# Cyclic sets from ribbon string links

Ivan Bartulović  Université de Lille, Laboratoire Paul Painlevé UMR CNRS 8524,
F-59000 Lille, France<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first endow the set of ribbon string links (up to isotopy)
with a structure of a cyclic and of a cocyclic set. Next, we relate these
(co)cyclic sets with those associated with the coend of a ribbon category. The
relationship is given by the universal quantum invariants à la Reshetikhin-
Turaev.

_Keywords_. Cyclic sets, string links, coend of a category.

2020 _Mathematics Subject Classification_. 18N50, 18M15, 57K16.

###### Contents

  1. 1 Introduction
  2. 2 Cyclic objects
  3. 3 Cyclic and cocyclic sets from ribbon string links
  4. 4 Ribbon categories and graphical calculus
  5. 5 Algebraic cyclic theories
  6. 6 Relation with quantum invariants

##  1\. Introduction

A (co)cyclic object in a category is, roughly speaking, a (co)simplicial
object with compatible actions of the cyclic groups. In particular, a
(co)cyclic set/vector space is a (co)cyclic object in the category of
sets/vector spaces. The first example arose in homological algebra: Connes [3,
4] associated to any algebra a cocyclic vector space whose cohomology is
called the cyclic cohomology of the algebra. Majid and Akrami [1] generalized
this construction by associating to any ribbon algebra (an algebra in a
braided monoidal category equipped with a ribbon automorphism) a cocyclic
vector space.

In this paper, we first endow the set of ribbon string links (which are framed
long knots with several components) with the structure of a cyclic set and of
a cocyclic set (see Theorem 1 and Section 3.3). Next, we prove that these
(co)cyclic sets universally “dominate” the (co)cyclic sets associated to the
coends of ribbon categories (see Theorem 2 and Corollary 1). To be more
specific, consider the category RSL of ribbon string links (where composition
is given by stacking). Following Bruguières-Virelizier [2], the quantum
invariants à la Reshetikhin-Turaev associated to a ribbon category
$\mathcal{B}$ give rise to a functor $\phi_{\mathcal{B}}$ from RSL to the
convolution category $\text{Conv}_{\mathcal{B}}(\mathbb{F},\mathbb{1})$, where
$\mathbb{F}$ is the coend of $\mathcal{B}$ (endowed with its canonical
coalgebra structure) and $\mathbb{1}$ is the monoidal unit of $\mathcal{B}$
(endowed with the trivial algebra structure). We show that the functor
$\phi_{{\mathcal{B}}}$ induces a morphism of (co)cyclic sets from the
(co)cyclic sets associated to ribbon string links to the (co)cyclic sets (à la
Akrami-Majid [1] and its variants, see Section 5) associated with the braided
Hopf algebra $\mathbb{F}$.

The paper is organized as follows. In Section 2, we recall the notions of
(co)simplicial and (co)cyclic objects in a category. In Section 3, we
construct (co)cyclic sets from ribbon string links. In Section 4, we review
ribbon categories and graphical calculus. Section 5 is dedicated to (co)cyclic
objects from categorical (co)algebras. In Section 6, we relate, via the
quantum invariants, the (co)cyclic sets from string links to those associated
to a coend of a ribbon category. Throughout the paper, the class of objects of
a category $\mathcal{C}$ is denoted by $\mathrm{Ob}(\mathcal{C})$.

### Acknowledgments

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). The author is particularly grateful for the valuable
comments and advice of his thesis supervisor Alexis Virelizier.

##  2\. Cyclic objects

In this section we review the cyclic category, which is an extension of the
simplicial category and is used to define (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 $n\in\mathbb{N}$. For $n\in\mathbb{N}$,
consider the ordered sets $[n]=\\{0,\dots,n\\}$. A morphism from $n$ to $m$ in
$\Delta$ is an increasing map $[n]\to[m]$. 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 [6, Lemma 5.1]) that morphisms in $\Delta$ are generated
by the cofaces $\\{\delta_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$ and
the codegeneracies $\\{\sigma_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$
subject to the following three 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) $\displaystyle\sigma_{j}^{n}\delta_{i}^{n+1}$
$\displaystyle=\begin{cases}\delta_{i}^{n}\sigma_{j-1}^{n-1}&\text{for all
}0\leq i<j\leq n,\\\ \mathrm{id}_{n}&\text{for }0\leq i=j\leq n\text{ or
}1\leq i=j+1\leq n+1,\\\ \delta_{i-1}^{n}\sigma_{j}^{n-1}&\text{for all }1\leq
j+1<i\leq n+1.\end{cases}$ (3)

In the opposite category $\Delta^{\mathrm{op}}$, the coface $\delta_{i}^{n}$
and the 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
the faces and the morphisms $\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$
are called the degeneracies.

###  2.2. The cyclic category

The  _cyclic category_ $\Delta C$ (introduced by Connes [3]) is defined as
follows. The objects of $\Delta C$ are the non-negative integers
$n\in\mathbb{N}$. The morphisms are generated by the morphisms
$\\{\delta_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, called cofaces, the
morphisms $\\{\sigma_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$, called
codegeneracies, and the isomorphisms $\\{\tau_{n}\colon n\to
n\\}_{n\in\mathbb{N}}$, called  _cocyclic operators_ , satisfying the
simplicial relations (1)-(3), the following four compatibility relations:

$\displaystyle\tau_{n}\delta_{i}^{n}=\delta_{i-1}^{n}\tau_{n-1}\text{\quad for
all }1\leq i\leq n,$ (4)
$\displaystyle\tau_{n}\delta_{0}^{n}=\delta_{n}^{n},\text{\quad for all }n\geq
1,$ (5)
$\displaystyle\tau_{n}\sigma_{i}^{n}=\sigma_{i-1}^{n}\tau_{n+1}\text{\quad for
all }1\leq i\leq n,$ (6)
$\displaystyle\tau_{n}\sigma_{0}^{n}=\sigma_{n}^{n}\tau_{n+1}^{2},\text{\quad
for all }n\geq 0,$ (7)

and the cocyclicity condition

$\tau_{n}^{n+1}=\mathrm{id}_{n}\quad\text{for all}\quad n\in\mathbb{N}.$ (8)

In the opposite category $\Delta C^{\mathrm{op}}$, the coface
$\delta_{i}^{n}$, the codegeneracy $\sigma_{j}^{n}$, and the 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
the faces, the morphisms $\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$ are
called the degeneracies, and the morphisms $\\{t_{n}\\}_{n\in\mathbb{N}}$ are
called the cyclic operators.

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

Let $\mathcal{C}$ be a category. A simplicial object in $\mathcal{C}$ is a
functor $\Delta^{\mathrm{op}}\to\mathcal{C}$. 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
morphism between two (co)simplicial/(co)cyclic objects is a natural
transformation between them. A (co)simplicial/(co)cyclic object in the
category of sets are called (co)simplicial/(co)cyclic sets.

Since the categories $\Delta$ and $\Delta C$ are defined by generators and
relations, a (co)simplicial/(co)cyclic object in a category is completely
determined by the images of the generators satisfying the corresponding
relations. As usual, we denote these images by the same letter. For example, a
cocyclic object $X$ in $\mathcal{C}$ may be described explicitly by a family
$X^{\bullet}=\\{X_{n}\\}_{n\in\mathbb{N}}$ of objects in $\mathcal{C}$ and by
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).
From this point of view, a morphism $\alpha^{\bullet}\colon X^{\bullet}\to
Y^{\bullet}$ between cocyclic 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 $\mathcal{C}$ such that

$\displaystyle\delta_{i}^{n}\alpha^{n-1}$
$\displaystyle=\alpha^{n}\delta_{i}^{n}\quad\text{for all }n\geq 1\text{ and
}0\leq i\leq n,$ $\displaystyle\sigma_{j}^{n}\alpha^{n+1}$
$\displaystyle=\alpha^{n}\sigma_{j}^{n}\quad\text{for all }n\geq 0\text{ and
}0\leq j\leq n,$ $\displaystyle\alpha^{n}\tau_{n}$
$\displaystyle=\tau_{n}\alpha^{n}\quad\text{for all }n\geq 0.$

Similarly as above, a cyclic object $X$ in $\mathcal{C}$ is described by a
family $X_{\bullet}=\\{X_{n}\\}_{n\in\mathbb{N}}$ of objects in $\mathcal{C}$
and by 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
relations

$\displaystyle d_{i}^{n}d_{j}^{n+1}$
$\displaystyle=d_{j-1}^{n}d_{i}^{n+1}\text{\quad for all }0\leq i<j\leq n+1,$
(9) $\displaystyle s_{i}^{n+1}s_{j}^{n}$
$\displaystyle=s_{j+1}^{n+1}s_{i}^{n}\text{\quad for all }0\leq i\leq j\leq
n,$ (10) $\displaystyle d_{i}^{n+1}s_{j}^{n}$
$\displaystyle=\begin{cases}s_{j-1}^{n-1}d_{i}^{n}&\text{for all }0\leq
i<j\leq n,\\\ \mathrm{id}_{X_{n}}&\text{for }0\leq i=j\leq n\text{ or }1\leq
i=j+1\leq n+1,\\\ s_{j}^{n-1}d_{i-1}^{n}&\text{for all }1\leq j+1<i\leq
n+1,\end{cases}$ (11) $\displaystyle d_{i}^{n}t_{n}$
$\displaystyle=t_{n-1}d_{i-1}^{n}\text{\quad for all }1\leq i\leq n,$ (12)
$\displaystyle d_{0}^{n}t_{n}$ $\displaystyle=d_{n}^{n}\text{\quad for all
}n\geq 1,$ (13) $\displaystyle s_{i}^{n}t_{n}$
$\displaystyle=t_{n+1}s_{i-1}^{n}\text{\quad for all }1\leq i\leq n,$ (14)
$\displaystyle s_{0}^{n}t_{n}$ $\displaystyle=t_{n+1}^{2}s_{n}^{n}\text{\quad
for all }n\geq 0,\hskip 19.91684pt$ (15)

and such that the cyclicity condition holds for any $n\in\mathbb{N}$:

