# Categorifying equivariant monoids

Daniel Graves  School of Mathematics, University of Leeds, Woodhouse, Leeds,
LS2 9JT, UK<EMAIL_ADDRESS>
###### Abstract.

Equivariant monoids are very important objects in many branches of
mathematics: they combine the notion of multiplication and the concept of a
group action. In this paper we will construct categories which encode the
structure borne by monoids with a group action by combining the theory of
PROPs and PROBs with the theory of crossed simplicial groups. PROPs and PROBs
are categories used to encode structures borne by objects in symmetric and
braided monoidal categories respectively, whilst crossed simplicial groups are
categories which encode a unital, associative multiplication and a compatible
group action. We will produce PROPs and PROBs whose categories of algebras are
equivalent to the categories of monoids, comonoids and bimonoids with group
action using extensions of the symmetric and braid crossed simplicial groups.
We will extend this theory to balanced braided monoidal categories using the
ribbon braid crossed simplicial group. Finally, we will use the
hyperoctahedral crossed simplicial group to encode the structure of an
involutive monoid with a compatible group action.

###### Key words and phrases:

PROP, PROB, monoid, comonoid, bimonoid, group action, symmetric monoidal
category, braided monoidal category, crossed simplicial group

###### 1991 Mathematics Subject Classification:

16T10, 16W22, 18M05, 18M15, 18M85

## Introduction

Equivariant monoids (also known as monoids with an action of a group $G$, or
$G$-monoids) are very important objects in many branches of mathematics: they
combine the notion of multiplication and the concept of a group action.
Important examples in homological algebra and algebraic topology include
equivariant algebras over a commutative ring and equivariant orthogonal ring
spectra. These examples will be discussed further in Section 1.

In this paper we construct categories which encode the structure inherent in a
$G$-monoid in any symmetric or braided monoidal category, by combining the
theories of PROPs, PROBs and crossed simplicial groups.

PROPs are “product and permutation” categories. They were originally
introduced by Mac Lane in collaboration with Adams [Mac65, 24], to encode
operations with multiple inputs and multiple outputs. PROPs come in a variety
of flavours: the topological version was defined by Boardman and Vogt [BV68];
coloured PROPs have been studied by Johnson and Yau [JY09]; simplicial PROPs
have been studied by Hackney and Robertson [HR17]; wheeled PROPs have studied
by Markl, Merkulov and Shadrin [MMS09]. The notion of a PROP is more general
than the notion of an operad, as we shall soon discuss. There is a rich theory
of using PROPs to encode and link structures that are of interest in topology
and algebra. For example, the papers [Abr96] and [RSW05] provide PROPs for
categories of Frobenius algebras (with extra structure) in terms of two-
dimensional cobordisms and cospans of finite sets. These have proved to have
applications further afield, such as in network theory [BCR18]. Another
example comes from the papers [MM20] and [MM21] where PROPs are used to study
$E_{\infty}$-structures.

Crossed simplicial groups were introduced independently by Fiedorowicz and
Loday [FL91] and Krasauskas [Kra87]. These papers explore the relationship
between crossed simplicial groups and equivariant homology. Crossed simplicial
groups have also been used to study marked surfaces equipped with group
actions [DK15]. A crossed simplicial group is a family of groups together with
face and degeneracy maps that satisfy “crossed” simplicial identities. Any
crossed simplical group has an associated category that encodes a unital,
associative multiplication compatible with group actions given by the family
of groups. The foundational example is Connes’ cyclic category [Con83]. In
this paper, we are going to consider the families of symmetric groups,
hyperoctahedral groups, braid groups and ribbon braid groups, all of which
have interesting applications in algebraic topology. The symmetric and
hyperoctahedral crossed simplicial groups are used in stable homotopy theory
and equivariant stable homotopy theory; see [Aul10], [Aul14], [Gra22c] and
[Gra22b]. The homology theory associated to the braid crossed simplicial group
was originally used by Fiedorowicz to study two-fold loop-suspension spaces
[Fie]. The braid crossed simplicial group also arises in the study of
configuration spaces [BCWW06]. The ribbon braid crossed simplicial group has
been considered by Krasauskas in connection with framed braids and framed
mapping class groups of smooth manifolds [Kra96].

The idea of combining the theory of PROPs and the theory of crossed simplicial
groups to categorify categories of monoids goes back to Pirashvili [Pir02].
Pirashvili uses the structure of the symmetric crossed simplicial groups to
construct PROPs whose categories of algebras are monoids, comonoids and
bimonoids in a symmetric monoidal category. These results provided a
generalization of previous results of Markl [Mar96], who constructed such
PROPs in terms of generators and relations in order to study the deformation
theory of algebras. Pirashvili’s result shows that PROPs offer a strict
generalization of operads. There is no operad whose algebras are bialgebras,
but Pirashvili shows that there is such a PROP. Lack [Lac04] showed that
Pirashvili’s result fits into a broader categorical framework; that of
composing PROPs via a distributive law. Using the techniques of Pirashvili and
Lack, the author extended the existing theory to construct PROPs for monoids,
comonoids and bimonoids equipped with an order-reversing involution, using the
hyperoctahedral crossed simplicial group [Gra20]. The author has also shown
that one can extend the results of Lack’s paper to the setting of braided
monoidal categories and prove analogues of Pirashvili’s result by using the
braid crossed simplicial group [Gra22a]. In this setting the categories we
consider are called PROBs; “product and braid” categories.

In this paper we will study equivariant extensions of the symmetric, braid and
hyperoctahedral crossed simplicial groups and, in doing so, we completely
generalize the existing results. In particular, our Theorems 5.3, 6.16 and 7.1
specialize to the theorems [Gra22a, 7.2], [Pir02, 5.2] and [Gra20, 5.3]
respectively, by taking the specific case of the trivial group. We also recall
what it means for a braided monoidal category to be balanced and extend the
results to this setting by considering the ribbon braid crossed simplicial
group and its equivariant extensions.

The paper is organized as follows. In Section 1 we recall the definitions of
$G$-monoids, $G$-comonoids and $G$-bimonoids in a symmetric or braided
monoidal category, together with examples of interest from topology and
homological algebra. In Section 2 we recall the definitions of PROPs and
PROBs. We recall the definition of an algebra for these categories in a
symmetric or braided monoidal category. We provide a list of examples that
will be used throughout the paper. In Section 3 we describe the procedure for
forming new PROPs and PROBs via a distributive law in two special cases. We
show that the group composition in certain families of semi-direct products
induces a distributive law between the associated groupoids. We also recall
how the composition law in a crossed simplicial group determines a
distributive law. We use these distributive laws to form new PROPs and PROBs
from the examples in Section 2. In Section 4 we prove that we have constructed
PROPs and PROBs whose categories of algebras are equivalent to the categories
of $G$-monoids and $G$-comonoids. We prove this for symmetric monoidal
categories, braided monoidal categories and balanced braided monoidal
categories. We also construct a PROP for $G$-commutative monoids and
$G$-cocommutative comonoids in a symmetric monoidal category. In Section 5 we
construct the PROBs for $G$-bimonoids in braided and balanced braided monoidal
categories by composing the PROBs for $G$-monoids and $G$-comonoids. In
Section 6 we prove similar results for $G$bimonoids in a symmetric monoidal
category. We construct the necessary PROPs in two different ways. In the first
instance we provide a proof that is similar to the one given for braided
monoidal categories; by composing the PROPs for $G$-monoids and $G$-comonoids.
However, in the symmetric monoidal case we have an interesting alternative
description of these PROPs in terms of categories of non-commutative sets and
double categories. Using this construction also allows us to give a nice
description of the PROPs for $G$-bimonoids that are commutative, cocommutative
or both. In Section 7 we note that in a symmetric monoidal category we have
monoids that possess both an involution and a group action in a compatible
way. Group algebras provide one important example. We use the equivariant
extensions of the hyperoctahedral crossed simplicial group to construct a PROP
which encodes this data.

### Acknowledgements

I would like to thank James Cranch and Sarah Whitehouse for interesting and
helpful conversations whilst writing this paper.

##  1\. Equivariant monoids

In this section we introduce some notation for categories of $G$-monoids,
$G$-comonoids and $G$-bimonoids. We also give some examples from homological
algebra and algebraic topology.

######  Definition 1.1.

Let $\mathbf{C}$ be a braided or symmetric monoidal category. We denote by
$\mathbf{Mon}\left(\mathbf{C}\right)$, $\mathbf{Comon}\left(\mathbf{C}\right)$
and $\mathbf{Bimon}\left(\mathbf{C}\right)$ the categories of monoids,
comonoids and bimonoids in $\mathbf{C}$ respectively.

######  Definition 1.2.

Let $\mathbf{C}$ be a braided or symmetric monoidal category and let $G$ be a
group. We denote the functor category
$\mathrm{Fun}\left(G,\mathbf{Mon}(\mathbf{C})\right)$ by $\mathbf{GMon}$ and
call it the _category of $G$-monoids in $\mathbf{C}$_. Similarly we denote the
functor categories $\mathrm{Fun}\left(G,\mathbf{Comon}(\mathbf{C})\right)$ and
$\mathrm{Fun}\left(G,\mathbf{Bimon}(\mathbf{C})\right)$ by
$\mathbf{GComon}\left(\mathbf{C}\right)$ and
$\mathbf{GBimon}\left(\mathbf{C}\right)$ respectively and call them the
_categories of $G$-comonoids and $G$-bimonoids in $\mathbf{C}$_.

