# Hopf Monads: A Survey with New Examples and Applications

Aryan Ghobadi
Queen Mary University of London
School of Mathematics, Mile End Road
London E1 4NS, UK
Email<EMAIL_ADDRESS>
###### Abstract

We survey the theory of Hopf monads on monoidal categories, and present new
examples and applications. As applications, we utilise this machinery to
present a new theory of cross products, as well as analogues of the
Fundamental Theorem of Hopf algebras and Radford’s biproduct Theorem for Hopf
algebroids. Additionally, we describe new examples of Hopf monads which arise
from Galois and Ore extensions of bialgebras. We also classify Lawvere
theories whose corresponding monads on the category of sets and functions
become Hopf, as well as Hopf monads on the poset of natural numbers.

2020 Mathematics Subject Classification: Primary 16T99, 18C15, 18M05;
Secondary: 03G30, 18D15
Keywords: monads, bimonads, Hopf algebras, monoidal categories, braided
categories, Hopf algebroid, bialgebroid

###### Contents

  1. 1 Introduction
  2. 2 Categorical Background
    1. 2.1 Monoidal Categories
    2. 2.2 Braided Monoidal Categories and Hopf Algebras
    3. 2.3 Dual of a Monoidal Functor and Center
    4. 2.4 Key Examples of Monoidal Categories
    5. 2.5 Monads
    6. 2.6 Ends and Coends
  3. 3 Hopf and Bimonads
    1. 3.1 Bimonads and Comonoidal Adjunctions
    2. 3.2 Hopf Monads on Rigid Monoidal Categories
    3. 3.3 Hopf Monads on General Monoidal Categories
    4. 3.4 Hopf Monads on Closed Monoidal Categories
  4. 4 Examples
    1. 4.1 Central Hopf Algebras as Hopf monads
    2. 4.2 Hopf Monads on $\mathbb{N}_{0}$
    3. 4.3 Hopf Monads on $\mathsf{Set}$
    4. 4.4 Galois and Ore Extensions of Bialgebras
    5. 4.5 Hopf Monads from Pivotal Pairs
    6. 4.6 Bialgebroids and Hopf Algebroids
  5. 5 Combining Hopf monads and Bimonads
    1. 5.1 Bosonisation and Cross Products
    2. 5.2 Distributive Laws
  6. 6 Central Coalgebras and Hopf Monads
    1. 6.1 From Central Coalgebras to Hopf Adjunctions
    2. 6.2 Induced Central Coalgebra of a Hopf Adjunction
    3. 6.3 Equivalence of Hopf Monads and Central Coalgebras
  7. 7 Classical Hopf-algebraic results for Hopf monads
    1. 7.1 Radford’s Biproduct Theorem
    2. 7.2 Hopf Modules
    3. 7.3 Quasitriangular Structures
    4. 7.4 Tannaka Reconstruction for Hopf monads
  8. 8 Other aspects of Hopf monads
  9. A Appendix: Hopf Adjunctions in Topos Theory

##  1 Introduction

Hopf monads were originally introduced to generalise the lifting properties of
ordinary Hopf algebras in the category $\mathsf{Vec}$ of vector spaces, to
arbitrary monoidal categories. In [51], Hopf monads were defined as monads
which lift the monoidal structure of a monoidal category to their category of
modules (or Eilenberg-Moore categories). These monads are now referred to as
_bimonads_ or _comonoidal monads_ , whereas monads which also lift the closed
structure (inner-homs) of a closed monoidal category are called Hopf monads
[12]. While the purpose of the latter definition was rooted in lifting the
closed structure, the ‘Hopf condition’ presented in [12] makes sense for
bimonads on arbitrary monoidal categories. Additionally, this theory has
garnered interest because many of the various categorical generalisations of
Hopf algebras including Hopf algebroids, Hopf Polyads and Hopf categories,
among others [8, 9] can all be viewed within this framework. Hence, any
results proved at the level of Hopf monads can have fruitful applications for
all these other structures.

Initially, Hopf monads were defined for rigid monoidal categories [14] and
various Hopf algebraic results were extended to this setting and applied in
the study of tensor categories [13] and topological field theories [64]. This
new point of view has also produced novel results for its simplest family of
examples, namely braided Hopf algebras in rigid monoidal categories, such as
the notions of Drinfeld double and quasitriangular structures [15, 16]. While
some of the classical Hopf algebraic results, including the Fundamental
Theorem of Hopf algebras and Radford’s biproduct Theorem, were also
generalised to the setting of Hopf monads on arbitrary monoidal categories
[12], other results such as the theory of integrals have not received the same
treatment. This is partly due to the lack of explicit examples when the
category is not rigid. In the same vein, even some of the results which have
been generalised to this setting have not been translated into the language of
interesting examples of Hopf monads, such as Hopf algebroids.

Here, we present a survey-style review of the theory of Hopf monads and
bimonads with a particular focus on constructing new examples of these
objects. While collecting the various results in the theory of Hopf monads,
which are spread between [14, 12, 16, 60] among other texts, we also compare
them with their classical analogues in the theory of ordinary Hopf algebras
and apply some of them in the setting of Hopf algebroids. We also aim to
provide simpler sketches of the proofs of these results, which provide the
reader with a better picture of the proof and avoid unnecessary details.

The book [9] along with several surveys including [65] formulate numerous
generalisations of Hopf algebraic structures as examples of Hopf monads, but
do not review the various results on Hopf monads themselves. In addition to
providing a survey of these results, we take a different approach to examples.
Throughout Section 4, we pick several base monoidal categories and then
construct examples of Hopf monads on these bases, which do not already appear
as Hopf algebra-like structures in the literature. The reader can also refer
to Chapter II of [64] as it contains a review of the theory of Hopf monads on
rigid monoidal categories. In this survey however, we primarily focus on
results and examples in the non-rigid setting.

Organisation: In Section 2, we review the basic categorical machinery which we
will use and review some key features of some well-known monoidal categories.
In Section 3, we recall the notion of bimonads and the various definitions of
Hopf monads and review their fundamental properties. In Section 5, we review
the different ways in which we can combine these structures to obtain new Hopf
monads, while in Section 7, we review some well-known results in the theory of
Hopf algebras which have been generalised to the setting of Hopf monads. In
Section 6 we recall the correspondence between Hopf monads and cocommutative
central coalgebras. The novel portions of our work mainly appear in Section 4
where we present several new examples of Hopf monads by looking at poset
categories, algebraic theories and Galois extensions of bialgebras. The
various new applications of Hopf monads to the theory of Hopf algebroids are
spread within Sections 5, 6 and 7 as examples. Finally, we mention some
aspects of Hopf monads which we have not discussed here in Section 8 and
present an account of where Hopf adjunctions appear in topos theory in
Appendix A.

