# Cyclic objects from surfaces

Ivan Bartulović  Technische Universität Dresden, Institut für Geometrie,
Zellescher Weg 12–14, 01062 Dresden, Germany<EMAIL_ADDRESS>
###### Abstract.

In this paper, we endow the family of closed oriented genus $g$ surfaces,
starting with torus, with a structure of a (co)cyclic object in the category
of $3$-dimensional cobordisms. As a corollary, any $3$-dimensional TQFT
induces a (co)cyclic module, which we compute algebraically for the
Reshetikhin-Turaev TQFT.

_Keywords_. Cyclic objects, cobordisms, topological quantum field theories
(TQFTs).

2020 _Mathematics Subject Classification_. 18N50, 57K16.

###### Contents

  1. 1 Introduction
  2. 2 Cyclic objects
  3. 3 $3$-cobordisms
  4. 4 Cyclic objects from surfaces
  5. 5 Preliminaries on monoidal categories and Hopf algebras
  6. 6 Cyclic modules from (co)algebras
  7. 7 Cyclic modules from TQFTs
  8. 8 Related work
  9. 9 Appendix

##  1\. Introduction

In this paper, we study cyclic objects and their interplay with topological
field theories. A (co)cyclic object in a category is, roughly speaking, a
(co)simplicial object with compatible actions of the cyclic groups. Cyclic
homology of algebras was independently introduced by Connes [Co85] and Tsygan
[Ts83]. To any algebra is associated a cocyclic vector space, that is, a
cocyclic object in the category of vector spaces. This construction was
generalized to braided setting by Akrami and Majid [AM04], who associate a
cocyclic vector space to any ribbon algebra in a braided monoidal category.

On the other hand, compact surfaces are relatively well-understood
$2$-manifolds which appear in many areas of mathematics. In particular, closed
oriented surfaces are objects of a symmetric monoidal category of
$3$-dimensional cobordisms. A morphism between two surfaces is given by a
homeomorphism class of $3$-cobordisms between the given surfaces. The category
of $3$-cobordisms is of great interest in quantum topology. Introduced by
Atiyah [At89], a $3$-dimensional topological quantum field theory (or shortly,
TQFT) is a strong symmetric monoidal functor from the category of
$3$-cobordisms to vector spaces. A fundamental construction of a
$3$-dimensional TQFT in this sense is the Reshetikhin-Turaev TQFT [RT91,
Tu10].

The main result of this paper is that the family of closed oriented surfaces
has a structure of a (co)cyclic object in the category of $3$-dimensional
cobordisms (see Theorem 4.1). As a corollary, any $3$-dimensional TQFT induces
a (co)cyclic vector space. We calculate this (co)cyclic vector space for the
Reshetikhin-Turaev TQFT (see Theorem 7.1). In a certain sense, the (co)cyclic
objects obtained from closed oriented surfaces universally calculate the
cyclic (co)homology of (co)cyclic modules (à la Akrami-Majid [AM04] and its
variants) associated to the coend (which is a braided Hopf algebra) of anomaly
free modular categories. Finally, we discuss some potentially related work in
the setting of category of connected cobordisms, which first appeared in
[Ke97] and is different from the cobordism category used throughout the paper.
For instance, it is a non-symmetric braided category. In this setting, we
outline a construction of the so called para(co)cyclic objects associated to
the one-holed torus (see sections 8.2 and 8.3). By composition, the braided
monoidal functor $J_{3}$ from [BDR23] universally induces para(co)cyclic
objects associated to the end of unimodular ribbon factorizable categories.

The paper is organized as follows. In Section 2, we recall the notion of a
(co)cyclic object in a category. In Section 3, we recall some facts about
$3$-cobordisms and their presentation via special ribbon graphs. In Section 4,
we construct (co)cyclic objects in the category of $3$-dimensional cobordisms.
In Section 5, we review ribbon categories and their graphical calculus,
braided Hopf algebras, and related concepts. Section 6 is dedicated to
(co)cyclic modules from categorical (co)algebras. In Section 7 we relate, via
the Reshetikhin-Turaev TQFT, the (co)cyclic objects from surfaces with
(co)cyclic modules associated to the coend of an anomaly free modular
category. In Section 8, we discuss paracyclic objects in the category of
connected cobordisms. In Appendix, we recall some known facts about (co)ends
of a braided pivotal category, thought as universal (co)modules.

Unless otherwise stated, by $\Bbbk$ we denote any commutative ring. The class
of objects in a category $\mathcal{C}$ is denoted by
$\mathrm{Ob}(\mathcal{C})$.

### Acknowledgments

Most of the content of this paper is part of the author’s PhD thesis. The
author would like to thank his supervisor Alexis Virelizier for his comments
and advice. He would also like to thank the Laboratoire Paul Painlevé of the
University of Lille and the Institut für Geometrie of the TU Dresden for their
hospitality. This work was supported by the Labex CEMPI (ANR-11-LABX-0007-01),
by the Région Hauts-de-France, and by the FNS-ANR OChoTop grant
(ANR-18-CE93-0002-01).

##  2\. Cyclic objects

In this section we recall the notions of (co)simplicial and (co)cyclic objects
in a category.

###  2.1. The simplicial category

The _simplicial category_ $\Delta$ is defined as follows. The objects of
$\Delta$ are the non-negative integers. For $n\in\mathbb{N}$, denote
$[n]=\\{0,\dots,n\\}$. A morphism $n\to m$ in $\Delta$ is an increasing map
between sets $[n]$ and $[m]$. For $n\in\mathbb{N}^{*}$ and $0\leq i\leq n$,
the $i$-th _coface_ $\delta_{i}^{n}\colon n-1\to n$ is the unique increasing
injection from $[n-1]$ into $[n]$ which misses $i$. For $n\in\mathbb{N}$ and
$0\leq j\leq n$, the $j$-th  _codegeneracy_ $\sigma_{j}^{n}\colon n+1\to n$ is
the unique increasing surjection from $[n+1]$ onto $[n]$ which sends both $j$
and $j+1$ to $j$.

It is well known (see [MCL95, Lemma 5.1]) that morphisms in $\Delta$ are
generated by cofaces $\\{\delta_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$
and codegeneracies $\\{\sigma_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$
subject to the simplicial relations:

$\displaystyle\delta_{j}^{n+1}\delta_{i}^{n}$
$\displaystyle=\delta_{i}^{n+1}\delta_{j-1}^{n}\text{\quad for all }0\leq
i<j\leq n+1,$ (1) $\displaystyle\sigma_{j}^{n}\sigma_{i}^{n+1}$
$\displaystyle=\sigma_{i}^{n}\sigma_{j+1}^{n+1}\text{\quad for all }0\leq
i\leq j\leq n,$ (2) $\displaystyle\sigma_{j}^{n}\delta_{i}^{n+1}$
$\displaystyle=\begin{cases}\delta_{i}^{n}\sigma_{j-1}^{n-1}&\text{for all
}0\leq i<j\leq n,\\\ \mathrm{id}_{n}&\text{for all }0\leq i=j\leq n\text{ or
}1\leq i=j+1\leq n+1,\\\ \delta_{i-1}^{n}\sigma_{j}^{n-1}&\text{for all }1\leq
j+1<i\leq n+1.\end{cases}$ (3)

In the opposite category $\Delta^{\mathrm{op}}$, every coface $\delta_{i}^{n}$
and every codegeneracy $\sigma_{j}^{n}$ are respectively denoted by

$d_{i}^{n}\colon n\to n-1\quad\text{and}\quad s_{j}^{n}\colon n\to n+1.$

The morphisms $\\{d_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$ are called
faces and the morphisms $\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$ are
called degeneracies.

###  2.2. The cyclic category

The _cyclic category_ $\Delta C$ is defined as follows. The objects of $\Delta
C$ are the non-negative integers. The morphisms in this category are generated
by morphisms $\\{\delta_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, called
cofaces, morphisms $\\{\sigma_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$,
called codegeneracies, and isomorphisms $\\{\tau_{n}\colon n\to
n\\}_{n\in\mathbb{N}}$, called  _cocyclic operators_ , satisfying the
simplicial relations and additionally:

$\displaystyle\tau_{n}\delta_{i}^{n}=\delta_{i-1}^{n}\tau_{n-1}\text{\quad for
all }1\leq i\leq n,$ (4)
$\displaystyle\tau_{n}\delta_{0}^{n}=\delta_{n}^{n}\quad\text{for all }n\geq
1,$ (5)
$\displaystyle\tau_{n}\sigma_{i}^{n}=\sigma_{i-1}^{n}\tau_{n+1}\text{\quad for
all }1\leq i\leq n,$ (6)
$\displaystyle\tau_{n}\sigma_{0}^{n}=\sigma_{n}^{n}\tau_{n+1}^{2}\quad\text{for
all }n\geq 0\quad\text{and }$ (7)
$\displaystyle\tau_{n}^{n+1}=\mathrm{id}_{n}\quad\text{for all}\quad
n\in\mathbb{N}.$ (8)

Note that $\tau_{0}=\mathrm{id}_{0}$. In the opposite category $\Delta
C^{\mathrm{op}}$, every coface $\delta_{i}^{n}$, every codegeneracy
$\sigma_{j}^{n}$, and every cocyclic operator $\tau_{n}$ are respectively
denoted by

$d_{i}^{n}\colon n\to n-1,\quad s_{j}^{n}\colon n\to n+1,\quad\text{and}\quad
t_{n}\colon n\to n.$

The morphisms $\\{d_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$ are called
faces, the morphisms $\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$ are
called degeneracies, and the morphisms $\\{t_{n}\\}_{n\in\mathbb{N}}$ are
called cyclic operators.

###  2.3. (Co)simplicial and (co)cyclic objects in a category

Let $\mathcal{C}$ be any category. A simplicial object in $\mathcal{C}$ is a
functor $\Delta^{\mathrm{op}}\to\mathcal{C}$ and a  _cyclic object_ in
$\mathcal{C}$ is a functor $\Delta C^{\mathrm{op}}\to\mathcal{C}.$ Dually, a
_cosimplicial object_ in $\mathcal{C}$ is a functor $\Delta\to\mathcal{C}$ and
a  _cocyclic object_ in $\mathcal{C}$ is a functor $\Delta C\to\mathcal{C}$. A
(co)simplicial/(co)cyclic object in the category of $\Bbbk$-modules is called
a (co)simplicial/(co)cyclic $\Bbbk$-module. A morphism between two
(co)simplicial/(co)cyclic objects is a natural transformation between them.
For shortness, one often denotes the image of a morphism $f$ under a
(co)simplicial/(co)cyclic object in $\mathcal{C}$ by the same letter $f$.

Since the categories $\Delta$ and $\Delta C$ are defined by generators and
relations, a (co)simplicial/(co)cyclic object in a category is entirely
determined by the images of the generators satisfying the corresponding
relations. For example, a cocyclic object $X$ in $\mathcal{C}$ may be
described explicitly as a family $X^{\bullet}=\\{X^{n}\\}_{n\in\mathbb{N}}$ of
objects in $\mathcal{C}$, equipped with morphisms $\\{\delta_{i}^{n}\colon
X^{n-1}\to X^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, called cofaces,
morphisms $\\{\sigma_{j}^{n}\colon X^{n+1}\to X^{n}\\}_{n\in\mathbb{N},0\leq
j\leq n}$, called codegeneracies, and isomorphisms $\\{\tau_{n}\colon X^{n}\to
X^{n}\\}_{n\in\mathbb{N}}$, called cocyclic operators, which satisfy (1)-(8).
Note that $\tau_{0}=\mathrm{id}_{X^{0}}$. A morphism $\alpha^{\bullet}\colon
X^{\bullet}\to Y^{\bullet}$ between cocyclic objects $X^{\bullet}$ and
$Y^{\bullet}$ in $\mathcal{C}$ is then described by a family
$\alpha^{\bullet}=\\{\alpha^{n}\colon X^{n}\to Y^{n}\\}_{n\in\mathbb{N}}$ of
morphisms in $\mathcal{C}$ such that

$\displaystyle\delta_{i}^{n}\alpha^{n-1}$
$\displaystyle=\alpha^{n}\delta_{i}^{n}\quad\text{for all
}n\in\mathbb{N}^{*}\text{ and }0\leq i\leq n,$ (9)
$\displaystyle\sigma_{j}^{n}\alpha^{n+1}$
$\displaystyle=\alpha^{n}\sigma_{j}^{n}\quad\text{for all
}n\in\mathbb{N}\text{ and }0\leq j\leq n,$ (10)
$\displaystyle\alpha^{n}\tau_{n}$
$\displaystyle=\tau_{n}\alpha^{n}\quad\text{for all }n\in\mathbb{N}.$ (11)

Similarly as above, a cyclic object $X$ in $\mathcal{C}$ may be seen as a
family $X_{\bullet}=\\{X_{n}\\}_{n\in\mathbb{N}}$ of objects in $\mathcal{C}$
equipped with morphisms $\\{d_{i}^{n}\colon X_{n}\to
X_{n-1}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, called faces, morphisms
$\\{s_{j}^{n}\colon X_{n}\to X_{n+1}\\}_{n\in\mathbb{N},0\leq j\leq n}$,
called degeneracies, and isomorphisms $\\{t_{n}\colon X_{n}\to
X_{n}\\}_{n\in\mathbb{N}}$, called cyclic operators, which satisfy the
corresponding relations in $\Delta C^{\mathrm{op}}$. It follows that
$t_{0}=\mathrm{id}_{X_{0}}$. Also, a morphism $\alpha_{\bullet}\colon
X_{\bullet}\to Y_{\bullet}$ between two cyclic objects $X_{\bullet}$ and
$Y_{\bullet}$ in $\mathcal{C}$ is described by a family
$\alpha_{\bullet}=\\{\alpha_{n}\colon X_{n}\to Y_{n}\\}_{n\in\mathbb{N}}$ of
morphisms in $\mathcal{C}$ commuting with faces, degeneracies and cyclic
operators of $X_{\bullet}$ and $Y_{\bullet}$.

###  2.4. Cyclic duality

It is well-known that cyclic category is isomorphic to its opposite category.
The isomorphism established by Connes in [Co83] is called cyclic duality. In
its version due to Loday [Lo98, Proposition 6.1.11], the cyclic duality
$L\colon\Delta C^{\mathrm{op}}\to\Delta C$ is identity on objects and it is
defined on morphisms as follows. For $n\in\mathbb{N}^{*}$ and $0\leq i\leq n$,

$L(d_{i}^{n})=\begin{cases}\sigma_{i}^{n-1}&\text{if }0\leq i\leq n-1,\\\
\sigma_{0}^{n-1}\tau_{n}^{-1}&\text{if }i=n,\end{cases}$

and for $n\in\mathbb{N}$ and $0\leq j\leq n$,

$L(s_{j}^{n})=\delta_{j+1}^{n+1}\quad\text{and}\quad L(t_{n})=\tau_{n}^{-1}.$

Given a category $\mathcal{C}$, the cyclic duality transforms any cocyclic
object $X\colon\Delta C\to{\mathcal{C}}$ in $\mathcal{C}$ into the cyclic
object $XL\colon\Delta C^{\mathrm{op}}\to\mathcal{C}$. Similarly, the opposite
functor $L^{\mathrm{op}}$ turns any cyclic object $Y\colon\Delta
C^{\mathrm{op}}\to\mathcal{C}$ in $\mathcal{C}$ into the cocyclic object
$YL^{\mathrm{op}}\colon\Delta C\to{\mathcal{C}}.$

###  2.5. Reindexing involution automorphism

Following Loday [Lo98, Section 6.1.14], we recall the reindexing involution
automorphism $\Phi$ of the cyclic category. It is identity on objects and it
is defined on morphisms as follows. For $n\in\mathbb{N}^{*}$ and $0\leq i\leq
n$,

$\Phi(\delta_{i}^{n})=\delta_{n-i}^{n},$

and for $n\in\mathbb{N}$ and $0\leq j\leq n$,

$\Phi(\sigma_{j}^{n})=\sigma_{n-j}^{n}\quad\text{and}\quad\Phi(\tau_{n})=\tau_{n}^{-1}.$

##  3\. $3$-cobordisms

In this section we recall some facts about the category $\textbf{Cob}_{3}$ of
$3$-dimensional cobordisms (or shortly, $3$-cobordisms) and their surgery
presentations via ribbon graphs. We denote by $D^{n}$ the closed unit ball in
$\mathbb{R}^{n}$. The $n$-dimensional sphere is denoted by $S^{n}$.

###  3.1. $3$-cobordisms

A  _$3$ -cobordism_ is a quadruple $(M,h,\Sigma_{0},\Sigma_{1})$, where $M$ is
a compact oriented $3$-manifold, $\Sigma_{0}$ and $\Sigma_{1}$ are two closed
oriented surfaces, and $h$ is an orientation preserving homeomorphism
$h\colon(-\Sigma_{0})\sqcup\Sigma_{1}\to\partial M$. The surface $\Sigma_{0}$
is called the bottom base and the surface $\Sigma_{1}$ is called the top base
of the cobordism $M$. Two cobordisms $(M,h,\Sigma_{0},\Sigma_{1})$ and
$(M^{\prime},h^{\prime},\Sigma_{0},\Sigma_{1})$ are homeomorphic, if there is
an orientation preserving homeomorphism $g\colon M\to M^{\prime}$ such that
$h=h^{\prime}g_{|\partial M}$. When clear, we will denote a cobordism
$(M,h,\Sigma_{0},\Sigma_{1})$ only by $M$.

The composition of two cobordisms $(M_{0},h_{0},\Sigma_{0},\Sigma_{1})$ and
$(M_{1},h_{1},\Sigma_{1},\Sigma_{2})$ is the cobordism
$(M,h,\Sigma_{0},\Sigma_{2})$, where $M$ is obtained by gluing $M_{0}$ to
$M_{1}$ along $h_{1}{h_{0}}^{-1}\colon h_{0}(\Sigma_{1})\to h_{1}(\Sigma_{1})$
and the homeomorphism $h$ is given by

$h=h_{0}|_{\Sigma_{0}}\sqcup
h_{1}|_{\Sigma_{2}}\colon(-\Sigma_{0})\sqcup\Sigma_{2}\to\partial M.$

We say that cobordism $M$ is obtained by gluing cobordisms $M_{0}$ and $M_{1}$
along $\Sigma_{1}$.

###  3.2. The category $\textbf{Cob}_{3}$

The category $\textbf{Cob}_{3}$ of $3$-cobordisms is defined as follows. The
objects are closed oriented surfaces. A morphism
$f\colon\Sigma_{0}\to\Sigma_{1}$ in $\textbf{Cob}_{3}$ is a homeomorphism
class of cobordisms between $\Sigma_{0}$ and $\Sigma_{1}$. In
$\textbf{Cob}_{3}$, the identity of a closed oriented surface $\Sigma$ is
represented by _identity cobordism_ $(C_{\Sigma},e,\Sigma,\Sigma)$, where
$C_{\Sigma}=\Sigma\times[0,1]$ is a cylinder over $\Sigma$ together with
product orientation, and $e\colon(-\Sigma)\sqcup\Sigma\to\partial C_{\Sigma}$
is the homeomorphism with $e|_{-\Sigma}(x,0)=(x,0)$ and
$e|_{\Sigma}(x,1)=(x,1)$. The composition of morphisms
$\Sigma_{0}\to\Sigma_{1}$ and $\Sigma_{1}\to\Sigma_{2}$ in $\textbf{Cob}_{3}$,
represented respectively by cobordisms $M$ and $N$, is represented by
cobordism obtained by gluing cobordisms $M$ and $N$ along $\Sigma_{1}$. The
category $\textbf{Cob}_{3}$ is symmetric monoidal (see Section 5.2). The
monoidal product is given by disjoint union and the monoidal unit is the empty
surface. For more details, see [TV17].

