with edge set $E$ and vertex set $V$.

* The edge reversal  reverses the orientation of an edge $\beta\in E$.

* The  contraction of an edge $\alpha\in E$ that is not a loop  removes $\alpha\in E$ and fuses the vertices $s(\alpha)$ and $t(\alpha)$.

* The  edge slide  slides an end of $\beta\in E$ that is a neighbour of an end of $\alpha\in E$ along $\alpha$.

* The loop deletion removes an isolated loop $\beta\in E$ from $\Gamma$.

In all cases except 2. the resulting ribbon graph inherits all cilia from $\Gamma$. In 2. one erases either the cilium of $t(\alpha)$ or  of $s(\alpha)$ and speaks of contracting $\alpha$ towards $t(\alpha)$ and $s(\alpha)$, respectively.

every picture/.style=line width=0.75pt

[color=rgb, 255:red, 65; green, 117; blue, 5   ,draw opacity=1 ]   (128.35,35.76) – (177.32,35.69) ;
[shift=(180.32,35.69), rotate = 179.92] [fill=rgb, 255:red, 65; green, 117; blue, 5   ,fill opacity=1 ][line width=0.08]  [draw opacity=0] (8.93,-4.29) – (0,0) – (8.93,4.29) – cycle    ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (92.53,37.4) – (41.93,37.74) ;
[shift=(39.93,37.75), rotate = 359.62] [color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(19.28,53.86) – (33.47,40.87) ;
[shift=(34.94,39.52), rotate = 137.54] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(37.94,15.19) – (37.26,33) ;
[shift=(37.18,35), rotate = 272.2] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(97.08,38.61) .. controls (131.42,59.54) and (119.33,5) .. (96.34,33.3) ;
[shift=(95.29,34.65), rotate = 307.06] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (34.43,37.75) .. controls (34.43,36.23) and (35.66,35) .. (37.18,35) .. controls (38.7,35) and (39.93,36.23) .. (39.93,37.75) .. controls (39.93,39.27) and (38.7,40.5) .. (37.18,40.5) .. controls (35.66,40.5) and (34.43,39.27) .. (34.43,37.75) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (92.53,37.4) .. controls (92.53,35.88) and (93.77,34.65) .. (95.29,34.65) .. controls (96.81,34.65) and (98.04,35.88) .. (98.04,37.4) .. controls (98.04,38.92) and (96.81,40.16) .. (95.29,40.16) .. controls (93.77,40.16) and (92.53,38.92) .. (92.53,37.4) – cycle ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (257.23,35.9) – (204.63,36.23) ;

[shift=(52, 0)]
[shift=(204.63,36.24), rotate = 180.62] [color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(183.48,52.85) – (197.67,39.86) ;
[shift=(199.14,38.51), rotate = 137.54] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(202.64,13.68) – (201.96,31.49) ;
[shift=(201.88,33.49), rotate = 272.2] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(261.33,37.1) .. controls (295.67,58.03) and (283.59,3.49) .. (260.6,31.79) ;
[shift=(259.54,33.14), rotate = 307.06] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (199.13,36.24) .. controls (199.13,34.72) and (200.36,33.49) .. (201.88,33.49) .. controls (203.4,33.49) and (204.63,34.72) .. (204.63,36.24) .. controls (204.63,37.76) and (203.4,39) .. (201.88,39) .. controls (200.36,39) and (199.13,37.76) .. (199.13,36.24) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (256.79,35.9) .. controls (256.79,34.38) and (258.02,33.14) .. (259.54,33.14) .. controls (261.06,33.14) and (262.29,34.38) .. (262.29,35.9) .. controls (262.29,37.42) and (261.06,38.65) .. (259.54,38.65) .. controls (258.02,38.65) and (256.79,37.42) .. (256.79,35.9) – cycle ;

[color=rgb, 255:red, 65; green, 117; blue, 5   ,draw opacity=1 ]   (128.13,149.69) – (177.1,149.62) ;
[shift=(180.1,149.62), rotate = 179.92] [fill=rgb, 255:red, 65; green, 117; blue, 5   ,fill opacity=1 ][line width=0.08]  [draw opacity=0] (8.93,-4.29) – (0,0) – (8.93,4.29) – cycle    ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (96,152.66) .. controls (97.71,154.27) and (97.76,155.94) .. (96.15,157.65) .. controls (94.54,159.37) and (94.59,161.04) .. (96.31,162.65) .. controls (98.02,164.27) and (98.07,165.94) .. (96.46,167.65) .. controls (94.85,169.36) and (94.9,171.03) .. (96.61,172.65) – (96.62,172.82) – (96.62,172.82) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (93.53,149.9) – (42.93,150.24) ;
[shift=(40.93,150.25), rotate = 359.62] [color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (21.57,140.93) .. controls (23.85,140.31) and (25.3,141.14) .. (25.92,143.41) .. controls (26.54,145.68) and (27.99,146.51) .. (30.26,145.89) .. controls (32.53,145.27) and (33.98,146.1) .. (34.6,148.37) – (35.39,148.82) – (35.39,148.82) ;

[color=rgb, 255:red, 65; green, 117; blue, 5   ,draw opacity=1 ]   (38.43,153) – (43.77,177.4) ;
[shift=(44.19,179.36), rotate = 257.66] [color=rgb, 255:red, 65; green, 117; blue, 5   ,draw opacity=1 ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(20.53,166.61) – (34.72,153.62) ;
[shift=(36.19,152.27), rotate = 137.54] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(38.94,127.69) – (38.26,145.5) ;
[shift=(38.18,147.5), rotate = 272.2] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[color=rgb, 255:red, 245; green, 166; blue, 35   ,draw opacity=1 ]   (44.93,158.7) – (52.86,158.45) ;
[shift=(55.86,158.36), rotate = 178.2] [fill=rgb, 255:red, 245; green, 166; blue, 35   ,fill opacity=1 ][line width=0.08]  [draw opacity=0] (5.36,-2.57) – (0,0) – (5.36,2.57) – cycle    ;