Novel Constructions: Our simplest family of examples appear in Section 4.2,
where we classify Hopf monads on the poset $(\mathbb{N}_{0},\leq)$ which is
viewed as a category with its monoidal structure given by $+$. We show that
bimonads on this category correspond to infinite submonoids of
$(\mathbb{N}_{0},+)$, while Hopf monads on this category correspond to
submonoids which are generated by a single positive number.

Monads on $\mathsf{Set}$ are said to be generalisation of algebraic theories,
in particular, because finitary monads on $\mathsf{Set}$ correspond exactly to
algebraic or Lawvere theories. In Section 4.3, we show that any finitary Hopf
monad must correspond to the theory of $G$-sets for a group $G$ (a Hopf
algebra in $\mathsf{Set}$).

In Section 4.4, we consider extensions of bialgebras $f:B\rightarrow H$ and
investigate when the induced adjunction given by restriction and extension of
scalars ${}_{H}{\mathcal{M}}\leftrightarrows{}_{B}{\mathcal{M}}$ gives rise to
a Hopf monad. We show that the (left) _pre-Hopf_ condition presented in [12]
corresponds to $H$ being a Galois extension [59] of $B$ over a coalgebra
determined by $f$. Furthermore, we present a generalised Galois condition (57)
for $f$ which determines when the adjunction is (left) Hopf. We then show that
any suitable Ore extension of bialgebras satisfies this condition.

In [27], we presented a construction for Hopf monads corresponding to pivotal
pairs $(P,Q)$ in suitable closed monoidal categories. In Section 4.5, we
review this construction and show that we still obtain a Hopf monad under more
general assumptions where the base category is not necessarily closed, Theorem
4.15. We already noted in [27], that this construction can be viewed as an
example of Shimizu’s Tannaka-Krein reconstruction for Hopf monads [60]. In
Section 7.4, we review the latter theory and use our example in Section 4.5 to
clarify different aspects of this complicated machinery.

In Section 4.6, we review how Hopf algebroids and bialgebroids over a base
algebra $A$ can be viewed as Hopf monads and bimonads over the category of
$A$-bimodules. While this point of view is well-known, various useful results
for Hopf monads have not been translated into the language of Hopf algebroids.
For instance, in Example 5.3 we translate the theory of cross products of Hopf
monads to the setting of Hopf algebroids and present an analogous construction
to the cross product of Hopf algebras in this setting. We present three more
such results in Examples 6.5, 7.6 and 7.4, which include the construction of a
lax braiding on the induced coalgebra of Hopf algebroids and analogues of the
Fundamental Theorem of Hopf algebras and Radford’s biproduct Theorem for these
structures.

Acknowledgements: The author would like to thank Shahn Majid for numerous
helpful conversations on all things Hopf-related. We would also like to show
gratitude to Ivan Tomas̆ić for useful discussions on Section 4.3 and Appendix
A, as well as Shu Sasaki and Tomasz Brzeziński for their examination of the
author’s dissertation [28], which contained a large portion of this work in
its first two chapters. The author is also grateful to the EPSRC for grant
number EP/W522508/1 which supported part of this work.

##  2 Categorical Background

In this section we recall some of the categorical concepts which we will use
in future chapters and set our notation. Our main references for basic
category theory will be [39, 54].

###  2.1 Monoidal Categories

In this section we briefly recall the theory of monoidal categories and set
our notation. We refer the reader to Section 1 of [64] for additional details.

Given a category $\mathcal{C}$, a quadruple $(\otimes,\mathtt{1},\alpha,l,r)$
is called a monoidal structure on $\mathcal{C}$, where $\mathtt{1}$ is an
object in $\mathcal{C}$,
$\otimes:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ a bifunctor and
$\alpha:(\mathrm{id}_{\mathcal{C}}\otimes\mathrm{id}_{\mathcal{C}})\otimes\mathrm{id}_{\mathcal{C}}\rightarrow\mathrm{id}_{\mathcal{C}}\otimes(\mathrm{id}_{\mathcal{C}}\otimes\mathrm{id}_{\mathcal{C}})$,
$l:\mathtt{1}\otimes\mathrm{id}_{\mathcal{C}}\rightarrow\mathrm{id}_{\mathcal{C}}$
and
$r:\mathrm{id}_{\mathcal{C}}\otimes\mathtt{1}\rightarrow\mathrm{id}_{\mathcal{C}}$
natural isomorphisms satisfying coherence axioms as presented in Section 1.2.1
of [64]. Given such a structure $(\mathcal{C},\otimes,\mathtt{1})$ is referred
to as a _monoidal category_ and $\mathtt{1}$ is called its _monoidal unit_.
The monoidal structure is said to be _strict_ if $\alpha,l$ and $r$ are all
identity morphisms. Given any monoidal category
$(\mathcal{C},\otimes,\mathtt{1})$ we obtain another monoidal category
$(\mathcal{C},\otimes^{\mathrm{op}},\mathtt{1})$ where the bifunctor
$\otimes^{\mathrm{op}}$ is defined by $X\otimes^{\mathrm{op}}Y=Y\otimes X$.
This construction is distinct to the opposite category
$\mathcal{C}^{\mathrm{op}}$ which has the same objects as $\mathcal{C}$ but
morphisms in reversed directions.

Throughout this work, every monoidal category will either be strict or the
coherence isomorphisms $\alpha,l,r$, while not the identity morphisms, will be
trivial e.g. in the case of the category of sets, $\mathsf{Set}$, the function
$\alpha_{X,Y,Z}:(X\times Y)\times Z\rightarrow X\times(Y\times Z)$ sends every
element $((x,y),z)$ to $(x,(y,z))$. Hence, the effect of these natural
isomorphisms will be negligible and we will not discuss them further.

A functor $F:\mathcal{C}\rightarrow\mathcal{D}$ between monoidal categories
$(\mathcal{C},\otimes_{\mathcal{C}},\mathtt{1}_{\mathcal{C}})$ and
$(\mathcal{D},\otimes_{\mathcal{D}},\mathtt{1}_{\mathcal{D}})$ is said to be
_(strong) monoidal_ if there exist a pair $(F_{2},F_{0})$ where
$F_{2}(-,-):F(-)\otimes_{\mathcal{D}}F(-)\rightarrow
F(-\otimes_{\mathcal{C}}-)$ is a natural (isomorphism) transformation and
$F_{0}:\mathtt{1}_{\mathcal{D}}\rightarrow F(\mathtt{1}_{\mathcal{C}})$ a
(isomorphism) morphism satisfying