######  Example 1.3.

We note that we can think of monoids in a symmetric or braided monoidal
category as $G$-monoids, with $G$ being the trivial group.

######  Example 1.4.

The following three examples demonstrate the importance of $G$-monoids in
algebraic topology and homological algebra.

  1. (1)

Let $k$ be a commutative ring and let $\mathbf{C}$ be the symmetric monoidal
category of $k$-modules, $\mathbf{Mod}_{k}$. In this case, a $G$-monoid is a
$G$-algebra. Explicitly, a $G$-algebra is an associative $k$-algebra $A$
equipped with a group homomorphism $G\rightarrow\mathrm{Aut}(A)$. These have
been studied by Inassaridze in connection with equivariant homological algebra
[Ina22].

  2. (2)

Let $G$ be a group. Consider the braided monoidal category of crossed
$G$-sets. This is described in [FY89, 4.2] and used to define invariants in
low dimensional topology. A monoid in this category is a crossed $G$-monoid.
These are described in [Bou03, Definition 4.1]. One can consider these as
$G$-monoids in the category of crossed $G$-sets via the multiplication in $G$.
Bouc uses crossed $G$-monoids to construct certain Green functors and Mackey
functors and uses these to study the Hochschild cohomology of $G$ over a ring.

  3. (3)

Consider the symmetric monoidal category of orthogonal spectra,
$\mathbf{Sp}^{O}$. This category first appeared in [MMSS01]. A monoid in
$\mathbf{Sp}^{O}$ is an orthogonal ring spectrum. A $G$-monoid in
$\mathbf{Sp}^{O}$ is a $G$-equivariant orthogonal ring spectrum. Equivariant
orthogonal spectra were first studied in [MM02]. These are important objects
in equivariant stable homotopy theory. For example, they are used in the
construction of the $\infty$-category of genuine $G$-equivariant spectra; see
[NS18, Section II.2] or [MNN17, Part 2] for example.

##  2\. PROPs and PROBs

In this section we recall the definitions of PROPs and PROBs. We recall the
definition of an algebra for such a category in a symmetric or braided
monoidal category. We give a list of the examples that will be used throughout
the paper.

######  Definition 2.1.

For $n\geqslant 1$ we define $\underline{n}$ to be the set
$\left\\{1,\dotsc,n\right\\}$. We define $\underline{0}=\emptyset$.

######  Definition 2.2.

Let $\mathbf{M}$ be a monoidal category, let $\mathbf{S}$ be a symmetric
monoidal category and let $\mathbf{B}$ be a braided monoidal category.

  1. (1)

A PRO $\mathbf{T}$ is a strict monoidal category whose objects are the sets
$\underline{n}$ for $n\geqslant 0$ and whose tensor product is given by
addition. An _algebra of $\mathbf{T}$ in $\mathbf{M}$_ is a strict monoidal
functor $\mathbf{T}\rightarrow\mathbf{M}$.

  2. (2)

A _PROP_ $\mathbf{P}$ is a symmetric strict monoidal category whose objects
are the sets $\underline{n}$ for $n\geqslant 0$ with tensor product given by
addition. A _$\mathbf{P}$ -algebra in $\mathbf{S}$_ is a symmetric strict
monoidal functor $\mathbf{P}\rightarrow\mathbf{S}$. We denote the category of
$\mathbf{P}$-algebras in $\mathbf{S}$ and natural transformations by
$\mathbf{Alg}\left(\mathbf{P},\mathbf{S}\right)$.

  3. (3)

A _PROB_ $\mathbf{Q}$ is a braided strict monoidal category whose objects are
the sets $\underline{n}$ for $n\geqslant 0$ with tensor product given by
addition. A _$\mathbf{Q}$ -algebra in $\mathbf{B}$_ is a braided strict
monoidal functor $\mathbf{Q}\rightarrow\mathbf{B}$. We denote the category of
$\mathbf{Q}$-algebras in $\mathbf{B}$ and natural transformations by
$\mathbf{Alg}\left(\mathbf{Q},\mathbf{B}\right)$.

######  Example 2.3.

The following examples are the building blocks for the results of this paper.

  1. (1)

Following [Lac04, 2.2] we denote by $\mathbb{D}$ the PRO of finite ordinals
and order-preserving maps. For a strict monoidal category $\mathbf{M}$, an
algebra of $\mathbb{D}$ in $\mathbf{M}$ is a monoid in $\mathbf{M}$, see
[ML98, VII 5].

  2. (2)

Following [Lac04, 5.1], let $\mathbb{F}$ denote the skeletal category of
finite sets. For a symmetric monoidal category $\mathbf{S}$, an algebra of
$\mathbb{F}$ in $\mathbf{S}$ is a commutative monoid.

  3. (3)

Let $G$ be a group. Let $\mathbb{G}$ be the PRO such that
$\mathrm{Hom}_{\mathbb{G}}\left(\underline{n},\underline{m}\right)$ is empty
if $n\neq m$ and
$\mathrm{Hom}_{\mathbb{G}}\left(\underline{n},\underline{n}\right)=G^{n}$. The
disjoint union of morphisms corresponds to the product of group elements.

  4. (4)

Following [Lac04, 2.4], let $\mathbb{P}$ denote the PROP of finite sets and
bijections. Observe that this is the subobject of $\mathbb{F}$ consisting of
the isomorphisms in the skeletal category of finite sets.

  5. (5)

Following [Gra20, 2.11], let $\mathbb{H}$ denote the PROP of hyperoctahedral
groups. Recall that the hyperoctahedral group $H_{n}$ is the semi-direct
product $C_{2}^{n}\rtimes\Sigma_{n}$.

  6. (6)

Following [JS93, Example 2.1], let $\mathbb{B}$ denote the PROB of braid
groups. The set
$\mathrm{Hom}_{\mathbb{B}}\left(\underline{n},\underline{m}\right)$ is empty
for $n\neq m$ and
$\mathrm{Hom}_{\mathbb{B}}\left(\underline{n},\underline{n}\right)=B_{n}$, the
braid group on $n$ strings. The strict monoidal structure is given by the
_addition of braids_.

  7. (7)

Following [JS93, Example 6.4], let $\mathbb{R}\mathbb{B}$ denote the PROB of
ribbon braid groups. The set
$\mathrm{Hom}_{\mathbb{R}\mathbb{B}}\left(\underline{n},\underline{m}\right)$
is empty for $n\neq m$ and
$\mathrm{Hom}_{\mathbb{\mathbb{R}\mathbb{B}}}\left(\underline{n},\underline{n}\right)=RB_{n}$,
the ribbon braid group on $n$ ribbons. The strict monoidal structure is given
by addition of ribbons. The ribbon braid group $RB_{n}$ can be identified with
the semi-direct product $\mathbb{Z}^{n}\rtimes B_{n}$ where the integer labels
indicate the number of full twists to perform on each ribbon.

##  3\. Distributive laws and crossed simplicial groups

We can form new PROs, PROPs and PROBs via the notion of a _distributive law_
as defined in [Lac04, Section 3] and [RW02, Section 2]. In particular we will
form composites from the the categories defined in Example 2.3.

###  3.1. Semi-direct products

Let $G$ be a group and consider the semi-direct products
$G^{n}\rtimes\Sigma_{n}$, $G^{n}\rtimes B_{n}$, $G^{n}\rtimes RB_{n}$ and
$G^{n}\rtimes H_{n}$ where in each case the tuple $G^{n}$ is acted on by the
underlying permutation. These groups are sometimes also called wreath
products. We can think of these as labelled permutations ([FL91, Theorem
3.10]), labelled braids ([JS93, Example 2.1]), labelled ribbons ([JS93,
Example 6.4]) and labelled hyperoctahedral group elements.

The definition of composition in these semi-direct products satisfies the data
of a distributive law. For example, if we consider the symmetric case, a pair
$\left(\mathbf{x},\sigma\right)$ where
$\sigma\in\mathrm{Hom}_{\mathbb{P}}\left(\underline{n},\underline{n}\right)$
and
$\mathbf{x}\in\mathrm{Hom}_{\mathbb{G}}\left(\underline{n},\underline{n}\right)$
determines a unique pair $\left(\sigma,\sigma\left(\mathbf{x}\right)\right)$
where $\sigma$ remains unchanged and $\sigma$ acts on the element
$\mathbf{x}\in G^{n}$ by permutation. One can readily check that this
assignment is compatible with the monoidal structure of both categories.
Following [Lac04, 3.8], we have a well-defined PRO
$\mathbb{P}\otimes\mathbb{G}$, with
$\mathrm{Hom}_{\mathbb{P}\otimes\mathbb{G}}\left(\underline{n},\underline{m}\right)=\emptyset$
if $n\neq m$ and
$\mathrm{Hom}_{\mathbb{P}\otimes\mathbb{G}}\left(\underline{n},\underline{n}\right)=G^{n}\rtimes\Sigma_{n}$.
In fact, this category has a canonical PROP structure induced from
$\mathbb{P}$

Mimicking this procedure we form a PROP $\mathbb{H}\otimes\mathbb{G}$ and
PROBs $\mathbb{B}\otimes\mathbb{G}$ and
$\mathbb{R}\mathbb{B}\otimes\mathbb{G}$.