###  3.3. Surgery presentation of closed 3-manifolds

Let $L$ be an $n$-component framed link in the $3$-sphere $S^{3}$. Pick a
closed tubular neighborhood $N_{L}$ of $L$. Since $N_{L}$ is homeomorphic to
$\sqcup_{i=1}^{n}S^{1}\times D^{2}$, the boundary of the $3$-manifold
$S^{3}\setminus\mathrm{Int}(N_{L})$ is homeomorphic to the disjoint union of
$n$-tori $S^{1}\times S^{1}$. The Dehn surgery on $S^{3}$ along $L$ is the
closed manifold

$S^{3}_{L}=(S^{3}\setminus
N_{L})\bigcup_{\phi}\left(\sqcup_{i=1}^{n}D^{2}\times S^{1}\right),$

where
$\phi\colon\partial(S^{3}\setminus\mathrm{Int}(N_{L}))\to\sqcup_{i=1}^{n}S^{1}\times
S^{1}$ is a homeomorphism exchanging meridians and parallels. Any connected,
oriented, closed $3$-manifold $M$ is, according to the Lickorish’s theorem
[Li97, Chapter 12], homeomorphic to $S^{3}_{L}$ for some framed link $L\subset
S^{3}$.

###  3.4. Ribbon graphs

A  _circle_ is a $1$-manifold homeomorphic to $S^{1}$. An _arc_ is a
$1$-manifold homeomorphic to the closed interval $[0,1]$. The boundary points
of an arc are called its _endpoints_. A _rectangle_ is a $2$-manifold with
corners homeomorphic to $[0,1]\times[0,1]$. The four corner points of a
rectangle split its boundary into four arcs called the  _sides_. A  _coupon_
is an oriented rectangle with a distinguished side called the  _bottom base_ ,
the opposite side being the  _top base_.

A  _plexus_ is a topological space obtained from a disjoint union of a finite
number of oriented circles, oriented arcs, and coupons by gluing some
endpoints of the arcs to the bases of the coupons. It is required that
different endpoints of the arcs are never glued to the same point of a (base
of a) coupon. The endpoints of the arcs that are not glued to coupons are
called  _free ends_. The set of free ends of a plexus $\gamma$ is denoted by
$\partial\gamma$.

Given non-negative integers $g$ and $h$, a ribbon $(g,h)$-graph $\Gamma$ is a
plexus $\Gamma$ embedded in $\mathbb{R}^{2}\times[0,1]$ and equipped with a
framing such that

$\partial\Gamma=\Gamma\cap\partial(\mathbb{R}^{2}\times[0,1])=\\{(1,0,0),\dots,(g,0,0)\\}\cup\\{(1,0,1),\dots,(h,0,1)\\}$

and such that the arcs of $\Gamma$ are transverse to
$\partial(\mathbb{R}^{2}\times[0,1])$ at all points of $\partial\Gamma$. The
free end $(i,0,0)$ is called the  _$i$ -th input_ and the free end $(j,0,1)$
is called the  _$j$ -th output_ of $\Gamma$. For example, ribbon graphs
without free ends and without coupons are nothing but framed oriented links in
$\mathbb{R}^{2}\times(0,1)\cong\mathbb{R}^{3}$.

We represent ribbon graphs by plane diagrams with blackboard framing. We
require that for each coupon, its orientation is that of the plane, its bases
are horizontal, and its bottom base is below its top base. By Reidemeister
theorem, two diagrams represent isotopic ribbon graphs if and only if they are
related by a finite sequence of plane isotopies, ribbon Reidemeister moves
R1-R3

0.85RI1.eps $\leftrightarrow$ 0.85RI3.eps $\leftrightarrow$0.85RI2.eps ,
$\textbf{R}1$ 0.85RII1.eps $\leftrightarrow$ 0.85RII2.eps
$\leftrightarrow$0.85RII3.eps , $\textbf{R}2$ 0.85RIII1.eps $\leftrightarrow$
0.85RIII2.eps , $\textbf{R}3$

and the following move:

$\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{slide1.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{slide2.eps}}}\;.$

###  3.5. Standard ribbon graphs and surfaces

For $g\geq 0$, consider the ribbon graph

$G_{g}^{+}=\raisebox{-0.45pt}{\scalebox{0.7}{\includegraphics{mirribbongraphRg.eps}}}$

consisting of one coupon and $g$ unknotted untwisted cups oriented from right
to left successively attached to the bottom base of the coupon. We fix a
closed regular neighborhood
$H_{g}^{+}\subset\mathbb{R}^{2}\times\left[0,1\right]$ of $G_{g}^{+}$. This is
a handlebody of genus $g$ and is provided with the right-handed orientation.
Consider also the ribbon graph

$G_{g}^{-}=\raisebox{-0.45pt}{\scalebox{0.7}{\includegraphics{ribbongraphRg.eps}}}$

consisting of one coupon and $g$ unknotted untwisted caps oriented from left
to right successively attached to the top base of the coupon. We fix a closed
regular neighborhood $H_{g}^{-}\subset\mathbb{R}^{2}\times\left[0,1\right]$ of
$G_{g}^{-}$. This is a handlebody of genus $g$ and is provided with the right-
handed orientation. We choose the neighborhoods $H_{g}^{+}$ and $H_{g}^{-}$ so
that the mirror reflection of $\mathbb{R}^{3}$ with respect to the line
$\mathbb{R}^{2}\times\\{\frac{1}{2}\\}$ induces an orientation preserving
homeomorphism between $-(H_{g}^{-})$ and $H_{g}^{+}$, where $-(H_{g}^{-})$ is
$H_{g}^{-}$ with opposite orientation.

The boundary $S_{g}$ of $H_{g}^{+}$ is a closed connected oriented surface of
genus $g$ called the standard surface of genus $g$. The orientation of $S_{g}$
is induced by that of $H_{g}^{+}$. (We use the “outward vector first”
convention for the induced orientation of the boundary.) The above mirror
reflection induces a canonical orientation preserving homeomorphism between
$\partial(H_{g}^{-})$ and $-S_{g}$.

###  3.6. Special ribbon graphs and $3$-cobordisms

Let $g$ and $h$ be non-negative integers. A _special ribbon $(g,h)$-graph_ is
a ribbon $(2g,2h)$-graph $\Gamma$ with no coupons such that:

  * •

for all $1\leq i\leq g$, the $(2i-1)$-th and $2i$-th inputs of $\Gamma$ are
connected by an arc oriented from the $(2i-1)$-th input to the $2i$-th input,

  * •

for all $1\leq j\leq h$, the $(2j-1)$-th and $2j$-th outputs of $\Gamma$ are
connected by an arc oriented from the $2j$-th output to the $(2j-1)$-th
output.

Any special ribbon $(g,h)$-graph $\Gamma$ gives rise to a connected
$3$-cobordism $M_{\Gamma}$ between the standard surfaces $S_{g}$ and $S_{h}$
(see Section 3.5). It is defined as follows. Attach coupons

$Q^{-}=\left[0,2g+1\right]\times\\{0\\}\times\left[-1,0\right]\quad\text{and}\quad
Q^{+}=\left[0,2h+1\right]\times\\{0\\}\times\left[1,2\right]$

to the bottom and the top of $\Gamma$, respectively. The result is a ribbon
graph without free ends $\tilde{\Gamma}$ in
$\mathbb{R}^{3}\cong\mathbb{R}^{2}\times(0,1)$ with no free ends, two coupons
$Q^{\pm}$, and finitely many circle components which form a framed link $L$ in
$S^{3}$. The arcs connecting the inputs of $\Gamma$ become caps attached on
the top base of $Q^{-}$, so there is an embedding $f^{-}\colon
G_{g}^{-}\to\tilde{\Gamma}$ mapping the coupon of $G_{g}^{-}$ to $Q^{-}$ and
mapping the caps attached to $G_{g}^{-}$ to those attached to $Q^{-}$.
Similarly, the arcs connecting the outputs of $\Gamma$ become cups attached on
the bottom base of $Q^{+}$, so there is an embedding $f^{+}\colon
G_{h}^{+}\to\tilde{\Gamma}$ mapping the coupon of $G_{h}^{+}$ to $Q^{+}$ and
mapping the cups attached to $G_{h}^{+}$ to those attached to $Q^{+}$.
Consider a tubular neighborhood $N_{L}$ of $L$ and embeddings
$\tilde{f}^{-}\colon H_{g}^{-}\to S^{3}\setminus N_{L}$ and
$\tilde{f}^{+}\colon H_{h}^{+}\to S^{3}\setminus N_{L}$ respectively extending
$f^{-}$ and $f^{+}$. Let $S^{3}_{L}$ be the Dehn surgery of $S^{3}$ along $L$
(see Section 3.3). The manifold

$S^{3}_{\Gamma}=S^{3}_{L}\setminus\left(\tilde{f}^{-}(\mathrm{Int}(H_{g}^{-}))\cup\tilde{f}^{+}(\mathrm{Int}(H_{h}^{+}))\right)$

is a connected oriented compact $3$-manifold. By Section 3.5, its boundary
$\tilde{f}^{-}(\partial(H_{g}^{-}))\sqcup\tilde{f}^{+}(\partial(H_{h}^{+}))$
is canonically homeomorphic to $(-S_{g})\sqcup S_{h}$. This gives rise to a
connected $3$-cobordism $M_{\Gamma}\colon S_{g}\to S_{h}$.

For example, by [Tu10, Chapter IV, Lemma 2.6.], the identity cobordism of the
standard surface $S_{g}$ (see Section 3.2) is represented by the following
special ribbon $(g,g)$-graph

$I_{g}=\,\raisebox{-0.55pt}{\scalebox{0.7}{\includegraphics{cylcansurf.eps}}}\;.$
(12)

The following lemma gives a presentation of the composition of 3-cobordisms
between standard surfaces.

######  Lemma 3.1 ([Tu10, Section IV.2.3]).

If $\Gamma$ is a special ribbon $(g,h)$-graph and $\Gamma^{\prime}$ is a
special ribbon $(h,k)$-graph, then the composition of $3$-cobordisms
$M_{\Gamma}\colon S_{g}\to S_{h}$ and $M_{\Gamma^{\prime}}\colon S_{h}\to
S_{k}$ is then the $3$-cobordism $M_{\Gamma^{\prime}\circ\Gamma}\colon
S_{g}\to S_{k}$, where $\Gamma^{\prime}\circ\Gamma$ is the special ribbon
$(g,k)$-graph obtained by stacking $\Gamma^{\prime}$ over $\Gamma$.

Note that the connectedness of top base of $M_{\Gamma}$ and bottom base of
$M_{\Gamma}^{\prime}$ is here important. For the case of $3$-cobordisms with
non-connected bases, see [Tu10, Section IV.2.8].

###  3.7. Extended Kirby calculus

Recall from Section 3.3 that any closed oriented 3-manifold can be obtained by
surgery of $S^{3}$ along a framed link. Kirby proved [Ki78] that two framed
links represent the same $3$-manifold (up to an orientation preserving
homeomorphism) if and only if they are related by a finite sequence of
isotopies and of the Kirby moves $\textbf{K}1$ and $\textbf{K}2$. The move
$\textbf{K}1$ consists in adding an unknot with framing number (which is the
self-linking number) $1$ or $-1$:

$L\sqcup\raisebox{-0.4pt}{\scalebox{0.7}{\includegraphics{MK11.eps}}}\longleftrightarrow
L\longleftrightarrow
L\sqcup\raisebox{-0.4pt}{\scalebox{0.7}{\includegraphics{MK1.eps}}}\hskip
2.84544pt.$

The move $\textbf{K}2$ consists in sliding a component over another component.
More precisely, given two distinct components $L_{i}$ and $L_{j}$ of a framed
link, this move replaces $L_{i}$ by the connected sum $L_{i}\\#L_{j}^{\prime}$
of $L_{i}$ with a copy $L^{\prime}_{j}$ of $L_{j}$ obtained by slightly
pushing $L_{j}$ along its framing. For example, sliding an unknot with framing
number $0$ over an unknot with framing number $1$ can be depicted as:

$\,\raisebox{-0.6pt}{\scalebox{0.7}{\includegraphics{K2example.eps}}}\;\longrightarrow\,\raisebox{-0.6pt}{\scalebox{0.7}{\includegraphics{K2example1.eps}}}\;.$

It follows from Reidemeister and Kirby theorems that the Kirby calculus
extends to the representation of 3-cobordisms by special ribbon graphs (see
[Tu10, Chapter I, Lemma 3.4]). More explicitly, special ribbon graphs
represent the same $3$-cobordism (up to an orientation preserving
homeomorphism) if and only if they are related by a finite sequence of
isotopies and of the following moves: the move $\textbf{K}1$, the generalized
Kirby move $\textbf{K}2^{\prime}$, the move OR, the move COUPON, and the move
TWIST. The move $\textbf{K}2^{\prime}$ consists in sliding an arc or circle
component of a special ribbon graph over a distinct circle component. The move
OR consists in reversing the orientation of a circle component. The COUPON
move consists in changing the type of a crossing of a component passing over
(or under) all its outputs:

$\raisebox{-0.4pt}{\scalebox{0.75}{\includegraphics{couponplusgore.eps}}}\;\hskip
22.76228pt\longleftrightarrow\,\raisebox{-0.4pt}{\scalebox{0.75}{\includegraphics{couponminusgore.eps}}}\;\hskip
28.45274pt,$

or all its inputs:

$\raisebox{-0.4pt}{\scalebox{0.75}{\includegraphics{couponplusdole.eps}}}\;\hskip
22.76228pt\longleftrightarrow\,\raisebox{-0.4pt}{\scalebox{0.75}{\includegraphics{couponminusdole.eps}}}\;\hskip
24.18501pt.$

The move TWIST consists in a simultaneous twist of the output components:

$\raisebox{-0.4pt}{\scalebox{0.65}{\includegraphics{zatvistangore.eps}}}\hskip
5.69046pt\;\longleftrightarrow\,\raisebox{-0.4pt}{\scalebox{0.65}{\includegraphics{normalangore.eps}}}\;\hskip
11.38092pt\;\longleftrightarrow\,\raisebox{-0.4pt}{\scalebox{0.65}{\includegraphics{zatvistangoreminus.eps}}}\hskip
5.69046pt,$

or the input components:

$\,\raisebox{-0.4pt}{\scalebox{0.65}{\includegraphics{zatvistandoleminus.eps}}}\;\hskip
5.69046pt\longleftrightarrow\,\raisebox{-0.4pt}{\scalebox{0.65}{\includegraphics{normalandole.eps}}}\;\hskip
5.69046pt\;\longleftrightarrow\,\raisebox{-0.4pt}{\scalebox{0.65}{\includegraphics{zatvistandoleplus.eps}}}\;.$

##  4\. Cyclic objects from surfaces

The first main result of this paper is that closed oriented surfaces can be
organized in a (co)cyclic object in the category $\textbf{Cob}_{3}$ of
$3$-dimensional cobordisms:

######  Theorem 4.1.

For $g\geq 1$, consider a closed oriented surface $\Sigma_{g}$ of genus $g$.
Then the family $\\{\Sigma_{g}\\}_{g\geq 1}$ has a structure of a cocyclic
object $X^{\bullet}$ in _Cob_ 3 and a structure of a cyclic object
$X_{\bullet}$ in _Cob_ 3.

We construct (co)cyclic objects in $\textbf{Cob}_{3}$ by means of surgery
presentation of $3$-cobordisms developed in [RT91, Tu10] and reviewed in the
Section 3. We prove Theorem 4.1 in sections 4.1–4.3. First, in Section 4.1 we
construct the functor $Y^{\bullet}\colon\Delta C\to\textbf{Cob}_{3}$. Next, in
Section 4.2, we construct the functor $Y_{\bullet}\colon\Delta
C^{\mathrm{op}}\to\textbf{Cob}_{3}$. Cobordisms in both of these constructions
have standard surfaces (see Section 3.5) as bases. Finally, in Section 4.3 we
pass from $Y^{\bullet}$ and $Y_{\bullet}$ to arbitrary $X^{\bullet}$ and
$X_{\bullet}$, as stated in Theorem 4.1.

The category $\textbf{Cob}_{3}$ is of great interest in quantum topology. For
instance, a $3$-dimensional TQFT is a symmetric monoidal functor from
$\textbf{Cob}_{3}$ to $\mathrm{Mod}_{\Bbbk}$. By composition, we have the
following:

######  Corollary 4.2.

If $Z$ is a $3$-dimensional TQFT, then $Z\circ X^{\bullet}$ is a cocyclic
$\Bbbk$-module and $Z\circ X_{\bullet}$ is a cyclic $\Bbbk$-module.

A fundamental construction of a $3$-dimensional TQFT is the Reshetikhin-Turaev
TQFT $\text{RT}_{\mathcal{B}}\colon\textbf{Cob}_{3}\to\mathrm{Mod}_{\Bbbk}$
associated to an anomaly free modular category $\mathcal{B}$. We postpone
calculations of $\text{RT}_{\mathcal{B}}\circ X^{\bullet}$ and
$\text{RT}_{\mathcal{B}}\circ X_{\bullet}$ to Section 7. Some algebraic
preparations for it are given in sections 5 and 6.

###  4.1. The construction $Y^{\bullet}$

Recall the standard surface $S_{g}$ (see Section 3.5) of genus $g$. For any
$n\in\mathbb{N}$, set $Y^{n}=S_{n+1}$. For $n\in\mathbb{N}^{*}$ and $0\leq
i\leq n$, the faces $Y^{\bullet}(\delta_{i}^{n})\colon S_{n}\to S_{n+1}$ are
defined as follows. The morphism $Y^{\bullet}(\delta_{0}^{n})\colon S_{n}\to
S_{n+1}$ is the cobordism class presented by the special ribbon graph
$G_{Y^{\bullet}(\delta_{0}^{n})}$:

$G_{Y^{\bullet}(\delta_{0}^{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{coface0Cob.eps}}}\;.$

For $1\leq i\leq n-1$, the morphism $Y^{\bullet}(\delta_{i}^{n})\colon
S_{n}\to S_{n+1}$ is defined as the cobordism class presented by the special
ribbon graph $G_{Y^{\bullet}(\delta_{i}^{n})}$:

$G_{Y^{\bullet}(\delta_{i}^{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{cofaceiCob.eps}}}\;.$

Finally, the morphism $Y^{\bullet}(\delta_{n}^{n})\colon S_{n}\to S_{n+1}$ is
defined as the cobordism class presented by the special ribbon graph
$G_{Y^{\bullet}(\delta_{n}^{n})}$:

$G_{Y^{\bullet}(\delta_{n}^{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{cofacenCob.eps}}}\;.$

For $n\in\mathbb{N}$ and $0\leq j\leq n$, the degeneracy
$Y^{\bullet}(\sigma_{j}^{n})\colon S_{n+2}\to S_{n+1}$ is the cobordism class
presented by the special ribbon graph $G_{Y^{\bullet}(\sigma^{n}_{j})}$:

$G_{Y^{\bullet}(\sigma_{j}^{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{codegjCob.eps}}}\;.$

The morphism $Y^{\bullet}(\tau_{0})\colon S_{1}\to S_{1}$ is the identity map
$\mathrm{id}_{S_{1}}$, which is represented by the special ribbon graph
$I_{1}$ depicted in (12). For $n\in\mathbb{N}^{*}$, the cocyclic operator
$Y^{\bullet}(\tau_{n})\colon S_{n+1}\to S_{n+1}$ is the cobordism class
presented by the special ribbon graph $G_{Y^{\bullet}(\tau_{n})}$:

$G_{Y^{\bullet}(\tau_{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{cocycopCob.eps}}}\;.$

######  Lemma 4.3.

The family $Y^{\bullet}=\\{S_{n+1}\\}_{n\in\mathbb{N}}$, equipped with the
cofaces $\\{Y^{\bullet}(\delta_{i}^{n})\\}_{n\in\mathbb{N}^{*},0\leq i\leq
n}$, the codegeneracies
$\\{Y^{\bullet}(\sigma_{j}^{n})\\}_{n\in\mathbb{N},0\leq j\leq n}$, and the
cocyclic operators $\\{Y^{\bullet}(\tau_{n})\\}_{n\in\mathbb{N}}$ is a
cocyclic object in $\textbf{\emph{Cob}}_{3}$.