$t_{n}^{n+1}=\mathrm{id}_{X_{n}}.$ (16)

In this characterization, 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 $\mathcal{C}$ satisfying

$\displaystyle\alpha_{n-1}d_{i}^{n}$
$\displaystyle=d_{i}^{n}\alpha_{n}\quad\text{for all }n\geq 1\text{ and }0\leq
i\leq n,$ $\displaystyle\alpha_{n+1}s_{j}^{n}$
$\displaystyle=s_{j}^{n}\alpha_{n}\quad\text{for all }n\geq 0\text{ and }0\leq
j\leq n,$ $\displaystyle\alpha_{n}t_{n}$
$\displaystyle=t_{n}\alpha_{n}\quad\text{for all }n\geq 0.$

###  2.4. Cyclic duality

Connes defined in [3] an isomorphism of categories $\Delta C\cong\Delta
C^{\mathrm{op}}$ called cyclic duality. This cyclic duality $L\colon\Delta
C^{\mathrm{op}}\to\Delta C$ (in its version due to Loday [8, Chapter 6]) is
identity on objects and it is defined on morphisms as follows. For $n\geq 1$
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 $n\geq 0$ 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 a cocyclic
object $X\colon\Delta C\to\mathcal{C}$ into the cyclic object $XL\colon\Delta
C^{\mathrm{op}}\to\mathcal{C}$ and its opposite $L^{\mathrm{op}}$ transforms a
cyclic object $Y\colon\Delta C^{\mathrm{op}}\to\mathcal{C}$ into the cocyclic
object $YL^{\mathrm{op}}\colon\Delta C\to\mathcal{C}$.

##  3\. Cyclic and cocyclic sets from ribbon string links

In this section, we construct (co)cyclic sets from ribbon string links.

###  3.1. Ribbon string links

A ribbon is a homeomorphic image of the rectangle $[0,1]\times[0,1]$. The
image of the segment $[0,1]\times\\{0\\}$ is called the bottom base and the
image of the segment $[0,1]\times\\{1\\}$ is called the top base of the
ribbon. The image of the segment $\\{\frac{1}{2}\\}\times[0,1]$ is called the
core of the ribbon. Let $n$ be a non-negative integer. A ribbon $n$-string
link is an oriented surface $T$ embedded in the strip
$\mathbb{R}^{2}\times[0,1]$ and decomposed into a disjoint union of $n$
ribbons such that $T$ meets the planes $\mathbb{R}^{2}\times\\{0\\}$ and
$\mathbb{R}^{2}\times\\{1\\}$ orthogonally as follows. For all $1\leq k\leq
n$, the bottom base and the top base of the $k$-th ribbon of $T$ are
respectively the segments
$\left[k-\frac{1}{4},k+\frac{1}{4}\right]\times\\{0\\}\times\\{0\\}$ and
$\left[k-\frac{1}{4},k+\frac{1}{4}\right]\times\\{0\\}\times\\{1\\}$, and in
the points of these segments, the orientation of $T$ is determined by the pair
of vectors $(1,0,0)$ and $(0,0,1)$ tangent to $T$.

Note that there is a unique ribbon $0$-string link, which is the empty set. By
an isotopy of ribbon string links, we mean isotopy in
$\mathbb{R}^{2}\times[0,1]$ constant on the boundary and preserving the
splitting into ribbons as well as the orientation of the surface $T$.

We represent a ribbon string link $T$ by a plane diagram with the blackboard
framing convention: the ribbons of $T$ should go close and parallel to the
plane $\mathbb{R}\times\\{0\\}\times\mathbb{R}$ and the orientation of $T$
corresponds to the counterclockwise orientation in
$\mathbb{R}\times\\{0\\}\times\mathbb{R}$. We represent then $T$ by the
projection of the cores of its ribbons onto the plane
$\mathbb{R}\times\\{0\\}\times\mathbb{R}$ so that there are only double
transversal crossings (with overcrossing and undercrossing information). For
example,

$\raisebox{-0.4pt}{\scalebox{0.44}{\includegraphics{ribbon.eps}}}\quad\text{is
isotopic
to}\quad\raisebox{-0.4pt}{\scalebox{0.44}{\includegraphics{ribbon2.eps}}},\quad\text{and
is then represented by the
diagram}\quad\raisebox{-0.4pt}{\scalebox{0.88}{\includegraphics{exsl2.eps}}}.$

Any diagram defines a ribbon string link (up to isotopy). Two planar diagrams
represent isotopic string links if and only if they are related by a finite
sequence of planar isotopies fixing the bases and the following ribbon
Reidemeister moves:

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

The category RSL of ribbon string links has as objects non-negative integers.
For any two non-negative integers $m$ and $n$, the set of morphisms from $m$
to $n$ is defined by

$\mathrm{Hom}_{\textbf{RSL}}(m,n)=\begin{cases}\text{isotopy classes of ribbon
$n$-string links}\quad&\text{if }m=n,\\\ \emptyset\quad&\text{if }m\neq
n.\end{cases}$

The composition $T^{\prime}\circ T$ of two ribbon $n$-string links is given by
stacking $T^{\prime}$ on the top of $T$ (i.e., with ascending convention) and
compressing the result into $\mathbb{R}^{2}\times\left[0,1\right]$:

$T^{\prime}\circ
T=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{RSLcompo.eps}}}\;.$

Identity of $n$ is the trivial ribbon $n$-string link

$\mathrm{id}_{n}=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{trivial.eps}}}\;$

As above, we number the ribbons of a ribbon string link from the left to the
right. For any $n\in\mathbb{N}$, we denote by $\mathcal{RSL}_{n}$ the monoid
$\text{End}_{\textbf{RSL}}(n+1)$ of the isotopy classes of ribbon
$(n+1)$-string links.

###  3.2. (Co)cyclic sets from ribbon string links

For any $n\in\mathbb{N}$, define $\mathcal{SL}^{n}=\mathcal{RSL}_{n}$ as a
set. Next, define the cofaces
$\\{\delta_{i}^{n}\colon\mathcal{SL}^{n-1}\to\mathcal{SL}^{n}\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, the codegeneracies
$\\{\sigma_{j}^{n}\colon\mathcal{SL}^{n+1}\to\mathcal{SL}^{n}\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and the cocyclic operators
$\\{\tau_{n}\colon\mathcal{SL}^{n}\to\mathcal{SL}^{n}\\}_{n\in\mathbb{N}}$ by
setting

$\displaystyle\delta_{0}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdelta0.eps}}}\;,\quad\delta_{i}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltai.eps}}}\;,\quad\delta_{n}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltan.eps}}}\;,$
$\displaystyle\sigma_{j}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLsigmaj.eps}}}\;,\quad\tau_{0}(T)=T,\quad\tau_{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLcyc.eps}}}\;.$

The string link $\delta_{i}^{n}(T)$ is obtained from $T$ by inserting from
behind a trivial component between the $i$-th and $(i+1)$-th component. The
string link $\sigma_{j}^{n}(T)$ is obtained from $T$ by connecting from behind
the $(j+1)$-th and $(j+2)$-th component.

Similarly as above, for any $n\in\mathbb{N}$, define
$\mathcal{SL}_{n}=\mathcal{RSL}_{n}$ as a set. Next, define the faces
$\\{d_{i}^{n}\colon\mathcal{SL}_{n}\to\mathcal{SL}_{n-1}\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, the degeneracies
$\\{s_{j}^{n}\colon\mathcal{SL}_{n}\to\mathcal{SL}_{n+1}\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and the cyclic operators
$\\{t_{n}\colon\mathcal{SL}_{n}\to\mathcal{SL}_{n}\\}_{n\in\mathbb{N}}$ by
setting

$\displaystyle
d_{i}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeletion.eps}}}\;,\hskip
7.11317pts_{j}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLduplication.eps}}}\;,\hskip
7.11317ptt_{0}(T)=T,\hskip
7.11317ptt_{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLcycneg.eps}}}\;.$

The string link $d_{i}^{n}(T)$ is obtained from $T$ by deleting the $(i+1)$-th
component. The string link $s_{j}^{n}(T)$ is obtained from $T$ by duplicating,
along the framing, the $(j+1)$-th component. Note that the removal and
duplication operations for string links appeared in the work of Habiro [5].

######  Theorem 1.

  * (a)

The family $\mathcal{SL}^{\bullet}=\\{\mathcal{SL}^{n}\\}_{n\in\mathbb{N}}$
endowed with the cofaces $\\{\delta_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq
n}$, the codegeneracies $\\{\sigma_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$
and the cocyclic operators $\\{\tau_{n}\\}_{n\in\mathbb{N}}$ is a cocyclic
set.