###  3.2. Crossed simplicial groups

Consider a family of groups $\left\\{J_{n}\right\\}$ indexed over the natural
numbers. Let $\mathbb{J}$ be the groupoid whose objects are the sets
$\underline{n}$ with
$\mathrm{Hom}_{\mathbb{J}}\left(\underline{n},\underline{n}\right)=J_{n}$. We
use $J$ for a group here because we are already using $G$ for a group, and $H$
for hyperoctahedral groups. One of the equivalent definitions for
$\left\\{J_{n}\right\\}$ to be a crossed simplicial group is the existence of
a distributive law
$\mathbb{J}\otimes\mathbb{D}\rightarrow\mathbb{D}\otimes\mathbb{J}$. In other
words, given a pair of morphisms $\left(j,\varphi\right)$ with
$\varphi\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{n},\underline{m}\right)$
and $j\in\mathrm{Hom}_{\mathbb{J}}\left(\underline{m},\underline{m}\right)$ we
have a unique pair $\left(j_{\star}(\varphi),\varphi^{\star}(j)\right)$ with
$\varphi^{\star}(j)\in\mathrm{Hom}_{\mathbb{J}}\left(\underline{n},\underline{n}\right)$
and
$j_{\star}(\varphi)\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{n},\underline{m}\right)$,
satisfying the relations of [RW02, 2.4] and compatible with the monoidal
structure of both categories. These assignments turn
$\mathbb{D}\otimes\mathbb{J}$ into a well-defined category. The objects are
the sets $\underline{n}$. An element of
$\mathrm{Hom}_{\mathbb{D}\otimes\mathbb{J}}\left(\underline{n},\underline{m}\right)$
is a pair $\left(\varphi,j\right)$ with
$j\in\mathrm{Hom}_{\mathbb{J}}\left(\underline{n},\underline{n}\right)$ and
$\varphi\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{n},\underline{m}\right)$,
with composition determined by the distributive law.

The descriptions of these maps for the families of symmetric groups,
hyperoctahedral groups and braid groups were given in [FL91, 3.1, 3.3, 3.7]
respectively. For the case of ribbon braids one follows [FL91, 3.7], replacing
the identity braid with the identity ribbon.

By a theorem of Fiedorowicz and Loday [FL91, Theorem 3.10], we know that the
family of groups $\left\\{G^{n}\rtimes\Sigma_{n}\right\\}$ form a crossed
simplicial group. In particular, the proof describes assignments
$\varphi^{\star}$ and $g_{\star}$ which describe a distributive law

$\left(\mathbb{P}\otimes\mathbb{G}\right)\otimes\mathbb{D}\rightarrow\mathbb{D}\otimes\left(\mathbb{P}\otimes\mathbb{G}\right),$

compatible with the monoidal structure. These assignments are constructed from
those in the symmetric crossed simplicial group by extending them to also take
into account the $G$-labels. We therefore have a well-defined PROP
$\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}$, whose set of morphisms
$\mathrm{Hom}_{\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}}\left(\underline{n},\underline{m}\right)$
consists of pairs $(\varphi,g)$ with
$g\in\mathrm{Hom}_{\mathbb{P}\otimes\mathbb{G}}\left(\underline{n},\underline{n}\right)=G^{n}\rtimes\Sigma_{n}$
and
$\varphi\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{n},\underline{m}\right)$.

We observe that the proof [FL91, Theorem 3.10] works equally well when we
replace $\mathbb{P}\otimes\mathbb{G}$ with the categories
$\mathbb{H}\otimes\mathbb{G}$, $\mathbb{B}\otimes\mathbb{G}$ and
$\mathbb{R}\mathbb{B}\otimes\mathbb{G}$. In this way we form a PROP
$\mathbb{D}\otimes\mathbb{H}\otimes\mathbb{G}$ and PROBs
$\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G}$ and
$\mathbb{D}\otimes\mathbb{R}\mathbb{B}\otimes\mathbb{G}$.

######  Remark 3.1.

Note that any morphism $\left(\varphi,j\right)$ can be decomposed in terms of
precisely three morphisms. We have the group element $j$ as an automorphism.
An order-preserving map $\varphi$ has a unique decomposition in terms of face
and degeneracy maps [Lod98, B.2]. However, using the monoidal structure in the
category $\mathbb{D}$, any face map can be written as a tensor product of the
unique morphism
$m\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{2},\underline{1}\right)$ with
identity morphisms and any degeneracy maps can be expressed as a tensor
product of the unique map
$u\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{0},\underline{1}\right)$ with
identity morphisms. Therefore, using the unique decomposition of order-
preserving maps and the monoidal structure we can write any morphism in a
crossed simplicial group as a finite expression in terms of $m$, $u$ and the
group elements.

###  3.3. A distributive law for skeletal finite sets

We define a distributive law between the PROP given by the skeletal category
of finite sets, $\mathbb{F}$, and the groupoid $\mathbb{G}$. Given a pair
$\left(\mathbf{x},f\right)$ with
$f\in\mathrm{Hom}_{\mathbb{F}}\left(\underline{n},\underline{m}\right)$ and

$\mathbf{x}=\left(g_{1},\dotsc,g_{m}\right)\in G^{m},$

we have a unique pair $\left(f,\mathbf{x}^{\prime}\right)$ where $f$ has
remained unchanged and

$\mathbf{x}^{\prime}=\left(g_{f(1)},\dotsc,g_{f(n)}\right)\in G^{n}.$

One can readily check that this satisfies the relations of [RW02, 2.4] and is
compatible with the monoidal structure of both categories. We therefore have a
well-defined PROP $\mathbb{F}\otimes\mathbb{G}$.

##  4\. Categorifying equivariant monoids and comonoids

We show that the PROP $\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}$ and the
PROB $\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G}$ encode the structures of
$G$-monoids and $G$-comonoids in symmetric and braided monoidal categories. We
show that if we take the PROP $\mathbb{F}\otimes\mathbb{G}$ we encode the
structures of $G$-commutative monoids and $G$-cocommutative comonoids. We also
recall the notion of a balanced braided monoidal category and show that the
PROBs $\mathbb{D}\otimes\mathbb{R}\mathbb{B}$ and
$\mathbb{D}\otimes\mathbb{R}\mathbb{B}\otimes\mathbb{G}$ encode the structures
of monoids and $G$-monoids in a balanced braided monoidal category.

######  Proposition 4.1.

Let $\mathbf{S}$ be a symmetric monoidal category, let $\mathbf{B}$ be a
braided monoidal category and let $G$ be a group. There are equivalences of
categories

  1. (1)

$\mathbf{Alg}\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G},\mathbf{S}\right)\simeq\mathbf{GMon}\left(\mathbf{S}\right)$;

  2. (2)

$\mathbf{Alg}\left(\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G},\mathbf{B}\right)\simeq\mathbf{GMon}\left(\mathbf{B}\right)$;

  3. (3)

$\mathbf{Alg}\left(\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}\right)^{op},\mathbf{S}\right)\simeq\mathbf{GComon}\left(\mathbf{S}\right)$
and

  4. (4)

$\mathbf{Alg}\left(\left(\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G}\right)^{op},\mathbf{B}\right)\simeq\mathbf{GComon}\left(\mathbf{B}\right)$.

###### Proof.

The proof for monoids in both cases is similar to [Lac04, 5.5], [Gra20, 2.13]
and [Gra22a, 6.2]. We will give the details of the symmetric monoidal case and
indicate the necessary changes for the braided monoidal case. Let
$U\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}\right)$ denote the
underlying PRO of the PROP $\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}$. By
[Lac04, 3.10], a
$U\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}\right)$-algebra structure
on an object $M$ of $\mathbf{S}$ consists of a $\mathbb{D}$-algebra structure
and a $\left(\mathbb{P}\otimes\mathbb{G}\right)$-algebra structure subject to
a compatibility condition. A $\mathbb{D}$-algebra structure is a monoid
structure. A $\left(\mathbb{P}\otimes\mathbb{G}\right)$-algebra is an object
$M$ together with a morphism $g\colon M\rightarrow M$ for each element $g\in
G$ and, for each element
$\left(\sigma,\left(g_{1},\dotsc,g_{n}\right)\right)$, a morphism $M^{\otimes
n}\rightarrow M^{\otimes n}$ given by applying $(g_{1},\dotsc,g_{n})$
component-wise followed by an isomorphism determined by $\sigma\in\Sigma_{n}$.
By [Lac04, 5.5] a
$U\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}\right)$-algebra structure
is a $\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}\right)$-algebra
structure if and only if the isomorphisms induced from the elements
$\sigma\in\Sigma_{n}$ are the symmetry isomorphisms. The compatibility
condition ensures that the morphisms $g\colon M\rightarrow M$ form a group
action compatible with the symmetric monoidal structure. Finally, a morphism
in $\mathbf{S}$ is a map of $G$-monoids if and only if it respects the
$\mathbb{D}$-algebra structure and the
$\left(\mathbb{P}\otimes\mathbb{G}\right)$-algebra structure. By [Lac04,
3.12], this is true if and only if it respects the
$\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}\right)$-algebra structure.
The braided monoidal case is similar, using [Gra22a, 5.10] in place of [Lac04,
3.12].