To prove the Lemma 4.3, we intensively use some well-known consequences of the
Kirby calculus from Section 3.7, which we recollect in the following lemma:

######  Lemma 4.4.

One has the following moves on special ribbon graphs:

$\displaystyle(1)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{Hopflinks.eps}}}\;\leftrightarrow\emptyset,\qquad(2)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{cyllocmove.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{cyllocmove1.eps}}}\;,\qquad(3)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{slamdunk1.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{slamdunk2.eps}}}\;,$
$\displaystyle(4)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbymultip.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbymultip1.eps}}}\;,\qquad(5)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbymultip2.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbymultip3.eps}}}\;,$
$\displaystyle(6)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbyass.eps}}}\;\leftrightarrow\hskip
14.22636pt\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbyass1.eps}}}\;\leftrightarrow\hskip
14.22636pt\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbyass3.eps}}}\;.$

###### Proof of Lemma 4.3.

In the proof, we shorten the notation by writing $\delta_{i}^{n}$,
$\sigma_{j}^{n}$, and $\tau_{n}$ instead of $Y^{\bullet}(\delta_{i}^{n})$,
$Y^{\bullet}(\sigma_{j}^{n})$, and $Y^{\bullet}(\tau_{n})$. Let us first
verify the relation (1). For $1\leq i<j\leq n$ we have:

$\displaystyle\delta_{j}^{n+1}\delta_{i}^{n}$
$\displaystyle\overset{(a)}{=}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{djdicob.eps}}}\;$
$\displaystyle\overset{(b)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{djdicob2.eps}}}\;$
$\displaystyle\overset{(c)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{djdicob3.eps}}}\;$
$\displaystyle\overset{(d)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{djdicob4.eps}}}\;$
$\displaystyle\overset{(e)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{djdicob5.eps}}}\;\overset{(f)}{=}\delta_{i}^{n+1}\delta_{j-1}^{n}.$

Here $(a)$ and $(f)$ follow from definitions and Lemma 3.1, $(b)$ and $(e)$
follow from Lemma 3.1 and the fact that the graph from equation (12)
represents the identity cobordism, $(c)$ from Lemma 4.4 $(3)$, and $(d)$ by
isotopy. The remaining cases are verified in a similar way.

Let us verify the relation (2).

  1. (1)

Suppose that $0\leq i<j\leq n$. We have:

$\displaystyle\sigma_{j}^{n}\sigma_{i}^{n+1}$
$\displaystyle\overset{(a)}{=}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjsicob.eps}}}\;$
$\displaystyle\overset{(b)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjsicob2.eps}}}\;$
$\displaystyle\overset{(c)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjsicob3.eps}}}\;$
$\displaystyle\overset{(d)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjsicob4.eps}}}\;$
$\displaystyle\overset{(e)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjsicob5.eps}}}\;,$

which equals to $\sigma_{i}^{n}\sigma_{j+1}^{n+1}$. Here $(a)$ follows from
definitions and Lemma 3.1, $(b)$ and $(d)$ follow from Lemma 3.1 and the fact
that the graph from equation (12) represents the identity cobordism, $(c)$
from Lemma 4.4 $(4)$, $(e)$ follows by applying the parts $(2),(4)$ and $(5)$
of Lemma 4.4 and isotopy.

  2. (2)

Suppose that $0\leq i=j\leq n$. By definition and Lemma 3.1, the composition
$\sigma_{i}^{n}\sigma_{i}^{n+1}$ is presented by

$\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sisicob.eps}}}\;.$ (13)

On the other hand, the composition $\sigma_{i}^{n}\sigma_{i+1}^{n+1}$ is
presented by

$\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sisi+1cob.eps}}}\;.$
(14)

By Lemma 4.4$(6)$, both special graphs from (13) and (14) are equivalent to
the special graph

$\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sisicob2.eps}}}\;.$

Consequently, the cobordisms $\sigma_{i}^{n}\sigma_{i}^{n+1}$ and
$\sigma_{i}^{n}\sigma_{i+1}^{n+1}$ are in the same class, that is, they are
equal in $\textbf{Cob}_{3}$.

Let us now verify the relation (3).

Suppose that $1\leq i=j\leq n-1$. We have

$\displaystyle\sigma_{i}^{n}\delta_{i}^{n+1}$
$\displaystyle\overset{(a)}{=}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sidicob.eps}}}\;$
$\displaystyle\overset{(b)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sidicob2.eps}}}\;\overset{(c)}{=}\mathrm{id}_{S_{n+1}}.$

Here $(a)$ follows from definition and Lemma 3.1, $(b)$ from Lemma 4.4 $(3)$,
and $(c)$ by isotopy, Lemma 3.1, and presentation of identity cobordism which
is given in equation (12). The cases when $i=0$ or $i=n$ are proven similarly.

Suppose now that $0\leq i<j\leq n$. In the case when $i\neq 0$, we have

$\displaystyle\sigma_{j}^{n}\delta_{i}^{n+1}$
$\displaystyle\overset{(a)}{=}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjdiiltjcob.eps}}}\;$
$\displaystyle\overset{(b)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjdiiltjcob2.eps}}}\;$
$\displaystyle\overset{(c)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjdiiltjcob3.eps}}}\;$
$\displaystyle\overset{(d)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjdiiltjcob4.eps}}}\;$
$\displaystyle\overset{(e)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{sjdiiltjcob5.eps}}}\;\overset{(f)}{=}\delta_{i}^{n}\sigma_{j-1}^{n-1}.$

Here $(a)$ and $(f)$ follow from definitions and Lemma 3.1, $(b)$ and $(e)$
follow from Lemma 3.1 and the fact that the graph from equation (12)
represents the identity cobordism, $(c)$ follows by applying twice the move
from Lemma 4.4 $(3)$, and $(d)$ by isotopy. The case when $i=0$ is proven
similarly.

The case when $1\leq i=j+1\leq n+1$ is verified in a similar way as the case
$0\leq i=j\leq n$. Also, the case when $1\leq j+1<i\leq n+1$ is verified in a
similar way as the case $0\leq i<j\leq n$.

It remains to show that relations (4), (6) and (8) hold. According to [Lo98,
Section 6.1.1], these relations imply relations (5) and (7). Let us show the
relation (4). Let $1\leq i\leq n$. In the case when $n\geq 3$ and $2\leq i\leq
n-1$, we have

$\displaystyle\tau_{n}\delta_{i}^{n}$
$\displaystyle\overset{(a)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{tndicob.eps}}}\;$
$\displaystyle\overset{(b)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{tndicob2.eps}}}\;$
$\displaystyle\overset{(c)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{tndicob4.eps}}}\;\overset{(d)}{\leftrightarrow}\delta_{i-1}^{n}\tau_{n-1}.$

Here $(a)$ and $(d)$ follow from definitions and Lemma 3.1, $(b)$ by isotopy,
$(c)$ follows from Lemma 4.4 $(3)$ and by isotopy. The other cases are proven
similarly.

We further show the relation (6). Assume that $1\leq j\leq n$. We have:

$\displaystyle\tau_{n}\sigma_{j}^{n}$
$\displaystyle\overset{(a)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{tnsjcob.eps}}}\;$
$\displaystyle\overset{(b)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{tnsjcob2.eps}}}\;$
$\displaystyle\overset{(c)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{tnsjcob5.eps}}}\;\overset{(d)}{\leftrightarrow}\sigma_{j-1}^{n}\tau_{n+1}.$

Here $(a)$ and $(d)$ follow from the definition and Lemma 3.1, $(b)$ follows
from Lemma 3.1 and the fact that the graph from equation (12) represents the
identity cobordism, $(c)$ follows by applying the parts $(2)$ and $(5)$ of
Lemma 4.4 and isotopy.

Finally, we check the relation (8) in the case $n=1$. The general case is
proven by the similar reasoning. We have:

$\tau_{1}^{2}\overset{(a)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{t_12Cob.eps}}}\;\overset{(b)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{t_12Cob2.eps}}}\;\overset{(c)}{\leftrightarrow}\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{t_12Cob4.eps}}}\;\overset{(d)}{\leftrightarrow}\mathrm{id}_{S_{2}}.$

Here $(a)$ follows from definition and Lemma 3.1, $(b)$ follows by the TWIST
move, $(c)$ follows by isotopy, $(d)$ follows by isotopy, Lemma 3.1, and
presentation of the identity cobordism, as depicted in (12). This finishes the
proof of Lemma 4.3. ∎

###  4.2. The construction $Y_{\bullet}$

Recall the standard surface $S_{g}$ (see Section 3.5) of genus $g$. For
$n\in\mathbb{N}$, set $Y_{n}=S_{n+1}$. For $n\in\mathbb{N}^{*}$ and $0\leq
i\leq n$, the face $Y_{\bullet}(d_{i}^{n})\colon S_{n+1}\to S_{n}$ is the
cobordism class presented by the special ribbon graph
$G_{Y_{\bullet}(d_{i}^{n})}$:

$G_{Y_{\bullet}(d_{i}^{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{faceiCob.eps}}}\;.$

Let $n\in\mathbb{N}$ and $0\leq j\leq n$. The morphism
$Y_{\bullet}(s_{j}^{n})\colon S_{n+1}\to S_{n+2}$ is the cobordism class
presented by the special ribbon graph $G_{Y_{\bullet}(s^{n}_{j})}$:

$G_{Y_{\bullet}(s_{j}^{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{degjCob.eps}}}\;.$

The morphism $Y_{\bullet}(t_{0})\colon S_{1}\to S_{1}$ equals the identity map
$\mathrm{id}_{S_{1}}$, which is represented by the graph $I_{1}$ depicted in
(12). For $n\in\mathbb{N}^{*}$, the morphism $Y_{\bullet}(t_{n})\colon
S_{n+1}\to S_{n+1}$ is the cobordism class presented by the special ribbon
graph $G_{Y_{\bullet}(t_{n})}$:

$G_{Y_{\bullet}(t_{n})}=\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{cycopCob.eps}}}\;.$

The proof of Theorem 4.1 in this case is similar to the proof of its version
with $X^{\bullet}$, which is detailed in Section 4.1. Namely, we have the
following lemma:

######  Lemma 4.5.

The family $Y_{\bullet}=\\{S_{n+1}\\}_{n\in\mathbb{N}}$, equipped with the
faces $\\{Y_{\bullet}(d_{i}^{n})\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the
degeneracies $\\{Y_{\bullet}(s_{j}^{n})\\}_{n\in\mathbb{N},0\leq j\leq n}$,
and the cyclic operators $\\{Y_{\bullet}(t_{n})\\}_{n\in\mathbb{N}}$ is a
cyclic object in $\textbf{\emph{Cob}}_{3}$.

To prove Lemma 4.5, one uses the following result, which is analogous to to
Lemma 4.4:

######  Lemma 4.6.

One has the following moves on special ribbon graphs:

$\displaystyle(1)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{slamdunk3.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{slamdunk4.eps}}}\;,\qquad(2)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{comultikirby1.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{comultikirby2.eps}}}\;,$
$\displaystyle(3)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{comultikirby3.eps}}}\;\leftrightarrow\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{comultikirby4.eps}}}\;,$
$\displaystyle(4)\quad\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbycoass1.eps}}}\;\leftrightarrow\hskip
14.22636pt\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbycoass2.eps}}}\;\leftrightarrow\hskip
14.22636pt\,\raisebox{-0.45pt}{\scalebox{0.5}{\includegraphics{kirbycoass.eps}}}\;.$

###  4.3. Passing to $X^{\bullet}$ and $X_{\bullet}$

Let $\\{\Sigma_{n+1}\\}_{n\geq 0}$ be any family of closed oriented surfaces.
By the classification theorem, for each $n$ there exists an orientation
preserving homeomorphism $f_{n}\colon\Sigma_{n+1}\to S_{n+1}.$ Denote by
$\mathrm{Cyl}({f_{n}})\colon\Sigma_{n+1}\to S_{n+1}$ the associated morphism
in $\textbf{Cob}_{3}$, given by the quadruple

$\mathrm{Cyl}({f_{n}})=(C_{S_{n+1}}=S_{n+1}\times\left[0,1\right],h_{n}\colon(-\Sigma_{n+1})\sqcup
S_{n+1}\to\partial(C_{S_{n+1}}),\Sigma_{n+1},S_{n+1}),$

where $h_{n}(x)=(f_{n}(x),0)$, if $x\in\Sigma_{n+1}$ and $h_{n}(x)=(x,1)$, if
$x\in S_{n+1}$. It follows from [Tu10, Chapter IV, Section 5.1.], that the
cobordism $\mathrm{Cyl}({f_{n}})$ is determined up to isotopy.

We pass from $Y^{\bullet}$ to the cocyclic object $X^{\bullet}$ in
$\textbf{Cob}_{3}$ as follows. For any $n\in\mathbb{N}$, define
$X^{n}=\Sigma_{n+1}$. Next, the cofaces
$\\{X^{\bullet}(\delta_{i}^{n})\colon\Sigma_{n}\to\Sigma_{n+1}\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, the codegeneracies
$\\{X^{\bullet}(\sigma_{j}^{n})\colon\Sigma_{n+2}\to\Sigma_{n+1}\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and the cocyclic operators
$\\{X^{\bullet}(\tau_{n})\colon\Sigma_{n+1}\to\Sigma_{n+1}\\}_{n\in\mathbb{N}}$
are defined by formulas

$\displaystyle X^{\bullet}(\delta_{i}^{n})$
$\displaystyle=(\mathrm{Cyl}({f_{n}}))^{-1}Y^{\bullet}(\delta_{i}^{n})\mathrm{Cyl}({f_{n-1}}),$
$\displaystyle X^{\bullet}(\sigma_{j}^{n})$
$\displaystyle=(\mathrm{Cyl}({f_{n}}))^{-1}Y^{\bullet}(\sigma_{j}^{n})\mathrm{Cyl}({f_{n+1}}),$
$\displaystyle X^{\bullet}(\tau_{n})$
$\displaystyle=(\mathrm{Cyl}({f_{n}}))^{-1}Y^{\bullet}(\tau_{n})\mathrm{Cyl}({f_{n}}).$

It follows from definitions that the family of cylinders
$\\{\mathrm{Cyl}({f_{n}})\colon\Sigma_{n+1}\to S_{n+1}\\}_{n\in\mathbb{N}}$ is
a natural isomorphism between cocyclic objects $X^{\bullet}$ and $Y^{\bullet}$
in $\textbf{Cob}_{3}$. One similarly passes from $Y_{\bullet}$ to a cyclic
object $X_{\bullet}$ in $\textbf{Cob}_{3}$. ∎

##  5\. Preliminaries on monoidal categories and Hopf algebras

In this section, we recall some algebraic preliminaries on ribbon categories
and their graphical calculus as well as categorical Hopf algebras and related
concepts. For more details, see [TV17].

###  5.1. Conventions

In what follows, we suppress in our formulas the associativity and unitality
constraints of the monoidal category. We denote by $\otimes$ and $\mathbb{1}$
the monoidal product and unit object of a monoidal category. For any objects
$X_{1},\dots,X_{n}$ of a monoidal category with $n\geq 2$, we set

$X_{1}\otimes X_{2}\otimes\cdots\otimes X_{n}=(\cdots((X_{1}\otimes
X_{2})\otimes X_{3})\otimes\cdots\otimes X_{n-1})\otimes X_{n}$

and similarly for morphisms. A monoidal category is $\Bbbk$-linear, if its Hom
sets have a structure of a $\Bbbk$-module such that the composition and
monoidal product of morphisms are $\Bbbk$-bilinear. For shortness, we often
use the term monoidal $\Bbbk$-category.

###  5.2. Braided categories

A braiding of a monoidal category $(\mathcal{B},\otimes,\mathbb{1})$ is a
family $\tau=\\{\tau_{X,Y}\colon X\otimes Y\to Y\otimes
X\\}_{X,Y\in\mathrm{Ob}(\mathcal{B})}$ of natural isomorphisms such that

$\displaystyle\tau_{X,Y\otimes Z}$
$\displaystyle=(\mathrm{id}_{Y}\otimes\tau_{X,Z})(\tau_{X,Y}\otimes\mathrm{id}_{Z})\text{
and }$ $\displaystyle\tau_{X\otimes Y,Z}$
$\displaystyle=(\tau_{X,Z}\otimes\mathrm{id}_{Y})(\mathrm{id}_{X}\otimes\tau_{Y,Z})$

for all $X,Y,Z\in\mathrm{Ob}(\mathcal{B})$. Note that the above axioms imply
that $\tau_{X,\mathbb{1}}=\tau_{\mathbb{1},X}=\mathrm{id}_{X}$ for any
$X\in\mathrm{Ob}(\mathcal{B})$. A braided category is a monoidal category
endowed with a braiding. A braiding $\tau$ of $\mathcal{B}$ is symmetric if
for all $X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\tau_{Y,X}\tau_{X,Y}=\mathrm{id}_{X\otimes Y}.$

A symmetric category is a monoidal category endowed with a symmetric braiding.
For example, the category of left $\Bbbk$-modules is symmetric.

###  5.3. Braided categories with a twist

A twist for a braided monoidal category $\mathcal{B}$ is a natural isomorphism
$\theta=\\{\theta_{X}\colon X\to X\\}_{X\in\mathrm{Ob}(\mathcal{B})}$ such
that for all $X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\theta_{X\otimes Y}=\tau_{Y,X}\tau_{X,Y}(\theta_{X}\otimes\theta_{Y}).$ (15)

Note that this implies $\theta_{\mathbb{1}}=\mathrm{id}_{\mathbb{1}}$. By a
braided category with a twist, we mean a braided category endowed with a
twist. For example, the family $\\{\mathrm{id}_{X}\colon X\to
X\\}_{X\in\mathrm{Ob}(\mathcal{B})}$ is a twist for $\mathcal{B}$ if and only
if $\mathcal{B}$ is symmetric. Also, any ribbon category (see Section 5.6) has
a canonical twist.