$\displaystyle
F_{2}(X\otimes_{\mathcal{C}}Y,Z)(F_{2}(X,Y)\otimes_{\mathcal{D}}\mathrm{id}_{F(Z)})=$
$\displaystyle
F_{2}(X,Y\otimes_{\mathcal{C}}Z)(\mathrm{id}_{F(X)}\otimes_{\mathcal{D}}F_{2}(Y,Z))$
(1) $\displaystyle
F_{2}(X,\mathtt{1}_{\mathcal{C}})(\mathrm{id}_{F(X)}\otimes_{\mathcal{D}}F_{0})=\mathrm{id}_{F(X)}$
$\displaystyle=F_{2}(\mathtt{1}_{\mathcal{C}},X)(F_{0}\otimes_{\mathcal{D}}\mathrm{id}_{F(X)})$
(2)

for any three objects $X,Y,Z$ in $\mathcal{C}$. The functor $F$ is said to be
_(strong) comonoidal_ if there exist a pair $(F_{2},F_{0})$ with arrows going
in the opposite direction. From here onwards, we will omit the subscript
denoting the ambient category e.g. replace $\otimes_{\mathcal{C}}$ and
$\otimes_{\mathcal{D}}$ by $\otimes$, since the choice of monoidal structure
will be clear from context. Note that a strong monoidal structure
$(F_{2},F_{0})$ on a functor $F$ is equivalent to a strong comonoidal
structure $(F_{0}^{-1},F_{2}^{-1})$ on $F$. A strong monoidal functor with
$F_{2}=\mathrm{id}_{-\otimes-}$ and $F_{0}=\mathrm{id}_{\mathtt{1}}$ is called
_strict monoidal_. The reader should also note that in many sources the term
monoidal functor refers to a strong monoidal functor, but we choose to match
the terminology used in the main Hopf monad literature [14, 12].

The composition $GF$ of two functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and
$G:\mathcal{D}\rightarrow\mathcal{E}$ with (co)monoidal structures
$(F_{2},F_{0})$ and $(G_{2},G_{0})$ obtains a natural (co)monoidal structure
by $(GF_{2})G_{2}(F,F)$ and $(GF_{0})G_{0}$ (resp. $G_{2}(F,F)GF_{2}$ and
$G_{0}(GF_{0})$). A natural transformation $\theta:F\Rightarrow G$ between two
(co)monoidal functors $(F,F_{2},F_{0})$ and $(G,G_{2},G_{0})$ is said to be
_(co)monoidal_ if $\theta F_{2}=G_{2}(\theta\otimes\theta)$ and
$\theta_{\mathtt{1}}F_{0}=G_{0}$ (resp.
$(\theta\otimes\theta)F_{2}=G_{2}\theta$ and $G_{0}\theta=F_{0}$) hold.

Given an object $X$ in a monoidal category $(\mathcal{C},\otimes,\mathtt{1})$,
we say an object ${}^{\vee}{X}$ is a _left dual_ of $X$, if there exist
morphisms $\mathrm{ev}_{X}:{}^{\vee}{X}\otimes X\rightarrow\mathtt{1}$ and
$\mathrm{coev}_{X}:\mathtt{1}\rightarrow X\otimes{}^{\vee}{X}$ such that

$(\mathrm{ev}_{X}\otimes\mathrm{id}_{{}^{\vee}{X}})(\mathrm{id}_{{}^{\vee}{X}}\otimes\mathrm{coev}_{X})=\mathrm{id}_{{}^{\vee}{X}},\quad(\mathrm{id}_{X}\otimes\mathrm{ev}_{X})(\mathrm{coev}_{X}\otimes\mathrm{id}_{X})=\mathrm{id}_{X}$

In such a case, we call $X$ a _right dual_ for ${}^{\vee}{X}$. Furthermore, a
right dual of an object $X$ is denoted by $X^{\vee}$, with evaluation and
coevaluation maps denoted by $\underline{\mathrm{ev}}_{X}:X\otimes
X^{\vee}\rightarrow\mathtt{1}$ and
$\underline{\mathrm{coev}}_{X}:\mathtt{1}\rightarrow X^{\vee}\otimes X$,
respectively. We will refer to evaluation and coevaluation maps as such, as
_duality morphisms_. We say an object $X$ is _dualizable_ if it has both a
left dual and a right dual. Note that if a left (or right) dual object to $X$
exists, then it is unique upto isomorphisms. Let $Y$ and $Z$ both be left dual
objects to $X$, with duality morphisms $(\mathrm{ev},\mathrm{coev})$ and
$(\mathrm{ev}^{\prime},\mathrm{coev}^{\prime})$, then
$(\mathrm{ev}\otimes\mathrm{id}_{Z})(\mathrm{id}_{Y}\otimes\mathrm{coev}^{\prime})$
and
$(\mathrm{ev}^{\prime}\otimes\mathrm{id}_{Y})(\mathrm{id}_{Z}\otimes\mathrm{coev})$
form an isomorphism between $Y$ and $Z$.

The monoidal category $\mathcal{C}$ is said to be left (right) _rigid_ or
_autonomous_ if all objects have left (right) duals. If a category is both
left and right rigid, we simply call it _rigid_. In the literature, when a
category is said to be left (or right rigid), it is assumed that we have
_chosen_ a left dual and a pair of duality morphisms for every object $X$ and
${}^{\vee}{X}$ denotes this specific choice of left dual. Given these choices,
we have a contravariant functor
${}^{\vee}{(-)}:\mathcal{C}\rightarrow\mathcal{C}$ which sends objects $X$ to
their left duals ${}^{\vee}{X}$ and morphisms $f:X\rightarrow Y$ to morphisms
$(\mathrm{ev}_{Y}\otimes\mathrm{id}_{{}^{\vee}{X}})(\mathrm{id}_{{}^{\vee}{Y}}\otimes
f\otimes\mathrm{id}_{{}^{\vee}{X}})(\mathrm{id}_{{}^{\vee}{Y}}\otimes\mathrm{coev}_{X})$.
Similarly, $(-)^{\vee}:\mathcal{C}\rightarrow\mathcal{C}$ defines a
contravariant functor on a right rigid category.

We call a monoidal category $\mathcal{C}$ _left (right) closed_ if for every
object $X$ there exists an endofunctor $[X,-]^{l}$ (resp. $[X,-]^{r}$) on
$\mathcal{C}$ which is right adjoint to $-\otimes X$ (resp. $X\otimes-$). We
will denote the unit and counit of these adjunctions by
$\mathrm{coev}^{-}_{X}$ (resp. $\underline{\mathrm{coev}}^{-}_{X}$) and
$\mathrm{ev}^{-}_{X}$ (resp. $\underline{\mathrm{ev}}^{-}_{X}$). By definition
$[-,-]^{l},[-,-]^{r}:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathcal{C}$
become bifunctors and we refer to them as _inner-homs_. If a category is left
and right closed we call it _closed_. Observe that if $X$ has a left (right)
dual ${}^{\vee}{X}$ (resp. $X^{\vee}$), the functor $-\otimes{}^{\vee}{X}$
(resp. $X^{\vee}\otimes-$) becomes right adjoint to $-\otimes X$ (resp.
$X\otimes-$) and therefore every left (right) rigid category is left (right)
closed. Furthermore, if $X$ has a left (right) dual,
${}^{\vee}{X}\cong[X,\mathtt{1}]^{l}$ (resp.
$X^{\vee}\cong[X,\mathtt{1}]^{r}$). We have adopted the notation of [12] here,
and what we refer to as a left closed structure is referred to as a right
closed structure in some of our other references such as [56].