For comonoids the proof is also similar for both the symmetric and braided
monoidal cases. We give the proof for the symmetric monoidal case. For
categories $\mathbf{C}$ and $\mathbf{D}$ there is an isomorphism of functor
categories
$\mathrm{Fun}\left(\mathbf{C}^{op},\mathbf{D}^{op}\right)\cong\mathrm{Fun}\left(\mathbf{C},\mathbf{D}\right)^{op}$
and so

$\mathrm{Fun}\left(G,\mathbf{Comon}(\mathbf{S})\right)=\mathrm{Fun}\left(G,\mathbf{Mon}(\mathbf{S}^{op})^{op}\right)\cong\mathrm{Fun}\left(G^{op},\mathbf{Mon}(\mathbf{S}^{op})\right)^{op}.$

Since a group is isomorphic to its opposite we see that

$\mathrm{Fun}\left(G^{op},\mathbf{Mon}(\mathbf{S}^{op})\right)^{op}\cong\mathrm{Fun}\left(G,\mathbf{Mon}(\mathbf{S}^{op})\right)^{op}=\mathbf{GMon}\left(\mathbf{S}^{op}\right)^{op}.$

Furthermore, by Proposition 4.1,

$\mathbf{GMon}\left(\mathbf{S}^{op}\right)^{op}\simeq\mathbf{Alg}\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G},\mathbf{S}^{op}\right)^{op}=\mathbf{Alg}\left(\left(\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}\right)^{op},\mathbf{S}\right)$

as required. ∎

######  Proposition 4.2.

Let $\mathbf{S}$ be a symmetric monoidal category and let $G$ be a group.

  1. (1)

The category $\mathbf{Alg}\left(\mathbb{F}\otimes\mathbb{G},\mathbf{S}\right)$
is equivalent to the category of $G$-commutative monoids in $\mathbf{S}$;

  2. (2)

the category
$\mathbf{Alg}\left(\left(\mathbb{F}\otimes\mathbb{G}\right)^{op},\mathbf{S}\right)$
is equivalent to the category of $G$-cocommutative comonoids in $\mathbf{S}$.

###### Proof.

The proof follows the same method as Proposition 4.1, noting that an algebra
for the PROP $\mathbb{F}$ is a commutative monoid, rather than a monoid. ∎

Recall from [JS93, Definition 6.1] that a braided monoidal category
$\mathbf{B}$ is called _balanced_ if for each object $A$ we have a _full twist
isomorphism_ $\Theta_{A}\colon A\rightarrow A$, such that the full twist on
the unit object is the identity and such that the full twists are compatible
with the braiding. The category of braids $\mathbb{B}$, the category of ribbon
braids $\mathbb{R}\mathbb{B}$ and the category of labelled ribbon braids
$\mathbb{R}\mathbb{B}\otimes\mathbb{G}$ are all examples of balanced braided
monoidal categories. Furthermore, any symmetric monoidal category is balanced.
We can use the PROB of ribbon braids to identify the extra structure present
in balanced braided monoidal categories.

######  Proposition 4.3.

Let $\mathbf{B}$ be a balanced braided monoidal category and let $G$ be a
group. There are equivalences of categories

  1. (1)

$\mathbf{Alg}\left(\mathbb{D}\otimes\mathbb{R}\mathbb{B},\mathbf{B}\right)\simeq\mathbf{Mon}\left(\mathbf{B}\right)$;

  2. (2)

$\mathbf{Alg}\left(\mathbb{D}\otimes\mathbb{R}\mathbb{B}\otimes\mathbb{G},\mathbf{B}\right)\simeq\mathbf{GMon}\left(\mathbf{B}\right)$;

  3. (3)

$\mathbf{Alg}\left(\left(\mathbb{D}\otimes\mathbb{R}\mathbb{B}\right)^{op},\mathbf{B}\right)\simeq\mathbf{Comon}\left(\mathbf{B}\right)$
and

  4. (4)

$\mathbf{Alg}\left(\left(\mathbb{D}\otimes\mathbb{R}\mathbb{B}\otimes\mathbb{G}\right)^{op},\mathbf{B}\right)\simeq\mathbf{GComon}\left(\mathbf{B}\right)$.

###### Proof.

The non-equivariant cases are similar to [Gra22a, 6.2] and [Gra22a, 6.3]. The
added data of twists in the ribbons corresponds precisely to applying the
twist isomorphisms in $\mathbf{B}$. The equivariant cases are similar to
Proposition 4.1. ∎

##  5\. Equivariant bimonoids in a braided monoidal category

In this section we construct a PROB for $G$-bimonoids in a braided monoidal
category. We do this by defining a distributive law between
$\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G}$ and its opposite, that is
between the PROBs for $G$-monoids and $G$-comonoids, and analysing the
composite PROB using the machinery introduced in [Gra22a, Section 5].

######  Definition 5.1.

For ease of notation, let $Q$ denote the category
$\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G}$. Let
$Q^{op}\otimes_{\mathbb{B}\otimes\mathbb{G}}Q$ denote the category of
equivalence classes of cospans in $Q$. Let
$Q\otimes_{\mathbb{B}\otimes\mathbb{G}}Q^{op}$ denote the category of
equivalence classes of spans in $Q$.

Following Remark 3.1 it suffices to define the necessary distributive law in
terms of the morphisms
$m\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{2},\underline{1}\right)$,
$u\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{0},\underline{1}\right)$ and
the morphisms of $\mathbb{B}\otimes\mathbb{G}$. In the following definition we
extend the distributive law of [Gra22a, 7.1] to take into account the
labelling data. Observe that this distributive law is completely determined by
the crossed simplicial group structure of
$\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G}$

######  Definition 5.2.

We define a distributive law of PROBs

$Q^{op}\otimes_{\mathbb{B}\otimes\mathbb{G}}Q\rightarrow
Q\otimes_{\mathbb{B}\otimes\mathbb{G}}Q^{op}$

to be determined as follows

  1. (1)

For $g$, $h\in G^{n}\rtimes B_{n}$

${\underline{n}}$${\underline{n}}$${\underline{n}}$${\mapsto}$${\underline{n}}$${\underline{n}}$${\underline{n}}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{g^{-1}}$$\scriptstyle{h^{-1}}$

  2. (2)

For
$\varphi\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{n},\underline{m}\right)$
and $g\in G^{m}\rtimes B_{m}$

${\underline{n}}$${\underline{m}}$${\underline{m}}$${\mapsto}$${\underline{n}}$${\underline{n}}$${\underline{m}}$${\underline{m}}$${\underline{m}}$${\underline{n}}$${\mapsto}$${\underline{m}}$${\underline{n}}$${\underline{n}}$$\scriptstyle{\varphi}$$\scriptstyle{g}$$\scriptstyle{g_{\star}(\varphi)}$$\scriptstyle{\varphi^{\star}(g)}$$\scriptstyle{g}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi^{\star}(g)}$$\scriptstyle{g_{\star}(\varphi)}$

where $g_{\star}(\varphi)$ and $\varphi^{\star}(g)$ are the assignments from
the crossed simplicial group structure of
$\mathbb{D}\otimes\mathbb{B}\otimes\mathbb{G}$.

  3. (3)

For $m\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{2},\underline{1}\right)$
and $u\in\mathrm{Hom}_{\mathbb{D}}\left(\underline{0},\underline{1}\right)$

${\underline{2}}$${\underline{1}}$${\underline{2}}$${\mapsto}$${\underline{2}}$${\underline{4}}$${\underline{2}}$${\underline{0}}$${\underline{1}}$${\underline{0}}$${\mapsto}$${\underline{0}}$${\underline{0}}$${\underline{0}}$${\underline{2}}$${\underline{1}}$${\underline{0}}$${\mapsto}$${\underline{2}}$${\underline{0}}$${\underline{0}}$${\underline{0}}$${\underline{1}}$${\underline{2}}$${\mapsto}$${\underline{0}}$${\underline{0}}$${\underline{2}}$$\scriptstyle{m}$$\scriptstyle{m}$$\scriptstyle{m^{\amalg
2}\circ\sigma_{2,3}}$$\scriptstyle{m^{\amalg
2}}$$\scriptstyle{u}$$\scriptstyle{u}$$\scriptstyle{id_{0}}$$\scriptstyle{id_{0}}$$\scriptstyle{m}$$\scriptstyle{u}$$\scriptstyle{u^{2}}$$\scriptstyle{id_{0}}$$\scriptstyle{u}$$\scriptstyle{m}$$\scriptstyle{id_{0}}$$\scriptstyle{u^{2}}$

where $\sigma_{2,3}$ is notation for $id_{1}\amalg\sigma\amalg id_{1}$ and
$\sigma$ is the generator of the braid group $B_{2}$.

######  Theorem 5.3.

Let $\mathbf{B}$ be a braided monoidal category. Let $G$ be a group. There is
an equivalence of categories
$\mathbf{Alg}\left(\mathbb{D}\otimes\left(\mathbb{B}\otimes\mathbb{G}\right)\otimes\mathbb{D}^{op},\mathbf{B}\right)\simeq\mathbf{GBimon}\left(\mathbf{B}\right)$.

###### Proof.

The distributive law of Definition 5.2 implies that
$Q\otimes_{\mathbb{B}\otimes\mathbb{G}}Q^{op}$ is a composite PROB by [Gra22a,
5.9]. By [Gra22a, 5.10] a
$\left(Q\otimes_{\mathbb{B}\otimes\mathbb{G}}Q^{op}\right)$-algebra structure
for an object $M$ in $\mathbf{B}$ is a $Q$-algebra structure, that is a
$G$-monoid structure, and a $Q^{op}$-algebra structure, that is a $G$-comonoid
structure, subject to compatibility conditions arising from the distributive
law of Definition 5.2. These compatibility conditions are precisely those
requiring $M$ to be a $G$-bimonoid. Finally we observe that a morphism in
$\mathbf{B}$ is a morphism of $G$-bimonoids if and only if it preserves the
$Q$-algebra structure and the $Q^{op}$-algebra structure. By [Gra22a, 5.10]
this is true if and only if it preserves the
$\left(Q\otimes_{\mathbb{B}\otimes\mathbb{G}}Q^{op}\right)$-algebra structure.