(97.79,151.11) .. controls (132.13,172.04) and (120.05,117.5) .. (97.06,145.8) ;
[shift=(96,147.15), rotate = 307.06] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (35.68,150.25) .. controls (35.68,148.73) and (36.91,147.5) .. (38.43,147.5) .. controls (39.95,147.5) and (41.18,148.73) .. (41.18,150.25) .. controls (41.18,151.77) and (39.95,153) .. (38.43,153) .. controls (36.91,153) and (35.68,151.77) .. (35.68,150.25) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (93.25,149.9) .. controls (93.25,148.38) and (94.48,147.15) .. (96,147.15) .. controls (97.52,147.15) and (98.75,148.38) .. (98.75,149.9) .. controls (98.75,151.42) and (97.52,152.66) .. (96,152.66) .. controls (94.48,152.66) and (93.25,151.42) .. (93.25,149.9) – cycle ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (272.86,150.7) .. controls (275.2,150.99) and (276.22,152.31) .. (275.93,154.65) .. controls (275.64,156.99) and (276.66,158.31) .. (279,158.6) .. controls (281.34,158.89) and (282.36,160.21) .. (282.06,162.55) .. controls (281.77,164.89) and (282.79,166.2) .. (285.13,166.49) – (286.68,168.48) – (286.68,168.48) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (270.1,147.95) – (219.5,148.29) ;
[shift=(217.5,148.3), rotate = 359.62] [color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (197.9,138.97) .. controls (200.17,138.36) and (201.62,139.19) .. (202.24,141.46) .. controls (202.86,143.73) and (204.31,144.56) .. (206.58,143.94) .. controls (208.85,143.32) and (210.3,144.15) .. (210.92,146.42) – (211.71,146.87) – (211.71,146.87) ;

[color=rgb, 255:red, 65; green, 117; blue, 5   ,draw opacity=1 ]   (271.32,150.2) – (269.16,172.2) ;
[shift=(268.96,174.19), rotate = 275.62] [color=rgb, 255:red, 65; green, 117; blue, 5   ,draw opacity=1 ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(196.85,164.4) – (211.04,151.42) ;
[shift=(212.52,150.07), rotate = 137.54] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(215.52,125.74) – (214.83,143.55) ;
[shift=(214.75,145.55), rotate = 272.2] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(274.65,149.16) .. controls (308.99,170.08) and (296.9,115.55) .. (273.91,143.85) ;
[shift=(272.86,145.2), rotate = 307.06] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (212,148.3) .. controls (212,146.78) and (213.23,145.55) .. (214.75,145.55) .. controls (216.27,145.55) and (217.5,146.78) .. (217.5,148.3) .. controls (217.5,149.82) and (216.27,151.05) .. (214.75,151.05) .. controls (213.23,151.05) and (212,149.82) .. (212,148.3) – cycle ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (270.1,147.95) .. controls (270.1,146.43) and (271.34,145.2) .. (272.86,145.2) .. controls (274.38,145.2) and (275.61,146.43) .. (275.61,147.95) .. controls (275.61,149.47) and (274.38,150.7) .. (272.86,150.7) .. controls (271.34,150.7) and (270.1,149.47) .. (270.1,147.95) – cycle ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (87.24,61.69) .. controls (88.95,63.31) and (89,64.98) .. (87.39,66.69) .. controls (85.78,68.41) and (85.83,70.08) .. (87.55,71.69) .. controls (89.26,73.3) and (89.31,74.97) .. (87.7,76.68) .. controls (86.09,78.4) and (86.14,80.07) .. (87.86,81.68) – (87.86,81.85) – (87.86,81.85) ;

[color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ]   (85.39,84.6) – (34.79,84.94) ;
[shift=(32.79,84.95), rotate = 359.62] [color=rgb, 255:red, 208; green, 2; blue, 27   ,draw opacity=1 ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (32.07,87.18) .. controls (34.42,87.36) and (35.51,88.63) .. (35.33,90.98) .. controls (35.15,93.33) and (36.23,94.59) .. (38.58,94.77) .. controls (40.93,94.95) and (42.02,96.22) .. (41.84,98.57) .. controls (41.66,100.92) and (42.75,102.19) .. (45.1,102.36) – (46.07,103.5) – (46.07,103.5) ;

(89.84,86.16) .. controls (109.95,112.34) and (131.18,69.57) .. (92.75,82.64) ;
[shift=(90.96,83.27), rotate = 339.82] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (27.29,84.95) .. controls (27.29,83.43) and (28.52,82.2) .. (30.04,82.2) .. controls (31.56,82.2) and (32.79,83.43) .. (32.79,84.95) .. controls (32.79,86.47) and (31.56,87.7) .. (30.04,87.7) .. controls (28.52,87.7) and (27.29,86.47) .. (27.29,84.95) – cycle ;

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Examples of graph transformations

These graph transformations are illustrated in Figure <ref>. Note that they are not independent. Contracting an edge $\alpha$ towards $t(\alpha)$ is the same as first sliding some edge ends  along $\alpha$ and then contracting $\alpha$ towards $t(\alpha)$.
Contracting an edge $\alpha$ towards $t(\alpha)$ is also the same as first reversing $\alpha$, then contracting $\alpha$ towards $s(\alpha)$ and then reversing $\alpha$.
By reversing $\alpha$ and $\beta$ before and after a slide, one can reduce all edge slides to the ones that slide the target end of $\beta$ along the left of $\alpha$.

There are of course other possible graph transformations such as deleting edges, which is dual to edge contractions.
However, the graph transformations in Definition <ref> are sufficient to transform any connected ribbon graph into a standard graph.
This  is well-known and appears implicitly in many publications. We summarise the argument for the convenience of the reader.

Every connected ribbon graph
can be transformed into the standard graph  (<ref>) by edge reversals,  edge slides, edge contractions and loop deletions.

Selecting a maximal tree in $\Gamma$ and contracting all edges in the tree  transforms $\Gamma$ into a graph $\Gamma'$ with a single vertex. By applying edge slides one can transform $\Gamma'$ into a graph $\Gamma''$ that coincides with (<ref>) up to edge orientation and up to the presence  a number of isolated loops between the cilium and the starting end of $\alpha_1$. This follows from an analogous statement for chord diagrams, which correspond to ribbon graphs with a single vertex, see for instance Chmutov, Duzhin and Mostovoy <cit.>. Deleting the isolated loops and reversing edges in $\Gamma''$ then yields the standard graph (<ref>).

§ (CO)MODULES FROM HOPF MONOIDS AND RIBBON GRAPHS

In this section we use involutive Hopf monoids in symmetric monoidal categories to assign (co)modules over  Hopf monoids to  ciliated  ribbon graphs. In Section <ref>
we then show that their biinvariants are topological invariants: their isomorphism classes depend only on the genus of the  surface obtained by attaching discs to the faces of the graph. In Sections <ref> and <ref> we determine these biinvariants for simplicial groups as Hopf monoids in $\SSet$ and for crossed modules as group objects in $\Cat$.