It is well-known that strong monoidal functors preserve dual objects i.e.
$F({}^{\vee}{X})\cong{}^{\vee}{F(X)}$ since
$F_{0}^{-1}F(\mathrm{ev})F_{2}({}^{\vee}{X},X)$ and
$F_{2}^{-1}(X,{}^{\vee}{X})F(\mathrm{coev})F_{0}$ act as the evaluation and
coevaluation morphisms making $F({}^{\vee}{X})$ a left dual object to $F(X)$.

Given a monoidal functor $F:\mathcal{C}\rightarrow\mathcal{D}$ between a pair
of left closed monoidal categories, $\mathcal{C}$ and $\mathcal{D}$, we obtain
a canonical family of morphisms
$\Phi^{l}_{X,Y}:=F(\mathrm{ev}_{X}^{Y})F_{2}([X,Y]^{l}_{\mathcal{C}},X):F[X,Y]_{\mathcal{C}}^{l}\otimes_{\mathcal{D}}FX\rightarrow
FY$ for every pair of objects $X,Y$ in $\mathcal{C}$. Consequently we obtain a
family of morphisms

$F_{X,Y}^{l}:=[F(X),\Phi^{l}_{X,Y}]^{l}_{\mathcal{D}}\mathrm{coev}_{F(X)}^{F[X,Y]^{l}_{\mathcal{C}}}:F[X,Y]^{l}_{\mathcal{C}}\rightarrow[F(X),F(Y)]^{l}_{\mathcal{D}}$

as the corresponding morphism to $\Phi^{l}_{X,Y}$ under the adjunction
$-\otimes_{\mathcal{D}}F(X)\dashv[F(X),-]^{l}_{\mathcal{D}}$. We say that the
monoidal functor $F$ is _left closed_ if the morphisms $F^{l}_{X,Y}$ are
isomorphisms for all pair of objects $X,Y$ in $\mathcal{C}$. A symmetric
definition can be made for _right closed_ functors. We refer the reader to
Section 3.2 of [12] for additional details.

###  2.2 Braided Monoidal Categories and Hopf Algebras

In this section we briefly recall the definitions of braided monoidal
categories and braided bialgebras and Hopf algebras in them, which appeared in
the work of Majid [41] under the name of braided groups. We refer the reader
to Section II.6 of [64] for additional details and a modern treatment.

A monoidal category is said to be (lax) _braided_ if there exists a natural
isomorphism (natural transformation) $\Psi_{X,Y}:X\otimes Y\rightarrow
Y\otimes X$ satisfying what we refer to as the braiding axioms:

$\displaystyle\Phi_{X,Y\otimes
Z}=(\mathrm{id}_{Y}\otimes\Phi_{X,Z})(\Phi_{X,Y}\otimes\mathrm{id}_{Z}),\quad\Phi_{X\otimes
Y,Z}=(\Phi_{X,Z}\otimes\mathrm{id}_{Y})(\mathrm{id}_{X}\otimes\Phi_{Y,Z})$ (3)

A braided monoidal category is said to be symmetric if
$\Phi_{Y,X}\Phi_{X,Y}=\mathrm{id}_{X\otimes Y}$ for every pair of objects
$X,Y$ in $\mathcal{C}$.

Note that in a braided monoidal category, an object is left dualisable if and
only if it is right dualisable. If ${}^{\vee}{X}$ denotes the left dual of $X$
in $\mathcal{C}$, with duality morphisms $\mathrm{coev}$ and $\mathrm{ev}$
then $\mathrm{ev}\Psi_{X,{}^{\vee}{X}}$ and
$\Psi^{-1}_{X,{}^{\vee}{X}}\mathrm{coev}$ make ${}^{\vee}{X}$ a right dual of
$X$. The converse argument also holds, making $X^{\vee}$ a left dual of $X$.
More generally, $\Psi_{-,X}:-\otimes X\rightarrow X\otimes-$ is an isomorphism
of functors and thereby a braided monoidal category is left closed if and only
if its is right closed.

An _algebra_ or _monoid_ in a monoidal category $\mathcal{C}$ consists of a
triple $(M,\mu,\eta)$, where $M$ is an object of $\mathcal{C}$ and
$\mu:M\otimes M\rightarrow M$ and $\eta:1\rightarrow M$ are morphisms in
$\mathcal{C}$ which satisfy
$\mu(\mathrm{id}_{M}\otimes\eta)=\mathrm{id}_{M}=\mu(\eta\otimes\mathrm{id}_{M})$
and $\mu(\mathrm{id}_{M}\otimes\mu)=\mu(\mu\otimes\mathrm{id}_{M})$. A
_coalgebra_ or _comonoid_ in $\mathcal{C}$ can be defined by simply reversing
the arrows in the definition of a monoid.

A _braided bialgebra_ in a braided monoidal category $(\mathcal{C},\Psi)$
consists of an object $B$ in $\mathcal{C}$, and quadruple of morphisms
$(m,\eta,\Delta,\epsilon)$, such that $(B,m,\eta)$ form a monoid in
$\mathcal{C}$, $(B,\Delta,\epsilon)$ form a comonoid in $\mathcal{C}$ and the
following axioms hold

$\displaystyle(m\otimes
m)(\mathrm{id}_{B}\otimes\Psi_{B,B}\otimes\mathrm{id}_{B})(\Delta\otimes\Delta)=\Delta$
$\displaystyle m$ (4) $\displaystyle\Delta\eta=\eta\otimes\eta,\quad\epsilon
m=(\epsilon\otimes\epsilon),\quad\epsilon\eta=\mathrm{id}_{\mathtt{1}}$ (5)

A braided bialgebra is called a _braided Hopf algebra_ if there exists a
morphism $S:B\rightarrow B$ called its _antipode_ such that
$m(\mathrm{id}_{B}\otimes
S)\Delta=\eta\epsilon=m(S\otimes\mathrm{id}_{B})\Delta$. If
$(\mathcal{C},\Psi)$ is a symmetric category then we simply omit the “braided”
prefix when referring to bialgebras and Hopf algebras in $\mathcal{C}$.