The isomorphisms in $Q$ and $Q^{op}$ are the labelled braid groups. Therefore,
when we take equivalence classes of spans we are identifying the two group
actions. We therefore have an isomorphism of PROBs

$Q\otimes_{\mathbb{B}\otimes\mathbb{G}}Q^{op}\cong\mathbb{D}\otimes\left(\mathbb{B}\otimes\mathbb{G}\right)\otimes\mathbb{D}^{op}$

from which the result follows. ∎

######  Theorem 5.4.

Let $\mathbf{B}$ be a balanced braided monoidal category. There are
equivalences of categories

  1. (1)

$\mathbf{Alg}\left(\mathbb{D}\otimes\mathbb{R}\mathbb{B}\otimes\mathbb{D}^{op},\mathbf{B}\right)\simeq\mathbf{Bimon}\left(\mathbf{B}\right)$.

  2. (2)

$\mathbf{Alg}\left(\mathbb{D}\otimes\left(\mathbb{R}\mathbb{B}\otimes\mathbb{G}\right)\otimes\mathbb{D}^{op},\mathbf{B}\right)\simeq\mathbf{GBimon}\left(\mathbf{B}\right)$.

###### Proof.

We define appropriate distributive laws by replacing the labelled braids in
Definition 5.2 by the ribbon braids and the labelled ribbon braids
respectively, and using the composition in the categories
$\mathbb{D}\otimes\mathbb{R}\mathbb{B}$ and
$\mathbb{D}\otimes\mathbb{R}\mathbb{B}\otimes\mathbb{G}$. The proof then
follows similarly to Theorem 5.3, making use of Proposition 4.3. ∎

##  6\. Equivariant bimonoids in a symmetric monoidal category

In this section we construct a PROP for $G$-bimonoids in a symmetric monoidal
category. We begin by stating this result directly, following Theorem 5.3.
However, in the symmetric monoidal case we have an interesting alternative
description of this PROP in terms of non-commutative sets after the fashion of
[Pir02] and [Gra20]. We also include results for $G$-commutative and
$G$-cocommutative bimonoids.

###  6.1. A composite PROP for equivariant bimonoids

We begin by stating the symmetric monoidal analogue of Theorem 5.3.

######  Theorem 6.1.

Let $\mathbf{S}$ be a symmetric monoidal category. Let $G$ be a group. There
is an equivalence of categories
$\mathbf{Alg}\left(\mathbb{D}\otimes\left(\mathbb{P}\otimes\mathbb{G}\right)\otimes\mathbb{D}^{op},\mathbf{S}\right)\simeq\mathbf{GBimon}\left(\mathbf{S}\right)$.

###### Proof.

We define the necessary distributive law by replacing the labelled braids by
labelled permutations in Definition 5.2 and using the crossed simplicial group
structure in $\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}$. The proof then
follows the same argument of Theorem 5.3, using [Lac04, 4.4, 4.8] in place of
[Gra22a, 5.9, 5.10]. ∎

###  6.2. Non-commutative sets

The category of non-commutative sets $\mathcal{F}(as)$, originally defined in
[FT87, A10], is used in [Pir02] to construct PROPs for monoids, comonoids and
bimonoids in a symmetric monoidal category. The objects are the sets
$\underline{n}$ for $n\geqslant 0$ and morphisms are maps of sets with total-
ordering data on the preimages. In this subsection we extend the definition of
non-commutative sets to include labelling data from a group $G$ on the pre-
existing preimage data in order to encode the added structure of a group
action.

######  Definition 6.2.

Let $G$ be a group. We define an _ordered $G$-labelled set_ to be a finite,
ordered subset of the sets $\underline{n}$, whose elements are adorned with
superscript labels from $G$. We define an action of $G$ on such sets by

$g\ast\left\\{j_{1}^{\alpha_{j_{1}}}<\cdots<j_{r}^{\alpha_{j_{r}}}\right\\}=\left\\{j_{1}^{g\alpha_{j_{1}}}<\cdots<j_{r}^{g\alpha_{j_{r}}}\right\\}.$

In other words, we multiply the labels by $g$ on the left.

If we remove the ordering, we call these objects _$G$ -labelled sets_. Note
that we have a similar action of $G$ for these objects.

######  Definition 6.3.

Let $G$ be a group. The _PROP of $G$-labelled non-commutative sets_, denoted
$G\mathcal{F}(as)$, has as objects the sets $\underline{n}$ for $n\geqslant
0$. An element
$f\in\mathrm{Hom}_{G\mathcal{F}(as)}\left(\underline{n},\underline{m}\right)$
is a map of finite sets such that for each $i\in\underline{m}$, $f^{-1}(i)$ is
an ordered $G$-labelled set. In particular,
$\mathrm{Hom}_{G\mathcal{F}(as)}\left(\underline{0},\underline{m}\right)$ is
the singleton set containing the map $\emptyset\rightarrow\underline{m}$ and
$\mathrm{Hom}_{G\mathcal{F}(as)}\left(\underline{n},\underline{0}\right)$ is
the empty set. For
$f_{1}\in\mathrm{Hom}_{G\mathcal{F}(as)}\left(\underline{n},\underline{m}\right)$
and
$f_{2}\in\mathrm{Hom}_{G\mathcal{F}(as)}\left(\underline{m},\underline{l}\right)$,
the composite $f_{2}\bullet f_{1}$ in $G\mathcal{F}(as)$ is the composite of
the underlying set maps $f_{2}\circ f_{1}$ with

$\left(f_{2}\bullet f_{1}\right)^{-1}(i)=\coprod_{j^{\alpha_{j}}\in
f_{2}^{-1}(i)}\left(\alpha_{j}\ast f_{1}^{-1}(j)\right).$

The symmetry isomorphisms are given by the block permutations.

######  Definition 6.4.

Let $G\mathcal{F}$ be the category whose objects are the sets $\underline{n}$.
A morphism in $G\mathcal{F}$ is a map of sets such that the preimage of each
singleton in the codomain is a $G$-labelled set. Composition is given by
composition of set maps and multiplication of labels.

######  Proposition 6.5.

There are isomorphisms of PROPs

  1. (1)

$G\mathcal{F}(as)\cong\mathbb{D}\otimes\mathbb{P}\otimes\mathbb{G}$ and

  2. (2)

$G\mathcal{F}\cong\mathbb{F}\otimes\mathbb{G}$.

###### Proof.

The proof of the first isomorphism follows [Gra20, Proposition 3.11], with
$C_{2}$ replaced by $G$, the PROP of hyperoctahedral groups $\mathbb{H}$
replaced with the PROP $\mathbb{P}\otimes\mathbb{G}$ and using the crossed
simplicial group of [FL91, Theorem 3.10] in place of [FL91, 3.1]. The second
isomorphism is similar. ∎

######  Corollary 6.6.

Let $\mathbf{S}$ be a symmetric monoidal category. Let $G$ be a group. There
are equivalences of categories

  1. (1)

$\mathbf{Alg}\left(G\mathcal{F}(as),\mathbf{S}\right)\simeq\mathbf{GMon}\left(\mathbf{S}\right)$,

  2. (2)

$\mathbf{Alg}\left(G\mathcal{F}(as)^{op},\mathbf{S}\right)\simeq\mathbf{GComon}\left(\mathbf{S}\right)$
and

  3. (3)

$\mathbf{Alg}\left(G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op},\mathbf{S}\right)\simeq\mathbf{GBimon}\left(\mathbf{S}\right)$.

###### Proof.

This follows from Proposition 4.1, Proposition 6.5 and Theorem 6.1. ∎

We will give an alternative proof of the final part of this corollary, using
the theory of double categories. This machinery will also allow us to give
nice descriptions of the PROPs for $G$-commutative and $G$-cocommutative
bimonoids in a symmetric monoidal category.

######  Corollary 6.7.

Let $\mathbf{S}$ be a symmetric monoidal category. Let $G$ be a group.

  1. (1)

The category $\mathbf{Alg}\left(G\mathcal{F},\mathbf{S}\right)$ is equivalent
to the category of $G$-commutative monoids in $\mathbf{S}$;

  2. (2)

the category $\mathbf{Alg}\left(G\mathcal{F}^{op},\mathbf{S}\right)$ is
equivalent to the category of $G$-cocommutative
comonoids in $\mathbf{S}$.

###### Proof.

This follows from Proposition 6.5 and Proposition 4.2. ∎

######  Remark 6.8.

Note that when $G=C_{2}$, the cyclic group of order two, the PROP
$G\mathcal{F}(as)$ is not isomorphic to the PROP $\mathcal{IF}(as)$ of [Gra20,
3.5]. In the latter category, the action of $C_{2}$ also reverses the total-
ordering data, encoding an order-reversing involution. However, the PROP
$G\mathcal{F}$ is isomorphic to the PROP $\mathcal{I}\mathcal{F}$ of [Gra20,
3.13], since these PROPs do not have any total-ordering data.

###  6.3. Double categories

In this subsection we follow the constructions used in [Pir02, Section 4] and
[Gra20, Section 4] to construct a double category from the PROP
$G\mathcal{F}(as)$, which satisfies the star condition. Recall from [FL91,
2.3] that a double category satisfies the star condition if a horizontal
morphism and a vertical morphism with the same codomain determine a unique
bimorphism.