  * (b)

The family $\mathcal{SL}_{\bullet}=\\{\mathcal{SL}_{n}\\}_{n\in\mathbb{N}}$
endowed with the faces $\\{d_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$,
the degeneracies $\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$ and the
cyclic operators $\\{t_{n}\\}_{n\in\mathbb{N}}$ is a cyclic set.

We prove Theorem 1 in Section 3.4.

###  3.3. Cyclic duals

By precomposing the cyclic duality $L\colon\Delta C^{\mathrm{op}}\to\Delta C$
from Section 2.4 with the cocyclic set ${\mathcal{SL}}^{\bullet}$ from Theorem
1$(a)$, we obtain the cyclic set ${\mathcal{SL}}^{\bullet}\circ L$. By
definitions, $\mathcal{SL}^{\bullet}\circ
L(n)=\mathcal{SL}^{n}=\mathcal{RSL}_{n}$ for all $n\in\mathbb{N}$. The faces
$\\{\tilde{d}_{i}\colon\mathcal{RSL}_{n}\to\mathcal{RSL}_{n-1}\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, the degeneracies
$\\{\tilde{s}_{j}^{n}\colon\mathcal{RSL}_{n}\to\mathcal{RSL}_{n+1}\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and the cyclic operators
$\\{\tilde{t}_{n}\colon\mathcal{RSL}_{n}\to\mathcal{RSL}_{n}\\}_{n\in\mathbb{N}}$
of the cyclic set ${\mathcal{SL}}^{\bullet}\circ L$ are computed by

$\displaystyle\tilde{d}_{i}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLsigmaj2.eps}}}\;,\quad\tilde{d}_{n}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{dualSLdn.eps}}}\;,\quad\tilde{s}_{j}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltai2.eps}}}\;,$
$\displaystyle\tilde{s}_{n}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltan.eps}}}\;,\quad\tilde{t}_{0}(T)=T,\quad\tilde{t}_{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLcycneg.eps}}}\;.$

Similarly as above, by precomposing the functor $L^{\mathrm{op}}\colon\Delta
C\to\Delta C^{\mathrm{op}}$ with the cyclic set ${\mathcal{SL}}_{\bullet}$
from Theorem 1$(b)$, we obtain the cocyclic set ${\mathcal{SL}}_{\bullet}\circ
L^{\mathrm{op}}$. It follows by definitions that $\mathcal{SL}_{\bullet}\circ
L^{\mathrm{op}}(n)=\mathcal{SL}_{n}=\mathcal{RSL}_{n}$ for all
$n\in\mathbb{N}$. The cofaces
$\\{\tilde{\delta}_{i}^{n}\colon\mathcal{RSL}_{n-1}\to\mathcal{RSL}_{n}\\}_{n\in\mathbb{N}^{*},0\leq
i\leq n}$, the codegeneracies
$\\{\tilde{\sigma}_{j}^{n}\colon\mathcal{RSL}_{n+1}\to\mathcal{RSL}_{n}\\}_{n\in\mathbb{N},0\leq
j\leq n}$, and the cocyclic operators
$\\{\tilde{\tau}_{n}\colon\mathcal{RSL}_{n}\to\mathcal{RSL}_{n}\\}_{n\in\mathbb{N}}$
of the cocyclic set ${\mathcal{SL}}_{\bullet}\circ L^{\mathrm{op}}$ are
computed by

$\displaystyle\tilde{\delta}_{i}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLduplication.eps}}}\;,\quad\tilde{\delta}_{n}^{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{dualtSLdeltan.eps}}}\;,\quad\tilde{\sigma}_{j}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeletion2.eps}}}\;,$
$\displaystyle\tilde{\tau}_{0}(T)=T,\quad\tilde{\tau}_{n}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLcyc.eps}}}\;.$

###  3.4. Proof of Theorem 1

We only prove the part $(a)$ of the theorem, by checking the relations
(1)-(8). The proof of the part $(b)$ is similar and is left to the reader (one
needs to verify relations (9)-(16)).

Let us prove the part $(a)$. We first verify (1). If $1\leq i<j\leq n-1$ and
$T\in\mathcal{RSL}_{n-1}$, then

$\displaystyle\delta_{j}^{n+1}\delta_{i}^{n}(T)$
$\displaystyle\overset{(i)}{=}\delta_{j}^{n+1}\left(\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltai.eps}}}\;\right)\overset{(ii)}{=}\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{djdi.eps}}}\;$
$\displaystyle\overset{(iii)}{=}\delta_{i}^{n+1}\left(\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltai.eps}}}\;\right)\overset{(iv)}{=}\delta_{i}^{n+1}\delta_{j-1}^{n}(T).$

Here $(i)$, $(iii)$, $(iv)$ follow from the definition and $(ii)$ follows from
the definition and the hypothesis that $i<j$. Indeed, since we count the
unlabeled trivial component, which is inserted between the components labeled
by $i$ and $i+1$, the $j$-th component of $\delta_{i}^{n}(T)$ is the one
labeled by $j-1$ on the string link $T$. The cases $i=0<j\leq n+1$ and
$i=n,j=n+1$ are checked in a similar way.

Next, we verify (2). Let $n\geq 0$. If $i<j$ and $T\in\mathcal{RSL}_{n+2}$,
then

$\displaystyle\sigma_{j}^{n}\sigma_{i}^{n+1}(T)$
$\displaystyle\overset{(i)}{=}\sigma_{j}^{n}\left(\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLsigmajv3.eps}}}\;\quad\right)\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{sjsi.eps}}}\;$
$\displaystyle\overset{(iii)}{=}\sigma_{i}^{n}\left(\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{SLsigmajv2.eps}}}\;\quad\right)\overset{(iv)}{=}\sigma_{i}^{n}\sigma_{j+1}^{n+1}(T).$

Here $(i),(iii),(iv)$ follow from the definition and $(ii)$ follows from the
definition and the hypothesis that $i<j$. Indeed, since one concatenates the
components labeled by $i$ and $i+1$, the $(j+1)$-th component of
$\sigma_{i}^{n+1}(T)$ is the one labeled by $j+1$ on the string link $T$. The
case $i=j$ is trivial to check.

Let us verify relations (3). Let $T\in\mathcal{RSL}_{n}$. If $i=j$ and $i\neq
0$, we have

$\sigma_{i}^{n}\delta_{i}^{n+1}(T)\overset{(i)}{=}\sigma_{i}^{n}\left(\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltaiv2.eps}}}\;\hskip
5.69046pt\right)\overset{(ii)}{=}\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{sidiv2.eps}}}\;\overset{(iii)}{=}T.$

Here $(i)$ follows from the definition, $(iii)$ follows by the isotopy, and
$(ii)$ follows from the definition and since the $(i+1)$-th component of the
string link $\delta_{i}^{n+1}(T)$ is the unlabeled component inserted between
the components labeled by $i$ and $i+1$ on the string link $T$. The case
$i=j=0$ is trivial to check. Next, consider the case when $i<j$. If $i\neq 0$,
we have

$\displaystyle\sigma_{j}^{n}\delta_{i}^{n+1}(T)$
$\displaystyle\overset{(i)}{=}\sigma_{j}^{n}\left(\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltaiv3.eps}}}\;\hskip
5.69046pt\right)\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{sjdiiltjv2.eps}}}\;$
$\displaystyle\overset{(iii)}{=}\delta_{i}^{n}\left(\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLsigmaj.eps}}}\;\hskip
5.69046pt\right)\overset{(iv)}{=}\delta_{i}^{n}\sigma_{j-1}^{n-1}(T).$

Here $(i)$, $(iii)$, $(iv)$ follow from the definition and $(ii)$ follows from
the definition and the hypothesis that $i<j$. Indeed, since we count the
unlabeled trivial component, which is inserted between the components labeled
by $i$ and $i+1$, the $(j+1)$-th component of the string link
$\delta_{i}^{n+1}(T)$ is the one labeled by $j$ on the string link $T$. We
proceed in the same way if $i=0$. The cases when $i>j+1$ or $i=j+1$ are proven
analogously.

Let us verify the relation (4). Assume that $2\leq i\leq n-1$. For
$T\in\mathcal{RSL}_{n-1}$, we have

$\displaystyle\tau_{n}\delta_{i}^{n}(T)$
$\displaystyle\overset{(i)}{=}\,\raisebox{-0.47pt}{\scalebox{1}{\includegraphics{tndi.eps}}}\;\overset{(ii)}{=}\,\raisebox{-0.47pt}{\scalebox{1}{\includegraphics{tndi2.eps}}}\;$
$\displaystyle\overset{(iii)}{=}\delta_{i-1}^{n}\left(\,\raisebox{-0.47pt}{\scalebox{1}{\includegraphics{tndi3.eps}}}\;\right)\overset{(iv)}{=}\delta_{i-1}^{n}\tau_{n-1}(T).$

Here $(i),(iii),(iv)$ follow from the definition and $(ii)$ follows by isotopy
and R3 move. The remaining cases are shown in the same manner.