Given any monoidal category $\mathcal{C}$, we can define the category of
monoids in $\mathcal{C}$ which has monoids $(M,m,\eta)$ in $\mathcal{C}$ as
objects and morphisms
$f:(M,m,\eta)\rightarrow(M^{\prime},m^{\prime},\eta^{\prime})$ are morphisms
$f:M\rightarrow M^{\prime}$ in $\mathcal{C}$ which commute with the structural
morphisms i.e. satisfy $m^{\prime}(f\otimes f)=fm$ and $f\eta=\eta^{\prime}$.
If $\mathcal{C}$ is endowed with a braiding $\Psi$, then the category of
monoids in $\mathcal{C}$ obtains a monoidal structure defined by

$(M,m,\eta)\otimes(M^{\prime},m^{\prime},\eta^{\prime})=\big{(}M\otimes_{\mathcal{C}}M^{\prime},(m\otimes
m^{\prime})(\mathrm{id}_{M}\otimes\Psi_{M,M^{\prime}}\otimes\mathrm{id}_{M^{\prime}}),\eta\otimes\eta^{\prime}\big{)}$

We can define the category of comonoids in $\mathcal{C}$ and its monoidal
structure, when $\mathcal{C}$ is braided, in a symmetric way. From this point
of view, picking out a bialgebra in $\mathcal{C}$ is equivalent to picking a
comonoid (monoid) in the monoidal category of monoids (comonoids) in
$\mathcal{C}$.

###  2.3 Dual of a Monoidal Functor and Center

In this section we recall the definition of the dual of a monoidal functor and
the center of a monoidal category with [40] as our main reference. We also
present some basic results regarding the dual construction which do not appear
together elsewhere.

If $U:\mathcal{D}\rightarrow\mathcal{C}$ is a strong monoidal functor between
monoidal categories, the _dual_ of the functor $U$ is defined as the category
whose objects are pairs $(X,\tau)$ with $X$ being an object of $\mathcal{C}$
and $\tau:X\otimes U(-)\rightarrow U(-)\otimes X$ a natural isomorphism
satisfying

$\displaystyle(\mathrm{id}_{U(M)}\otimes\tau_{N})(\tau_{M}\otimes\mathrm{id}_{U(N)})=$
$\displaystyle(U_{2}(M,N)^{-1}\otimes\mathrm{id}_{X})\tau_{M\otimes
N}(\mathrm{id}_{X}\otimes U_{2}(M,N))$ (6) $\displaystyle(U_{0}^{-1}\otimes$
$\displaystyle\mathrm{id}_{X})\tau_{\mathtt{1}}(\mathrm{id}_{X}\otimes
U_{0})=\mathrm{id}_{X}$ (7)

with morphisms $f:(X,\tau)\rightarrow(X^{\prime},\tau^{\prime})$ being
morphisms $f:X\rightarrow X^{\prime}$ in $\mathcal{C}$ which satisfy
$\tau^{\prime}(f\otimes\mathrm{id}_{\mathcal{C}})=(\mathrm{id}_{\mathcal{C}}\otimes
f)\tau$. We denote the dual by
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}$. The dual of the
identity functor
$\mathrm{id}_{\mathcal{C}}:\mathcal{C}\rightarrow\mathcal{C}$, is called the
_center_ of $\mathcal{C}$ and denoted by $Z(\mathcal{C})$. This construction
is often referred to as the Drinfeld-Majid center.

The _lax left_ dual of $U$, denoted by
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$ is
defined exactly as $\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}$
but where the natural transformations $\tau$ are not assumed to be
isomorphisms. In a symmetric manner, we can define the _lax right_ dual of
$U$, whose objects are pairs $(X,\tau)$ where $\tau:U(-)\otimes X\rightarrow
X\otimes U(-)$ is a natural transformation. The lax left (right) dual of the
identity functor on $\mathcal{C}$ is called the lax left (right) center of
$\mathcal{C}$ and is denoted by $Z_{l,\mathrm{lax}}(\mathcal{C})$
($Z_{r,\mathrm{lax}}(\mathcal{C})$). In [57],
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$ is
referred to as the _weak left centralizer_ of $U$ and denoted by
$\mathcal{W}_{l}(U)$.

It is easy to check that the dual
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}$ carries a monoidal
structure via

$(X,\tau)\otimes(X^{\prime},\tau^{\prime}):=(X\otimes
X^{\prime},(\tau\otimes\mathrm{id}_{X^{\prime}})(\mathrm{id}_{X}\otimes\tau^{\prime}))$
(8)

with the pair $(\mathtt{1}_{\mathcal{C}},\mathrm{id}_{U(-)})$ acting as the
monoidal unit. Furthermore, the forgetful functor
$\overline{U}:\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}\rightarrow\mathcal{C}$
defined by $\overline{U}(X,\tau)=X$ becomes strict monoidal. Thereby, the dual
construction actually sends a strong monoidal functor
$U:\mathcal{D}\rightarrow\mathcal{C}$ with codomain $\mathcal{C}$ to another
such functor
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}\rightarrow\mathcal{C}$.

The notion of lax left (right) dual of a functor $U$ becomes equivalent to the
dual of $U$ when $\mathcal{D}$ is right (left) rigid:

######  Theorem 2.1.

If $\mathcal{D}$ is a right rigid category then
$(_{\mathcal{D}}U_{\mathcal{C}})^{\circ}=(_{\mathcal{D}}U_{\mathcal{C}})^{\circ}_{l,\mathrm{lax}}$.

###### Proof.

Let $M$ be an object of $\mathcal{D}$ and $M^{\vee}$ denote its right dual.
Since $U$ is strong monoidal then following our discussion in Section 2.1,
$U(M^{\vee})$ becomes a right dual of $U(M)$ via
$\underline{\mathrm{ev}}^{\prime}=U_{0}^{-1}U(\underline{\mathrm{ev}}_{M})U_{2}(M,M^{\vee})$
and
$\underline{\mathrm{coev}}^{\prime}=U_{2}^{-1}(M^{\vee},M)U(\underline{\mathrm{coev}}_{M})U_{0}$.
Hence, for any object $(X,\tau)$ in
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$, we
can define $\tau^{-1}_{M}$ by
$(\underline{\mathrm{ev}}^{\prime}\otimes\mathrm{id}_{X\otimes
M})\tau_{M^{\vee}}(\mathrm{id}_{M\otimes
X}\otimes\underline{\mathrm{coev}}^{\prime})$. It follows from $\tau$ being
natural and satisfying (6) and (7) that $\tau_{M}$ and $\tau_{M}^{-1}$ are
inverses. Consequently, for any $(X,\tau)$ in
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$, the
transformation $\tau$ is an isomorphism and
$(_{\mathcal{D}}U_{\mathcal{C}})^{\circ}=(_{\mathcal{D}}U_{\mathcal{C}})^{\circ}_{l,\mathrm{lax}}$
∎

Now we observe that dualities between objects in the base category lift to the
monoidal dual.

######  Corollary 2.2.

Let $X$ be a left dualizable object of $\mathcal{C}$ and ${}^{\vee}{X}$ denote
its left dual. If $(X,\tau)$ is an object of
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}$, then there exists a
natural braiding on $\rho$ such that $({}^{\vee}{X},\rho)$ becomes an object
of $\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}$ and the left dual
of $(X,\tau)$.