######  Definition 6.9.

The double category $G\mathcal{F}(as)_{2}$ has as objects the sets
$\underline{n}$ for $n\geqslant 0$. The sets of horizontal and vertical
morphisms in $G\mathcal{F}(as)_{2}$ are both equal to the set of all morphisms
in $G\mathcal{F}(as)$. A bimorphism in $G\mathcal{F}(as)_{2}$ is a square

${\underline{n}}$${\underline{p}}$${\underline{m}}$${\underline{q}}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{f_{1}}$$\scriptstyle{\varphi}$$\scriptstyle{f}$

of morphisms in $G\mathcal{F}(as)$ such that

  * •

the underlying diagram of finite sets is a pullback square,

  * •

for all $x\in\underline{m}$ the map
$\varphi_{1}^{-1}(x)\rightarrow\varphi^{-1}(f(x))$ induced by $f_{1}$ is an
isomorphism of ordered $G$-labelled sets and

  * •

for all $y\in\underline{p}$ the map $f_{1}^{-1}(y)\rightarrow
f^{-1}(\varphi(y))$ induced by $\varphi_{1}$ is an isomorphism of ordered
$G$-labelled sets.

######  Remark 6.10.

Note that the squares in the definition of a bimorphisms need not necessarily
be commutative.

######  Definition 6.11.

We define the double category $G\mathcal{F}_{2}$ similarly to
$G\mathcal{F}(as)_{2}$; the objects are those of $G\mathcal{F}$, the sets of
horizontal and vertical morphisms are the set of morphisms in $G\mathcal{F}$
and the bimorphisms are defined similarly to the bimorphisms of
$G\mathcal{F}(as)_{2}$, with ordered $G$-labelled sets replaced by
$G$-labelled sets.

We define two further double categories, which combine the PROPs
$G\mathcal{F}(as)$ and $G\mathcal{F}$.

######  Definition 6.12.

The double category $\mathcal{V}$ has as objects the objects of
$G\mathcal{F}(as)$. The set of vertical morphisms is the set of morphisms in
$G\mathcal{F}(as)$. The set of horizontal morphisms is the set of morphisms in
$G\mathcal{F}$. The bimorphisms are defined similarly to those of
$G\mathcal{F}(as)_{2}$ except that the horizontal morphisms are now in
$G\mathcal{F}$.

The double category $\mathcal{H}$ is defined similarly; the set of horizontal
morphisms is the set of morphisms in $G\mathcal{F}(as)$, the set of vertical
morphisms is the set of morphisms in $G\mathcal{F}$ and the bimorphisms are
defined similarly to those of $G\mathcal{F}(as)_{2}$ except that the vertical
morphisms are in $G\mathcal{F}$.

######  Remark 6.13.

Given a horizontal morphism $f\colon\underline{m}\rightarrow\underline{q}$ and
a vertical morphism $\varphi\colon\underline{p}\rightarrow\underline{q}$ in
$G\mathcal{F}(as)_{2}$ we determine a unique bimorphism by first taking the
pullback of the underlying maps of sets. As for the double category of Example
2 in [Pir02, Section 4] and the double category $\mathcal{IF}(as)_{2}$ of
[Gra20, 4.1], the maps of the pullback have a unique lift to the category
$G\mathcal{F}(as)$ where the preimage data is induced from $f$ and $\varphi$
using the conditions on bimorphisms. Hence the double category
$G\mathcal{F}(as)_{2}$ satisfies the star condition. A similar justification
show that the double categories $G\mathcal{F}_{2}$, $\mathcal{V}$ and
$\mathcal{H}$ also satisfy the star condition.

###  6.4. An alternative description and commutativity

We can use the double category $G\mathcal{F}(as)_{2}$ and Lack’s machinery of
composing PROPs [Lac04] to give an alternative construction of the PROP for
$G$-bimonoids in a symmetric monoidal category.

######  Proposition 6.14.

There exist composite PROPs

  1. (1)

$G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op}$;

  2. (2)

$G\mathcal{F}\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op}$;

  3. (3)

$G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}^{op}$;

  4. (4)

$G\mathcal{F}\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}^{op}$.

###### Proof.

We give the proof of the first case. The remaining three are similar.
Following the method of [Gra20, Proposition 5.1] we use the star condition for
the double category $G\mathcal{F}(as)_{2}$ to define a distributive law of
PROs

$G\mathcal{F}(as)^{op}\otimes G\mathcal{F}(as)\rightarrow
G\mathcal{F}(as)\otimes G\mathcal{F}(as)^{op}.$

Both $G\mathcal{F}(as)$ and $G\mathcal{F}(as)^{op}$ are PROPs and so
$G\mathcal{F}(as)\otimes G\mathcal{F}(as)^{op}$ has a PROP structure.

A morphism in $G\mathcal{F}(as)\otimes G\mathcal{F}(as)^{op}$ from
$\underline{n}$ to $\underline{m}$ can be written as a span

${\underline{n}}$${\underline{p}}$${\underline{m}}$$\scriptstyle{\varphi}$$\scriptstyle{f}$

in $G\mathcal{F}(as)$. Two spans

${\underline{n}}$${\underline{p}}$${\underline{m}}$and${\underline{n}}$${\underline{p}}$${\underline{m}}$$\scriptstyle{\varphi}$$\scriptstyle{f}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{f_{1}}$

are said to be _equivalent_ if there exists a bijection
$h\colon\underline{p}\rightarrow\underline{p}$ in $G\mathcal{F}(as)$ such that
$\varphi_{1}\circ h=\varphi$ and $f_{1}\circ h=f$. Note that the bijections in
the category $G\mathcal{F}(as)$ correspond precisely to the morphisms in the
PROP $\mathbb{P}\otimes\mathbb{G}$.

Let
$G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op}$
be the category obtained from $G\mathcal{F}(as)\otimes G\mathcal{F}(as)^{op}$
by identifying equivalent spans. It follows from [Lac04, Theorem 4.6] that
$G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op}$
is a composite PROP defined via a distributive law induced from the one
defined for PROs. ∎

######  Definition 6.15.

Let
$Q_{\mathcal{V}}=G\mathcal{F}\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op}$.
Let
$Q_{\mathcal{H}}=G\mathcal{F}^{op}\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)$.
Let
$Q_{G\mathcal{F}}=G\mathcal{F}\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}^{op}$.

The following theorem gives an alternative proof of the third part of
Corollary 6.6.

######  Theorem 6.16.

Let $\mathbf{S}$ be a symmetric monoidal category. There is an equivalence of
categories

$\mathbf{Alg}\left(G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op},\mathbf{S}\right)\simeq\mathbf{GBimon}\left(\mathbf{S}\right).$

###### Proof.

We argue analogously to [Gra20, 5.3]. By [Lac04, Proposition 4.7], an algebra
for
$G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op}$
in $\mathbf{S}$ consists of an object $M$ with a $G\mathcal{F}(as)$-algebra
structure and a $G\mathcal{F}(as)^{op}$-algebra structure subject to
compatibility conditions. A $G\mathcal{F}(as)$-algebra structure is a
$G$-monoid structure and a $G\mathcal{F}(as)^{op}$-algebra structure is a
$G$-comonoid structure. Arguing analogously to [Lac04, 5.3] and [Gra20, 5.3],
the compatibility conditions arising from the distributive law are precisely
the conditions requiring $M$ to be a $G$-bimonoid. Finally we observe that a
morphism in $\mathbf{S}$ is a morphism of $G$-bimonoids if and only if it
preserves the $G\mathcal{F}(as)$-algebra structure and the
$G\mathcal{F}(as)^{op}$-algebra structure. By [Lac04, 4.8] this is true if and
only if it preserves the
$G\mathcal{F}(as)\otimes_{\mathbb{P}\otimes\mathbb{G}}G\mathcal{F}(as)^{op}$-algebra
structure. ∎

######  Theorem 6.17.

Let $\mathbf{S}$ be a symmetric monoidal category. Let $G$ be a group.

  1. (1)

The category $\mathbf{Alg}\left(Q_{\mathcal{V}},\mathbf{S}\right)$ is
equivalent to the category of $G$-commutative bimonoids in $\mathbf{S}$.

  2. (2)

The category $\mathbf{Alg}\left(Q_{\mathcal{H}},\mathbf{S}\right)$ is
equivalent to the category of $G$-cocommutative bimonoids in $\mathbf{S}$.

  3. (3)

The category $\mathbf{Alg}\left(Q_{G\mathcal{F}},\mathbf{S}\right)$ is
equivalent to the category of $G$-commutative and cocommutative bimonoids in
$\mathbf{S}$.

###### Proof.

These equivalences are proved similarly to Theorem 6.16. ∎

##  7\. Group actions and involutions in symmetric monoidal categories

In symmetric monoidal categories there are examples of monoids that come
equipped with an involution and an action of a group $G$ which commute with
one another.

For example, if we take a commutative ring $k$ and a group $G$, the group
algebra, $k[G]$, comes equipped with an involution (given by the $k$-linear
extension of the group homomorphism $g\mapsto g^{-1}$) and an action of $G$
given by the $k$-linear extension of the conjugation action on $G$. The
following result shows that we can extend the techniques of this paper to
encode this structure using PROPs.

Let $\mathbf{S}$ be a symmetric monoidal category. Recall from [Gra20, 1.1]
that we denote the categories of involutive monoids, involutive comonoids and
involutive bimonoids in $\mathbf{S}$ by
$\mathbf{IMon}\left(\mathbf{S}\right)$,
$\mathbf{IComon}\left(\mathbf{S}\right)$ and
$\mathbf{IBimon}\left(\mathbf{S}\right)$ respectively.