Let us check the relation (6). If $1\leq i\leq n$ and
$T\in\mathcal{RSL}_{n+1}$, then

$\displaystyle\tau_{n}\sigma_{i}^{n}(T)$
$\displaystyle\overset{(i)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{tnsj1.eps}}}\;\hskip
36.98866pt\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{tnsj2.eps}}}\;$
$\displaystyle\overset{(iii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{tnsj3.eps}}}\;\hskip
14.22636pt\overset{(iv)}{=}\sigma_{i-1}^{n}\left(\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{tnsj4.eps}}}\;\hskip
5.69046pt\right)$
$\displaystyle\overset{(v)}{=}\sigma_{i-1}^{n}\tau_{n+1}(T).$

Here $(i),(iv),(v)$ follow from the definition, $(ii)$ follows by isotopy, and
$(iii)$ follows by isotopy and R3 move.

According to [8, Section 6.1.1], the relation (5) is a consequence of
relations (8) and (4). Similarly, the relation (7) is a consequence of
relations (8) and (6). Hence, it suffices to show that the relation (8) holds.
We show it in the case $n=1$. The general case is treated similarly. If
$T\in\mathcal{RSL}_{1}$, then

$\tau_{1}^{2}(T)\overset{(i)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{cycrel1.eps}}}\;\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{cycrel2.eps}}}\;\overset{(iii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{cycrel3.eps}}}\;\overset{(iv)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{cycrel4.eps}}}\;\overset{(v)}{=}T.$

Here $(i)$ follows from the definition, $(ii)$ by adding one positive and one
negative left hand twist on each component and by using the naturality of
twists, $(iii)$ by isotopy and $\textbf{R}3$ move, $(iv)$ by isotopy, $(v)$ by
isotopy, $\textbf{R}2$ move, and $\textbf{R}3$ move. The general case is
treated similarly.

##  4\. Ribbon categories and graphical calculus

In this section, we recall some algebraic preliminaries on ribbon categories
and their graphical calculus used in the remaining sections of the paper. For
more details, see [12].

###  4.1. 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.

###  4.2. 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 $X,Y,Z\in\mathrm{Ob}(\mathcal{B})$. 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.

###  4.3. 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 $X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\theta_{X\otimes Y}=\tau_{Y,X}\tau_{X,Y}(\theta_{X}\otimes\theta_{Y}).$ (17)

Note that (17) implies $\theta_{\mathbb{1}}=\mathrm{id}_{\mathbb{1}}$. For
example, the family $\mathrm{id}_{\mathcal{B}}=\\{\mathrm{id}_{X}\colon X\to
X\\}_{X\in\mathrm{Ob}(\mathcal{B})}$ is a twist for $\mathcal{B}$ if and only
if $\mathcal{B}$ is symmetric. Also, any ribbon category (see Section 4.6) has
a canonical twist. By a braided category with a twist, we mean a braided
category endowed with a twist.

###  4.4. Graphical calculus

In this paper, we intensively use the Penrose graphical calculus, which allows
us to avoid lengthy algebraic computations by using 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

$\tau_{X,Y}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{brcgCMunori.eps}}}\;\quad\text{and}\quad\tau_{X,Y}^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{invbrcgCMunori.eps}}}\;.$

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

$\theta_{X}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{twistX.eps}}}\;\quad\text{and}\quad\left(\theta_{X}\right)^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{twistXinv.eps}}}\;.$

The defining condition (17) 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 4.6). We choose this notation
since any ribbon category is an important particular example of a braided
category with a twist.

###  4.5. 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 (see Section 4.4) 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

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

For more details, see [12, Chapter 1].

###  4.6. Ribbon categories

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

$\theta_{X}^{l}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{ribbonlplus.eps}}}\;=(\mathrm{id}_{X}\otimes\widetilde{\text{ev}}_{X})(\tau_{X,X}\otimes\mathrm{id}_{X^{*}})(\mathrm{id}_{X}\otimes\text{coev}_{X})\colon
X\to X,$

while the _right twist_ of $X$ is defined by

$\theta_{X}^{r}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{ribbonrplus.eps}}}\;=(\text{ev}_{X}\otimes\mathrm{id}_{X})(\mathrm{id}_{X^{*}}\otimes\tau_{X,X})(\widetilde{\text{coev}}_{X}\otimes\mathrm{id}_{X})\colon
X\to X.$

The left and the right twist are natural isomorphisms with inverses

$(\theta_{X}^{l})^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{ribbonlminus.eps}}}\;\quad\text{and}\quad(\theta_{X}^{r})^{-1}=\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{ribbonrminus.eps}}}\;.$

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 4.3
and is called the twist of $\mathcal{B}$.

##  5\. Algebraic cyclic theories

In this section, $\mathcal{B}$ denotes a braided category with a twist
$\theta$. We review some constructions of (co)cyclic sets from (co)algebras in
$\mathcal{B}$.

###  5.1. A cocyclic set from coalgebras

A coalgebra in $\mathcal{B}$ is a triple $(C,\Delta,\varepsilon)$, where $C$
is an object of $\mathcal{B}$, $\Delta\colon C\to C\otimes C$ and
$\varepsilon\colon C\to\mathbb{1}$ are morphisms in $\mathcal{B}$, 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

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

Any coalgebra $C$ in $\mathcal{B}$ gives rise to a cocyclic set $C^{\bullet}$
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

$\displaystyle\delta_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Edeltai.eps}}}\;,\quad\sigma_{j}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{HDsj.eps}}}\;,\quad\tau_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{HDtn.eps}}}\;.$

Here $\tau_{0}=\mathrm{id}_{C^{0}}$. An integer $k$ below an arc denotes the
$k$-th tensorand of a tensor power of $C$.

######  Lemma 1.

The family $C^{\bullet}=\\{C^{n}\\}_{n\in\mathbb{N}}$ endowed with the cofaces
$\\{\delta_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the codegeneracies
$\\{\sigma_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$, and the cocyclic
operators $\\{\tau_{n}\\}_{n\in\mathbb{N}}$ is a cocyclic set.

Lemma 1 is proved in Section 5.4.

###  5.2. A cyclic set from algebras

An algebra in $\mathcal{B}$ is a triple $(A,m,u)$, where $A$ is an object of
$\mathcal{B}$, $m\colon A\otimes A\to A$ and $u\colon\mathbb{1}\to A$ are
morphisms in $\mathcal{B}$, 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

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

Any algebra $A$ in $\mathcal{B}$ gives rise to a cyclic set $A_{\bullet}$ 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

$\displaystyle
d_{0}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Ed0.eps}}}\;,\quad
d_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Edi.eps}}}\;,\quad
d_{n}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Edn.eps}}}\;,$
$\displaystyle
s_{j}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Esj.eps}}}\;,\quad
t_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Etn.eps}}}\;.$

Here $t_{0}=\mathrm{id}_{A_{0}}$.

######  Lemma 2.

The family $A_{\bullet}=\\{A_{n}\\}_{n\in\mathbb{N}}$ endowed with the faces
$\\{d_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the degeneracies
$\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$, and the cyclic operators
$\\{t_{n}\\}_{n\in\mathbb{N}}$ is a cyclic set.

The proof of Lemma 2 is similar to the proof of Lemma 1.

###  5.3. Cyclic duals

The cyclic duality $L\colon\Delta C^{\mathrm{op}}\to\Delta C$ from Section 2.4
transforms the cocyclic set $C^{\bullet}$ from Lemma 1 into the cyclic set
$C^{\bullet}\circ L$. 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
$\\{\tilde{d}_{i}^{n}\colon\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+1},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n},\mathbb{1})\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the degeneracies
$\\{\tilde{s}_{j}^{n}\colon\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+1},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+2},\mathbb{1})\\}_{n\in\mathbb{N},0\leq j\leq n}$, and the cyclic operators
$\\{\tilde{t}_{n}\colon\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+1},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+1},\mathbb{1})\\}_{n\in\mathbb{N}}$ are computed by setting

$\displaystyle\tilde{d}_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{HDsj.eps}}}\;,\quad\tilde{d}_{n}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{EdeltanFASO.eps}}}\;,$
$\displaystyle\tilde{s}_{j}^{n}(f)=\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Edeltai.eps}}}\;,\quad\tilde{t}_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Etn.eps}}}\;.$

Similarly as above, the functor $L^{\mathrm{op}}\colon\Delta C\to\Delta
C^{\mathrm{op}}$ transforms the cyclic set $A_{\bullet}$ from Lemma 2 into the
cocyclic set $A_{\bullet}\circ L^{\mathrm{op}}$. 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
$\\{\tilde{\delta}_{i}^{n}\colon\mathrm{Hom}_{\mathcal{B}}(A^{\otimes
n},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(A^{\otimes
n+1},\mathbb{1})\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the codegeneracies
$\\{\tilde{\sigma}_{j}^{n}\colon\mathrm{Hom}_{\mathcal{B}}(A^{\otimes
n+2},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(A^{\otimes
n+1},\mathbb{1})\\}_{n\in\mathbb{N},0\leq j\leq n}$, and the cocyclic
operators $\\{\tilde{\tau}_{n}\colon\mathrm{Hom}_{\mathcal{B}}(A^{\otimes
n+1},\mathbb{1})\to\mathrm{Hom}_{\mathcal{B}}(A^{\otimes
n+1},\mathbb{1})\\}_{n\in\mathbb{N}}$ are computed by

$\displaystyle\tilde{\delta}_{i}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Esj.eps}}}\;,\quad\tilde{\delta}_{n}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{EdeltanAKMA.eps}}}\;,\quad$
$\displaystyle\tilde{\sigma}_{j}^{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{Edi.eps}}}\;,\quad\tilde{\tau}_{n}(f)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{HDtn.eps}}}\;.$

Note that the construction $A_{\bullet}\circ L^{\mathrm{op}}$ is a particular
case of the work of Akrami and Majid [1] (since any algebra in a braided
category with a twist is a ribbon algebra in the sense of [1]).