The construction generalises Kitaev's quantum double model and the toric code from [25],  which was first formulated for the  group algebra of a finite group over $\C$ and  then generalised by Buerschaper et al. in [11] to finite-dimensional semisimple $C^*$-Hopf algebras. A very similar construction to the one in this article is used in  [35] to obtain mapping class group actions from pivotal  Hopf  monoids in symmetric monoidal categories. The work [35] considers the biinvariants of a Yetter-Drinfeld module structure assigned to the standard graph
(<ref>), but it does not establish  that the biinvariants are graph-independent.

The construction of the (co)module structures from an involutive Hopf monoid  and  a ciliated ribbon graph in this section is directly analogous to the one in [35], which in turn is a straightforward generalisation of  [25, 11]. The only difference is that $H^*$-modules in [11] are replaced by $H$-comodules and $D(H)$-modules by Yetter-Drinfeld modules over $H$.

What differs substantially from [25, 11] are the notions of (co)invariants, biinvariants and the construction of the topological invariant. The works in [25, 11]  rely on  the normalised Haar integral  of a finite-dimensional semisimple complex Hopf algebra,  which is not available in our setting. Our construction is more general, as the only
assumptions are that the underlying symmetric monoidal category is complete and finitely cocomplete and the Hopf monoid involutive.
The article [35] also allows pivotal Hopf monoids. The involutive Hopf monoids in this article are examples of pivotal Hopf monoids,  with their unit as  pivotal structure.

Let $H$ be an involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category $\mac$ and $\Gamma$ a ciliated ribbon graph  with vertex set $V$, edge set $E$ and face set $F$.

We consider the $|E|$-fold tensor product of $H$ with itself, together with an assignment of the copies of $H$ in this tensor product to the  edges of $\Gamma$, which we emphasise by writing $H^{\oo E}$. If $E=\emptyset$, we set $H^{\oo E}=e$.  The object $H^{\oo E}$ can be viewed as the counterpart of the Hilbert space of Kitaev's quantum double model in [25, 11].

We assign to each edge $\alpha\in E$ two $H$-module structures $\rhd_{\alpha\pm}: H\oo H^{\oo E}\to H^{\oo E}$ and  $H$-comodule structures $\delta_{\alpha\pm}: H^{\oo E}\to H\oo H^{\oo E}$. The $H$-module structures $\rhd_{\alpha+}$ and $\rhd_{\alpha-}$ are assigned to the target and starting end of $\alpha$ and the $H$-comodule structures to its left and right side, respectively.  They are induced by the standard  $H$-(co)module  structures on $H$ via left (co)multiplication.

This requires some notation. Given a morphism $f: H\to K$ in $\mac$ and an edge $\alpha\in E$ we write  $f_\alpha$  for the morphism that applies $f$ to the copy of $H$ in $H^{\oo E}$ that belongs to $\alpha$  and the identity morphism to the other copies. We write $\tau_{\alpha}: H^{\oo E}\to H^{\oo E}$ or $\tau_{\alpha}: H\oo H^{\oo E}\to H\oo H^{\oo E}$ for the composite of braidings that moves the copy of $H$ for $\alpha$ to the left.
We denote by  $m_{\alpha}: H \oo H^{\oo E} \to H^{\oo E}$   the morphism that moves the first copy of $H$   to the left of the one for $\alpha$ and then applies $m$ to them.

The $H$-module structures $\rhd_{\alpha\pm}: H \oo H^{\oo E}\to H^{\oo E}$ and $H$-comodule structures $\delta_{\alpha\pm}:H^{\oo E}\to H \oo H^{\oo E}$ for an edge $\alpha\in E$ are

\begin{align*}
\rhd_{\alpha +} := m_{\alpha}, \quad
\rhd_{\alpha -}:= S_{\alpha}\circ \rhd_{\alpha + }\circ (1_H \oo S_{\alpha}), \quad
\delta_{\alpha +}:= \tau_{\alpha}\circ \Delta_{\alpha},  \quad
\delta_{\alpha -}:= (1_H \oo S_{\alpha}) \circ \delta_{\alpha+}\circ S_{\alpha}.
\end{align*}

By definition, the (co)module structures assigned to different edges of a graph commute, since they (co)act on different copies of $H$ in the tensor product $H^{\oo E}$. A direct computation using (<ref>) and (<ref>)  shows that the two $H$-(co)module  structures assigned to a given edge commute  as well. The proof is directly analogous to the ones for Hopf algebras in [11].

For any edge $\alpha\in E$ the $H$-module structures $\rhd_{\alpha \pm}$ and the $H$-comodule structures $\delta_{\alpha \pm}$ commute:

\begin{align*}
\rhd_{\alpha -}\circ (1_H \oo \rhd_{\alpha +}) &=\rhd_{\alpha +}\circ (1_H \oo \rhd_{\alpha -}) \circ (\tau_{H, H}\oo 1_{H^{\oo E}})\text{,}\\
(1_H \oo \delta_{\alpha -})\circ \delta_{\alpha +} &= (\tau_{H, H}\oo 1_{H^{\oo E}}) \circ (1_H \oo \delta_{\alpha +})\circ \delta_{\alpha -}\text{.}
\end{align*}

The (co)module structures from Definition <ref> define an $H$-module structure on $H^{\oo E}$ for each ciliated vertex $v$  and an $H$-comodule structure  on $H^{\oo E}$ for each  ciliated face $f$  of $\Gamma$. The former applies the comultiplication to $H$,  distributes the resulting copies of $H$ to the  edge ends at $v$ according to their ordering  and  acts on them with $\rhd_{\alpha\pm}$ according to their orientation. Dually, the coaction applies the $H$-coaction $\delta_{\alpha\pm}$ to each edge $\alpha$ in $f$, depending on its orientation relative to $f$, and multiplies the resulting copies of $H$ according to  the order of the edge sides in $f$.