######  Theorem 7.1.

Let $\mathbf{S}$ be a symmetric monoidal category. Let $G$ be a group. There
are equivalences of categories

  1. (1)

$\mathbf{Alg}\left(\mathbb{D}\otimes\mathbb{H}\otimes\mathbb{G},\mathbf{S}\right)\simeq\mathrm{Fun}\left(G,\mathbf{IMon}\left(\mathbf{S}\right)\right)$;

  2. (2)

$\mathbf{Alg}\left(\left(\mathbb{D}\otimes\mathbb{H}\otimes\mathbb{G}\right)^{op},\mathbf{S}\right)\simeq\mathrm{Fun}\left(G,\mathbf{IComon}\left(\mathbf{S}\right)\right)$
and

  3. (3)

$\mathbf{Alg}\left(\mathbb{D}\otimes(\mathbb{H}\otimes\mathbb{G})\otimes\mathbb{D}^{op},\mathbf{S}\right)\simeq\mathrm{Fun}\left(G,\mathbf{IBimon}\left(\mathbf{S}\right)\right)$.

###### Proof.

We define the necessary distributive law by replacing the labelled braids by
labelled hyperoctahedral group elements in Definition 5.2 and using the
crossed simplicial group structure in
$\mathbb{D}\otimes\mathbb{H}\otimes\mathbb{G}$. The proof then follows the
same argument of Theorem 5.3, using [Lac04, 4.4, 4.8] in place of [Gra22a,
5.9, 5.10]. ∎

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#  Analyzing sports commentary in order to automatically recognize events and
extract insights

Yanis Miraoui
ETH Zürich
<EMAIL_ADDRESS>
January, 2022

###### Abstract

In this paper, we carefully investigate how we can use multiple different
Natural Language Processing techniques and methods in order to automatically
recognize the main actions in sports events. We aim to extract insights by
analyzing live sport commentaries from different sources and by classifying
these major actions into different categories. We also study if sentiment
analysis could help detect these main actions.

##  1 Introduction

This research report aims to study how to recognize and analyze the main
actions in sports events.
Indeed, detecting important actions and extracting insights of sports events
has become of great interest in the recent years. Extracting these insightful
moments allows for example sports teams to analyze their performance
meticulously, media broadcasters to repurpose and redistribute their video
content easily and fans to get more engaged.
On the other hand, during most sports events, media broadcasters provide their
audience with very explanatory live commentaries through different means (TV,
radio, newspaper, website, …). These live commentaries have been particularly
well developed for Football (Soccer) : Almost every sports newspaper or _live
score website_ 111 E.g., livescore.com, flashscore.com, sofascore.com, … has
now a section dedicated to live football commentary.
Therefore, in this study, we intend to use these large amount of audio and
textual live commentaries to detect the main actions in sports events. This
would allow us to automatically extract the most insightful moments in sports
games.
The first step of our study is to build our dataset. In order to use audio
live commentary, we have to transcribe the sports commentary audio into
exploitable textual data. We would then be able to perform a thorough analysis
on textual content. Moreover, we also make use of textual live commentaries
from live score websites. Equipped with this textual data, we apply the main
preprocessing techniques learned in this course. We then train different
classification models able to detect and classify actions during a sport
event. Finally, we evaluate the metrics of these models on different types of
live sports commentaries and determine how this model could have a real impact
on the performance of an already existing event recognition AI. We also
explore if sentiment analysis could help recognize important actions during
some sports events.

Figure 1: Schema showing how an ideal simple example of audio live sports
commentary would be analyzed and processed.

##  2 Literature review and related work

Only a very few works and research have been done on this subject. This is not
very surprising as this topic emerged only recently and remains very niche and
specialized.
However, one research paper in particular, aims to study these live sports
commentaries with a similar method (Minard et al., 2016). Indeed, this study
mainly focus on trying to understand the content of live sports commentaries
by detecting and classifying relevant events in football games. This paper
mainly used newspapers live commentaries that they manually annotated.
Moreover, they used a SVM model to classify the events in different categories
and sub-categories. As this research paper has a very similar intended
objective, we will make use of its methods and its model as a baseline for our
study.
It is also possible to find several research papers that aim to study the
automatic generation of live sports commentaries for video games (Zheng and
Kudenko, 2010) or media broadcasters (Nijholt et al., 2003). Some research
papers study as well how to automatically generate sports news article by
summarizing the main actions of sports events using pre-existing datasets
(Huang et al., 2020) and (Zhang et al., 2016) or by analyzing Tweets (Nichols
et al., 2012). These studies remain relevant regarding the semantic analysis
they perform. Nevertheless, the methods and techniques used do not fit well
with our intended purpose.
On the other hand, we can notice that there has been an important interest
from the NLP community for event recognition and classification through the
organization of several hackathons:

  * •

Hackathon organized by Google in Paris in 2016,
http://hackatal.github.io/2016/

  * •

Hackathon organized by the United States Soccer Federation in 2015,
http://www.ussoccer.com/official-us-soccer-hackathon/

##  3 Background

This study is carried out in link with a company (_Egoli Media_ 222 _Egoli
Media_ is a proprietary AI-enabled video annotation technology which enables
real-time personalized delivery of video content. This AI aims to annotate
videos by detecting the main events and athletes in sports content. For more
information: egolimedia.com.) which kindly provided us an important amount of
data.
Indeed, the need of such a study arises as the company is currently trying to
improve its event recognition pipeline. One of the ways to reach that goal
would be to optimize the computer vision AI inspecting video content. However,
this improvement is very limited. On the other hand, we also have access to
the English sports commentary of most of the sports events. Within the
company, no work has already been done regarding how to make use of this large
amount of sports commentaries to improve the event recognition.
Moreover, from a wider viewpoint, many different live scores websites have
been recently developing and using these live sports commentaries to provide
their audience with a more detailed and accurate description of the main
actions of a sports event. These live commentaries aim to truly engage fans by
giving them more absorbing insights.

##  4 Data sources

###  4.1 Audio dataset

We obtained from the company _Egoli Media_ a rich audio dataset. This dataset
is extracted from real live sports commentary of the 2021 Paralympic games and
of a few English Premier League games. To transcribe the audio into text, we
used the Speech-to-Text API developed by Google (Chiu et al., 2018). This tool
aims to convert speech into text by using Google’s AI technologies. However,
when inspecting the transcription, the dataset seems to be very noisy. It
would be very laborious to exploit it, for example:

  * •

_“Like that shot that got put in on him by the three americans as a work in a
dormant volcano cause he’s beginning to bubble”_

  * •

_“Babies and kids the and some of the united states of america so do about it
goes as you could be easily”_

  * •

_“The first minute so gym roberts surveys the scene the usa team happy it’s a
cluster the key and make things awkward for gb but robots finds is running
part of our and fits”_

We notice that this transcription looks very inaccurate and unreliable.
Building a high-performing event recognition and classification model with
this data seems very difficult.

###  4.2 Textual dataset

Therefore, we decide to mostly use other sources of data for building and
training a model that would perform well.
Firstly, we build a complete dataset of clean text commentaries by scraping
live scores websites such as bbc.com, espn.com and onefootball.com. These live
commentaries are only extracted from football events as they are the most
enriched textual commentaries. These commentaries have the advantage to be as
well accompanied with a timeline of events that would represent the labels for
our training process. The meaning and definition of these labels can be found
in Appendix A.

League | Number of games
---|---
Serie A (Italy) | 2,152
Ligue 1 (France) | 2,076
La Liga (Spain) | 1,939
Bundesliga (Germany) | 1,608
Premier League (UK) | 1,299
TOTAL | 9074
Table 1:  Distribution of the number of games of our dataset in terms of the
different football leagues.

Moreover, we can also consider building the training dataset by scraping the
automatic subtitles add-on of YouTube. Indeed, it is also possible to obtain
the textual live commentaries from various sports events through the YouTube
API. Nonetheless, this method remains laborious as it would imply to manually
label a large amount of sentences for the training of our model. Therefore, we
only use this method to test and evaluate the accuracy of our model on a
different dataset.

##  5 Modeling techniques explored

###  5.1 Cleaning and preprocessing of the textual data

The dataset scraped from the live score websites contains $\approx$ 941,000
sentences labeled with one of the 12 categories (Attempt, Corner, Foul, Yellow
Card, …). These textual commentaries are extracted from 9074 real games that
occurred since 2011 in the five biggest European football leagues. An
essential step in the preprocessing of this dataset was to check that the
dataset was balanced enough to train a classification model on:

Figure 2: Bar plot showing the distribution of the different categories of the
dataset of live textual commentaries.

As we can notice on Figure 2, even if only a few commentaries belong to the
categories “Offside”, “Hand ball” and “Penalty conceded”, there is not one
category that holds more than 50% of the commentaries. However, we could still
consider re-balancing the dataset using oversampling (Chawla et al., 2002).
Furthermore, we apply some simple textual cleaning techniques:

  * •

Removing punctuation, special characters, leading, trailing and extra white
spaces/tabs

  * •

Stop-word removal

  * •

Stemming

  * •

Lemmatization

Statistics | Avg. words | Avg. characters
---|---|---
Live commentaries | 12.05 | 73.459
Table 2:  Statistics of our scraped textual dataset.