###  5.4. Proof of Lemma 1

The cofaces $\\{\delta_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$, the
codegeneracies $\\{\sigma_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$, and the
cocyclic operators $\\{\tau_{n}\\}_{n\in\mathbb{N}}$ of $C^{\bullet}$ are
given by formulas $\delta_{i}^{n}(f)=fd_{i}^{n}$,
$\sigma_{j}^{n}(f)=fs_{j}^{n}$, and $\tau_{n}(f)=ft_{n}$, where

$\displaystyle
d_{i}^{n}=\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{faceAC.eps}}}\;,\quad
s_{j}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{degenAC.eps}}}\;,\quad
t_{0}=\theta_{C}^{-1},\quad
t_{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{cycopnegAC.eps}}}\;.$

We claim that morphisms $\\{d_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$,
$\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$, and
$\\{t_{n}\\}_{n\in\mathbb{N}}$ satisfy (9)-(15) and for all $n\in\mathbb{N}$,
the following “twisted cyclicity condition”:

$t_{n}^{n+1}=(\theta_{C^{\otimes n+1}})^{-1}.$ (18)

This indeed implies that $C^{\bullet}$ is a cocyclic set. Here, the
cocyclicity condition (8) for $\tau_{n}$ follows by the condition (18),
naturality of the twist of $\mathcal{B}$, and the fact that
$\theta_{\mathbb{1}}=\mathrm{id}_{\mathbb{1}}.$ Indeed, for any
$n\in\mathbb{N}$ and any $f\in\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n+1},\mathbb{1})$,

$\tau_{n}^{n+1}(f)=ft_{n}^{n+1}=f(\theta_{C^{\otimes
n+1}})^{-1}=(\theta_{\mathbb{1}})^{-1}f=f.$

Let us now check the fact that morphisms
$\\{d_{i}^{n}\\}_{n\in\mathbb{N}^{*},0\leq i\leq n}$,
$\\{s_{j}^{n}\\}_{n\in\mathbb{N},0\leq j\leq n}$, and
$\\{t_{n}\\}_{n\in\mathbb{N}}$ satisfy (9)-(15) and (18). Let $n\geq 1$ and
$0\leq i<j\leq n+1$. By the level-exchange property, we have

$d_{i}^{n}d_{j}^{n+1}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{didjAC.eps}}}\;\quad=\quad\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{dj-1diAC.eps}}}\;=d_{j-1}^{n}d_{i}^{n+1},$

whence the relation (9).

We now check the relation (10). Let $n\geq 0$ and $i\leq j$. If $i<j$, we have
by the level-exchange property that

$s_{i}^{n+1}s_{j}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{sisjAC.eps}}}\;\quad=\quad\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{sj+1siAC.eps}}}\;=s_{j+1}^{n+1}s_{i}^{n}.$

In the case when $i=j$, we have by the coassociativity of $C$ that

$s_{i}^{n+1}s_{i}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{sisiAC.eps}}}\;\quad=\quad\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{si+1siAC.eps}}}\;=s_{i+1}^{n+1}s_{i}^{n}.$

Let us now show that relations (11) hold. If $i<j$ or $i>j+1$, this follows by
the level-exchange property. Further, if $i=j$, we have by the counitality of
$C$ that

$d_{i}^{n+1}s_{i}^{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{disiAC.eps}}}=\mathrm{id}_{C^{\otimes
n+1}}.\;$

The case when $i=j+1$ also follows by the counitality of $C$.

Let us now check the relation (12). Let $n\geq 1$ and $1\leq i\leq n$. By the
naturality of the braiding in $\mathcal{B}$, we have

$d_{i}^{n}t_{n}=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{ditnAC1.eps}}}\;\quad=\quad\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{ditnAC2.eps}}}\;=t_{n-1}d_{i-1}^{n},$

whence the relation (12). The relation (13) follows by the naturality of the
braiding, naturality of the twist morphism, and the fact that
$\theta_{\mathbb{1}}=\mathrm{id}_{\mathbb{1}}$. The relation (14) follows by
the naturality of the braiding and the level-exchange property.

By the naturality of the braiding and the equation (17), we have

$\displaystyle s_{0}^{n}t_{n}$
$\displaystyle=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{s0tnAC.eps}}}\;\quad=\quad\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{s0tnAC2.eps}}}\;$
$\displaystyle=\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{s0tnAC3.eps}}}\;\quad=\quad\,\raisebox{-0.55pt}{\scalebox{0.75}{\includegraphics{s0tnAC4.eps}}}\;\quad=\quad
t_{n+1}^{2}s_{n}^{n},$

whence the relation (15) in the case when $n\geq 1$. For $n=0$, this follows
by the equation (17).

Finally, let us verify the “twisted cyclicity condition” (18). In the case
$n=1$ (the general case is treated similarly), this follows by naturality of
the braiding and by equation (17):

$t_{1}^{2}=\quad\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{tn_n+1.eps}}}\;\quad=\quad\,\raisebox{-0.45pt}{\scalebox{0.75}{\includegraphics{tn_n+1AC2.eps}}}\;=\left(\theta_{C^{\otimes
2}}\right)^{-1}.$

∎

##  6\. Relation with quantum invariants

In this section we relate the (co)cyclic sets constructed via ribbon string
links (in Section 3) to (co)cyclic sets (as in Section 5) associated to the
coend of a ribbon category (which is a Hopf algebra object). The relationship
is given by the quantum invariants à la Reshetikhin-Turaev.

###  6.1. Ribbon handles

Recall the notion of a ribbon from Section 3.1. Let $n$ be a non-negative
integer. A ribbon $n$-_handle_ is an oriented surface $H$ embedded in the
strip $\mathbb{R}^{2}\times\left[0,1\right]$ and decomposed into a disjoint
union of $n$ ribbons such that $H\cap\mathbb{R}^{2}\times\\{1\\}=\emptyset$
and so that $H$ meets $\mathbb{R}^{2}\times\\{0\\}$ orthogonally as follows.
For all $1\leq k\leq n$, the bottom base of the $k$-th ribbon of $H$ is the
segment
$\left[2k-1-\frac{1}{4},2k-1+\frac{1}{4}\right]\times\\{0\\}\times\\{0\\}$,
the top base of the $k$-th ribbon of $H$ is the segment
$\left[2k-\frac{1}{4},2k+\frac{1}{4}\right]\times\\{0\\}\times\\{0\\}$, and in
the points of these segments, the orientation of $T$ is determined by the
vector $(1,0,0)$ tangent to $H$. By an isotopy of ribbon handles, we mean
isotopy in $\mathbb{R}^{2}\times[0,1]$ constant on the boundary and preserving
splitting into ribbons as well as the orientation of the surface. As in
Section 3.1, we present a ribbon $n$-handle by a planar diagram with a
blackboard framing convention:

$\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{RHplanar.eps}}}\;.$

As shown, we number its ribbons from left to the right.

The category RH of ribbon handles has as objects non-negative integers. For
two non-negative integers $m$ and $n$, the set of morphisms from $m$ to $n$ is
defined by

$\mathrm{Hom}_{\textbf{RH}}(m,n)=\begin{cases}\text{isotopy classes of ribbon
$n$-handles}\quad&\text{if }m=n,\\\ \emptyset\quad&\text{if }m\neq
n.\end{cases}$

The composition $H\circ H^{\prime}$ of two ribbon $n$-handles $H$ and
$H^{\prime}$ and the identity for this composition are defined by

$H\circ
H^{\prime}=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{RHcompo.eps}}}\;\quad\text{and}\quad\mathrm{id}_{n}=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{RHid.eps}}}\;.$

Let us recall the construction of the mutually inverse functors
$\mathfrak{F}\colon\textbf{RSL}\to\textbf{RH}$ and
$\mathfrak{G}\colon\textbf{RH}\to\textbf{RSL}$ from [2]. For any non-negative
integer $n$, set $\mathfrak{F}(n)=n$ and $\mathfrak{G}(n)=n$. For an isotopy
class of a ribbon $n$-string link $T$ and an isotopy class of a ribbon
$n$-handle $H$, set

$\mathfrak{F}(T)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLtoRH.eps}}}\;\quad\text{and}\quad\mathfrak{G}(H)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{RHtoSL.eps}}}\;.$

###  6.2. Convolution category

Let $A=(A,\mu,\eta)$ be an algebra and $C=(C,\Delta,\varepsilon)$ a coalgebra
in a braided category $\mathcal{B}$ (see Sections 5.1 and 5.2). The
convolution category $\text{Conv}_{\mathcal{B}}(C,A)$ is defined as follows.
Its objects are the non-negative integers. For two non-negative integers $m$
and $n$, the set of morphism from $m$ to $n$ is defined by

$\mathrm{Hom}_{\text{Conv}_{\mathcal{B}}(C,A)}(m,n)=\begin{cases}\mathrm{Hom}_{\mathcal{B}}(C^{\otimes
n},A)\quad&\text{if }m=n,\\\ \emptyset\quad&\text{if }m\neq n.\end{cases}$

The composition of morphisms is given by the convolution product $*$, which is
defined as follows. For two morphisms
$f,g\in\mathrm{Hom}_{\mathcal{B}}(C^{\otimes n},A)$, we set

$f*g=\mu(f\otimes g)\Delta_{C^{\otimes n}},$

where $\Delta_{C^{\otimes n}}$ denotes the coproduct on $C^{\otimes n}$ (see
[12, Exercise 6.1.7]). The identity of an object $n\in\mathbb{N}$ is given by
$\mathrm{id}_{n}=\eta\varepsilon^{\otimes n}$.