###### Proof.

Let $\mathrm{ev}_{X}$ and $\mathrm{coev}_{X}$ denote the relevant duality
morphisms between $X$ and ${}^{\vee}{X}$. We define
$\rho_{M}:=(\mathrm{ev}_{X}\otimes\mathrm{id}_{M\otimes{}^{\vee}{X}})(\mathrm{id}_{{}^{\vee}{X}}\otimes\tau^{-1}_{M}\otimes\mathrm{id}_{{}^{\vee}{X}})(\mathrm{coev}_{X})$.
It follows by definition that $\mathrm{ev}_{X}$ and $\mathrm{coev}_{X}$ become
morphisms in $\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}$. ∎

In [57], a stronger statement is proved:

######  Theorem 2.3 (Proposition 3.1 [57]).

If $U:\mathcal{D}\rightarrow\mathcal{C}$ is a strong monoidal functor,
$\mathcal{C}$ is left closed and $\mathcal{D}$ right rigid then
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}$ has a left closed
structure which $U$ preserves.

The center of a monoidal category $\mathcal{C}$ is of particular interest
since it has a braided structure $\Psi$ defined by
$\Psi_{(X,\tau),(X^{\prime},\tau^{\prime})}=\tau_{X^{\prime}}$. A _central
bialgebra (Hopf algebra)_ in $\mathcal{C}$ refers to an object $(B,\tau)$ in
$Z(\mathcal{C})$ with a braided bialgebra (Hopf algebra) structure.

At this point we reflect on the fact that, under suitable conditions, the dual
of a monoidal functor lifts colimits from the base category.

######  Theorem 2.4.

If $-\otimes-$ in $\mathcal{C}$ preserves colimits in both entries, the
forgetful functor
$\overline{U}:\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}\rightarrow\mathcal{C}$
creates colimits which exist in $\mathcal{C}$. The same statement holds if we
consider
$\overline{U}:\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}\rightarrow\mathcal{C}$

###### Proof.

Consider a diagram
$\mathbb{D}:\mathcal{J}\rightarrow\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$
so that the diagram $\overline{U}\mathbb{D}:\mathcal{J}\rightarrow\mathcal{C}$
has a colimit $A$ in $\mathcal{C}$ with a family of universal morphisms
$\pi_{j}:\overline{U}\mathbb{D}(j)\rightarrow A$ for objects $j$ in
$\mathcal{J}$. Since
$\mathbb{D}:\mathcal{J}\rightarrow\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$
is a functor, for any object $X\in\mathcal{D}$ we have a family of morphisms
$\sigma^{j}_{X}:\mathbb{D}(j)\otimes U(X)\rightarrow U(X)\otimes\mathbb{D}(j)$
which are natural with respect to $\mathcal{J}$ and thereby form a natural
transformation $\sigma:\mathbb{D}\otimes U(X)\Rightarrow
U(X)\otimes\mathbb{D}$. Because the bifunctor $-\otimes-$ in $\mathcal{C}$
preserves colimits in both components, the diagrams
$\overline{U}\mathbb{D}\otimes U(X)$ and $U(X)\otimes\overline{U}\mathbb{D}$
admit colimits $A\otimes U(X)$ and $U(X)\otimes A$, respectively. By the
universal property of $A\otimes U(X)$, there exists a unique morphism
$\sigma^{A}_{X}$ such that $\sigma^{A}_{X}(\pi_{j}\otimes
U(X))=(U(X)\otimes\pi_{j})\sigma^{j}_{X}$. If we show that $(A,\sigma_{A})$ is
indeed an object of
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$,
then it follows that $\pi:\mathbb{D}\Rightarrow(A,\sigma_{A})$ becomes a
cocone of the diagram $\mathbb{D}$.

Let $f:X\rightarrow Y$ be a morphism in $\mathcal{D}$. To show that
$(U(f)\otimes\mathrm{id}_{A})\sigma^{A}_{X}=\sigma^{A}_{Y}(\mathrm{id}_{A}\otimes
U(f))$, we observe that

$\displaystyle\sigma^{A}_{Y}$ $\displaystyle(\mathrm{id}_{A}\otimes
U(f))(\pi_{j}\otimes\mathrm{id}_{U(X)})=\sigma^{A}_{Y}(\pi_{j}\otimes\mathrm{id}_{U(Y)})(\mathrm{id}_{\mathbb{D}(j)}\otimes
U(f))$
$\displaystyle=(\mathrm{id}_{U(Y)}\otimes\pi_{j})\sigma^{j}_{Y}(\mathrm{id}_{\mathbb{D}(j)}\otimes
U(f))=(\mathrm{id}_{U(Y)}\otimes\pi_{j})(U(f)\otimes\mathrm{id}_{\mathbb{D}(j)})\sigma^{j}_{X}$
$\displaystyle=(U(f)\otimes
A)(\mathrm{id}_{U(X)}\otimes\pi_{j})\sigma^{j}_{X}=(U(f)\otimes\mathrm{id}_{A})\sigma^{A}_{X}(\pi_{j}\otimes\mathrm{id}_{U(X)})$

holds and use the universal property of $(\pi_{j}\otimes\mathrm{id}_{U(X)})$.

Hence, $(A,\sigma_{A})$ is an object of
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$ and
a cocone of the diagram $\mathbb{D}$ via $\pi_{j}$. Consider another cocone
$\kappa:\mathbb{D}\Rightarrow(B,\sigma_{B})$. Since $A$ is a colimit of
$\overline{U}\mathbb{D}$, there exists a unique morphism $t:A\rightarrow B$
such that $\overline{U}(\kappa)=t\overline{U}(\pi)$. What remains to be shown
is whether $t$ is a morphism in
$\left({}_{\mathcal{D}}{U}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$.
This also follows from the universality of $A\otimes U(X)$ and the calculation
below

$\displaystyle(\mathrm{id}_{U}\otimes
t)\sigma^{A}(\overline{U}(\pi)\otimes\mathrm{id}_{U})$
$\displaystyle=(\mathrm{id}_{U}\otimes
t\overline{U}(\pi))\sigma^{j}=(\mathrm{id}_{U}\otimes\overline{U}(\kappa))\sigma^{j}$
$\displaystyle=\sigma^{B}(\overline{U}(\kappa)\otimes\mathrm{id}_{U})=\sigma^{B}(t\otimes\mathrm{id}_{U})(\overline{U}(\pi)\otimes\mathrm{id}_{U})$

Hence $(\mathrm{id}_{U}\otimes
t)\sigma^{A}=\sigma^{B}(t\otimes\mathrm{id}_{U})$ and thereby,
$(A,\sigma_{A})$ is a colimit of the original diagram $\mathbb{D}$.