###  5.4. Graphical calculus

In this paper, we use the Penrose graphical calculus, which allows us to avoid
lengthy algebraic computations by using simple topological arguments. The
diagrams read from bottom to top. In a monoidal category $\mathcal{B}$, the
diagrams are made of arcs colored by objects of $\mathcal{B}$ and of boxes,
colored by morphisms of $\mathcal{B}$. Arcs colored by $\mathbb{1}$ may be
omitted in the pictures. The identity morphism of an object $X$, a morphism
$f\colon X\to Y$ in $\mathcal{B}$, and its composition with a morphism
$g\colon Y\to Z$ in $\mathcal{B}$ are represented respectively as

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{idXunori.eps}}}\;,\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{fXYunori.eps}}}\;,\quad\text{and}\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{morcompounori.eps}}}\;.$

The tensor product of two morphisms $f\colon X\to Y$ and $g\colon U\to V$ is
represented by placing a picture of $f$ to the left of the picture of $g$:

$f\otimes
g=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{tensfg1unori.eps}}}\;.$

Any diagram represents a morphism. The latter depends only on the isotopy
class of the diagram representing it. For example, the _level-exchange
property_

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{levelexchange1.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{levelexchange3.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{levelexchange2.eps}}}\;$

reflects the formula

$f\otimes g=(f\otimes\mathrm{id}_{V})(\mathrm{id}_{X}\otimes
g)=(\mathrm{id}_{Y}\otimes g)(f\otimes\mathrm{id}_{U}).$

When $\mathcal{B}$ is braided with a braiding $\tau$, we depict

$\tau_{X,Y}=\,\raisebox{-0.45pt}{\scalebox{0.8}{\includegraphics{brcgCMunori.eps}}}\;\quad\text{and}\quad\tau_{X,Y}^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.8}{\includegraphics{invbrcgCMunori.eps}}}\;.$

When $\mathcal{B}$ is a braided category with a twist
$\theta=\\{\theta_{X}\colon X\to X\\}_{X\in\mathrm{Ob}(\mathcal{B})}$, we
depict

$\theta_{X}=\,\raisebox{-0.45pt}{\scalebox{0.8}{\includegraphics{twistX.eps}}}\;\quad\text{and}\quad\left(\theta_{X}\right)^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.8}{\includegraphics{twistXinv.eps}}}\;.$

For example, the defining condition (15) for the twist says that for all
$X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{twistcondcg0.eps}}}\;\hskip
18.49411pt=\hskip
5.69046pt\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{twistcondcg.eps}}}\;.$

We warn the reader that this notation should not be confused with notation of
a left twist in a ribbon category (see Section 5.6). We made this choice of
notation since any ribbon category is a particularly important example of a
braided category with a twist.

###  5.5. Pivotal categories

A pivotal category is a monoidal category $\mathcal{C}$ such that to any
object $X$ of $\mathcal{C}$ is associated a dual object
$X^{*}\in\mathrm{Ob}(\mathcal{C})$ and four morphisms

$\displaystyle\text{ev}_{X}\colon X^{*}\otimes X\to\mathbb{1},$
$\displaystyle\quad\text{coev}_{X}\colon\mathbb{1}\to X\otimes X^{*},$
$\displaystyle\widetilde{\text{ev}}_{X}\colon X\otimes X^{*}\to\mathbb{1},$
$\displaystyle\quad\widetilde{\text{coev}}_{X}\colon\mathbb{1}\to X^{*}\otimes
X,$

satisfying several conditions and such that the so called left and right
duality functors coincide as monoidal functors. The latter implies in
particular that the dual morphism $f^{*}\colon Y^{*}\to X^{*}$ of a morphism
$f\colon X\to Y$ in $\mathcal{C}$ is computed by

$\displaystyle f^{*}$
$\displaystyle=(\mathrm{id}_{X^{*}}\otimes\widetilde{\text{ev}}_{Y})(\mathrm{id}_{X^{*}}\otimes
f\otimes\mathrm{id}_{Y^{*}})(\widetilde{\text{coev}}_{X}\otimes\mathrm{id}_{Y^{*}})=$
$\displaystyle=(\text{ev}_{Y}\otimes\mathrm{id}_{{X^{*}}})(\mathrm{id}_{{Y^{*}}}\otimes
f\otimes\mathrm{id}_{{X^{*}}})(\mathrm{id}_{{Y^{*}}}\otimes\text{coev}_{X}).$

The graphical calculus for monoidal categories from Section 5.4 is extended to
pivotal categories by orienting arcs. If an arc colored by $X$ is oriented
upwards, the represented object in source/target of corresponding morphism is
$X^{*}$. For example, $\mathrm{id}_{X},\mathrm{id}_{X^{*}}$, and a morphism
$f\colon X\otimes Y^{*}\otimes Z\to U\otimes V^{*}$ are depicted by

$\mathrm{id}_{X}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{idX.eps}}}\;,\quad\mathrm{id}_{X^{*}}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{idXD.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{idX.eps}}}\;\hskip
5.69046pt,\quad\text{and}\quad
f=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{exmorcg.eps}}}\;.$

The morphisms $\text{ev}_{X},\widetilde{\text{ev}}_{X},\text{coev}_{X}$, and
$\widetilde{\text{coev}}_{X}$ are respectively depicted by

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{evlcg.eps}}}\;,\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{evrcg.eps}}}\;,\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{coevlcg.eps}}}\;,\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{coevrcg.eps}}}\;.$

###  5.6. Ribbon categories

Let $\mathcal{B}$ be a braided pivotal category. The _left twist_ of an object
$X$ of $\mathcal{B}$ is defined by

$\theta_{X}^{l}=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{ribbonlplus.eps}}}\;=(\mathrm{id}_{X}\otimes\widetilde{\text{ev}}_{X})(\tau_{X,X}\otimes\mathrm{id}_{X^{*}})(\mathrm{id}_{X}\otimes\text{coev}_{X})\colon
X\to X,$

while the _right twist_ of $X$ is defined by

$\theta_{X}^{r}=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{ribbonrplus.eps}}}\;=(\text{ev}_{X}\otimes\mathrm{id}_{X})(\mathrm{id}_{X^{*}}\otimes\tau_{X,X})(\widetilde{\text{coev}}_{X}\otimes\mathrm{id}_{X})\colon
X\to X.$

The left and the right twist are natural isomorphisms with inverses

$(\theta_{X}^{l})^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{ribbonlminus.eps}}}\;\quad\text{and}\quad(\theta_{X}^{r})^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{ribbonrminus.eps}}}\;.$

A ribbon category is a braided pivotal category $\mathcal{B}$ such that
$\theta_{X}^{l}=\theta_{X}^{r}$ for all $X\in\mathrm{Ob}(\mathcal{B})$. In
this case, the family
$\theta=\\{\theta_{X}=\theta_{X}^{l}=\theta_{X}^{r}\colon X\to
X\\}_{X\in\mathrm{Ob}(\mathcal{B})}$ is a twist in the sense of Section 5.3
and is called the twist of $\mathcal{B}$.

###  5.7. Colored ribbon graphs

Let $\mathcal{B}$ be a ribbon category. A ribbon graph $\Gamma$ (recall
Section 3.4) is  _$\mathcal{B}$ -colored_ if each arc and circle of $\Gamma$
is endowed with an object of $\mathcal{B}$ and each coupon of $\Gamma$ is
endowed with a morphism in $\mathcal{B}$ from the object determined by its
bottom base (as in Section 5.4) to the object determined by its top base. By
[Tu10, Theorem 2.5], any $\mathcal{B}$-colored ribbon graph determines a
morphism in $\mathcal{B}$. The colors of $\mathcal{B}$-colored ribbon graphs
are shown on their diagrams. For example, given objects
$X,Y,Z,T\in\mathrm{Ob}(\mathcal{B})$ and a morphism $f\colon Y^{*}\otimes X\to
Z$ in ${\mathcal{B}}$, the diagram

0.85example1.eps

represents a $\mathcal{B}$-colored ribbon $(3,2)$-graph whose associated
morphism in $\mathcal{B}$ is

$(f\circ\tau_{X,Y^{*}})\otimes\theta_{T}\colon X\otimes Y^{*}\otimes T\to
Z\otimes T.$

Here, $\tau$ and $\theta$ are the braiding and the twist of $\mathcal{B}$.

###  5.8. Categorical traces and dimensions

Let $\mathcal{C}$ be a pivotal category. The left trace $\text{tr}_{l}(f)$ and
the right trace $\text{tr}_{r}(f)$ of a morphism $f\colon X\to X$ in
$\mathcal{C}$, are respectively defined as

$\displaystyle\text{tr}_{l}(f)=\text{ev}_{X}(\mathrm{id}_{X^{*}}\otimes
f)\widetilde{\text{coev}}_{X}\in\text{End}(\mathbb{1})\quad\text{and}$
$\displaystyle\text{tr}_{r}(f)=\widetilde{\text{ev}}_{X}(f\otimes\mathrm{id}_{X^{*}})\text{coev}_{X}\in\text{End}(\mathbb{1}).$

The left dimension $\dim_{l}(X)$ and the right dimension $\dim_{r}(X)$ of an
object $X$ of $\mathcal{C}$ are respectively defined as

$\dim_{l}(X)=\text{tr}_{l}(\mathrm{id}_{X})\quad\text{and}\quad\dim_{r}(X)=\text{tr}_{r}(\mathrm{id}_{X}).$

###  5.9. Spherical categories

A spherical category is a pivotal category $\mathcal{C}$ such that left and
right trace of any endomorphism in $\mathcal{C}$ coincide. In particular,
$\dim_{l}(X)=\dim_{r}(X)$ holds for all $X\in\mathrm{Ob}(\mathcal{C})$. In a
spherical category, we omit $l$ and $r$ from the notation for trace and
dimension. Any ribbon category is spherical (see [TV17, Corollary 3.4.]).

###  5.10. (Pre)fusion categories

An object $X$ of a $\Bbbk$-linear category $\mathcal{C}$ is simple if the
$\Bbbk$-module $\text{End}_{\mathcal{C}}(X)$ is free of rank $1$. In that
case, the map $\Bbbk\to\text{End}_{\mathcal{C}}(X)$, $k\mapsto
k\mathrm{id}_{X}$ is an isomorphism of $\Bbbk$-algebras.

A prefusion $\Bbbk$-category is a monoidal $\Bbbk$-linear category
$\mathcal{C}$ such that there is a set $I$ of simple objects of $\mathcal{C}$
satisfying the following conditions:

  * (a)

For any distinct elements $i,j$ of $I$, $\mathrm{Hom}_{\mathcal{C}}(i,j)=0$;

  * (b)

The unit object $\mathbb{1}$ of $\mathcal{C}$ is an element of $I$;

  * (c)

Any object of $\mathcal{C}$ is a finite direct sum of elements of $I$.

The set $I$ is called a representative set of simple objects of $\mathcal{C}$.
A fusion $\Bbbk$-category is a rigid prefusion $\Bbbk$-category such that the
set of isomorphism classes of simple objects is finite.

###  5.11. Dimension of a fusion category

Let $\mathcal{C}$ be a pivotal fusion $\Bbbk$-category. We identify $\Bbbk$
and $\text{End}_{\mathcal{C}}(\mathbb{1})$ via the $\Bbbk$-linear isomorphism
$k\mapsto k\mathrm{id}_{\mathbb{1}}$. Pick a representative set $I$ of simple
objects of $\mathcal{C}$. The _dimension of the category_ $\mathcal{C}$ is an
element of $\text{End}_{\mathcal{C}}(\mathbb{1})\cong\Bbbk$ defined by

$\dim(\mathcal{C})=\sum_{i\in I}\dim_{l}(i)\dim_{r}(i).$

The dimension of $\mathcal{C}$ does not depend on the choice of $I$ since
isomorphic objects of $\mathcal{C}$ have the same left/right dimensions. Note
that if $\mathcal{C}$ is spherical, then

$\dim(\mathcal{C})=\sum_{i\in I}(\dim(i))^{2}.$

###  5.12. Modular categories

Let $\mathcal{C}$ be a ribbon fusion $\Bbbk$-category. Pick a representative
set $I$ of simple objects of $\mathcal{C}$. The scalars

$\Delta_{\pm}=\sum_{i\in I}\dim(i)\,\mathrm{tr}(\theta_{i}^{\pm
1})\in\text{End}_{\mathcal{C}}(\mathbb{1})\cong\Bbbk,$

where $\theta$ is the twist of $\mathcal{C}$, do not depend on the choice of
$I$. The $S$-matrix $\left[S_{i,j}\right]_{i,j\in I}$ of $\mathcal{C}$ is
defined by

$S_{i,j}=\text{tr}(\tau_{i,j}\tau_{j,i})\in\text{End}_{\mathcal{C}}(\mathbb{1})\cong\Bbbk.$

Note that the invertibility of $S$ does not depend on the choice of $I$.

A modular $\Bbbk$-category is a ribbon fusion $\Bbbk$-category whose
$S$-matrix is invertible. The scalars $\Delta_{+}$, $\Delta_{-}$, and
$\dim(\mathcal{C})$ associated with a modular $\Bbbk$-category $\mathcal{C}$
are invertible in $\Bbbk$ and are related by
$\dim(\mathcal{C})=\Delta_{-}\Delta_{+}$ (see [Tu10, p. 89]). A modular
$\Bbbk$-category is anomaly free if $\Delta_{+}=\Delta_{-}$.

###  5.13. Categorical algebras

An algebra in a monoidal category $\mathcal{C}$ is a triple $(A,m,u)$, where
$A$ is an object of $\mathcal{C}$, $m\colon A\otimes A\to A$ and
$u\colon\mathbb{1}\to A$ are morphisms in $\mathcal{C}$, called multiplication
and unit respectively, which satisfy

$m(m\otimes\mathrm{id}_{A})=m(\mathrm{id}_{A}\otimes m)\quad\text{and}\quad
m(u\otimes\mathrm{id}_{A})=\mathrm{id}_{A}=m(\mathrm{id}_{A}\otimes u).$

The multiplication and the unit are depicted by

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{multiplication.eps}}}\;\quad\text{and}\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unit.eps}}}\;.$

An algebra morphism between two algebras $(A,m_{A},u_{A})$ and
$(B,m_{B},u_{B})$ in a monoidal category $\mathcal{C}$ is a morphism $f\colon
A\to B$ in $\mathcal{C}$ such that $fm_{A}=m_{B}(f\otimes f)$ and
$fu_{A}=u_{B}$.

###  5.14. Categorical coalgebras

A coalgebra in a monoidal category $\mathcal{C}$ is a triple
$(C,\Delta,\varepsilon)$, where $C$ is an object of $\mathcal{C}$,
$\Delta\colon C\to C\otimes C$ and $\varepsilon\colon C\to\mathbb{1}$ are
morphisms in $\mathcal{C}$, called comultiplication and counit respectively,
which satisfy

$(\Delta\otimes\mathrm{id}_{C})\Delta=(\mathrm{id}_{C}\otimes\Delta)\Delta\quad\text{and}\quad(\mathrm{id}_{C}\otimes\varepsilon)\Delta=\mathrm{id}_{C}=(\varepsilon\otimes\mathrm{id}_{C})\Delta.$

The comultiplication and the counit are depicted by

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{comultiplication.eps}}}\;\quad\text{and}\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{counit.eps}}}\;.$

A coalgebra morphism between two coalgebras $(C,\Delta_{C},\varepsilon_{C})$
and $(D,\Delta_{D},\varepsilon_{D})$ in a monoidal category $\mathcal{C}$ is a
morphism $f\colon C\to D$ in $\mathcal{C}$ such that $\Delta_{D}f=(f\otimes
f)\Delta_{C}$ and $\varepsilon_{D}f=\varepsilon_{C}$.

###  5.15. Categorical bialgebras

A bialgebra in a braided monoidal category $\mathcal{B}$ is a quintuple
$(A,m,u,\Delta,\varepsilon)$ such that $(A,m,u)$ is an algebra in
$\mathcal{B}$, $(A,\Delta,\varepsilon)$ is a coalgebra in $\mathcal{B}$, and
such that the following additional relations hold:

$\Delta m=(m\otimes
m)(\mathrm{id}_{A}\otimes\tau_{A,A}\otimes\mathrm{id}_{A})(\Delta\otimes\Delta),\quad\varepsilon
m=\varepsilon\otimes\varepsilon,\quad\Delta u=u\otimes
u,\quad\text{and}\quad\varepsilon u=\mathrm{id}_{\mathbb{1}}.$

A bialgebra morphism between two bialgebras $A$ and $B$ in a braided monoidal
category $\mathcal{B}$ is a morphism $A\to B$ in $\mathcal{B}$, which is both
an algebra and a coalgebra morphism.

###  5.16. Categorical Hopf algebras

A Hopf algebra in a braided monoidal category $\mathcal{B}$ is a sextuple
$(A,m,u,\Delta,\varepsilon,S)$, where $(A,m,u,\Delta,\varepsilon)$ is a
bialgebra in $\mathcal{B}$ and $S\colon A\to A$ is an isomorphism in
$\mathcal{B}$, called the antipode, which satisfies the equation

$m(S\otimes\mathrm{id}_{A})\Delta=u\epsilon=m(\mathrm{id}_{A}\otimes
S)\Delta.$

The antipode and its inverse are depicted by

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{antipode.eps}}}\;\quad\text{and}\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{antipodeinv.eps}}}\;.$

A Hopf algebra morphism between two Hopf algebras is a bialgebra morphism
between them.

###  5.17. Integrals and cointegrals

Consider a bialgebra $(A,m,\Delta,u,\varepsilon)$ in a braided category
$\mathcal{B}$. A left (respectively right) _integral_ of $A$ is a morphism
$\Lambda\colon\mathbb{1}\to A$ in $\mathcal{B}$ such that

$m(\mathrm{id}_{A}\otimes\Lambda)=\Lambda\varepsilon,\quad\text{respectively}\quad
m(\Lambda\otimes\mathrm{id}_{A})=\Lambda\varepsilon.$

A left (respectively right) cointegral of $A$ is a morphism $\lambda\colon
A\to\mathbb{1}$ in $\mathcal{B}$ such that

$(\mathrm{id}_{A}\otimes\lambda)\Delta=u\lambda,\quad\text{respectively}\quad(\lambda\otimes\mathrm{id}_{A})\Delta=u\lambda.$

###  5.18. Bialgebra pairings

Let $(A,m,u,\Delta,\varepsilon)$ be a bialgebra in a braided category
$\mathcal{B}$. A bialgebra pairing for $A$ is a morphism $\omega\colon
A\otimes A\to\mathbb{1}$ in $\mathcal{B}$ such that:

$\displaystyle\omega(m\otimes\mathrm{id}_{A})=\omega(\mathrm{id}_{A}\otimes\omega\otimes\mathrm{id}_{A})(\mathrm{id}_{A\otimes
A}\otimes\Delta)$
$\displaystyle,\quad\quad\omega(u\otimes\mathrm{id}_{A})=\varepsilon,$
$\displaystyle\omega(\mathrm{id}_{A}\otimes
m)=\omega(\mathrm{id}_{A}\otimes\omega\otimes\mathrm{id}_{A})(\Delta\otimes\mathrm{id}_{A\otimes
A})$ $\displaystyle,\quad\quad\omega(\mathrm{id}_{A}\otimes u)=\varepsilon.$

A bialgebra pairing $\omega$ for $A$ is non-degenerate if there exists a
morphism $\Omega\colon\mathbb{1}\to A\otimes A$ in $\mathcal{B}$ such that

$(\omega\otimes\mathrm{id}_{A})(\mathrm{id}_{A}\otimes\Omega)=\mathrm{id}_{A}\quad\text{and}\quad(\mathrm{id}_{A}\otimes\omega)(\Omega\otimes\mathrm{id}_{A})=\mathrm{id}_{A}.$

The morphism $\Omega$ is called the inverse of the pairing $\omega$. If $A$ is
Hopf algebra, the pairing $\omega$ for the underlying bialgebra is called a
Hopf pairing.