<cit.>$ $

* The $H$-module structure $\rhd_v: H\oo H^{\oo E}\to H^{\oo E}$ for a ciliated vertex $v$ with incident edge ends $\alpha_1<\alpha_2<\ldots<\alpha_n$ is

\begin{align}
\rhd_v=\rhd_{\alpha_1}\circ (1_H\oo \rhd_{\alpha_2})\circ \ldots \circ  (1_{H^{\oo (n-1)}}\oo  \rhd_{\alpha_n}) \circ (\Delta^{(n-1)} \oo 1_{H^{\oo E}}),
\end{align}

where $\rhd_\alpha=\rhd_{e(\alpha)+}$ if $\alpha$ is incoming, $\rhd_{\alpha}=\rhd_{e(\alpha)-}$ if $\alpha$ is outgoing  and $e(\alpha)$ is the edge of  $\alpha$.

* The $H$-comodule structure $\delta_f: H^{\oo E}\to H\oo H^{\oo E}$ for a ciliated face  $f$ that traverses the edges $\alpha_n,\alpha_{n-1},\ldots,\alpha_1$  in this order

\begin{align}
\delta_f= (m^{(n-1)}\oo 1_{H^{\oo E}}) \circ (1_{H^{\oo(n-1)}}\oo \delta_{\alpha_r})\circ \ldots\circ (1_H\oo\delta_{\alpha_2})\circ \delta_{\alpha_1},
\end{align}

where $\delta_\alpha=\delta_{e(\alpha)+}$ if $\alpha$ is traversed with, $\delta_{\alpha}=\delta_{e(\alpha)-}$ if $\alpha$ is traversed against its orientation  and $e(\alpha)$ is the edge of  $\alpha$.

To an isolated vertex  and face we assign the (co)module  structures $\rhd_v=\epsilon\oo 1_{H^{\oo E}}$ and $\delta_f=\eta\oo 1_{H^{\oo E}}$.

To avoid heavy notation we use Sweedler notation and   describe these (co)module structures by labelling edges of  a graph  with letters representing the associated copies of $H$.

$ $

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The $H$-module structure $\rhd_v$ for the ciliated vertex $v$ with incident edge ends $t(a)< s(b)< t(b)< t(c)< s(d)$ and the $H$-comodule structure $\delta_f$ for the ciliated face $f= e \circ e^\inv \circ d \circ c^\inv \circ  b \circ a$  are

\begin{align*}
h \rhd_v \, ( a\oo b\oo c\oo d) &= \low h 1 a \oo \low h 3 b S( \low h 2 ) \oo \low h 4 c \oo d S( \low h 5 ),\\
\delta_f \, (a\oo b\oo c\oo d\oo e) &= \low e 1 S( \low e 3 ) \low d 1 S( \low c 2 ) \low b 1 \low a 1 \oo  \low a 2 \oo \low b 2 \oo \low c 1 \oo \low d 2 \oo \low e 2.
\end{align*}

The interaction of the $H$-module and $H$-comodule structures assigned to ciliated vertices and faces of the graph is investigated in [25, 11, 35].
They are local in the sense that the $H$-(co)module structure for a vertex (face) affects only those copies of $H$ that belong to their incident edges. As the action $\rhd_{\alpha+}$ for an edge $\alpha\in E$ acts by left- and $\rhd_{\alpha-}$  by right-multiplication, the $H$-module structures for different vertices commute. The same holds for the $H$-comodule structures at different faces. Moreover, $H$-module structures commute with $H$-comodule structures unless  their cilia share a vertex or a face. The $H$-module and $H$-comodule structure for each cilium define a Yetter-Drinfeld module structure.

* The $H$-left module structures for distinct vertices $v\neq v'\in V$ and the $H$-left comodule structures for distinct  faces $f\neq f'\in F$ commute for all choices of cilia:

\begin{align}
\rhd_{v'}\circ (1_H \oo \rhd_v) &=\rhd_v \circ (1_H \oo \rhd_{v'})\circ (\tau_{H, H}\oo 1_{H^{\oo E}}), \label{eq:VertexActionsCommute} \\
(1_H \oo \delta_{f'}) \circ \delta_f &= (\tau_{H, H}\oo 1_{H^{\oo E}})\circ (1_H \oo \delta_f)\circ \delta_{f'} \text{.}\label{eq:FaceCoactionsCommute}
\end{align}

* If two cilia are at distinct vertices and distinct faces, the $H$-module structure for one of them commutes with the $H$-comodule structure for the other:

\begin{align}
\delta_f \circ \rhd_v= (1_H \oo  \rhd_v)\circ (\tau_{H, H}\oo 1_{H^{\oo E}}) \circ (1_H \oo \delta_f)\text{.}\label{eq:VertexAndFaceActionsCommute}
\end{align}

* If $v\in V$ and $f\in F$ share a cilium, then $(H^{\oo E}, \rhd_v, \delta_f)$ is a Yetter-Drinfeld module over $H$.

Let $H$ be an involutive Hopf monoid in $\mac$ and $\Gamma$ the standard graph (<ref>)  on a surface $\Sigma$ of genus $g\geq 1$.
Then the associated Yetter-Drinfeld module structure on  $H^{\oo E}$ is

\begin{align}\label{eq:ydgen}
&h\rhd(a^1\oo b^1\oo \ldots\oo a^g\oo b^g)\\
&\qquad=\low h 3 a^1 S(\low h 1)\oo \low h 4 b^1 S(\low h 2)\oo\ldots\oo \low h {4g-1} a^g S(\low h {4g-3}) \oo \low h {4g}b^gS(\low h {4g-2})\nonumber\\
&\delta(a^1\oo b^1\oo \ldots\oo a^g\oo b^g)\nonumber\\
&\qquad=S(b^g_{(3)}) a^g_{(1)} b^g_{(1)}  S(a^g_{(3)})\cdots S(b^1_{(3)}) a^1_{(1)} b^1_{(1)}  S(a^1_{(3)})\oo a^1_{(2)}\oo b^1_{(2)}\oo\ldots\oo a^g_{(2)}\oo b^g_{(2)}.\nonumber
\end{align}

If $H$ is a group object in a cartesian monoidal category, this reduces to

\begin{align}\label{eq:YDmodulegroup}
&h\rhd (a_1,b_1,\ldots, a_g,b_g)=(h a_1 h^\inv, hb_1 h^\inv,\ldots, ha_g h^\inv, hb_g h^\inv)\\
&\delta(a_1,b_1,\ldots, a_g, b_g)=([b_g^\inv, a_g]\cdots[b_1^\inv, a_1], a_1,b_1,\ldots, a_g, b_g).\nonumber
\end{align}

If each vertex and face of $\Gamma$ is  equipped with a  cilium,  then Definition <ref> assigns
an $H$-(co)module structure on $H^{\oo E}$ to each vertex (face) of $\Gamma$. By Lemma <ref> these (co)module structures  commute and hence  combine into  $H^{\oo E}$-module and $H^{\oo F}$-comodule structures on $H^{\oo E}$.