###  6.3. 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

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

The coend of $\mathcal{B}$, if it exists, is a coalgebra in $\mathcal{B}$ with
comultiplication $\Delta\colon\mathbb{F}\to\mathbb{F}\otimes\mathbb{F}$ and
counit $\varepsilon\colon\mathbb{F}\to\mathbb{1}$, which are unique morphisms
such that, for all $X\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dindelta1.eps}}}\;=~{}~{}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dindelta2.eps}}}\;\quad\text{and}\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dineps1.eps}}}\;=~{}~{}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dineps3.eps}}}\;.$

An important factorization property is given in the following lemma.

######  Lemma 3 (Fubini theorem for coends, [7]).

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

If $\mathcal{B}$ is a braided pivotal category, coend $\mathbb{F}$ of
$\mathcal{B}$ is a Hopf algebra in $\mathcal{B}$ (see [9, 10, 12]), which
means that the coproduct and the counit are algebra morphisms and that there
is an antipode. The 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 antipode
$S\colon\mathbb{F}\to\mathbb{F}$ are unique morphisms such that for all
$X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dinmult1.eps}}}\;=~{}~{}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dinmult2.eps}}}\;,\qquad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dinanti1.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{dinanti2.eps}}}\;.$

Note that $S^{2}=\theta_{\mathbb{F}}^{r}$.

###  6.4. Evaluations of ribbon string links

Let $\mathcal{B}$ be a ribbon category with a coend $\mathbb{F}$. We recall
the construction of the functor

$\phi_{\mathcal{B}}\colon\textbf{RSL}\to\text{Conv}_{\mathcal{B}}(\mathbb{F},\mathbb{1})$

from [2], which is important in the sequel. It is identity on objects. Let $n$
be a non-negative integer. For an $n$-string link $T$, the morphism
$\phi_{\mathcal{B}}(T)\colon\mathbb{F}^{\otimes n}\to\mathbb{1}$ is defined as
follows. 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 $\mathbb{F}$. First, we orient the
ribbon $n$-handle $\mathfrak{F}(T)$ as prescribed in Section 6.1. Coloring the
$k$-th ribbon of $\mathfrak{F}(T)$ by an object $X_{k}$ of $\mathcal{B}$, we
obtain a family of morphisms

$\mathfrak{F}(T)_{X_{1},\dots,X_{n}}\colon X_{1}^{*}\otimes
X_{1}\otimes\cdots\otimes X_{n}^{*}\otimes X_{n}\to\mathbb{1},$

which is dinatural in each variable. Hence, it factorizes by Lemma 3:

$\mathfrak{F}(T)_{X_{1},\dots,X_{n}}=\phi_{\mathcal{B}}(T)\circ(i_{X_{1}}\otimes\cdots\otimes
i_{X_{n}})$ (19)

for a unique morphism $\phi_{\mathcal{B}}(T)\colon\mathbb{F}^{\otimes
n}\to\mathbb{1}$. Note that $\mathfrak{F}(T)_{X_{1},\dots,X_{n}}$ is the value
of the $\mathcal{B}$-colored (as above) ribbon $n$-handle $\mathfrak{F}(T)$
under the Reshetikhin-Turaev functor (see [11, Theorem 2.5]) and the morphism
$\phi_{{\mathcal{B}}}(T)$ is the universal quantum invariant derived from
$\mathcal{B}$ of the ribbon $n$-string link $T$.

###  6.5. Cyclic sets from string links and quantum invariants

Let $\mathcal{B}$ be a ribbon category with a coend $\mathbb{F}$. The coend
$\mathbb{F}$ is a Hopf algebra in $\mathcal{B}$ and so gives rise to the
cocyclic set $\mathbb{F}^{\bullet}$ and the cyclic set $\mathbb{F}_{\bullet}$
(see Section 5). Recall from Section 3 the cocyclic set
$\mathcal{SL}^{\bullet}$ and cyclic set $\mathcal{SL}_{\bullet}$ defined
geometrically via ribbon string links. Consider the evaluation functor
$\phi_{\mathcal{B}}\colon\textbf{RSL}\to\text{Conv}_{\mathcal{B}}(\mathbb{F},\mathbb{1})$
from Section 6.4. The next theorem relates these (co)cyclic sets via quantum
invariants.

######  Theorem 2.

The evaluation functor $\phi_{\mathcal{B}}$ induces a morphism of cocyclic
sets from ${\mathcal{SL}}^{\bullet}$ to $\mathbb{F}^{\bullet}$ and a morphism
of cyclic sets from ${\mathcal{SL}}_{\bullet}$ to $\mathbb{F}_{\bullet}$.

Theorem 2 says that in a sense, $\mathcal{SL}^{\bullet}$ is an initial
cocyclic set, which is universal with respect to ribbon categories with a
coend. Similarly, $\mathcal{SL}_{\bullet}$ is an initial cyclic set, which is
universal with respect to ribbon categories with a coend.

Recall the cyclic (respectively, cocyclic) sets ${\mathcal{SL}}^{\bullet}\circ
L$ and $\mathbb{F}^{\bullet}\circ L$ (respectively,
${\mathcal{SL}}_{\bullet}\circ{L^{\mathrm{op}}}$ and
$\mathbb{F}_{\bullet}\circ L^{\mathrm{op}}$) from Sections 3.3 and 5.3. An
immediate corollary of Theorem 2 is the following:

######  Corollary 1.

The evaluation functor $\phi_{\mathcal{B}}$ induces a morphism of cyclic sets
from ${\mathcal{SL}}^{\bullet}\circ L$ to $\mathbb{F}^{\bullet}\circ L$ and a
morphism of cocyclic sets from
${\mathcal{SL}}_{\bullet}\circ{L^{\mathrm{op}}}$ to $\mathbb{F}_{\bullet}\circ
L^{\mathrm{op}}$.

###  6.6. Proof of Theorem 2

We first prove two lemmas.

######  Lemma 4.

Let $n\in\mathbb{N}^{*}$. For any $(n+1)$-ribbon handle $H$, the ribbon
handles

1cyclicRH.eps and 1cyclicRH7.eps

are isotopic.

###### Proof.

By isotopy and Reidemeister moves, we have

$\displaystyle\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{cyclicRH.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{cyclicRH2.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{cyclicRH3.eps}}}\;$
$\displaystyle=$
$\displaystyle\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{cyclicRH4.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{cyclicRH5.eps}}}\;=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{cyclicRH7.eps}}}\;.$

∎

Recall the isomorphism $\mathfrak{F}\colon\textbf{RSL}\to\textbf{RH}$ from
Section 6.1.

######  Lemma 5.

Let $n\in\mathbb{N}^{*}$. For any ribbon $(n+1)$-string link $T$, the ribbon
handle $\mathfrak{F}(\mathcal{SL}^{\bullet}(\tau_{n})(T))$ is isotopic to the
ribbon handle

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{cyclicRH7v2.eps}}}\;.$

###### Proof.

For any $T\in\mathcal{RSL}_{1}$, we have

$\mathfrak{F}(\mathcal{SL}_{\bullet}(\tau_{1})(T))=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{RHcyc.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{RHcyc2.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{RHcyc3.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{RHcyc4.eps}}}\;.$

This proves the lemma for $n=1$. The general case is similar and is left to
the reader.

∎

Let us prove Theorem 2. Since $\phi_{\mathcal{B}}$ is a functor, it induces
for any $n\in\mathbb{N}$ the map

$\phi^{n}\colon\text{End}_{\textbf{RSL}}(n+1)\to\text{End}_{\text{Conv}_{\mathcal{B}}(\mathbb{F},\mathbb{1})}(n+1).$

This defines a family of maps
$\phi^{\bullet}=\\{\phi^{n}\colon\mathcal{RSL}_{n}\to\mathrm{Hom}_{\mathcal{B}}(\mathbb{F}^{\otimes
n+1},\mathbb{1})\\}_{n\in\mathbb{N}}.$ To show that $\phi^{\bullet}$ defines a
morphism of cocyclic sets from $\mathcal{SL}^{\bullet}$ to
$\mathbb{F}^{\bullet}$, we have to verify that

$\displaystyle\phi^{n}(\delta_{i}^{n}(T))$
$\displaystyle=\delta_{i}^{n}(\phi^{n-1}(T))\quad\text{for all}\quad n\geq
1,0\leq i\leq n,\quad\text{and}\quad T\in{\mathcal{RSL}}_{n-1},$ (20)
$\displaystyle\phi^{n}(\sigma_{j}^{n}(T))$
$\displaystyle=\sigma_{j}^{n}(\phi^{n+1}(T))\quad\text{for all}\quad n\geq
0,0\leq j\leq n,\quad\text{and}\quad T\in{\mathcal{RSL}}_{n+1},$ (21)
$\displaystyle\phi^{n}(\tau_{n}(T))$
$\displaystyle=\tau_{n}(\phi^{n}(T))\quad\text{for all}\quad n\geq
0\quad\text{and}\quad T\in{\mathcal{RSL}}_{n}.$ (22)

By abuse, we use here the same notation for cofaces, codegeneracies, and
cocyclic operators of $\mathcal{SL}^{\bullet}$ and $\mathbb{F}^{\bullet}$.