###  5.19. Coend of a category

Let $\mathcal{B}$ be a pivotal category. Let
$F_{\mathcal{B}}\colon\mathcal{B}^{\mathrm{op}}\times\mathcal{B}\to\mathcal{B}$
be the functor defined by $F_{\mathcal{B}}(X,Y)=X^{*}\otimes Y$. A dinatural
transformation from $F_{\mathcal{B}}$ to an object $D$ of $\mathcal{B}$ is a
function $d$ that assigns to any object $X$ of $\mathcal{B}$ a morphism
$d_{X}\colon X^{*}\otimes X\to D$ such that for all morphisms $f\colon X\to Y$
in $\mathcal{B}$,

$d_{X}(f^{*}\otimes\mathrm{id}_{X})=d_{Y}(\mathrm{id}_{Y^{*}}\otimes f).$

The coend of $\mathcal{B}$, if it exists, is a pair $(\mathbb{F},i)$ where
$\mathbb{F}$ is an object of $\mathcal{B}$ and $i$ is a dinatural
transformation from $F_{\mathcal{B}}$ to $\mathbb{F}$, which is universal
among all dinatural transformations. More precisely, for any dinatural
transformation $d$ from $F_{\mathcal{B}}$ to $D$, there exists a unique
morphism $\varphi\colon\mathbb{F}\to D$ in $\mathcal{B}$ such that
$d_{X}=\varphi i_{X}$ for all $X\in\mathrm{Ob}(\mathcal{B})$. A coend
$(\mathbb{F},i)$ of a category $\mathcal{B}$, if it exists, is unique up to a
unique isomorphism commuting with the dinatural transformation. We depict the
dinatural transformation $i=\\{i_{X}\colon X^{*}\otimes
X\to\mathbb{F}\\}_{X\in\mathrm{Ob}(\mathcal{B})}$ as

$i_{X}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dincoend.eps}}}\;.$

An important factorization property is given in the following lemma.

######  Lemma 5.1 (Fubini theorem for coends, [MCL98]).

Let $(\mathbb{F},i)$ be a coend of a braided pivotal category $\mathcal{B}$.
If $d=\\{d_{X_{1},\dots,X_{n}}\colon X_{1}^{*}\otimes
X_{1}\otimes\cdots\otimes X_{n}^{*}\otimes X_{n}\to
D\\}_{X_{1},\dots,X_{n}\in\mathrm{Ob}(\mathcal{B})}$ is a family of morphisms
in $\mathcal{B}$, which is dinatural in each $X_{i}$ for $1\leq i\leq n$, then
there exists a unique morphism $\varphi\colon\mathbb{F}^{\otimes n}\to D$ in
$\mathcal{B}$ such that

$d_{X_{1},\dots,X_{n}}=\varphi(i_{X_{1}}\otimes\cdots\otimes i_{X_{n}})$

for all $X_{1},\dots,X_{n}\in\mathrm{Ob}(\mathcal{B})$.

According to [Ly95, Ma93], coend of a braided pivotal category $\mathcal{B}$
is a Hopf algebra in $\mathcal{B}$ endowed with a canonical Hopf pairing. Its
unit is $u=(\mathrm{id}_{\mathbb{1}}\otimes
i_{\mathbb{1}})(\text{coev}_{\mathbb{1}}\otimes\mathrm{id}_{\mathbb{1}})\colon\mathbb{1}\to\mathbb{F}$.
Multiplication $m\colon\mathbb{F}\otimes\mathbb{F}\to\mathbb{F}$ and canonical
pairing $\omega\colon\mathbb{F}\otimes\mathbb{F}\to\mathbb{1}$ are unique
morphisms such that for all $X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinmult1.eps}}}\;=~{}~{}\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinmult2.eps}}}\;,\qquad\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinomega1.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinomega2.eps}}}\;.$

Its comultiplication $\Delta\colon\mathbb{F}\to\mathbb{F}\otimes\mathbb{F}$,
counit $\varepsilon\colon\mathbb{F}\to\mathbb{1}$, and antipode
$S\colon\mathbb{F}\to\mathbb{F}$ are unique morphisms such that for all
$X\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dindelta1.eps}}}\;=~{}~{}\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dindelta2.eps}}}\;,\quad\quad\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dineps1.eps}}}\;=~{}~{}\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dineps3.eps}}}\;,\quad\text{and}\quad\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinanti1.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinanti2.eps}}}\;.$

Two useful properties of antipode of coend $\mathbb{F}$ are

$\displaystyle S^{2}$ $\displaystyle=\theta_{\mathbb{F}}^{r}\quad\text{and}$
(16) $\displaystyle\omega(S\otimes\mathrm{id}_{\mathbb{F}})$
$\displaystyle=\omega(\mathrm{id}_{\mathbb{F}}\otimes S).$ (17)

Also, we have by definitions of $\omega$ and $S$ that
$X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinomega5.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinomega3.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.95}{\includegraphics{dinomega4.eps}}}\;.$

##  6\. Cyclic modules from (co)algebras

Let $\mathcal{B}$ be a braided $\Bbbk$-linear category with a twist. In this
section, we first review some (co)cyclic $\Bbbk$-modules from coalgebras and
algebras in $\mathcal{B}$. Proofs can be found in [Ba22b], where these were
merely (co)cyclic sets, since one dropped the hypothesis of $\Bbbk$-linearity
of $\mathcal{B}$. Next, we outline some basic computations of cyclic
(co)homology of some introduced (co)cyclic $\Bbbk$-modules. Finally, in
Section 6.6, we explicit (co)faces, (co)degeneracies, and (co)cyclic operators
of (co)cyclic $\Bbbk$-vector spaces associated to the coend of the
representation category of a finite dimensional ribbon Hopf algebra over a
field $\Bbbk$. For more details on Hochschild and cyclic (co)homology of
(co)cyclic $\Bbbk$-modules, see [We94, Section 9.6].

###  6.1. Cocyclic modules from coalgebras

Any coalgebra $C$ in $\mathcal{B}$ gives rise to a cocyclic $\Bbbk$-module
$C^{\bullet}$ as follows. For any $n\in\mathbb{N}$, define
$C^{n}=\mathrm{Hom}_{\mathcal{B}}(C^{\otimes n+1},\mathbb{1}).$ Next, define
the cofaces $\\{\delta_{i}^{n}\colon C^{n-1}\to
C^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the codegeneracies
$\\{\sigma_{j}^{n}\colon C^{n+1}\to C^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$,
and the cocyclic operators $\\{\tau_{n}\colon C^{n}\to
C^{n}\\}_{n\in\mathbb{N}}$ by setting

$\displaystyle\delta_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Edeltai.eps}}}\;,\quad\sigma_{j}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{HDsj.eps}}}\;,\quad\tau_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{HDtn.eps}}}\;.$

An integer $k$ below an arc denotes the $k$-th tensorand of a tensor power of
$C$. This construction is functorial in $C$, that is, a morphism between
coalgebras in $\mathcal{B}$ induces the morphism of corresponding cocyclic
$\Bbbk$-modules.

###  6.2. Cyclic modules from algebras

Any algebra $A$ in $\mathcal{B}$ gives rise to a cyclic $\Bbbk$-module
$A_{\bullet}$ as follows. For any $n\in\mathbb{N}$, define
$A_{n}=\mathrm{Hom}_{\mathcal{B}}(A^{\otimes n+1},\mathbb{1})$. Next, define
the faces $\\{d_{i}^{n}\colon A_{n}\to A_{n-1}\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, the degeneracies $\\{s^{n}_{j}\colon A_{n}\to
A_{n+1}\\}_{n\in\mathbb{N},0\leq j\leq n}$, and cyclic operators
$\\{t_{n}\colon A_{n}\to A_{n}\\}_{n\in\mathbb{N}}$ by setting

$\displaystyle
d_{0}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Ed0.eps}}}\;,\quad
d_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Edi.eps}}}\;,\quad
d_{n}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Edn.eps}}}\;,$
$\displaystyle
s_{j}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Esj.eps}}}\;,\quad
t_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Etn.eps}}}\;.$

This construction is functorial in $A$, that is, a morphism between algebras
in $\mathcal{B}$ induces, in a functorial way, the morphism of corresponding
cyclic $\Bbbk$-modules.

###  6.3. Cyclic duals

The cyclic duality $L$ from Section 2.4 transforms the cocyclic $\Bbbk$-module
$C^{\bullet}$ from Section 6.1 into the cyclic $\Bbbk$-module
$C^{\bullet}\circ L$. For any $n\in\mathbb{N}$, $C^{\bullet}\circ
L(n)=C^{n}=\mathrm{Hom}_{\mathcal{B}}(C^{\otimes n+1},\mathbb{1})$. The faces
$\\{\tilde{d}_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the degeneracies
$\\{\tilde{s}_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$, and the cyclic
operators $\\{\tilde{t}_{n}\\}_{n\in\mathbb{N}}$ are computed by formulas

$\displaystyle\tilde{d}_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{HDsj.eps}}}\;,\quad\tilde{d}_{n}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{EdeltanFASO.eps}}}\;,$
$\displaystyle\tilde{s}_{j}^{n}(f)=\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Edeltai.eps}}}\;,\quad\tilde{t}_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Etn.eps}}}\;.$

Similarly, the functor $L^{\mathrm{op}}$ transforms the cyclic $\Bbbk$-module
$A_{\bullet}$ from Section 6.2 into the cocyclic $\Bbbk$-module
$A_{\bullet}\circ L^{\mathrm{op}}$. By definitions, $A_{\bullet}\circ
L^{\mathrm{op}}(n)=A_{n}=\mathrm{Hom}_{\mathcal{B}}(A^{\otimes
n+1},\mathbb{1})$ for all $n\in\mathbb{N}$. The cofaces
$\\{\tilde{\delta}_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the
codegeneracies $\\{\tilde{\sigma}_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$,
and the cocyclic operators $\\{\tilde{\tau}_{n}\\}_{n\in\mathbb{N}}$ are
computed by formulas

$\displaystyle\tilde{\delta}_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Esj.eps}}}\;,\quad\tilde{\delta}_{n}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{EdeltanAKMA.eps}}}\;,\quad$
$\displaystyle\tilde{\sigma}_{j}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Edi.eps}}}\;,\quad\tilde{\tau}_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{HDtn.eps}}}\;.$

Note that the construction $A_{\bullet}\circ L^{\mathrm{op}}$ is a particular
case of the work of Akrami and Majid [AM04] (since any algebra in a braided
category with a twist is a ribbon algebra in the sense of [AM04]).

###  6.4. On the (co)homology of $C^{\bullet}$ and $C_{\bullet}$

Let $\Bbbk$ be a commutative ring, $\mathcal{B}$ a braided $\Bbbk$-category
with a twist and $C$ any coalgebra in $\mathcal{B}$. Let
$\alpha\colon\mathbb{1}\to C$ be a morphism such that
$\varepsilon\alpha=\mathrm{id}_{\mathbb{1}}$. For example, if $C$ is a
bialgebra in $\mathcal{B}$, then we can take $\alpha$ to be the unit of $C$.
By expanding and writing in the categorical setting [KR05, Remark 1.], we
obtain that Hochschild (co)homology of underlying (co)simplicial modules of
$C^{\bullet}$ and $C_{\bullet}$ appears only in degree zero. Indeed, the
$n$-th Hochschild cohomology $HH^{n}(C^{\bullet})$ of $C^{\bullet}$ is the
$n$-th cohomology of the cochain complex

$\textstyle{\mathrm{Hom}_{\mathcal{B}}(C,\mathbb{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
2},\mathbb{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\textstyle{\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
3},\mathbb{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
22.76228pt\beta_{3}}$$\textstyle{\cdots}$

where $\beta_{n}=\sum_{i=0}^{n}(-1)^{i}\delta_{i}^{n}.$ Then the family
$\\{h_{n}\colon\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+1},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n},\mathbb{1})\\}_{n\in\mathbb{N}^{*}},$ defined by setting for any
$f\in\mathrm{Hom}_{\mathcal{B}}(C^{\otimes n+1},\mathbb{1})$,

$h_{n}(f)=~{}~{}\,\raisebox{-0.55pt}{\scalebox{0.95}{\includegraphics{Ed02.eps}}}\;,$

satisfies the equalities

$\displaystyle\beta_{n}h_{n}+h_{n+1}\beta_{n+1}$
$\displaystyle=\mathrm{id}_{\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+1},\mathbb{1})}\quad\text{for }n\geq 1\quad\text{and}$ $\displaystyle
h_{1}\beta_{1}+\mathrm{Hom}_{\mathcal{B}}(\alpha\varepsilon,\mathbb{1})$
$\displaystyle=\mathrm{id}_{\mathrm{Hom}_{\mathcal{B}}(C,\mathbb{1})}.$

As a corollary, $HH^{0}(C^{\bullet})=\ker(\beta_{1})$ and
$HH^{n}(C^{\bullet})=0$ for $n>0$. From the cohomological form of the Connes’
long exact sequence [We94, Proposition 9.6.11] and the fact that Hochschild
and cyclic (co)homology always agree in degree $0$, we easily obtain that
$HC^{n}(C^{\bullet})\cong\ker(\beta_{1})$ for even $n$ and
$HC^{n}(C^{\bullet})\cong 0$ for odd $n$. A similar statement can be derived
for Hochschild and cyclic homologies of $C_{\bullet}.$ This calculation shows
that in a sense, $C^{\bullet}$ and $C_{\bullet}$ are not interesting from the
homological point of view. Therefore, we focus on their cyclic duals in
sections that follow.

###  6.5. Internal characters

Let $\mathcal{B}$ be a ribbon $\Bbbk$-category with a coend $\mathbb{F}$. For
any object $X\in\mathrm{Ob}(\mathcal{B})$, the morphism
$\chi_{X}=i_{X}\widetilde{\text{coev}}_{X}\colon\mathbb{1}\to\mathbb{F}$, also
known as internal character [FGR19, Sh17], enjoys the following trace-like
property:

$\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{coHoch.eps}}}\;\overset{(a)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{coHoch2.eps}}}\;\overset{(b)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{coHoch3.eps}}}\;.$
(18)

Here $(a)$ follows by definition of comultiplication of $\mathbb{F}$,
naturality and definition of twists, braidings and the isotopy invariance of
graphical calculus. Note that a pictorial proof of this fact is given in
[Ba22a, p.22]. The equality $(b)$ is obtained by composing both sides of $(a)$
with
$\tau_{\mathbb{F},\mathbb{F}}^{-1}(\mathrm{id}_{\mathbb{F}}\otimes{\theta}_{\mathbb{F}}^{-1})$.

Next, the morphism $\psi_{X}=\omega(\chi_{X}\otimes\mathrm{id}_{\mathbb{F}})$
satisfies

$\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{Hoch2.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{Hoch3.eps}}}\;.$
(19)

Indeed, we have

$\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{hoch4.eps}}}\;\overset{(a)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{hoch5.eps}}}\;\overset{(b)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{hoch6.eps}}}\;\overset{(c)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{hoch7.eps}}}\;\overset{(d)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{hoch8.eps}}}\;.$

Here $(a)$ and $(d)$ follow by the axioms of Hopf pairing $\omega$, $(b)$ from
the equation (18), $(c)$ by equations (16), (17), naturality of braiding and
isotopy.

In the case when $\mathcal{B}$ is a ribbon fusion category with the
representative set $I$ of simple objects of $\mathcal{B}$, the family
$\\{\chi_{i}\\}_{i\in I}$ is a basis for
$\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F})$. Next, if the pairing
$\omega$ is non-degenerate,
$\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F})$ is isomorphic to
$\mathrm{Hom}_{\mathcal{B}}(\mathbb{F},\mathbb{1})$ as $\Bbbk$-module.
Combining this with the equation (19), we get that
$HC^{0}(\mathbb{F}_{\bullet}\circ
L^{\mathrm{op}})=HH^{0}(\mathbb{F}_{\bullet}\circ
L^{\mathrm{op}})=\mathrm{Hom}_{\mathcal{B}}(\mathbb{F},\mathbb{1})$.

######  Remark 6.1.

Let $\Bbbk$ be an algebraically closed field and $\mathcal{B}$ a finite tensor
$\Bbbk$-category in the sense of [EO04] with the representative set $I$ of
simple objects. In particular, $I$ is finite. By [Sh17, Theorem 4.1.], the
characters $\\{\chi_{i}\\}_{i\in I}$ form a linearly independent set in
$\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F})$. Under additional
unimodularity hypothesis, it is also shown that $\\{\chi_{i}\\}_{i\in I}$ is a
basis for $\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F})$ if and only if
$\mathcal{B}$ is semisimple. Note that Shimizu works with ends and not coends
(see Appendix). However, these theories are essentially the same (see [Sh17,
Remark 3.12.]).

###  6.6. The coend of the representation category of a Hopf algebra

Let $\Bbbk$ be a field and $H$ a finite dimensional ribbon Hopf algebra over
$\Bbbk$. For the comultiplication, we will use the usual Sweedler notation,
i.e., $\Delta(x)=x_{(1)}\otimes x_{(2)}$ for any $x\in H$. Denote by $R$ and
$\theta$ the $R$-matrix and the twist element of $H$. To recall these notions
and their properties one may consult [Vi06, sections 4.2-4.6], which we
closely follow. We write

$R=\sum_{i}a_{i}\otimes b_{i}\in H\otimes H\quad\text{ and
}R^{-1}=\sum_{i}\alpha_{i}\otimes\beta_{i}\in H\otimes H.$

The category $\text{rep}_{H}$ is a ribbon category. Let $\mathbb{F}$ be the
coend of the category $\text{rep}_{H}$. As a vector space, it is equal to
$H^{*}$ and as a left $H$-module, it is given by the coadjoint action, that
is, for all $h\in H$ and $f\in H^{*}$,

$h\triangleright f=f(S(h_{(1)}\\_h_{(2)}).$

Consider the evaluation $\text{ev}\colon H^{\otimes
n}\to\mathrm{Hom}_{\Bbbk}({{H^{*}}^{\otimes n}},\Bbbk)$ defined by setting for
all $X\in H^{\otimes n}$ and $f\in{{H^{*}}^{\otimes n}}$,

$\text{ev}(X)(f)=\langle f,X\rangle$

Following [Vi06, Lemma 4.5(d)],
$\mathrm{Hom}_{\text{rep}_{H}}(\mathbb{F}^{\otimes n},\Bbbk)$ is isomorphic to
the $\Bbbk$-vector space

$V_{n}(H)=\\{X\in H^{\otimes n}~{}|~{}X\triangleleft h=\varepsilon(h)X\text{
for any }h\in H\\}.$

We recall for any $n\in\mathbb{N}^{*}$, the right $H$-action $\triangleleft$
on $H^{\otimes n}$ is defined by the formula

$X\triangleleft h=S(h_{(1)})X_{1}h_{(2)}\otimes
S(h_{(3)})X_{2}h_{(4)}\otimes\cdots\otimes S(h_{(2n-1)})X_{n}h_{(2n)},$

where $h\in H$ and $X=X_{1}\otimes\cdots\otimes X_{n}\in H^{\otimes n}$ is an
elementary tensor. Remark that the vector space $V_{n}(H)$ is exactly the
$0$-th usual Hochschild homology $HH_{0}(H,H^{\otimes n})$ of $H$ with
coefficients in $H^{\otimes n}$, where $H^{\otimes n}$ is the bimodule over
$H$ (the left action is given by trivial action via counit).