The $H^{\oo n}$-module  structure for a subset $\emptyset \neq \mathcal V:=\{v_1, \dots, v_n\}\subset V$ and  $H^{\oo m}$-comodule structure for a subset $\emptyset \neq \mathcal F:=\{f_1, \dots, f_m\} \subset F$  are

\begin{align}
\rhd_{\mathcal V} &:= \rhd_{v_1}\circ (1_H \oo \rhd_{v_2}) \circ \dots \circ (1_{H^{\oo (n-2)}} \oo \rhd_{v_{n-1}})\circ (1_{H^{\oo (n-1)}}\oo \rhd_{v_n}): H^{\oo n}\oo H^{\oo E} \to H^{\oo E}, \label{eq:ComposedVertexAction}\\
\delta_{\mathcal F} &:= (1_{H^{\oo (m-1)}} \oo \delta_{f_m})\circ (1_{H^{\oo (m-2)}} \oo \delta_{f_{m-1}})\circ \dots \circ (1_H \oo \delta_{f_2}) \circ \delta_{f_1}: H^{\oo E} \to H^{\oo m}\oo H^{\oo E}.\nonumber
\end{align}

Equations (<ref>) and (<ref>) ensure that the (co)actions do not depend on the numbering of vertices or faces in Definition <ref>. That (<ref>) defines an $H^{\oo n}$-module structure  follows from the

\begin{align*}
\rhd_{\mathcal V'} \circ (1_{H^{\oo \vert \mathcal V'\vert }} \oo \rhd_v) \circ (\tau_{H,H^{\oo \vert \mathcal V'\vert}} \oo 1_{H^{\oo E}}) = \rhd_v \circ (1_H \oo \rhd_{\mathcal V'}),
\end{align*}

valid for any subset $\emptyset \neq \mathcal V'\subset V$, $v\in V\setminus \mathcal V'$. The dual statement for $\delta_{\mathcal F}$  follows analogously.

The module and comodule structure  from Definition <ref>  define the categorical counterpart of the  protected space or ground state in Kitaev's quantum double model. In the models based on a  finite-dimensional semisimple complex Hopf algebras in [25, 11] the ground state  is  an eigenspace  of a Hamiltonian that combines these $H$-(co)module structures. The normalised Haar integral defines a projector on the ground state.
In our setting these structures are not available. Instead, we consider the biinvariants from Definition <ref> for  the action and coaction from (<ref>).

$ $
The protected object for an involutive Hopf monoid $H$ and  a ciliated ribbon graph $\Gamma$ are the biinvariants
$M_{inv}=\mathrm{Im}(\pi \circ \iota)$ of  $H^{\oo E}$ with the module structure $\rhd_V$ and comodule structure $\delta_F$  from (<ref>).

In the quantum double models for  a finite-dimensional semisimple complex Hopf algebra
it is directly apparent that imposing (co)invariance under all  individual (co)actions at the vertices (faces) of a graph is the same as imposing (co)invariance under the combined action in Definition <ref>. In this setting the (co)invariants for the individual (co)actions are linear subspaces of $H^{\oo E}$ and the (co)invariants of the combined (co)actions their intersections.
In our setting an analogous statement  follows from the universal properties of the
coequaliser $\pi_{\mathcal V}: H^{\oo E}\to M^H_{\mathcal V}$  for the action $\rhd_{\mathcal V}$  and the  equaliser  $\iota_{\mathcal F}: M^{coH}_{\mathcal F}\to H^{\oo E}$ for the coaction $\delta_{\mathcal F}$, as given in Definition <ref>.

$ $
Let $\emptyset\neq \mathcal V \subset V$, $\emptyset\neq \mathcal F\subset F$ be subsets.

* For any subset $\emptyset\neq \mathcal V'\subset\mathcal V$ the morphism $\pi_{\mathcal V}: H^{\oo E}\to M^H_{\mathcal V}$ satisfies

\begin{align}
\pi_{\mathcal V}\circ \rhd_{\mathcal V'}= \pi_{\mathcal V}\circ (\epsilon^{\vert \mathcal V'\vert}\oo 1_{H^{\oo E}})\label{eq:BigInvariantsCoequalise}.
\end{align}

There is a unique morphism $\chi_{\mathcal V', \mathcal V}: M_{\mathcal V'}^H\to M_{\mathcal V}^H$ with $\chi_{\mathcal V', \mathcal V} \circ \pi_{\mathcal V'}=\pi_{\mathcal V}$.
is an epimorphism.

* For any subset $\emptyset \neq \mathcal F'\subset \mathcal F$ the morphism $\iota_{\mathcal F}: M^{coH}_{\mathcal F}\to H^{\oo E}$ satisfies

\begin{align}
\delta_{\mathcal F'}\circ \iota_{\mathcal F}= (\eta^{\vert \mathcal F'\vert} \oo 1_{H^{\oo E}})\circ \iota_{\mathcal F}\label{eq:BigCoinvariantsEqualise}.
\end{align}

There is a unique morphism $\xi_{\mathcal F', \mathcal F}: M_{\mathcal F}^{coH}\to M_{\mathcal F'}^{coH}$ with $\iota_{\mathcal F'}\circ \xi_{\mathcal F', \mathcal F}= \iota_{\mathcal F}$.
It is a monomorphism.