###  5.2 Modeling using tf-Idf

On one hand, for our word embedding, we build vectors from this cleaned
textual data using the Term Frequency-Inverse Document Frequencies (tf-Idf).
Using this method seems logical because in order to classify the textual live
commentaries, it is essential to encode the frequencies of the most important
term in each sentence with their relevancy (Liu et al., 2018). This is also
the method that gives us the best results in building an accurate
classification model. Equipped with these embedded sentences with tf-Idf, we
are now able to train, evaluate and compare different classification
algorithms.

###  5.3 Modeling using BERT

On the other hand, we also consider using BERT Transformers to model our text
classification task. Indeed, BERT is designed to help computers understand the
meaning of ambiguous language in text by using the bidirectional surrounding
text to establish context (Devlin et al., 2019). Therefore, BERT is a State of
the Art language model that distinguishes itself from previous language models
such as word2vec and GloVe, which are limited when interpreting context and
polysemous words. This is very pertinent to our task. To classify the live
commentaries efficiently, we would want our model to take into account the
surrounding context of each word. This is why we will explore and evaluate
this modeling technique more thoroughly.

##  6 Experimental findings and evaluation

###  6.1 Baseline : SVM model

The research paper (Minard et al., 2016) studies how an SVM model could be
trained in order to detect and classify the main events in newspapers
commentaries. As this study represents our baseline, we try to reproduce its
methods and techniques using our large live commentaries dataset.
Firstly, we randomly split the tf-Idf embedded vectors into a train and a test
subset using ratios of 80% / 20% respectively. We decide to shuffle the data
when splitting, as we want our model to be well-functioning on every type of
game and for every commentary made at anytime during the game.
We then train our SVM classifier model and evaluate its metrics on the test
set:

Metrics: | Accuracy | Precision | Recall | F1 Score
---|---|---|---|---
SVM | 97.30% | 88.64% | 83.60% | 0.85
Table 3:  Evaluation metrics of the SVM classifier model.

We also build the confusion matrix comparing the predictions made by the SVM
model on the test set and the true labels:

Figure 3: Confusion matrix of the evaluation of the SVM model.

At first sight, we can observe on Figure 3 that most of the errors concern
commentaries that are mislabeled as _"Foul"_. For example 26% of the true
labels _"Handball"_ are being mislabeled as _"Foul"_. However, these errors
can be caused by the fact that most handballs in football are considered to be
fouls as well. Hence, the clear distinction between the categories _"Foul"_
and _"Handball"_ could be quite ambiguous in football.
On the other hand, we also notice that our SVM is still very powerful and can
very accurately classify the main actions of various football games. The
baseline model built in the research paper achieved an F1 score of 0.71. In
comparison, our SVM model attains an F1 score of 0.85 (Table 3). This
important improvement could be explained by the fact that we have trained our
model on a much larger dataset.

###  6.2 Other classification models using tf-Idf

In order to extend our analysis, we build various other classification models
using the same embedded vectors with tf-Idf (Shaikh, 2017). We also use the
same train/test splitting methods for training and evaluating the different
classifiers.
To evaluate and compare these different classification models, we use the
accuracy metric ($accuracy=\frac{\text{Number of correct
predictions}}{\text{Total number of predictions}}$). For our task, we are
willing to get the most right predictions for the commentary of each game.
Therefore, we build and train the different classifications models. Firstly,
we compare the accuracy of each model on the same type of live sports
commentaries:

Figure 4: Histogram comparing the accuracy of different classification models.

On the other hand, we want our classification model to be able to generalize
well and to perform well on different types of live commentaries structures.
For example, we would like our model to perform effectively on commentaries
extracted from other sources like Youtube or livescore.com. Therefore, we also
compare the accuracy of each model on new these types of live sports
commentaries which have different semantic structures and syntactic
expressions:

Figure 5: Histogram comparing the accuracy of different classification models
on the new dataset scraped from livescore.com and Youtube.

We deduce from this comparison that most classification models generalize
relatively well to new types of sports commentaries (Figure 5). Moreover, the
XGBoost model may represent the best performing model for our task by
accurately predicting the label of most of the events and from different types
of live commentaries structures.

###  6.3 BERT Classification model

In addition, to model our classification task, we study how implementing a
BERT classification model could improve our results. Indeed, we expect that
the use of BERT Transformers would confer us a better accuracy as this
modeling technique is taking into account the surrounding context of each
word. Therefore, we train on _Google Collab_ 333Google Collab provide us
powerful CPU computation resources, essential for training a BERT
classification model. our model with our dataset using the existing
transformer pretrained on a large corpus of texts (_’bert-base-uncased’_). The
dataset which we use to fine-tune the BERT model is similar to the one used
previously. However, we do not proceed to the tf-Idf vectorizing step in this
case.
We evaluate our classification model on unseen live sports commentaries and
obtain a very high accuracy of 99.8%. This new model is also evaluated on new
types of sentences structures using the dataset scraped from livescore.com and
Youtube. It attains a also very high accuracy of 92% in average.

Figure 6: Confusion matrix of the evaluation of the fine-tuned BERT model.

Hence, modeling our classification task using a fine-tuned BERT transformer
seems to be a very high-performing and powerful solution. This modeling
technique, by accurately grasping the context of each word, seems to
significantly improve our results.

###  6.4 Sentiment Analysis for Event Recognition

Another less obvious alternative to detect the main actions during sports
events is to perform a sentiment analysis of the live commentaries. In fact,
we could think that scoring a goal (positive) does not convey the same
emotions as conceding a red card (negative). This is why, we make use of
_’cardiffnlp/twitter-roberta-base-sentiment’_ model to detect if some event
types convey more positive or negative emotions when being commented. If this
is the case, this would confirm our hypothesis that sentiment analysis of live
commentaries could help detect and classify the main actions in many sports
events.
Therefore, we predict the sentiment conveyed by each sentence of a sample of
our live sports commentaries ($\approx$10,000 sentences) using the pretrained
model _’cardiffnlp/twitter-roberta-base-sentiment’_. This model is a roBERTa-
base model trained on $\approx$58M tweets and finetuned for sentiment analysis
(Barbieri et al., 2020). This pretrained model seems to be the best performing
model for our task as there does not exist any pretrained sentiment analysis
model specifically for sports. Moreover, this model is known for being very
powerful for many general tasks444Note that we could have built our own
sentiment analysis model for sports. However, building this model would have
been very time consuming as we would have to label manually an important
amount of data.. We obtain the following results by disposing of the "Neutral"
labels and reporting only the predominant "Positive" or "Negative" label:

Event type | Sentiment
---|---
Attempt | Negative (6.04%)
Corner | Positive (2.59%)
Foul | Negative (18.77%)
Yellow card | Negative (74.01%)
Second yellow card | Negative (38.32%)
Red card | Negative (70%)
Substitution | Positive (1.37%)
Free kick won | Positive (0.49%)
Offside | Positive (2.97%)
Handball | None
Penalty conceded | Negative (100%)
Table 4:  Results of the sentiment analysis peformed on the different event
types.

We first observe from Table 4 that the results obtained seem quite convincing.
Indeed, a "Foul", a "Yellow card" or a "Red card" for example are events that
convey a negative sentiment when commented.
On the other hand, we can also notice that most of the "Positive" or
"Negative" labels have a very small percentage of appearance. This means that
most live commentaries are predicted as being "Neutral". If most of the live
sports commentaries are predicted as being "Neutral", the information gained
by this sentiment analysis becomes very limited. Perhaps, building a
specialized sentiment analysis model for sports could help improve our results
and the relevancy of performing a sentiment analysis.

##  7 Conclusion

By carrying out this study, we analyzed how different Natural Language
Processing techniques and methods could be used in order to automatically
recognize events and extract insights during sports events. We mainly worked
with live textual commentaries to build different classification models. In
particular, we explored mainly two different techniques in order to embed the
textual commentaries into numerical vectors.
Using the tf-Idf embeddings, we built an SVM model following the same method
as in the baseline (Minard et al., 2016). This model obtained an F1 score of
0.85, outperforming the baseline model. Moreover, we built and compared
different classification models before deciding on the most powerful one: the
XGBoost model.
On the other hand, we studied two different approaches using the BERT
Transformers. The first one aimed to label the live commentaries by using the
bidirectional surrounding text to establish context. The fine-tuned
classification model outperformed all the models previously built, becoming
therefore the best model to solve our task with an accuracy of 92% in average.
The second approach was to determine if sentiment analysis could help detect
the main actions in sports events. However, this analysis is much more
nuanced: the information gained by a sentiment analysis of the live
commentaries could be very limited.

## Acknowledgements

We especially thank _Egoli Media_ for providing us with an important amount of
data and for their feedback.

## References

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##  Appendix A Appendix

Dictionary of the event categories and their corresponding labels :

Labels | Type of event
---|---
0 | No event
1 | Attempt
2 | Corner
3 | Foul
4 | Yellow card
5 | Second yellow card
6 | Red card
7 | Substitution
8 | Free kick won
9 | Offside
10 | Handball
11 | Penalty conceded
Table 5:  Table of the different classes of our classification model and their
labels.

##  Appendix B Appendix

You can find here the code that has been done during the elaboration of this
study. This code has been cleaned and annotated with comments explaining the
methods and techniques used. Please read the README.md file for more
information.

##  Appendix C Appendix

Moreover, you can also find here a web application simulating some live
commentary examples and their predictions based on the chosen model (SVM,
XGBoost, …). Please note: username is "admin" and password is "admin". The
code of this web application is available here and has been deployed and
hosted on a standard virtual machine.