Under the above isomorphism, the cyclic $\Bbbk$-vector space
$\mathbb{F}^{\bullet}\circ L$ is identified with the cyclic $\Bbbk$-vector
space $\textbf{W}_{\bullet}$ which is defined as follows. For any
$n\in\mathbb{N}$, set $\textbf{W}_{n}=V_{n+1}(H)$. The faces
$\\{d_{i}^{n}\colon V_{n+1}(H)\to V_{n}(H)\\}_{n\in\mathbb{N}^{*},0\leq i\leq
n}$ are given by setting for any elementary tensor $h_{1}\otimes\cdots\otimes
h_{n+1}\in V_{n+1}(H)$,

$\displaystyle d_{i}^{n}(h_{1}\otimes\cdots\otimes h_{n+1})$
$\displaystyle=h_{1}\otimes h_{2}\otimes\cdots\otimes
h_{i+1}h_{i+2}\otimes\cdots\otimes h_{n+1}\quad\text{ and }$ $\displaystyle
d_{n}^{n}(h_{1}\otimes\cdots\otimes h_{n+1})$
$\displaystyle=\sum_{i}(h_{n+1}\triangleleft
a_{i}\theta)(h_{1}\triangleleft(b_{i})_{(1)})\otimes
h_{2}\triangleleft(b_{i})_{(2)}\otimes\cdots\otimes
h_{n}\triangleleft(b_{i})_{(n)}.$

The degeneracies $\\{s_{j}^{n}\colon V_{n+1}(H)\to
V_{n+2}(H)\\}_{n\in\mathbb{N},0\leq j\leq n}$ are given by setting for any
elementary tensor $h_{1}\otimes\cdots\otimes h_{n+1}\in V_{n+1}(H)$,

$s_{j}(h_{1}\otimes\cdots\otimes h_{n+1})=h_{1}\otimes\cdots\otimes
h_{j+1}\otimes 1_{H}\otimes h_{j+2}\otimes\cdots\otimes h_{n+1}.$

The cyclic operators $\\{t_{n}\colon V_{n+1}(H)\to
V_{n+1}(H)\\}_{n\in\mathbb{N}}$ are given by setting for any elementary tensor
$h_{1}\otimes\cdots\otimes h_{n+1}\in V_{n+1}(H)$,

$t_{n}(h_{1}\otimes\cdots\otimes h_{n+1})=\sum_{i}h_{n+1}\triangleleft
a_{i}\theta\otimes h_{1}\triangleleft(b_{i})_{(1)}\otimes\cdots\otimes
h_{n}\triangleleft(b_{i})_{(n)}.$

Similarly, the cocyclic $\Bbbk$-vector space $\mathbb{F}_{\bullet}\circ
L^{\mathrm{op}}$ is identified with the cocyclic $\Bbbk$-vector space
$\textbf{W}^{\bullet}$ which is defined as follows. For any $n\in\mathbb{N}$,
set $\textbf{W}^{n}=V_{n+1}(H)$. The cofaces $\\{\delta_{i}^{n}\colon
V_{n}(H)\to V_{n+1}(H)\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$ are given by
setting for any elementary tensor $h_{1}\otimes\cdots\otimes h_{n}\in
V_{n}(H),$

$\displaystyle\delta_{i}^{n}(h_{1}\otimes\cdots\otimes h_{n})$
$\displaystyle=h_{1}\otimes
h_{2}\otimes\cdots\otimes\Delta^{\text{Bd}}(h_{i+1})\otimes\cdots\otimes
h_{n}\quad\text{and}$ $\displaystyle\delta_{n}^{n}(h_{1}\otimes\cdots\otimes
h_{n})$
$\displaystyle=\sum_{i}(h_{1})_{(2)}^{\text{B}d}\triangleleft(\beta_{i})_{(1)}\otimes
h_{2}\triangleleft(\beta_{i})_{(2)}\otimes\cdots\otimes
h_{n}\triangleleft(\beta_{i})_{(n)}\otimes(h_{1})_{(1)}^{\text{B}d}\triangleleft(\alpha_{i}\theta^{-1}),$

where $\Delta^{\text{B}d}$ is a comultiplication on the braided Hopf algebra
$H^{\text{B}d}$ (see [Vi06, Lemma 4.4]) associated to $H$. As algebras
$H^{\text{B}d}=H$, but the comultiplication and antipode in $H^{\text{B}d}$
are different. Explicitly, for each $x\in H$,

$\Delta^{\text{B}d}(x)=x_{(2)}a_{i}\otimes
S((b_{i})_{(1)})x_{(1)}(b_{i})_{(2)}.$

The codegeneracies $\\{\sigma_{j}^{n}\colon V_{n+2}(H)\to
V_{n+1}(H)\\}_{n\in\mathbb{N},0\leq j\leq n}$ are given by setting for any
elementary tensor $h_{1}\otimes\cdots\otimes h_{n+2}\in V_{n+2}(H)$,

$\sigma_{j}(h_{1}\otimes\cdots\otimes h_{n+2})=h_{1}\otimes\cdots\otimes
h_{j+1}\otimes\varepsilon(h_{j+2})\otimes\cdots\otimes h_{n+2}.$

The cocyclic operators $\\{\tau_{n}\colon V_{n+1}(H)\to
V_{n+1}(H)\\}_{n\in\mathbb{N}}$ are given by setting for any elementary tensor
$h_{1}\otimes\cdots\otimes h_{n+1}\in V_{n+1}(H)$,

$\tau_{n}(h_{1}\otimes\cdots\otimes
h_{n+1})=\sum_{i}h_{2}\triangleleft(\beta_{i})_{(1)}\otimes\cdots\otimes
h_{n+1}\triangleleft(\beta_{i})_{(n)}\otimes
h_{1}\triangleleft(\alpha_{i}\theta^{-1}).$

##  7\. Cyclic modules from TQFTs

In Theorem 4.1 says that there exist (co)cyclic objects in the category of
$3$-cobordisms. By composition, any $3$-dimensional TQFT induces a (co)cyclic
$\Bbbk$-module. In this section, we compute it for the Reshetikhin-Turaev TQFT
$\text{RT}_{\mathcal{B}}\colon\textbf{Cob}_{3}\to\mathrm{Mod}_{\Bbbk}$
associated to an anomaly free modular category $\mathcal{B}$. Note that the
coend $\mathbb{F}$ of $\mathcal{B}$ exists and is a Hopf algebra in
$\mathcal{B}$. Recall the cocyclic $\Bbbk$-module $\mathbb{F}^{\bullet}$ and
the cyclic $\Bbbk$-module $\mathbb{F}_{\bullet}$ (see Section 6) associated to
$\mathbb{F}$, as well as the reindexing involution $\Phi\colon\Delta
C\to\Delta C$ (see Section 2.5). The second main result of this paper is the
following:

######  Theorem 7.1.

The cocyclic $\Bbbk$-modules $\emph{\text{RT}}_{\mathcal{B}}\circ X^{\bullet}$
and $\mathbb{F}^{\bullet}\circ\Phi$ are isomorphic. The cyclic $\Bbbk$-modules
$\emph{\text{RT}}_{\mathcal{B}}\circ X_{\bullet}$ and
$\mathbb{F}_{\bullet}\circ\Phi^{\mathrm{op}}$ are isomorphic.

Since, $\Phi$ does not affect the associated differentials, for all
$n\in\mathbb{N}$, the $n$-th Hochschild cohomology of
$\mathbb{F}^{\bullet}\circ\Phi$ and $\mathbb{F}^{\bullet}$ are equal. The
cohomological form of the Connes’ long exact sequence [We94, Proposition
9.6.11] then implies that this is also the case for their cyclic cohomology.
The proof of Theorem 7.1 is given in sections 7.1–7.5.

The cyclic duality $L\colon\Delta C^{\mathrm{op}}\to\Delta C$ and reindexing
involution automorphism $\Phi$ transform the cocyclic object $X^{\bullet}$ in
$\textbf{Cob}_{3}$ into a cyclic object $X^{\bullet}\circ\Phi\circ L$ in
$\textbf{Cob}_{3}$. Similarly, the functors $L^{\mathrm{op}}\colon\Delta
C\to\Delta C^{\mathrm{op}}$ and $\Phi^{\mathrm{op}}$ transform the cyclic
object $X_{\bullet}$ in $\textbf{Cob}_{3}$ into a cocyclic object
$X_{\bullet}\circ\Phi^{\mathrm{op}}\circ L^{\mathrm{op}}$ in
$\textbf{Cob}_{3}$. By Theorem 7.1 and the fact that $\Phi$ is involutive, we
obtain the following:

######  Corollary 7.2.

The cyclic $\Bbbk$-modules $\emph{\text{RT}}_{\mathcal{B}}\circ
X^{\bullet}\circ\Phi\circ L$ and $\mathbb{F}^{\bullet}\circ L$ are isomorphic.
The cocyclic $\Bbbk$-modules $\emph{\text{RT}}_{\mathcal{B}}\circ
X_{\bullet}\circ\Phi^{\mathrm{op}}\circ L^{\mathrm{op}}$ and
$\mathbb{F}_{\bullet}\circ L^{\mathrm{op}}$ are isomorphic.

Recall that the cyclic $\Bbbk$-module $\mathbb{F}^{\bullet}\circ L$ and the
cocyclic $\Bbbk$-module $\mathbb{F}_{\bullet}\circ L^{\mathrm{op}}$ (see
Section 6.3) are cyclic duals of $\mathbb{F}^{\bullet}$ and the cyclic
$\Bbbk$-module $\mathbb{F}_{\bullet}$.

Another fundamental construction of a $3$-dimensional TQFT is the Turaev-Viro
TQFT $\text{TV}_{\mathcal{C}}\colon\textbf{Cob}_{3}\to\mathrm{Mod}_{\Bbbk}$
associated to a spherical fusion $\Bbbk$-category $\mathcal{C}$ with
invertible dimension (for details, see [TV17]). Moreover, in the case when
$\mathcal{C}$ is additive and $\Bbbk$ is an algebraically closed field, the
center $\mathcal{Z}(\mathcal{C})$ of $\mathcal{C}$ is an anomaly free modular
category (see [TV17, Theorem 5.3., Theorem 5.4.]). In this case, according to
[TV17, Theorem 17.1.], the TQFTs $\text{RT}_{\mathcal{Z}(\mathcal{C})}$ and
$\text{TV}_{\mathcal{C}}$ are isomorphic. Denote by $\mathbb{G}$ the coend of
$\mathcal{Z}(\mathcal{C})$. These results and Theorem 7.1 imply the following
corollary:

######  Corollary 7.3.

The cocyclic $\Bbbk$-modules $\emph{\text{TV}}_{\mathcal{C}}\circ X^{\bullet}$
and $\mathbb{G}^{\bullet}\circ\Phi$ are isomorphic. The cyclic $\Bbbk$-modules
$\emph{\text{TV}}_{\mathcal{C}}\circ X_{\bullet}$ and
$\mathbb{G}_{\bullet}\circ\Phi^{\mathrm{op}}$ are isomorphic.

By using cyclic duality $L$ and reindexing involution $\Phi$, we obtain:

######  Corollary 7.4.

The cyclic $\Bbbk$-modules $\emph{\text{TV}}_{\mathcal{C}}\circ
X^{\bullet}\circ\Phi\circ L$ and $\mathbb{G}^{\bullet}\circ L$ are isomorphic.
The cocyclic $\Bbbk$-modules $\emph{\text{TV}}_{\mathcal{C}}\circ
X_{\bullet}\circ\Phi^{\mathrm{op}}\circ L^{\mathrm{op}}$ and
$\mathbb{G}_{\bullet}\circ L^{\mathrm{op}}$ are isomorphic.

To show the claim from Theorem 7.1, it follows from Section 4.3 that it
suffices to compute $\text{RT}_{\mathcal{B}}\circ Y^{\bullet}$ and
$\text{RT}_{\mathcal{B}}\circ Y_{\bullet}$. In Section 7.1, we give some
algebraic preliminaries on coend of a modular category. Next, in Section 7.2,
we describe the Reshetikhin-Turaev TQFT $\text{RT}_{\mathcal{B}}$ via the
coend $\mathbb{F}$ of $\mathcal{B}$. Then, in Section 7.3, we compute the
cocyclic $\Bbbk$-module $\text{RT}_{\mathcal{B}}\circ Y^{\bullet}$. In Section
7.4 we prove that the latter is isomorphic to $\mathbb{F}^{\bullet}\circ\Phi$,
as stated in Theorem 7.1. Finally, in Section 7.5, we sketch the computation
of $\text{RT}_{\mathcal{B}}\circ Y_{\bullet}$ and the proof of the fact that
it is isomorphic to the cyclic $\Bbbk$-module
$\mathbb{F}_{\bullet}\circ\Phi^{\mathrm{op}}$.

Remember that $\mathbb{F}$ is a Hopf algebra in $\mathcal{B}$ endowed with a
Hopf pairing $\omega\colon\mathbb{F}\otimes\mathbb{F}\to\mathbb{1}$. Here, we
denote by $m$, $u$, $\Delta$, $\varepsilon$, and $S$ multiplication, unit,
comultiplication, counit, and antipode of $\mathbb{F}$, respectively. While
using graphical calculus, we often drop $\mathbb{F}$ from notation.

###  7.1. Modularity and pairing of a coend

In this section, we provide some algebraic preliminaries needed for
computation of the Reshetikhin-Turaev TQFT $\text{RT}_{\mathcal{B}}$ via the
coend of anomaly-free modular category. In the following lemma, we compute the
inverse, under some conditions, of the pairing of the coend. Note that the
statement and the proof of Lemma 7.5$(b)$ is similar to [TV17, Lemma 6.2.].

######  Lemma 7.5.

Let $\mathcal{B}$ be a ribbon $\Bbbk$-category with a coend $\mathbb{F}$ and
suppose that the canonical pairing
$\omega\colon\mathbb{F}\otimes\mathbb{F}\to\mathbb{1}$ associated to the coend
is non-degenerate.

  * (a)

If $\Lambda\colon\mathbb{1}\to\mathbb{F}$ is a right integral of a coend, then
$\omega(\Lambda\otimes\mathrm{id}_{\mathbb{F}})\colon\mathbb{F}\to\mathbb{1}$
is a left cointegral of $\mathbb{F}.$

  * (b)

Let $\Lambda$ be a right integral of the coend $\mathbb{F}$ of $\mathcal{B}$.
Suppose that the element $\omega(\Lambda\otimes\Lambda)$ is invertible. The
inverse of the pairing $\omega$ is given by the morphism
$\Omega\colon\mathbb{1}\to\mathbb{F}\otimes\mathbb{F}$, which is computed by

$\Omega=\left(\,\raisebox{-0.4pt}{\scalebox{0.85}{\includegraphics{omegaalphaalpha.eps}}}\;\right)^{-1}\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral.eps}}}\;.$

###### Proof.

  * (a)

By the property of the bialgebra pairing and the fact that $\Lambda$ is a
right integral for $C$, we have

$\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{waidcoint.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{waidcoint1.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{waidcoint2.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{waidcoint3.eps}}}\;.$

The conclusion follows, since $\omega$ is non-degenerate.

  * (b)

We have

$\displaystyle\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral0.eps}}}\;\overset{(i)}{=}\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral1.eps}}}\;\overset{(ii)}{=}\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral2.eps}}}\;\overset{(iii)}{=}\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral3.eps}}}\;$
$\displaystyle\overset{(iv)}{=}$
$\displaystyle\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral4.eps}}}\;\overset{(v)}{=}\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral5.eps}}}\;\overset{(vi)}{=}\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral6.eps}}}\;\overset{(vii)}{=}\,\raisebox{-0.55pt}{\scalebox{0.85}{\includegraphics{inverseintegral8.eps}}}\;,$

and so the claim follows by invertibility of $\omega(\Lambda\otimes\Lambda)$.
Here $(i)$ follows by property of bialgebra pairing, $(ii)$ by (co)unitality,
$(iii)$ by the part $(a)$, $(iv)$ from bialgebra axioms, $(v)$ by
(co)associativity, $(vi)$ by the antipode axiom and (co)unitality, $(vii)$ by
the naturality of the braiding, the fact that $\Lambda$ is a right integral
for $\mathbb{F}$ and counitality. ∎

######  Corollary 7.6.

Let $\mathcal{B}$ be a ribbon $\Bbbk$-category with a coend $\mathbb{F}$ and
$\Lambda\colon\mathbb{1}\to\mathbb{F}$ a right integral for the coend
$\mathbb{F}$. If the pairing
$\omega\colon\mathbb{F}\otimes\mathbb{F}\to\mathbb{1}$ is non-degenerate, then

$\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{cylopentan5copy.eps}}}\;=\omega(\Lambda\otimes\Lambda)\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{idhigh.eps}}}\;.$
(20)

###### Proof.

By Lemma 7.5$(b)$ and invertibility of the antipode, we have

$\omega(\Lambda\otimes\Lambda)\mathrm{id}_{\mathbb{F}}\overset{(i)}{=}\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{invpetitcalcul1.eps}}}\;\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{invpetitcalcul2.eps}}}\;\overset{(iii)}{=}\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{cylopentan5copy.eps}}}\;.$

Here $(i)$ and $(iii)$ follow from invertibility of the antipode, $(ii)$
follows by Lemma 7.5$(b)$. ∎

If $\mathcal{B}$ is an additive ribbon fusion $\Bbbk$-category with a coend
$\mathbb{F}$, then, according to [TV17, Theorem 6.6.], the category
$\mathcal{B}$ is modular (in the sense of Section 5.12) if and only if the
canonical pairing $\omega\colon\mathbb{F}\otimes\mathbb{F}\to\mathbb{1}$
associated to $\mathbb{F}$ is non-degenerate. Moreover, the coend $\mathbb{F}$
is given by $\mathbb{F}=\bigoplus_{i\in I}i^{*}\otimes i$, where $I$ is a
representative set of simple objects of $\mathcal{B}$. For $i\in I$, we denote
the projection associated with the direct sum decomposition by
$p_{i}\colon\mathbb{F}\to i^{*}\otimes i$. However, we drop inclusions
$i^{*}\otimes i\to\mathbb{F}$ in our notation. By [TV17, Theorem 6.4.], any
integral of $\mathbb{F}$ is a scalar multiple of the universal integral