We prove 1., as 2. is the dual statement.  It suffices to verify (<ref>) for $\mathcal V=\{v_1, \ldots, v_n\}$, $\mathcal V'=\{v_j\}$,  and the claim follows by induction over $\vert \mathcal V'\vert$.  For this note first that Definition <ref> implies

\begin{align}
\rhd_{\mathcal V}\circ (\eta^{\oo (j-1)}\oo 1_H \oo \eta^{\oo (n-j)}\oo 1_{H^{\oo E}})&= \rhd_{v_j} & &\forall j \in \{1, \dots, n\} \label{eq:GrosseUndEinzelneDreiecke}.
\end{align}

As $\pi_{\mathcal V}$ is the coequaliser of $\rhd_{\mathcal V}$ and $\epsilon^{\oo n}\oo 1_{H^{\oo E}}$ one obtains

\begin{align*}
\pi_{\mathcal V}\circ \rhd_{v_j}&\overset{\eqref{eq:GrosseUndEinzelneDreiecke}}{=} \pi_{\mathcal V}\circ \rhd_{\mathcal V}\circ (\eta^{\oo (j-1)}\oo 1_H\oo \eta^{\oo (n-j)}\oo 1_{H^{\oo E}})\\
&= \pi_{\mathcal V}\circ (\epsilon^{\oo n}\oo 1_{H^{\oo E}})\circ (\eta^{\oo (j-1)}\oo 1_H\oo \eta^{\oo (n-j)}\oo 1_{H^{\oo E}})\, = \,  \pi_{\mathcal V} \circ (\epsilon\oo 1_{H^{\oo E}}).
\end{align*}

Equation (<ref>) and the universal property of the coequaliser $ \pi_{\mathcal V'}$ imply the existence of a unique morphism $\chi_{\mathcal V', \mathcal V}: M_{\mathcal V'}^H\to M_{\mathcal V}^H$ with $\chi_{\mathcal V', \mathcal V}\circ \pi_{\mathcal V'}=\pi_{\mathcal V}$. For any two morphisms $q_1, q_2: M_{\mathcal V}^H\to X$  with $q_1 \circ \chi_{\mathcal V', \mathcal V}=q_2 \circ \chi_{\mathcal V', \mathcal V}$ one  has
$q_1 \circ \chi_{\mathcal V', \mathcal V}\circ \pi_{\mathcal V'}= q_1\circ \pi_{\mathcal V}= q_2 \circ \pi_{\mathcal V}= q_2 \circ \chi_{\mathcal V', \mathcal V} \circ \pi_{\mathcal V'}$.
As $\pi_{\mathcal V}$ is a coequaliser and hence an epimorphism, this implies $q_1=q_2$, and $\chi_{\mathcal V', \mathcal V}$ is an epimorphism.

It is also directly apparent from Definition <ref> that (co)module morphisms with respect to all  individual (co)module structures at vertices  and faces in $\mathcal V$ and $\mathcal F$ are also (co)module morphisms with respect to the (co)actions  $\rhd_{\mathcal V}$ and $\delta_{\mathcal F}$. More precisely,
for ciliated ribbon graphs
$\Gamma, \Gamma'$, subsets $\emptyset\neq \mathcal V \subset V$, $\emptyset\neq \mathcal V' \subset V'$ and a bijection $\varphi:\mathcal V \to \mathcal V'$, $v\mapsto v'$,    any  morphism $g: H^{\oo E}\to H^{\oo E'}$ that is a module morphism with respect
to $\rhd_v$ and $\rhd_{v'}$ for all $v \in V$ is also a module morphism with respect to $\rhd_{\mathcal V}$ and $\rhd_{\mathcal V'}$. An analogous statement holds for $\delta_{\mathcal F}$ and  comodule morphisms.

§ GRAPH INDEPENDENCE

In this section we show that the protected object from Definition <ref> is a topological invariant:  Although its definition requires a ciliated ribbon graph $\Gamma$, its isomorphism class depends only on the homeomorphism class of the surface obtained by attaching  discs to the faces of $\Gamma$.

To prove this, we show first in Section <ref>  that  the (co)invariants associated to the (co)module structures at the vertices (faces) of $\Gamma$ depend neither on the edge orientation nor on the choices of the cilia. Reversing the orientation of edges and different choices of cilia yield  isomorphisms between these (co)invariants and hence also between the biinvariants.
We then show in Section <ref> and  <ref> that the other graph transformations from Definition <ref> induce isomorphisms between the protected objects, although not necessarily between the (co)invariants. In Section <ref> we combine these results to obtain topological invariance and treat some simple examples.

As in Section <ref> we consider a complete and finitely cocomplete symmetric monoidal category $\mathcal C$,  an involutive Hopf monoid $H$  in $\mathcal C$ and a ciliated ribbon graph $\Gamma$.

§.§ Edge orientation reversal and  moving the cilium

As  edge orientation reversal switches the start and target and the left and right side of an edge $\alpha\in E$, it  exchanges the associated actions $\rhd_{\alpha\pm}$ and coactions $\delta_{\alpha\pm}$  from Definition <ref>. It is directly apparent from their definitions that this is achieved by applying the antipode.

The automorphism of $H^{\oo E}$ associated to the reversal of an edge $\alpha\in E$ is $S_\alpha: H^{\oo E}\to H^{\oo E}$.

For any ciliated vertex $v\in V$,   ciliated face $f\in F$ and edge $\beta\in E$  the edge reversal $S_{\beta}$  is an isomorphism of $H$-modules and $H$-comodules with respect to $\rhd_v$ and  $\delta_f$.

We denote by $\rhd_v'$ and $\delta_f'$ the module and comodule structure in the graph where the  orientation of $\beta$ is reversed and verify that  $\rhd_v' \circ (1_H\oo S_{\beta})=S_{\beta} \circ \rhd_v$ and $\delta_f'\circ S_\beta=(1_H\oo S_\beta)\circ\delta_f$. If $\beta$ is not incident at $v$ and $f$, the copy of $H$ in $H^{\oo E}$ assigned to $\beta$ is not affected by $\rhd_v,\rhd_v'$ and $\delta_f,\delta'_f$, and
the identity follows directly. If $\beta$ is incident at $v$ or $f$, it follows from the expressions for the (co)actions in Definitions <ref> and <ref>.

As a direct consequence of  Lemma <ref>,  Lemma <ref> and Lemma <ref> one has

Reversing the orientation of an edge  in $\Gamma$ to obtain  $\Gamma'$  induces isomorphisms between the invariants, coinvariants and protected objects of $\Gamma$ and $\Gamma'$.

$ $
The (co)invariants for the $H$-(co)module structure at a given vertex (face) do not depend on the choice of cilia: moving the position of the cilium yields isomorphic (co)invariants. This induces isomorphisms of the protected objects.

We  focus on the $H$-module structure and its invariants. We consider a fixed position of the cilium at a vertex $v$ with associated vertex action $\rhd_v$ and coequaliser $\pi_v: H^{\oo E}\to M_v^H$ and compare it to the action $\rhd_v'$ and coequaliser $\pi_v': H^{\oo E}\to M'^H_v$ obtained by  rotating the cilium counterclockwise by one position. We first show that the coequaliser
$\pi_v: H^{\oo E}\to M_v^H$  satisfies

\begin{align}\label{eq:rotate}
\pi_v \circ \rhd_v'=\pi_v \circ (\epsilon\oo 1_{H^{\oo E}}).
\end{align}

By definition of the $H$-module structure $\rhd_v$  and by  Lemma <ref> it is sufficient to prove this for a vertex with $n$ incoming edges. The computations for vertices with incident loops are analogous.