Let us prove (20). Assume that $1\leq i\leq n-1$ and
$T\in\mathcal{RSL}_{n-1}$. We have:

$\mathfrak{F}(\delta_{i}^{n}(T))=\mathfrak{F}\left(\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{SLdeltai.eps}}}\;\right)=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{RHdi.eps}}}\;.$

Consequently, by equation (19) and by definition of the counit of coend of
$\mathcal{B}$ (see Section 6.3),

$\phi^{n}(\delta_{i}^{n}(T))=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{HDdi.eps}}}\;=\delta_{i}^{n}(\phi^{n-1}(T)).$

The cases when $n\geq 1$ and $i=0$ or $i=n$ are verified analogously.

Next, let us prove (21). Let $n\geq 0,0\leq j\leq n$ and
$T\in\mathcal{RSL}_{n+1}$. We have:

$\mathfrak{F}(\sigma_{j}^{n}(T))=\mathfrak{F}\left(\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{SLsigmajv2.eps}}}\;\right)\;=\,\raisebox{-0.5pt}{\scalebox{1}{\includegraphics{RHsj2.eps}}}=\,\raisebox{-0.5pt}{\scalebox{1}{\includegraphics{RHsj.eps}}}\;.$

Consequently, by equation (19) and by definition of the comultiplication of
coend of $\mathcal{B}$ (see Section 6.3),

$\phi^{n}(\sigma_{j}^{n}(T))=\,\raisebox{-0.55pt}{\scalebox{1}{\includegraphics{HDsj.eps}}}\;=\sigma_{j}^{n}(\phi^{n+1}(T)).$

Finally, let us prove (22). Let $n\geq 0$ and $T\in\mathcal{RSL}_{n}$. If
$n=0$, then by definition, we have for any $1$-string link $T$,
$\phi^{0}(\tau_{0}(T))=\phi^{0}(T)=\tau_{0}(\phi^{0}(T)).$ Assume that $n\geq
1$. We have

$\mathfrak{F}(\tau_{n}(T))\overset{(i)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{cyclicRH7v2.eps}}}\;\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{cyclicRHv2.eps}}}\;.$

Here $(i)$ follows by Lemma 5 and $(ii)$ follows by Lemma 4. Consequently, by
equation (19) and by naturality of the twist, we have

$\phi^{n}(\tau_{n}(T))=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{HDtn.eps}}}\;=\tau_{n}(\phi^{n}(T)).$

This completes the proof of the fact that the evaluation $\phi_{\mathcal{B}}$
induces a morphism of cocyclic sets from $\mathcal{SL}^{\bullet}$ to
$\mathbb{F}^{\bullet}$.

Let us prove that that $\phi_{\mathcal{B}}$ induces a morphism between cyclic
sets $\mathcal{SL}_{\bullet}$ and $\mathbb{F}_{\bullet}$. As above, the
functor
$\phi_{\mathcal{B}}\colon\textbf{RSL}\to\text{Conv}_{\mathcal{B}}(\mathbb{F},\mathbb{1})$
induces for any $n\in\mathbb{N}$ the map

$\phi_{n}\colon\text{End}_{\textbf{RSL}}(n+1)\to\text{End}_{\text{Conv}_{\mathcal{B}}(\mathbb{F},\mathbb{1})}(n+1).$

This defines a family of maps
$\phi_{\bullet}=\\{\phi_{n}\colon\mathcal{RSL}_{n}\to\mathrm{Hom}_{\mathcal{B}}(\mathbb{F}^{\otimes
n+1},\mathbb{1})\\}_{n\in\mathbb{N}}$. Recall that for any $n\in\mathbb{N}$,

$\mathcal{SL}_{n}=\mathcal{RSL}_{n}\quad\text{and}\quad\mathbb{F}_{n}=\mathbb{F}^{n}=\mathrm{Hom}_{\mathcal{B}}(\mathbb{F}^{\otimes
n+1},\mathbb{1})\quad\text{as sets}.$

To show that $\phi_{\bullet}$ defines a morphism of cyclic sets from
$\mathcal{SL}_{\bullet}$ to $\mathbb{F}_{\bullet}$, we have to verify that

$\displaystyle\phi_{n-1}(d_{i}^{n}(T))$
$\displaystyle=d_{i}^{n}(\phi_{n}(T))\quad\text{for all}\quad n\geq 1,0\leq
i\leq n,\quad\text{and}\quad T\in{\mathcal{RSL}}_{n},$ (23)
$\displaystyle\phi_{n+1}(s_{j}^{n}(T))$
$\displaystyle=s_{j}^{n}(\phi_{n}(T))\quad\text{for all}\quad n\geq 0,0\leq
j\leq n,\quad\text{and}\quad T\in{\mathcal{RSL}}_{n},$ (24)
$\displaystyle\phi_{n}(t_{n}(T))$
$\displaystyle=t_{n}(\phi_{n}(T))\quad\text{for all}\quad n\geq
0\quad\text{and}\quad T\in{\mathcal{RSL}}_{n}.$ (25)

To prove (23) and (24), we use the fact that the coend $\mathbb{F}$ of
$\mathcal{B}$ coacts on each object $X$ of $\mathcal{B}$ via the universal
coaction $\delta_{X}\colon X\to X\otimes\mathbb{F}$, defined by
$\delta_{X}=(\mathrm{id}_{X}\otimes
i_{X})(\text{coev}_{X}\otimes\mathrm{id}_{X})$. We depict $\delta_{X}$ by

$\delta_{X}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unicoact.eps}}}\;,\quad\text{
so by
definition}\quad\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unicoact.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unicoact2.eps}}}\;.$

Next, by definition of unit $u$ of coend $\mathbb{F}$ (see Section 6.3), we
have $u=\delta_{\mathbb{1}}$. Similarly, the multiplication $m$ of coend
$\mathbb{F}$ is characterized by the following universal property: for all
$X,Y\in\mathrm{Ob}(\mathcal{B})$,

$\delta_{X}=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unimult.eps}}}\hskip
2.84544pt=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{unimult1.eps}}}\hskip
5.69046pt.$

Let us express the universal isotopy invariant $\phi_{B}(T)$ of an $n$-string
link $T$ in terms of the universal coaction. By coloring the $k$-th ribbon of
$T$ by an object $X_{k}$ of $\mathcal{B}$, we obtain a family of morphisms

$T_{X_{1},\cdots,X_{n}}\colon X_{1}\otimes\cdots\otimes X_{n}\to
X_{1}\otimes\cdots\otimes X_{n},$

which is natural in each variable. It factorizes as follows:

$T_{X_{1},\dots,X_{n}}\overset{(i)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{RHtoSL1.eps}}}\;\overset{(ii)}{=}\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{RHtoSL2.eps}}}\;.$
(26)

Here $(i)$ follows by definition of mutually inverse functors $\mathfrak{F}$
and $\mathfrak{G}$ (see Section 6.1) and by isotopy invariance of graphical
calculus, $(ii)$ follows by equation (19), naturality of the braiding, and
definition of the universal coaction.

Let us prove (23). Assume that $1\leq i\leq n-1$, and $T\in\mathcal{RSL}_{n}$.
For all objects
$X_{0},\dots,X_{i-1},X_{i+1},\dots,X_{n}\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{diTeval.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{diTeval2.eps}}}\;.$

Hence

$\phi_{n-1}(d_{i}^{n}(T))=d_{i}^{n}(\phi_{n}(T)).$

The cases when $n\geq 1$ and $i=0$ or $i=n$ are verified analogously.

Let us prove (24). Let $n\geq 0$, $0\leq j\leq n$, and
$T\in\mathcal{RSL}_{n}$. For all
$X_{0},\dots,X_{n+1}\in\mathrm{Ob}(\mathcal{B})$,

$\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{sjTeval1.eps}}}\;=\,\raisebox{-0.45pt}{\scalebox{1}{\includegraphics{sjTeval.eps}}}\;$

Hence

$\phi_{n+1}(s_{j}^{n}(T))=s_{j}^{n}(\phi_{n}(T)).$

Note that (25) follows from the equation (22) combined with the fact that the
cyclic operators of $\mathcal{SL}_{\bullet}$ are inverse to cocyclic operators
of $\mathcal{SL}^{\bullet}$ and the fact that for any $n\in\mathbb{N}$,
$\phi_{n}=\phi^{n}$ as functions. This completes the proof of the fact that
the evaluation $\phi_{\mathcal{B}}$ induces a morphism of cyclic sets from
$\mathcal{SL}_{\bullet}$ to $\mathbb{F}_{\bullet}$.

## References

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  * [9] V. Lyubashenko.  Modular transformations for tensor categories.  Journal of Pure and Applied Algebra, 98(3):279 – 327, 1995.
  * [10] S. Majid.  Braided groups.  Journal of Pure and Applied Algebra, 86(2):187 – 221, 1993.
  * [11] V. Turaev.  Quantum Invariants of Knots and 3-manifolds.  De Gruyter, 2010.
  * [12] V. Turaev and A. Virelizier.  Monoidal Categories and Topological Field Theory.  Birkhäuser, 2017.