$\Lambda=\sum_{i\in
I}\dim(i)\widetilde{\text{coev}}_{i}\colon\mathbb{1}\to\mathbb{F}.$ (21)

Similarly, one can show that any cointegral of $\mathbb{F}$ is a scalar
multiple of the universal cointegral

$\lambda=\text{ev}_{\mathbb{1}}p_{\mathbb{1}}\colon\mathbb{F}\to\mathbb{1}.$
(22)

Universal (co)integrals $\lambda$ and $\Lambda$ satisfy
$\lambda\Lambda=\mathrm{id}_{\mathbb{1}}.$

######  Remark 7.7.

Let $\mathcal{B}$ be an additive ribbon fusion $\Bbbk$-category such that the
canonical pairing $\omega$ associated to the coend $\mathbb{F}$ is non-
degenerate. Recall the universal integral $\Lambda$ and the universal
cointegral $\lambda$ of the coend $\mathbb{F}$, defined in equations (21) and
(22), respectively. Let us calculate $\omega(\Lambda\otimes\Lambda)$. By Lemma
7.5 $(a)$, $\omega(\Lambda\otimes\mathrm{id}_{\mathbb{F}})$ is a left
cointegral of $\mathbb{F}$. By universality of $\lambda$,
$\omega(\Lambda\otimes\mathrm{id}_{\mathbb{F}})=k\lambda$, for some
$k\in\Bbbk\cong\text{End}(\mathbb{1}).$ This further implies that

$\omega(\Lambda\otimes\Lambda)=k\lambda\Lambda=k\mathrm{id}_{\mathbb{1}}.$

Now, remark also that

$\omega(\Lambda\otimes u)=\varepsilon\Lambda=\dim(\mathcal{B}).$

These two properties together with the fact that
$p_{\mathbb{1}}i_{\mathbb{1}}=\mathrm{id}_{\mathbb{1}^{*}\otimes\mathbb{1}}$,
with definition of unit $u$ of the coend $\mathbb{F}$, and definition of
$\lambda$, give that

$\displaystyle\omega(\Lambda\otimes\Lambda)=$ $\displaystyle
k\mathrm{id}_{\mathbb{1}}=k(\mathrm{id}_{\mathbb{1}}\otimes\text{ev}_{\mathbb{1}})(\text{coev}_{\mathbb{1}}\otimes\mathrm{id}_{\mathbb{1}})=k(\mathrm{id}_{\mathbb{1}}\otimes\text{ev}_{\mathbb{1}}p_{\mathbb{1}})(\mathrm{id}_{\mathbb{1}}\otimes
i_{\mathbb{1}})(\text{coev}_{\mathbb{1}}\otimes\mathrm{id}_{\mathbb{1}})=$
$\displaystyle=$ $\displaystyle
k(\mathrm{id}_{\mathbb{1}}\otimes\lambda)u=k\lambda u=\omega(\Lambda\otimes
u)=\dim(\mathcal{B}).$

###  7.2. The Reshetikhin-Turaev TQFT via coends

In this section we compute the Reshetikhin-Turaev TQFT
$\text{RT}_{\mathcal{B}}$ associated with an anomaly free modular category
$\mathcal{B}$. More precisely, given a special ribbon graph $\Gamma$, we
compute $\text{RT}_{\mathcal{B}}(M_{\Gamma})$ in terms of the coend of
${\mathcal{B}}$, where $M_{\Gamma}$ is the 3-cobordism represented by
$\Gamma$.

In [Tu10], Turaev associates to any modular category ${\mathcal{B}}$ a
3-dimensional TQFT $\text{RT}_{\mathcal{B}}$. There, a precise definition of a
$3$-dimensional TQFT involves Lagrangian spaces in homology of surfaces and
$p_{1}$-structures in cobordisms. However, if the modular category
${\mathcal{B}}$ is anomaly free (see Section 5.12), then the TQFT
$\text{RT}_{\mathcal{B}}$ does not depend on this additional data and is a
genuine symmetric monoidal functor
$\text{RT}_{\mathcal{B}}\colon\textbf{Cob}_{3}\to\mathrm{Mod}_{\Bbbk}$.

Let ${\mathcal{B}}$ be an anomaly free modular $\Bbbk$-category. Recall from
Section 5.12 that the scalar $\Delta=\Delta_{+}=\Delta_{-}$ is invertible and
satisfies $\Delta^{2}=\dim(\mathcal{B})$. By Section 3.6, any special ribbon
$(g,h)$-graph $\Gamma$ represents a $3$-cobordism $M_{\Gamma}\colon S_{g}\to
S_{h}$. Our goal is to compute the $\Bbbk$-linear homomorphism

$\text{RT}_{\mathcal{B}}(M_{\Gamma})\colon\text{RT}_{\mathcal{B}}(S_{g})\to\text{RT}_{\mathcal{B}}(S_{h})$

in terms of the coend $\mathbb{F}$ of ${\mathcal{B}}$ (which always exists,
see Section 7.1). First, it follows from the definition of
$\text{RT}_{\mathcal{B}}$ and the computation of the coend $\mathbb{F}$ in
terms of a representative set $I$ of simple objects of ${\mathcal{B}}$ that

$\text{RT}_{\mathcal{B}}(S_{g})=\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F}^{\otimes
g})\quad\text{and}\quad\text{RT}_{\mathcal{B}}(S_{h})=\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F}^{\otimes
h}).$

Next, the formula $(2.3)(a)$ from [Tu10, Section IV.2.3], which computes
$\text{RT}_{\mathcal{B}}(M_{\Gamma})$, rewrites in our setting as

$\text{RT}_{\mathcal{B}}(M_{\Gamma})=\Delta^{-n-h}\,\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\lvert\Gamma\rvert),$
(23)

where $\lvert\Gamma\rvert\colon\mathbb{F}^{\otimes g}\to\mathbb{F}^{\otimes
h}$ is a morphism in $\mathcal{B}$ defined as follows. By pulling down some
part of each circle component of $\Gamma$ and of each arc connecting the
outputs of $\Gamma$, we obtain that the ribbon graph $\Gamma$ is isotopic to

0.8invariantRibGrphcolorfree.eps

where the cups $L_{1},\dots,L_{n}$ correspond to the circle components of
$\Gamma$ and the cups $C_{1},\dots,C_{h}$ correspond to the upper arcs of
$\Gamma$. Here, $\widetilde{\Gamma}$ is a ribbon graph with $(g+n+2h)$ arcs,
$(2g+2n+2h)$ inputs, $2h$ outputs, no coupons, no circle components, and such
that:

  * •

for all $1\leq i\leq g+n$, an arc $a_{i}$ connects the $(2i-1)$-th input to
the $(2i)$-th input of $\widetilde{\Gamma}$,

  * •

for all $1\leq j\leq h$, an arc $u_{j}$ connects the $(2g+2n+2j-1)$-th input
to the $(2j-1)$-th output of $\widetilde{\Gamma}$, and an arc $v_{j}$ connects
the $(2j)$-th output to the $(2g+2n+2j)$-th input of $\widetilde{\Gamma}$.

The ribbon graph ${\Gamma}$ is called closure of the ribbon graph
$\widetilde{\Gamma}$. Coloring the arc $a_{i}$ by an object $X_{i}$ of
$\mathcal{B}$ and coloring both the arcs $u_{j},v_{j}$ by an object $Y_{j}$ of
$\mathcal{B}$, we obtain a $\mathcal{B}$-colored ribbon graph representing a
morphism $\phi_{X_{1},\dots,X_{g+n},Y_{1},\dots Y_{h}}$. Let $i=\\{i_{X}\colon
X^{*}\otimes X\to\mathbb{F}\\}_{X\in\mathrm{Ob}(\mathcal{B})}$ be the
universal dinatural transformation associated to the coend $\mathbb{F}$. Then
the family of morphisms

$(i_{Y_{1}}\otimes\cdots\otimes
i_{Y_{h}})\circ\phi_{X_{1},\dots,X_{g+n},Y_{1},\dots Y_{h}}$

from $X_{1}^{*}\otimes X_{1}\otimes\cdots\otimes X_{g+n}^{*}\otimes
X_{g+n}\otimes Y_{1}^{*}\otimes Y_{1}\otimes\cdots\otimes Y_{h}^{*}\otimes
Y_{h}$ to $\mathbb{F}^{\otimes h}$ is dinatural in each variable and so, by
Lemma 5.1, it factorizes as

$\phi_{\Gamma}\circ(i_{X_{1}}\otimes\cdots\otimes i_{X_{g+n}}\otimes
i_{Y_{1}}\otimes\cdots\otimes i_{Y_{h}})$

for a unique morphism $\phi_{\Gamma}\colon\mathbb{F}^{\otimes
g+n+h}\to\mathbb{F}^{\otimes h}$. Then

$\lvert\Gamma\rvert=\phi_{\Gamma}\circ(\mathrm{id}_{\mathbb{F}^{\otimes
g}}\otimes\Lambda^{\otimes(n+h)})\colon\mathbb{F}^{\otimes
g}\to\mathbb{F}^{\otimes h},$

where $\Lambda$ is the universal integral defined in the equation (21). It
follows from the fact that $\Lambda$ is a right integral for $\mathbb{F}$ (see
Section 7.1) that the morphism $\lvert\Gamma\rvert$ is an isotopy invariant of
$\Gamma$. This invariant is multiplicative:

$\lvert\Gamma\sqcup\Gamma^{\prime}\rvert=\lvert\Gamma\rvert\otimes\lvert\Gamma^{\prime}\rvert,$
(24)

for all special ribbon graphs, where $\Gamma\sqcup\Gamma^{\prime}$ is obtained
by concatenating $\Gamma^{\prime}$ to the right of $\Gamma$.

###  7.3. Computation of $\text{RT}_{\mathcal{B}}\circ Y^{\bullet}$

We remind that $\mathcal{B}$ denotes an anomaly-free modular category. In the
following lemma, we calculate the isotopy invariant $\lvert\cdot\rvert$ (see
Section 7.2) of particular special graphs:

######  Lemma 7.8.

If $T_{i}$ for $i=1,2,3$ are the following special ribbon graphs

$T_{1}=\,\raisebox{-0.45pt}{\scalebox{0.70}{\includegraphics{cyl.eps}}}\;,\qquad
T_{2}=\,\raisebox{0.13pt}{\scalebox{0.70}{\includegraphics{unitpiece.eps}}}\;,\qquad
T_{3}=\,\raisebox{-0.45pt}{\scalebox{0.70}{\includegraphics{mult.eps}}}\;,$

then

  * (a)

$\lvert T_{1}\rvert=\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}}$,

  * (b)

$\lvert T_{2}\rvert=\dim(\mathcal{B})u$,

  * (c)

$\lvert T_{3}\rvert=\dim(\mathcal{B})m$.

###### Proof.

  * (a)

A ribbon graph whose closure is isotopic to $T_{1}$ is

$\widetilde{T_{1}}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{cylopentan1.eps}}}\;.$

For all objects $X$, $Y$, and $Z$ in $\mathcal{B}$, we have:

0.75cylopentan2.eps
$\displaystyle=\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{cylopentan3.eps}}}\;$
$\displaystyle=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{cylopentan4.eps}}}\;.$

Hence, by definition of $\lvert\cdot\rvert$ given in Section 7.2, Corollary
7.6, and Remark 7.7, we have:

$\lvert
T_{1}\rvert=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{cylopentan5copy.eps}}}\;=\omega(\Lambda\otimes\Lambda)\mathrm{id}_{\mathbb{F}}=\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}}.$

  * (b)

A ribbon graph whose closure is isotopic to $T_{2}$ is

$\widetilde{T_{2}}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{unitopentangle1.eps}}}\;.$

For all objects $X,Y$ in $\mathcal{B}$, we have:

$\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{unitopentangle2.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{unitopentangle3.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{unitopentangle4.eps}}}\;.$

Hence, by definition of $\lvert\cdot\rvert$ given in Section 7.2, by Lemma 7.5
$(a)$, and Remark 7.7, we have:

$\lvert
T_{2}\rvert=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{unitopentangle5.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.85}{\includegraphics{unitopentangle6.eps}}}\;=\omega(\Lambda\otimes\Lambda)u=\dim(\mathcal{B})u.$

  * (c)

A ribbon graph whose closure is isotopic to $T_{3}$ is

$\widetilde{T_{3}}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{multopentan1.eps}}}\;.$

For all objects $X,Y,$ and $Z$ in $\mathcal{B}$, we have:

$\displaystyle\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{multopentan2.eps}}}\;=$
0.75multopentan3.eps $\displaystyle=$
$\displaystyle\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{multopentan4.eps}}}\;.$

Hence, by definition of $\lvert\cdot\rvert$ given in Section 7.2, axioms of a
Hopf pairing, the equation (17), Corollary 7.6, and Remark 7.7, we have

$\lvert
T_{3}\rvert=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{multopentan5.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{multopentan6.eps}}}\;=\omega(\Lambda\otimes\Lambda)m=\dim(\mathcal{B})m.$

∎

The previously proved Lemma 7.8 implies the following result:

######  Lemma 7.9.

We have:

  * (a)

$\lvert
G_{Y^{\bullet}(\delta_{i}^{n})}\rvert=\begin{cases}\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{facemAC1.eps}}},&\mbox{if
}i=0,\\\ &\\\
\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{facemAC2.eps}}},&\mbox{if
}1\leq i\leq n-1,\\\ &\\\
\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{facemAC3.eps}}}\;,&\mbox{if
}i=n,\end{cases}$

  * (b)

$\lvert
G_{Y^{\bullet}(\sigma_{j}^{n})}\rvert=\dim(\mathcal{B})^{n+1}\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenmAC.eps}}}\;$,

  * (c)

$\lvert
G_{Y^{\bullet}(\tau_{n})}\rvert=\begin{cases}\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}},&\mbox{if
}n=0,\\\ &\\\
\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopmAC.eps}}}\;&\mbox{if
}n\geq 1.\end{cases}$

###### Proof.

  * (a)

Recall the special ribbon graphs $T_{1}$ and $T_{2}$ from Lemma 7.8. Let
$n\in\mathbb{N}^{*}$ and $1\leq i\leq n-1$. By Lemma 7.8 and multiplicativity
of $\lvert\cdot\rvert$ (see equation (24)),

$\displaystyle\lvert G_{Y^{\bullet}(\delta_{i}^{n})}\rvert=$
$\displaystyle\left(\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}}\right)^{\otimes
i}\otimes\dim(\mathcal{B})u\otimes\left(\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}}\right)^{\otimes
n-i}=\hskip 18.49411pt$ $\displaystyle=$
$\displaystyle\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{facemAC2.eps}}}\;.$

The cases $i=0$ and $i=n$ are proven in a similar way.

  * (b)

Recall the special ribbon graphs $T_{1}$ and $T_{3}$ from Lemma 7.8. Let
$n\in\mathbb{N}$ and $0\leq j\leq n$. By Lemma 7.8,

$\displaystyle\lvert G_{Y^{\bullet}(\sigma_{j}^{n})}\rvert=$
$\displaystyle\left(\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}}\right)^{\otimes
j}\otimes\dim(\mathcal{B})m\otimes\left(\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}}\right)^{\otimes
n-j}$
$\displaystyle=\dim(\mathcal{B})^{n+1}\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenmAC.eps}}}\;.$

  * (c)

Recall that $Y^{\bullet}(\tau_{0})$ is the identity morphism
$\mathrm{id}_{S_{1}}$. Hence

$\lvert
G_{Y^{\bullet}(\tau_{0})}\rvert=\dim({\mathcal{B}})\mathrm{id}_{\mathbb{F}},$

by Lemma 7.8 (a). Let us show the statement in the case $n=1$. The general
case is verified similarly. The special ribbon graph
$G_{Y^{\bullet}(\tau_{1})}$ depicts as

$\,\raisebox{-0.55pt}{\scalebox{0.7}{\includegraphics{cycopentan1.eps}}}\;.$

A ribbon graph, whose closure is isotopic to the special ribbon graph
$G_{Y^{\bullet}(\tau_{1})}$ depicts as

$\widetilde{G_{Y^{\bullet}(\tau_{1})}}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopentan2.eps}}}\;.$

For all objects $X,Y,Z,U,V,$ and $W$ in $\mathcal{B}$, we have:

$\displaystyle\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopentan3.eps}}}\;=$
0.75cycopentan4.eps $\displaystyle=$ 0.75cycopentan5.eps

Hence, by definition of $\lvert\cdot\rvert$ given in Section 7.2, the
equations (17) and (16), Corollary 7.6 and Remark 7.7, we have:

$\displaystyle\lvert G_{Y^{\bullet}(\tau_{1})}\rvert=$
$\displaystyle\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopentan6.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopentan7.eps}}}\;$
$\displaystyle=$
$\displaystyle\left(\omega(\Lambda\otimes\Lambda)\right)^{2}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycop1mAC.eps}}}\;=\dim(\mathcal{B})^{2}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycop1mACleft.eps}}}\;.$

∎

The computation of $\mathrm{RT}_{\mathcal{B}}\circ Y^{\bullet}$ follows from
Lemma 7.9:

######  Lemma 7.10.

The cocyclic $\Bbbk$-module $\mathrm{RT}_{\mathcal{B}}\circ Y^{\bullet}$
equals to the cocyclic $\Bbbk$-module given by the family
$\\{\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F}^{n+1})\\}_{n\in\mathbb{N}}$,
equipped with the cofaces
$\\{\mathrm{RT}_{\mathcal{B}}(Y^{\bullet}(\delta_{i}^{n}))\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, codegeneracies
$\\{\mathrm{RT}_{\mathcal{B}}(Y^{\bullet}(\sigma_{j}^{n}))\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and cocyclic operators
$\\{\mathrm{RT}_{\mathcal{B}}(Y^{\bullet}(\tau_{n}))\\}_{n\in\mathbb{N}}$
given by formulas

$\displaystyle\delta_{0}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Fd0.eps}}}\;\quad\hskip
2.84544pt,\quad\delta_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Fdi.eps}}}\;\quad\hskip
2.84544pt,\quad\delta_{n}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Fdn.eps}}}\;\hskip
2.84544pt,$
$\displaystyle\sigma_{j}^{n}(f)=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{Fsj.eps}}}\;\quad\hskip
2.84544pt,\quad\tau_{n}(f)=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{Ftn.eps}}}\;.$

###### Proof.

By using the formula (23) from Section 7.2, the construction of $Y^{\bullet}$
from Section 4.1, and the Lemma 7.9, we calculate
$\\{\text{RT}_{\mathcal{B}}(Y^{\bullet}(\delta_{i}^{n}))\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$,
$\\{\text{RT}_{\mathcal{B}}(Y^{\bullet}(\sigma_{j}^{n}))\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and
$\\{\text{RT}_{\mathcal{B}}(Y^{\bullet}(\tau_{n}))\\}_{n\in\mathbb{N}}$. Let
$n\in\mathbb{N}^{*}$. In the case when $1\leq i\leq n-1$,

$\displaystyle\text{RT}_{\mathcal{B}}(Y^{\bullet}(\delta_{i}^{n}))=$
$\displaystyle\Delta^{-(n+1)-(n+1)}\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\lvert
G_{Y^{\bullet}(\delta_{i}^{n})}\rvert\right)$ $\displaystyle=$
$\displaystyle\dim(\mathcal{B})^{-(n+1)}\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{facemAC2.eps}}}\;\hskip
8.5359pt\right)$ $\displaystyle=$
$\displaystyle\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{facemAC2.eps}}}\;\hskip
8.5359pt\right).$

The cases $i=0$ or $i=n$ are verified analogously.