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(69.48,17.03) node [anchor=north west][inner sep=0.75pt]  [font=]  $a^2$;

(22.71,13.65) node [anchor=north west][inner sep=0.75pt]  [font=]  $\ldots$;

(19.43,39.94) node [anchor=north west][inner sep=0.75pt]  [font=]  $a^n$;

(54,52.94) node [anchor=north west][inner sep=0.75pt]  [font=]  $a^1$;

(104,25.86) node [anchor=north west][inner sep=0.75pt]   [align=left] shift;

(102,45) node [anchor=north west][inner sep=0.75pt]   [align=left] cilium;

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(36.86,70.8) node [anchor=north west][inner sep=0.75pt]  [font=]  $\rhd_v$;

(166.29,71.37) node [anchor=north west][inner sep=0.75pt]  [font=]  $\rhd'_v$;

For a vertex with $n$ incoming edges we have

\begin{align*}
&\pi_v \circ \rhd_v'\, (h \oo a^1\oo a^2 \oo \ldots \oo a^n)\, =\, \pi_v\, (\low h n a^1 \oo \low h 1 a^2 \oo \ldots \oo \low h {n-1} a^n)\\
&=\pi_v \, (\low {\low h 2} 1 S(\low h 1 ) \low h 3 a^1 \oo \low {\low h 2} 2 a^2 \oo \ldots \oo \low {\low h 2 } n a^n)\\
\pi_v \circ \rhd_v \, (\low h 2 \oo S(\low h 1 ) \low h 3 a^1 \oo a^2\oo \ldots \oo a^n)
=\pi_v \, (\epsilon(\low h 2 )\oo S(\low h 1) \low h 3 a^1 \oo  a^2 \oo \ldots \oo a^n)\\
&=\pi_v \circ (\epsilon\oo 1_{H^{\oo n}}) \, (h \oo a^1\oo a^2 \oo \ldots \oo a^n),
\end{align*}

where we used first the definition of $\rhd_v'$, then the defining property of the antipode and that $S\circ S=1_H$, then the definition of $\rhd_v$, the fact that $\pi_v$ coequalises $\rhd_v$ and $\epsilon\oo 1_{H^{\oo E}}$ and then again the defining properties of the antipode and the counitality of $H$.

Inductively, we obtain (<ref>) for all positions of the cilium at $v$ and the same identity with $\pi_v,\rhd_v$ and $\pi'_v,\rhd'_v$ swapped. With the universal property of the coequalisers $\pi_v$, $\pi'_v$ this yields unique morphisms $\phi: M^H_v \to M'^H_v$, $\phi': M'^H_v\to M^H_v$ with $\phi\circ \pi_v=\pi_v'$ and $\phi'\circ \pi_v'=\pi_v$. As $\pi_v$, $\pi_v'$ are epimorphisms,  this implies $\phi'=\phi^\inv$.

The dual claim for the comodule structure and its coinvariants follows analogously. For all positions of the cilium at $f$ with associated coaction $\delta'_f$, there is a unique morphism $\psi: M^{coH}\to M'^{coH}$ with  $\iota'_f\circ \psi=\iota_f$, and $\psi$ is an isomorphism. Combining these statements for the (co)invariants of all vertices (faces) and using Lemmas
<ref> and
<ref>  yields isomorphisms of the protected objects.

§.§ Edge slides and edge contractions

We now consider the edge slides and edge contractions from Definition <ref>. Edge slides were already investigated in [35], where it was shown that they define   mapping class group actions. They yield automorphisms of the object $H^{\oo E}$ that are  morphisms of $H$-modules and $H$-comodules as long as no edge ends slide over cilia.

Let $\alpha\neq \beta$ be edges of $\Gamma$ with the starting end of $\alpha$  directly before the target end of $\beta$ in the ordering at $s(\alpha)=t(\beta)$.
The edge slide of the target end of $\beta$ along $\alpha$ corresponds to the isomorphism

$$S_{\alpha,\beta}:= \rhd_{\beta +}\circ \delta_{\alpha +}: H^{\oo E}\to H^{\oo E}\text{ with } S_{\alpha,\beta}^\inv=\rhd_{\beta +}\circ (S \oo 1_{H^{\oo E}})\circ \delta_{\alpha +}: H^{\oo E}\to H^{\oo E}.$$

Edge slides for other edge orientations are defined by reversing edge orientations with the antipode.

The isomorphisms induced by the edge slides

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(40.57,21.11) node [anchor=north west][inner sep=0.75pt]  [font=]  $\textcolor[rgb]{0.25,0.46,0.02}{\beta}$;

(15.71,19.97) node [anchor=north west][inner sep=0.75pt]  [font=]  $\gamma$;

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(-25, 40) node [anchor=north west][inner sep=0.75pt]  [font=]  (a);

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(229.14,66.54) node [anchor=north west][inner sep=0.75pt]  [font=]  $\nu$;
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are obtained by (a) applying Definition <ref>  and (b) first reversing the orientation of $\alpha$,  applying the inverse edge slide from Definition <ref> and then reversing the orientation of $\alpha$. This yields

\begin{align*}
&(a) &&S_{\alpha,\beta}(\alpha\oo \beta\oo \gamma \oo \delta\oo \mu\oo \nu)=\low \alpha 2\oo \low \alpha 1 \beta\oo \gamma \oo \delta\oo \mu\oo \nu,\\
&(b) & &S_{\alpha,\beta}(\alpha\oo \beta\oo \gamma \oo \delta\oo \mu\oo \nu)=\low \alpha 1\oo  \beta\oo \gamma \oo \low \alpha 2\delta\oo \mu\oo \nu.\\
\end{align*}

By construction, edge slides affect only the two copies of $H$  in $H^{\oo E}$ of the edges involved in the slide and  commute with  edge orientation reversals.
Moreover, they respect the module and comodule structures at vertices and faces and hence induce isomorphisms between the  protected objects.