# Examining the Effect of Implementation Factors on Deep Learning
Reproducibility

Kevin Coakley12, Christine R. Kirkpatrick1, Odd Erik Gundersen2 Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1San Diego
Supercomputer Center, Univ. of California San Diego, La Jolla, USA  2Norwegian
University of Science and Technology, Trondheim, Norway

###### Abstract

Reproducing published deep learning papers to validate their conclusions can
be difficult due to sources of irreproducibility. We investigate the impact
that implementation factors have on the results and how they affect
reproducibility of deep learning studies. Three deep learning experiments were
ran five times each on 13 different hardware environments and four different
software environments. The analysis of the 780 combined results showed that
there was a greater than 6% accuracy range on the same deterministic examples
introduced from hardware or software environment variations alone. To account
for these implementation factors, researchers should run their experiments
multiple times in different hardware and software environments to verify their
conclusions are not affected.

###### Index Terms:

deep learning, machine learning, reproducibility

##  I Introduction

Artificial Intelligence has a reproducibility crisis, much like psychology,
medicine, and other fields over the past decade[1]. A study of 255 machine
learning (ML) papers from 2012 to 2017 showed that only 63.5% were
reproducible[2]. There are many factors for why papers cannot be reproduced.
Gundersen et al. divides the sources that cause irreproducibility into six
categories: study design factors, algorithmic factors, implementation factors,
observation factors, evaluation factors, and documentation factors[3]. To help
with the reproducibility crisis, the effect of these sources of
irreproducibility on the conclusions of ML research needs to be studied. We
focus on the implementation factors of deep learning (DL) experiments.

##  II Reproducibility

Reproducibility, a central concept of the scientific method, is often poorly
understood. Terms like repeatability and reproducibility are both used
interchangeably, even though they have different definitions and have
different implications for the conclusions of an experiment.

Using the definitions from analytical chemistry[4]:

  * •

Repeatability is when an experiment is carried out by the same staff, in the
same lab, with the same equipment, using the same procedure, within a short
period of time.

  * •

Reproducibility is when an experiment is carried out by different staff, in a
different lab, with different equipment, using the same procedure (or as close
as possible), at a different time.

The lab in analytical chemistry is equivalent to the hardware and software
environment in computing. When a deterministic experiment is repeated using
the same hardware and software, the results should be the same and therefore
the experiment is outcomes reproducible. However, when an experiment is
reproduced using a different hardware and software environment by different
researchers the outcomes can be different. If the same analysis of the
outcomes leads to the same conclusion then the experiment is analysis
reproducible. If neither the outcomes nor the analysis are the same as the
original, but the interpretation of the analysis leads to the same conclusion
then the experiment is interpretation reproducible[5].

##  III Implementation Factors

The hardware and software environments used in ML experiments are
implementation factors that can contribute to irreproducible results. The
following implementation factors are sources of irreproducibility: ancillary
software, ancillary software versions, auto-selection of primitive operations,
software bugs, compiler settings, initialization seeds, non-deterministic
ordering of floating-point operations, parallel execution, and processing
unit[3].

Not all of the implementation factors can be easily controlled for. However,
the effects of the implementation factors: ancillary software, ancillary
software versions, software bugs, compiler settings, and processing unit —can
be examined by executing multiple runs of the experiment in multiple hardware
and software environments.

##  IV Examining the Effects of Hardware and Software Environments on Deep
Learning Results

###  IV-A Experiment Setup

TABLE I: Container Major Software Versions Container Name | Operating System | Python | TensorFlow | CUDA | cuDNN
---|---|---|---|---|---
tensorflow/tensorflow:2.8.0 | Ubuntu 20.04.3 | 3.8.10 | 2.8.0 | 11.2 | 8.1.0
tensorflow/tensorflow:2.9.1 | Ubuntu 20.04.4 | 3.8.10 | 2.9.1 | 11.2 | 8.1.0
nvcr.io/nvidia/tensorflow:22:03 | Ubuntu 20.04.4 | 3.8.10 | 2.8.0 | 11.6.1 | 8.3.3
nvcr.io/nvidia/tensorflow:22:06 | Ubuntu 20.04.4 | 3.8.10 | 2.9.1 | 11.7 | 8.4.1

The following three DL examples were chosen from the Keras GitHub
repository111https://github.com/keras-team/keras-io/blob/master/examples/: 1)
Simple convolutional neural network classifying handwritten digits using MNIST
(simple MNIST), 2) a 2-layer bidirectional LSTM classifying sentiments
utilizing the IMDB movie review dataset (bidirectional LSTM), 3) a dense
neural network performing binary classification on the Kaggle Credit Card
Fraud Detection dataset (binary classification). These examples represent
three different disciplines within DL, computer vision, natural language
processing, and structured data, respectively. They use the TensorFlow library
and were modified to run deterministically. The examples were ran five times
in each hardware and software environment to verify the deterministic results.

The hardware environments used consisted of resources from the Open Science
Grid (OSG). The OSG was chosen because it has many heterogeneous hardware
environments. Based on the resources available, runs were completed with four
different Intel CPUs, six different AMD CPUs and three different NVIDIA GPUs.
The only controlling factor for the placement of the runs was the type of
processor available.

Docker containers were used to keep the software environment common across the
heterogeneous hardware environments. The four containers that were used are
listed in Table I. The tensorflow containers were developed by the TensorFlow
team and the nvcr.io containers developed by NVIDIA. The Docker containers
were converted to Singularity containers to run on the OSG.

###  IV-B Results

The results showed a greater than 6% accuracy range of the bidirectional LSTM
and binary classification examples when varying the hardware and software
environments. The simple MNIST example didn’t show significant accuracy
differences as all the runs returned a greater than 99% accuracy.

The accuracy range of the bidirectional LSTM example exceeded 8% using the
tensorflow/tensorflow containers on GPU hardware; the cause was a software bug
(see Conclusions). The bug in the verison of cuDNN included in the
tensorflow/tensorflow containers caused runs using a Bi-directional RNN
network to return non-deterministic
outputs222https://docs.nvidia.com/deeplearning/cudnn/release-
notes/rel_8.html#rel-821. Rebuilding the tensorflow/tensorflow:2.9.1-gpu
container with CUDA 11.3 and cuDNN 8.2.1 and rerunning the bidirectional LSTM
example eliminated the software bug and produced deterministic results for all
three examples.

##  V Conclusions

Running the same DL experiment in different hardware and software environments
can produce results that are varied enough that a researcher may come to a
different conclusion than they would have if they had only ran the experiment
in one environment. Hence, claims about the relative performance of DL
algorithms such as ”DL model A is better than DL model B on task A” cannot be
fully trusted if the experiments have not been executed on several hardware
and software environments.

When selecting which Keras examples to test, it wasn’t known that the
tensorflow/tensorflow containers had a version of the cuDNN with a software
bug that would affect the results when ran on a GPU. This was discovered only
when we ran the bidirectional LSTM example multiple times in different
software environments while controlling for other ML implementation factors. A
researcher may not expect a TensorFlow Docker container released by the
TensorFlow team in July 2022 to have a software bug that was fixed by NVIDIA
in June 2021.

ML researchers should not focus on the variation in the outcomes but how the
variation affects the analysis and interpretation. Borrowing again from
analytical chemistry, international bodies recommend reproducing experiments
in at least 8 different labs. We suggest a similar strategy for ML experiments
that rely on stochasticity and floating point calculations, which is the case
for DL algorithms. However, exactly what the strategy should be is the
objective of future work. These three examples were outcomes reproducible only
when deterministic and using the same hardware and software environment.
Researchers should test their hypothesis on multiple hardware and software
environments to validate their experiments are analysis or interpretation
reproducible.

## Acknowledgment

This research was done using services provided by the OSG Consortium [6][7],
which is supported by the National Science Foundation awards #2030508 and
#1836650.

## References

  * [1] M. Hutson, “Artificial intelligence faces reproducibility crisis,” 2018.
  * [2] E. Raff, “A step toward quantifying independently reproducible machine learning research,” _Advances in Neural Information Processing Systems_ , vol. 32, 2019.
  * [3] O. E. Gundersen, K. Coakley, and C. Kirkpatrick, “Sources of irreproducibility in machine learning: A review,” _arXiv preprint arXiv:2204.07610_ , 2022\.
  * [4] J. Miller and J. C. Miller, _Statistics and chemometrics for analytical chemistry_. Pearson education, 2018.
  * [5] O. E. Gundersen, “The fundamental principles of reproducibility,” _Philosophical Transactions of the Royal Society A_ , vol. 379, no. 2197, p. 20200210, 2021.
  * [6] R. Pordes, D. Petravick, B. Kramer, D. Olson, M. Livny, A. Roy, P. Avery, K. Blackburn, T. Wenaus, F. Würthwein _et al._ , “The open science grid,” in _Journal of Physics: Conference Series_ , vol. 78. IOP Publishing, 2007, p. 012057.
  * [7] I. Sfiligoi, D. C. Bradley, B. Holzman, P. Mhashilkar, S. Padhi, and F. Wurthwein, “The pilot way to grid resources using glideinwms,” in _2009 WRI World congress on computer science and information engineering_ , vol. 2. IEEE, 2009, pp. 428–432.