For $n\in\mathbb{N}$ and $0\leq j\leq n$,

$\displaystyle\text{RT}_{\mathcal{B}}(Y^{\bullet}(\sigma_{j}^{n}))=$
$\displaystyle\Delta^{-(n+1)-(n+1)}\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\lvert
G_{Y^{\bullet}(\sigma_{j}^{n})}\rvert\right)$ $\displaystyle=$
$\displaystyle\dim(\mathcal{B})^{-(n+1)}\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenmAC.eps}}}\;\hskip
5.69046pt\right)$ $\displaystyle=$
$\displaystyle\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenmAC.eps}}}\;\hskip
5.69046pt\right).$

Finally,

$\displaystyle\text{RT}_{\mathcal{B}}(Y^{\bullet}(\tau_{0}))=$
$\displaystyle\Delta^{-1-1}\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\lvert
G_{Y^{\bullet}(\tau_{0})}\rvert)$ $\displaystyle=$
$\displaystyle\dim(\mathcal{B})^{-1}\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}})=$
$\displaystyle=$
$\displaystyle\mathrm{id}_{\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F})}.$

In the case when $n\geq 1$,

$\displaystyle\text{RT}_{\mathcal{B}}(Y^{\bullet}(\tau_{n}))=$
$\displaystyle\Delta^{-(n+1)-(n+1)}\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\lvert
G_{Y^{\bullet}(\tau_{n})}\rvert)$ $\displaystyle=$
$\displaystyle\dim(\mathcal{B})^{-(n+1)}\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopmAC.eps}}}\;\right)$
$\displaystyle=$
$\displaystyle\mathrm{Hom}_{\mathcal{B}}\left(\mathbb{1},\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopmAC.eps}}}\;\right).$

∎

###  7.4. The final step

In Section 4.1, we constructed the cocyclic object $Y^{\bullet}$ in the
category of $3$-cobordisms. Next, in Lemma 7.10, we computed the cocyclic
$\Bbbk$-module $\text{RT}_{\mathcal{B}}\circ Y^{\bullet}$. To prove the first
part of Theorem 7.1, it suffices to show that the latter is isomorphic to
$\mathbb{F}^{\bullet}\circ\Phi$, where $\Phi\colon\Delta C\to\Delta C$ is the
reindexing involution from Section 2.5. Recall that the coend $\mathbb{F}$ is
a Hopf algebra in $\mathcal{B}$, equipped with a non-degenerate Hopf pairing
$\omega$. Denote its inverse by $\Omega$. The isomorphism between cocyclic
$\Bbbk$-modules $\text{RT}_{\mathcal{B}}\circ Y^{\bullet}$ and
$\mathbb{F}^{\bullet}\circ\Phi$ is provided by the family
$\omega^{\bullet}=\\{\omega^{n}\colon\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F}^{\otimes
n+1})\to\mathrm{Hom}_{\mathcal{B}}(\mathbb{F}^{\otimes
n+1},\mathbb{1})\\}_{n\in\mathbb{N}}$, which is defined by setting for any
$n\in\mathbb{N}$ and
$f\in\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F}^{\otimes n+1})$,

$\omega^{n}(f)=\,\raisebox{-0.5pt}{\scalebox{0.85}{\includegraphics{isococyc1.eps}}}\;.$
(25)

The inverse of $\omega^{\bullet}$ is given by the family
$\Omega^{\bullet}=\\{\Omega_{n}\colon\mathrm{Hom}_{\mathcal{B}}(\mathbb{F}^{\otimes
n+1},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F}^{\otimes
n+1})\\}_{n\in\mathbb{N}}$, which is defined by setting for any
$n\in\mathbb{N}$ and $f\in(\mathbb{F}^{\otimes n+1},\mathbb{1})$,

$\Omega^{n}(f)=\,\raisebox{-0.5pt}{\scalebox{0.85}{\includegraphics{isococyc.eps}}}\;.$

It remains to check that $\omega^{\bullet}$ is a natural transformation
between cocyclic $\Bbbk$-modules $\text{RT}_{\mathcal{B}}\circ Y^{\bullet}$
and $\mathbb{F}^{\bullet}\circ\Phi$. The equations (9) and (10) follow from
axioms of a Hopf pairing $\omega$ and isotopy invariance of graphical
calculus. The equation (11) follows from equations (16) and (17), naturality
of braiding, and isotopy invariance of graphical calculus. $\square$

###  7.5. Sketch of computation of $\text{RT}_{\mathcal{B}}\circ Y_{\bullet}$

Let us sketch the computation of $\text{RT}_{\mathcal{B}}\circ Y_{\bullet}$,
which is similar to computation of $\text{RT}_{\mathcal{B}}\circ Y^{\bullet}$
given in detail in Section 7.3 and 7.4. The following lemma is analogously
proved as the Lemma 7.8.

######  Lemma 7.11.

If $T_{4}$ and $T_{5}$ are the following special ribbon graphs

$T_{4}=\,\raisebox{-1.2pt}{\scalebox{0.7}{\includegraphics{counitpiece.eps}}}\;,\qquad
T_{5}=\,\raisebox{-0.45pt}{\scalebox{0.7}{\includegraphics{comult.eps}}}\;,$

then

  * (a)

$\lvert T_{4}\rvert=\varepsilon$,

  * (b)

$\lvert T_{5}\rvert=\dim(\mathcal{B})^{2}\Delta$.

Next, by using Lemma 7.11, one obtains an analogue of Lemma 7.9:

######  Lemma 7.12.

We have:

  * (a)

$\lvert
G_{Y_{\bullet}(d_{i}^{n})}\rvert=\dim(\mathcal{B})^{n}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{faceAC.eps}}}\;$,

  * (b)

$\lvert
G_{Y_{\bullet}(s_{j}^{n})}\rvert=\dim(\mathcal{B})^{n+2}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenAC.eps}}}\;$,

  * (c)

$\lvert
G_{Y_{\bullet}(t_{n})}\rvert=\begin{cases}\dim(\mathcal{B})\mathrm{id}_{\mathbb{F}},&\mbox{if
}n=0,\\\
\dim(\mathcal{B})^{n+1}\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopnegAC.eps}}}\;&\mbox{if
}n\geq 1.\end{cases}$

By using Lemma 7.12, one can compute that $\text{RT}_{\mathcal{B}}\circ
Y_{\bullet}$ is equal to the cyclic $\Bbbk$-module given by the family
$\\{\mathrm{Hom}_{\mathcal{B}}(\mathbb{1},\mathbb{F}^{\otimes
n+1})\\}_{n\in\mathbb{N}}$, equipped with the faces
$\\{\mathrm{RT}_{\mathcal{B}}(Y_{\bullet}(d_{i}^{n}))\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, codegeneracies
$\\{\mathrm{RT}_{\mathcal{B}}(Y_{\bullet}(s_{j}^{n}))\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and cocyclic operators
$\\{\mathrm{RT}_{\mathcal{B}}(Y_{\bullet}(t_{n}))\\}_{n\in\mathbb{N}}$ given
by formulas

$\displaystyle
d_{i}^{n}(f)=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{Fdeltai.eps}}}\;,\quad
s_{j}^{n}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{Fsigmaj.eps}}}\;,\quad
t_{n}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{Ftaun.eps}}}\;.$

The latter cyclic $\Bbbk$-module is isomorphic to
$\mathbb{F}_{\bullet}\circ\Phi^{\mathrm{op}}$, where $\Phi$ is the reindexing
involution automorphism from Section 2.5. This follows from the fact that the
coend $\mathbb{F}$ is equipped with a non-degenerate Hopf pairing $\omega$.
Indeed, the isomorphism is given by the family $\omega_{\bullet}$, defined by
the formula (25). This completes our proof of Theorem 7.1. $\square$

##  8\. Related work

In this Section, we discuss some possibly related perspectives. From algebraic
perspective, we review some constructions of paracyclic objects in a braided
$\Bbbk$-category and outline their connections to (co)cyclic $\Bbbk$-modules
from Section 6. These paracyclic objects can be further used to construct
$r$-cyclic $\Bbbk$-modules in the sense of Feigin and Tsygan [FT87]. From
topological perspective, we observe a relevance of the one-holed torus, which
is a Hopf algebra in $\text{Cob}_{3}(1)^{\text{conn}}$, a cobordism category
appearing in [Ke97] and different that the one from Section 3.

###  8.1. Variations on cyclic category

For $r\in\mathbb{N}^{*}$, the $r$-cyclic category $\Delta C_{r}$ is defined in
the same as cyclic category, except that the cyclic operators are replaced
with $r$_-cyclic operators_ $\\{\tau_{n}\colon n\to n\\}_{n\in\mathbb{N}}$,
which satisfy (4)-(7) and additionally, for any $n\in\mathbb{N}$,

$\tau_{n}^{r(n+1)}=\mathrm{id}_{n}.$ (26)

Notice that $\Delta C_{1}=\Delta C$. Another notion similar to that of the
cyclic category is the paracyclic category $\Delta C_{\infty}$. Here the
cyclic operators are replaced with the _paracyclic operators_
$\\{\tau_{n}\colon n\to n\\}_{n\in\mathbb{N}}$, which are isomorphisms
satisfying (4)-(7). Mutatis mutandi (see Section 2.3) one defines
$r$-(co)cyclic and para(co)cyclic objects in a category.

The cyclic duality from Section 2.4 descends to the paracyclic category.
Indeed, the key ingredient of proof in [Lo98] is the existence of the so
called extra degeneracies, which only relies on the invertibility of
(co)cyclic operators. Since the same formulas apply, we abusively denote the
paracyclic duality with $L\colon\Delta C^{\mathrm{op}}_{\infty}\to\Delta
C_{\infty}$.

###  8.2. Paracyclic and $r$-cyclic objects from braided Hopf algebras

Any Hopf algebra $H$ in a braided monoidal category $\mathcal{B}$ gives rise
to a paracyclic object $\textbf{C}_{\bullet}(H)\colon\Delta
C_{\infty}^{\mathrm{op}}\to\mathcal{B}$, given by setting for any
$n\in\mathbb{N}$, $\textbf{C}_{n}(H)=H^{\otimes n+1}$ and

$d_{i}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{faceAC.eps}}}\;,\quad
s_{j}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenAC.eps}}}\;,\quad
t_{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopnegAC2.eps}}}\;.$

This construction is functorial, that is, any morphism between coalgebras in
$\mathcal{B}$ induces, in a functorial way, the morphism of corresponding
paracyclic objects in $\mathcal{B}$.

Similarly, one can associate to $H$ the paracocyclic object
$\textbf{A}_{\bullet}(H)\colon\Delta C_{\infty}\to\mathcal{B}$, given by
setting for any $n\in\mathbb{N}$, $\textbf{A}_{n}(H)=H^{\otimes n+1}$ and

$\delta_{i}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{facemAC2.eps}}}\;,\quad\sigma_{j}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenmAC.eps}}}\;,\quad\tau_{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopmac2.eps}}}\;.$

This construction is functorial, that is, any algebra morphism in
$\mathcal{B}$ induces, in a functorial way, the morphism of corresponding
paracocyclic objects in $\mathcal{B}$.

Notice that if $\mathcal{B}$ is a ribbon category and if $H$ is an involutive
Hopf algebra in $\mathcal{B}$ (that is, $S^{2}=\theta_{H}$, where $\theta$ is
the canonical twist of $\mathcal{B}$), then the paracocyclic operators
$\\{t_{n}\\}_{n\in\mathbb{N}}$ of $\textbf{C}_{\bullet}(H)$ satisfy the
“twisted cyclicity condition”, that is, for all $n\in\mathbb{N}$,

$t_{n}^{n+1}=(\theta_{H^{\otimes n+1}})^{-1}.$ (27)

If $\mathcal{B}$ is additionally $\Bbbk$-linear, then the cocyclic
$\Bbbk$-module $H^{\bullet}$ from Section 6.1 can be obtained by composing
$\textbf{C}_{\bullet}(H)$ with the hom-functor
$\mathrm{Hom}_{\mathcal{B}}(-,\mathbb{1})$. Here, the cocyclicity condition
(8) for $H^{\bullet}$ follows by naturality of twists and the fact that
$\theta_{\mathbb{1}}=\mathrm{id}_{\mathbb{1}}$. In this vein, one derives from
$\textbf{C}_{\bullet}(H)$ an $r$-cocyclic $\Bbbk$-module as follows. Namely,
if $i$ is a simple object of $\mathcal{B}$, then the twist $\theta_{i}$ is a
scalar multiple of the identity morphism, still denoted by $\theta_{i}$. If
$\theta_{i}$ is of finite order $r$, then the composition of
$\textbf{C}_{\bullet}(H)$ with the hom-functor
$\mathrm{Hom}_{\mathcal{B}}(-,i)$ induces an $r$-cocyclic (respectively,
cyclic) $\Bbbk$-module. In a similar way, one obtains $r$-cyclic modules from
$\textbf{A}_{\bullet}(H)$. For instance, according to [BJT18], and built on
theorems of Vafa [Va88] and Müger [Mü03], all the twists on simple objects of
a ribbon fusion category over $\mathbb{C}$ are roots of unity.

Finally, we point out that the above construction of $\textbf{C}_{\bullet}(H)$
(respectively, $\textbf{A}_{\bullet}(H))$ may be restated for coalgebra $H$
(respectively, algebra) in a braided category with a twist $\mathcal{B}$, just
like we did in Section 6. Namely, in the above formulas, the square of the
antipode should be replaced with the twists. As already remarked in Section
6.3, by passing to $\textbf{C}_{\bullet}(H)\circ L$ (respectively,
$\textbf{A}_{\bullet}(H)\circ L^{\mathrm{op}}$), these constructions fit into
a more general framework of Akrami and Majid in [AM04], who considered ribbon
algebras (or dually, coribbon coalgebras) in a braided category.

###  8.3. Paracyclic and $r$-cyclic objects from Crane-Yetter Hopf algebra

Another relevant category in quantum topology is that of connected
$3$-cobordisms, further denoted as 3Cob, which first appeared in [Ke97], which
was studied over the last three decades, notably in [Ha06, Ke03, As11, BP12],
and most recently in [BDR23]. For each $g\in\mathbb{N}$, fix a compact
connected oriented genus $g$-surface $\Sigma_{g,1}$ with one boundary
component. The objects of 3Cob are surfaces $\Sigma_{g,1}$ for each
$g\in\mathbb{N}$. A morphism $\Sigma_{g,1}\to\Sigma_{h,1}$ is given by the
connected cobordism from $\Sigma_{g,1}$ to $\Sigma_{h,1}$. In contrast to the
category $\textbf{Cob}_{3}$ which is recalled in Section 3.2 and used
throughout the paper, the category 3Cob is a braided monoidal category, the
monoidal product on objects is given by connected sum of surfaces, on
morphisms it is given by connected sum of cobordisms, the and the unit object
is $\Sigma_{0,1}$.

It is a result of Crane and Yetter [CY99], that the one-holed torus
$\Sigma_{1,1}$ has a structure of a Hopf algebra in 3Cob. By the general
construction from Section 8.2, one can organize the family of surfaces
$\\{\Sigma_{1,1}^{\otimes
g}\\}_{g\in\mathbb{N}^{*}}=\\{\Sigma_{g,1}\\}_{g\in\mathbb{N}^{*}}$ into a
paracyclic object $C_{\bullet}(\Sigma_{1,1})$ (respectively, paracocyclic
object $A_{\bullet}(\Sigma_{1,1}))$ in 3Cob. The existence of
$C_{\bullet}(\Sigma_{1,1})$ (respectively, $A_{\bullet}(\Sigma_{1,1}))$
implies that any braided monoidal functor from 3Cob to a braided monoidal
category $\mathcal{B}$ induces a para(co)cyclic object in $\mathcal{B}$. Let
$\mathcal{B}$ be a ribbon unimodular factorizable category and $\Bbbk$ an
algebraically closed field. Here the unimodularity means that $\mathcal{B}$ is
a finite $\Bbbk$-category in the sense of [EGNO16] and that the projective
cover of $\mathbb{1}$ is self-dual. By [BDR23, Theorem 1.2.], there is a
braided monoidal functor $J_{3}\colon\textbf{3Cob}\to\mathcal{B}$, which, in
particular, sends the one-holed torus to the end $\mathbb{A}$ of category
$\mathcal{B}$, that is, an end of functor $(X,Y)\mapsto X\otimes Y^{*}$.
Furthermore,

$J_{3}\circ\textbf{C}_{\bullet}(\Sigma_{1,1})=\textbf{C}_{\bullet}(\mathbb{A})\quad\text{and}\quad
J_{3}\circ\textbf{A}_{\bullet}(\Sigma_{1,1})=\textbf{A}_{\bullet}(\mathbb{A}).$

Note that factorizability in the sense [BDR23] is equivalent to non-degeneracy
of coend of $\mathcal{B}$ (see Remark 9.1). In this setting, the end
$\mathbb{A}$ and the coend $\mathbb{F}$ are isomorphic Hopf algebras (see
Section 9.2). By functoriality, paracyclic objects
$\textbf{C}_{\bullet}(\mathbb{A})$ and $\textbf{C}_{\bullet}(\mathbb{F})$
(respectively, paracocyclic objects $\textbf{A}_{\bullet}(\mathbb{A})$ and
$\textbf{A}_{\bullet}(\mathbb{F})$) in $\mathcal{B}$ are isomorphic. Finally,
since $S^{2}_{\mathbb{A}}=\theta_{\mathbb{A}}$, paracyclic operators of
$\textbf{C}_{\bullet}(\mathbb{A})$ (respectively, paracocyclic operators of
$\textbf{A}_{\bullet}(\mathbb{A})$) satisfy the twisted cyclicity
(respectively, cocyclicity) condition (27). Hence, by composing
$J_{3}\circ\textbf{C}_{\bullet}(\Sigma_{1,1})$ with appropriate hom-functors
(see Section 8.2), one obtains $r$-(co)cyclic $\Bbbk$-modules.

######  Remark 8.1.

Cobordisms in 3Cob admit surgery presentation by certain tangles (for a
review, see [BDR23, Section 4]). On one hand, there is a resemblance between
tangles which present structural morphisms of the Hopf algebra $\Sigma_{1,1}$
and the special ribbon graphs which present generating morphisms of (co)cyclic
objects in $\textbf{Cob}_{3}$ (see Theorem 4.1). Note also that in the
construction of the latter objects, we do not use the monoidal structure of
$\textbf{Cob}_{3}$. One the other hand, in the contrast with the Kirby
calculus on special ribbon graphs from Section 3.7, the moves COUPON and TWIST
do not appear as moves between tangles presenting morphisms in 3Cob. It would
be interesting to explore relationships between (co)cyclic objects in
$\textbf{Cob}_{3}$ which were constructed in Section 4 and para(co)cyclic
objects in 3Cob from Section 8.3.

##  9\. Appendix

In this Section, we discuss (co)end of a braided pivotal category
$\mathcal{B}$ via universal (co)actions, the Drinfeld map as in [FGR19], and
factorizability.

###  9.1. (Co)ends via universal (co)actions