If $\sigma^{j}$ were invertible, it follows from the universal property of
$U(X)\otimes A$ that there exists a unique morphism $(\sigma^{A})^{-1}$ such
that
$(\sigma^{A})^{-1}(\mathrm{id}_{U}\otimes\pi)=(\pi\otimes\mathrm{id}_{U})(\sigma^{j})^{-1}$.
It follows from the universal properties of $U(X)\otimes A$ and $A\otimes
U(X)$ that $\sigma^{A}_{X}$ and $(\sigma^{A}_{X})^{-1}$ are inverses. ∎

The reader should note that the proof of Theorem 2.4 can be replicated in a
symmetric manner for any limits in $\mathcal{C}$ which are preserved in both
entries of $-\otimes-$.

###  2.4 Key Examples of Monoidal Categories

In this section we review some key examples of monoidal categories and their
properties.

######  Example 2.5.

Given any category $\mathcal{C}$, we obtain a new category denoted by
$\mathrm{End}(\mathcal{C})$, which has endofunctors
$F:\mathcal{C}\rightarrow\mathcal{C}$ as objects and natural transformations
between them as morphisms. The category of endofunctors has a canonical
monoidal structure via composition of functors i.e. $F\otimes G=FG$ for
endofunctors $F,G$ and the identity functor $\mathrm{id}_{\mathcal{C}}$ acting
as the monoidal unit. In this monoidal category, a functor $F$ being left
(right) dual to $G$, is exactly equivalent to $F$ being left (right) adjoint
to $G$. Additionally, the right adjoint functor $[F,-]^{l}$ to $-\otimes
F:\mathrm{End}(\mathcal{C})\rightarrow\mathrm{End}(\mathcal{C})$, if it
exists, is usually denoted by $\mathrm{Ran}_{F}$ and $[F,G]^{l}$ is called the
_right Kan extension_ of $G$ along $F$, see Proposition 6.1.5 of [54]. We also
have a trivial description of the center of any endofunctor category:

######  Proposition 2.6.

Given any non-empty category $\mathcal{C}$, the center of the monoidal
category $\mathrm{End}(\mathcal{C})$ is equivalent to the trivial category.

###### Proof.

For any object $X$ in $\mathcal{C}$, there exists an endofunctor $F_{X}$ which
sends all objects to $X$ and all morphisms to $\mathrm{id}_{X}$. Let
$(G,\tau)$ belong to $Z(\mathrm{End}(\mathcal{C}))$. It follows that for every
object $X$ in $\mathcal{C}$ we obtain an isomorphism
$\tau_{F_{X}}:GF_{X}\rightarrow F_{X}G=F_{X}$. Hence, we have a family of
isomorphisms $\tau_{F_{X}}:G(X)\cong X$ for objects $X$ in $\mathcal{C}$.
Moreover, natural transformations $F_{f}:F_{X}\rightarrow F_{Y}$ are in
correspondence with morphisms $f:X\rightarrow Y$. Since $\tau$ is a natural
then for any morphism $f$ the equality $f\tau_{F_{X}}=\tau_{F_{Y}}G(f)$ holds
and $\tau_{F_{-}}:G\rightarrow\mathrm{id}_{\mathcal{C}}$ becomes a natural
isomorphism of functors. Hence, every pair $(G,\tau)$ is isomorphic to
$(\mathrm{id}_{\mathcal{C}},\mathrm{id})$ via $\tau_{F_{-}}$ and the skeleton
of $Z(\mathrm{End}(\mathcal{C}))$ is the trivial category. ∎

######  Example 2.7.

Any category with finite products obtains a natural symmetric monoidal
structure with the product acting as $\otimes$ and the final object acting as
the monoidal unit. Such monoidal structures are called _cartesian_. In
particular, the category of sets and functions $\mathsf{Set}$ has a monoidal
structure via the product of sets $\times$ and the set with one element
$\mathtt{1}=\\{\star\\}$ acting as the monoidal unit. The symmetric structure
is given by the flip map $\Psi_{X,Y}=\mathfrak{flip}$ which sends a pair
$(x,y)\in X\times Y$ to $(y,x)\in Y\times X$. The only dualizable object in
$\mathsf{Set}$ is $\mathtt{1}$ since $\mathrm{coev}:\mathtt{1}\rightarrow
X\times{}^{\vee}{X}$ would have to be surjective on its projections to
${}^{\vee}{X}$ and $X$ but its image contains exactly one element, so
$X\cong{}^{\vee}{X}\cong\mathtt{1}$.

######  Proposition 2.8.

There exists a monoidal equivalence $Z(\mathsf{Set})\simeq\mathsf{Set}$.

###### Proof.

Let $(X,\tau)$ be an object of $Z(\mathsf{Set})$. For any set $Y$ and element
$y\in Y$, we have a unique morphism $f_{y}:\mathtt{1}\rightarrow Y$ sending
the element $\star$ to $y$. By the naturality of $\tau$ and (7) it follows
that $\tau_{Y}(\mathrm{id}_{X}\times
f_{y})=(f_{y}\times\mathrm{id}_{X})\mathrm{id}_{\mathtt{1}\times X}$ and
thereby $\tau_{Y}(x,y)=(y,x)$ for any $x\in X$. Hence, $\tau$ must act as the
flip map and the only objects in $Z(\mathsf{Set})$ are of the form
$(X,\mathfrak{flip})$. ∎

Since the category $(\mathsf{Set},\times,\mathtt{1})$ is symmetric monoidal,
we can describe a Hopf algebra object in $\mathsf{Set}$: Let
$(X,\Delta,\epsilon)$ be a comonoid in $(\mathsf{Set},\times,\mathtt{1})$.
Since $\epsilon:X\rightarrow\mathtt{1}$ is the unique map sending all elements
of $X$ to the unique element in $\mathtt{1}$, then it follows from
$(\epsilon\times\mathrm{id}_{X})\Delta=\mathrm{id}_{X}=(\mathrm{id}_{X}\times\epsilon)\Delta$
that $\Delta(x)=(x,x)\in X\times X$ for all $x\in X$. Consequently, any monoid
$(X,m,\eta)$ becomes a bialgebra $(X,m,\eta,\Delta,\epsilon)$ with this
trivial comonoid structure. The existence of an antipode for such a bialgebra
is precisely equivalent to the existence of inverses for all elements. Hence,
Hopf algebras in $\mathsf{Set}$ are precisely groups.