Let $v$ and $f$ be a ciliated vertex and face in a ribbon graph $\Gamma$ with associated $H$-module structure $\rhd_v$ and $H$-comodule structure $\delta_f$. Any edge slide that does not slide edge ends over their cilia is an isomorphism of $H$-left modules and $H$-left comodules with respect to $\rhd_v$ and  $\delta_f$.

$ $

Edge slides
from a ribbon graph $\Gamma$ to a ribbon graph $\Gamma'$ induce isomorphisms between the invariants, coinvariants and  protected objects of $\Gamma$ and $\Gamma'$.

For edge slides that do not slide edge ends over cilia, this follows directly from Lemmas <ref>,
<ref> and Proposition <ref>. If an edge end slides over a cilium, we can apply Lemma <ref> to move the cilium and obtain the same  result.

We now consider edge contractions. Recall from  Definition <ref> that an edge $\alpha\in E$ may only be contracted if its starting and target vertex differ and that contracting $\alpha$ towards  $v\in\{s(\alpha),t(\alpha)\}$ erases the cilium at $v$, while the cilium at the other vertex is preserved.

The morphism $c_{\alpha,v}: H^{\oo E}\to H^{\oo (E-1)}$ induced by an edge contraction of an edge $\alpha$ towards $v\in \{s(\alpha), t(\alpha)\}$ is

\begin{align*}
\rhd_{v, \alpha}\circ \tau_{\alpha}\circ S_{\alpha} &\text{ if } v=t(\alpha)\\
\rhd_{v, \alpha} \circ \tau_{\alpha} &\text{ if } v=s(\alpha)
\end{cases}
\end{align*}

where $\rhd_{v, \alpha}: H^{\oo E}\to H^{\oo (E-1)}$ denotes the $H$-module structure  from Definition <ref> at $v$, where $\alpha$ is replaced by a cilium and $\tau_\alpha$ is given before Definition <ref>. If $v$ is univalent, then $c_{\alpha,v}=\epsilon_{\alpha}$.

Contracting the edge $\alpha$ towards $v$ in

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[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (101.28,38.58) .. controls (101.28,37.06) and (102.51,35.83) .. (104.03,35.83) .. controls (105.55,35.83) and (106.78,37.06) .. (106.78,38.58) .. controls (106.78,40.1) and (105.55,41.33) .. (104.03,41.33) .. controls (102.51,41.33) and (101.28,40.1) .. (101.28,38.58) – cycle ;

(106.7,17.15) – (104.31,33.85) ;
[shift=(104.03,35.83), rotate = 278.15] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (52.7,52.87) .. controls (50.5,52.02) and (49.82,50.5) .. (50.67,48.3) .. controls (51.52,46.1) and (50.84,44.58) .. (48.64,43.73) – (47.47,41.08) – (47.47,41.08) ;

(44.71,38.61) .. controls (11.95,38.89) and (32.97,7.31) .. (46.64,34.14) ;
[shift=(47.47,35.86), rotate = 245.47] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(50.29,40.43) .. controls (64.98,49.82) and (76.79,56.87) .. (100.93,42.08) ;
[shift=(102.42,41.15), rotate = 147.53] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[color=rgb, 255:red, 74; green, 144; blue, 226   ,draw opacity=1 ]   (226.42,38.61) .. controls (224.63,40.15) and (222.97,40.03) .. (221.43,38.24) .. controls (219.9,36.45) and (218.24,36.33) .. (216.45,37.87) .. controls (214.66,39.41) and (213,39.29) .. (211.46,37.5) – (211.07,37.47) – (211.07,37.47) ;

(228.42,53.15) – (212.76,41.15) ;
[shift=(211.17,39.94), rotate = 37.46] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(185.56,43.15) – (202.86,39.29) ;
[shift=(204.81,38.86), rotate = 167.42] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[fill=rgb, 255:red, 0; green, 0; blue, 0   ,fill opacity=1 ] (205.56,37.47) .. controls (205.56,35.95) and (206.79,34.72) .. (208.31,34.72) .. controls (209.83,34.72) and (211.07,35.95) .. (211.07,37.47) .. controls (211.07,38.99) and (209.83,40.22) .. (208.31,40.22) .. controls (206.79,40.22) and (205.56,38.99) .. (205.56,37.47) – cycle ;

(225.28,24.3) – (212.25,33.3) ;
[shift=(210.6,34.43), rotate = 325.36] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(205.56,35.44) .. controls (172.8,35.72) and (193.82,6.09) .. (207.49,33) ;
[shift=(208.31,34.72), rotate = 245.47] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

(205.56,40.58) .. controls (185.05,64.05) and (215.67,62.67) .. (208.91,41.86) ;
[shift=(208.31,40.22), rotate = 68.16] [color=rgb, 255:red, 0; green, 0; blue, 0   ][line width=0.75]    (6.56,-1.97) .. controls (4.17,-0.84) and (1.99,-0.18) .. (0,0) .. controls (1.99,0.18) and (4.17,0.84) .. (6.56,1.97)   ;

[color=rgb, 255:red, 128; green, 128; blue, 128   ,draw opacity=1 ]   (142.7,38.13) – (160.56,38.03) ;
[shift=(163.56,38.01), rotate = 179.67] [fill=rgb, 255:red, 128; green, 128; blue, 128   ,fill opacity=1 ][line width=0.08]  [draw opacity=0] (8.93,-4.29) – (0,0) – (8.93,4.29) – cycle    ;

(65.99,22.49) node [anchor=north west][inner sep=0.75pt]  [font=]  $\textcolor[rgb]{0.82,0.01,0.11}{\alpha }$;

(47.14,56.83) node [anchor=north west][inner sep=0.75pt]  [font=]  $\textcolor[rgb]{0.29,0.56,0.89}{v}$;

(23.43,45.65) node [anchor=north west][inner sep=0.75pt]  [font=]  $b$;

(66.57,51.94) node [anchor=north west][inner sep=0.75pt]  [font=]  $c$;

(14.86,18.8) node [anchor=north west][inner sep=0.75pt]  [font=]  $d$;

(108.86,14.8) node [anchor=north west][inner sep=0.75pt]  [font=]  $k$;

(123.43,27.4) node [anchor=north west][inner sep=0.75pt]  [font=]  $\textcolor[rgb]{0.29,0.56,0.89}{w}$;

(121.14,53.69) node [anchor=north west][inner sep=0.75pt]  [font=]  $l$;

(175.71,36.54) node [anchor=north west][inner sep=0.75pt]  [font=]  $b$;