######  Example 2.9.

The category of vectorspaces $\mathsf{Vec}$ over a base field $\mathbb{K}$ is
endowed with a symmetric monoidal structure via the tensor product of
vectorspaces $\otimes_{\mathbb{K}}$ and the one dimensional vectorspace
$\mathbb{K}$ acting as the unit. The symmetric structure
$\Psi_{V,W}:V\otimes_{\mathbb{K}}W\rightarrow W\otimes_{\mathbb{K}}V$ is the
flip map sending $v\otimes_{\mathbb{K}}w\in V\otimes_{\mathbb{K}}W$ to
$v\otimes_{\mathbb{K}}w$. The famous hom-tensor adjunction illustrates that
this monoidal structure is closed with inner-homs $[V,W]$ given by spaces of
$\mathbb{K}$-linear maps $\mathrm{Hom}_{\mathbb{K}}(V,W)$. It is well-known
that a vectorspace $V$ is dualizable in $\mathsf{Vec}$ if and only if $V$ is
finite dimensional. This is because $\mathrm{coev}:\mathbb{K}\rightarrow
V\otimes_{\mathbb{K}}{}^{\vee}{V}$ is uniquely determined by the choice of
$\mathrm{coev}(1)$ which must be of the form
$\sum_{i=1}^{n}v_{i}\otimes_{\mathbb{K}}v^{*}_{i}$ for some number $n$. It
then follows by the definition of the duality morphisms that
$\\{v_{i}\\}_{i=1}^{n}$ must form a finite basis for $V$. A similar argument
to Proposition 2.8 with maps $f_{v}:\mathbb{K}\rightarrow V$ corresponding to
$v\in V$ proves that $Z(\mathsf{Vec})\simeq\mathsf{Vec}$.

Monoids in $\mathsf{Vec}$ are simply $\mathbb{K}$-algebras. Comonoids
$(C,\Delta,\epsilon)$ in $\mathsf{Vec}$ are called $\mathbb{K}$-coalgebras, or
coalgebras for simplicity. We will use _Sweedler’s notation_ for the
_coproduct_ $\Delta:C\rightarrow C\otimes_{\mathbb{K}}C$ and write
$\Delta(c)=c_{(1)}\otimes_{\mathbb{K}}c_{(2)}$ where the right hand side
depicts the finite sum of elements of the form
$c_{1}\otimes_{\mathbb{K}}c_{2}\in C\otimes_{\mathbb{K}}C$ corresponding to
$c$. We recover the theory of ordinary bialgebras and Hopf algebras when
looking at bialgebras and Hopf algebras in $\mathsf{Vec}$ with this symmetric
monoidal structure and refer the reader to [46] for more details on these.

######  Example 2.10.

Let $A$ be a $\mathbb{K}$-algebra. The category of $A$-bimodules has a natural
monoidal structure by tensoring bimodules over the algebra $A$, denoted by
$\otimes_{A}$, and the algebra $A$, regarded as an $A$-bimodule, acting as the
unit object. It is well-known that a bimodule has a left (right) dual in the
monoidal category ${}_{A}{\mathcal{M}}_{A}$ if and only if it is finitely
generated and projective, _fgp_ for short, as a right (left) $A$-module. In
particular, ${}_{A}{\mathcal{M}}_{A}$ is closed with

$\displaystyle[M,N]^{l}:=\mathrm{Hom}_{A}(M,N),\hskip 28.45274pt$
$\displaystyle[afb](m)=af(bm),\quad f\in\mathrm{Hom}_{A}(M,N)$
$\displaystyle[M,N]^{r}:=\prescript{}{A}{\mathrm{Hom}}(M,N),\hskip 28.45274pt$
$\displaystyle[agb](m)=g(ma)b,\quad g\in\prescript{}{A}{\mathrm{Hom}}(M,N)$

where $a,b\in A$ and $\mathrm{Hom}_{A}(M,N)$ and
$\prescript{}{A}{\mathrm{Hom}(M,N)}$ denote the vectorspaces of right and left
$A$-module morphisms from $M$ to $N$, respectively. Explicitly, the units and
counits of the adjunctions for the left and right closed structures are given
by

(13) (18)

for any pair of $A$-bimodules $M$ and $N$. Consequently, for a right or left
fgp bimodule $M$, we identify ${}^{\vee}{M}$ by $\mathrm{Hom}_{A}(M,A)$ and
$M^{\vee}$ by $\prescript{}{A}{\mathrm{Hom}(M,A)}$. Note that the category
${}_{A}{\mathcal{M}}_{A}$ is generally not braided. For a discussion of when
${}_{A}{\mathcal{M}}_{A}$ admits a braiding we refer the reader to [2].
Several characterisations of the center of ${}_{A}{\mathcal{M}}_{A}$ are also
provided in [1].

Since the category of bimodules does not admit a braiding in general, we can
not discuss Hopf algebra objects in this category. However, the role of Hopf
algebras and bialgebras in this setting is taken by Hopf algebroids and
bialgebroids which we will discuss further in Section 4.6.

###  2.5 Monads

In this section we recall the theory of monads, with Chapter VI of [39] as our
main reference.

A _monad_ on a category $\mathcal{C}$ is a monoid in its monoidal category of
endofunctors $\mathrm{End}(\mathcal{C})$. Explicitly, a monad consists of a
triple $(T,\mu,\eta)$, where $T$ is an endofunctor on $\mathcal{C}$ and
$\mu:TT\rightarrow T$ and $\eta:\mathrm{id}_{\mathcal{C}}\rightarrow T$ are
natural transformations satisfying $\mu T\mu=\mu\mu_{T}$, $\mu
T\eta=\mathrm{id}_{T}=\mu\eta_{T}$. Morphisms of monads
$\theta:(T,\mu,\eta)\rightarrow(T,\mu^{\prime},\eta^{\prime})$ can be defined
accordingly as natural transformations $\theta:T\rightarrow T^{\prime}$
satisfying $\theta\eta=\eta^{\prime}$ and
$\theta\mu=\mu^{\prime}(T^{\prime}\theta)\theta_{T}$.

Any monad gives rise to an adjunction $F_{T}\dashv
U_{T}:\mathcal{C}^{T}\leftrightarrows\mathcal{C}$, where $\mathcal{C}^{T}$ is
the _Eilenberg-Moore category_ associated to $T$. The category
$\mathcal{C}^{T}$ has pairs $(X,r)$, where $X$ is an object in $\mathcal{C}$
and $r:TX\rightarrow X$ a $T$-_action_ satisfying $r\mu_{X}=rTr$ and
$r\eta=\mathrm{id}_{X}$, as its objects and morphisms of $\mathcal{C}$ which
commute with the $T$-actions as its morphisms. The _free_ functor $F_{T}$ acts
as $F_{T}(X)=(TX,\mu_{X})$ and the _forgetful_ functor $U_{T}$ acts by
$U_{T}(X,r)=X$. We will call every pair $(X,r)$ a $T$-_module_ and will
sometimes refer to the Eilenberg-Moore category as the category of
$T$-modules. In the converse direction, any adjunction $F\dashv
G:\mathcal{D}\leftrightarrows\mathcal{C}$ gives rise to a monad
$(GF,\eta,G\epsilon_{F})$ where $\eta:\mathrm{id}_{\mathcal{C}}\rightarrow GF$
and $\epsilon:FG\rightarrow\mathrm{id}_{\mathcal{D}}$ denote the unit and
counit of the adjunction, respectively. We also obtain a natural functor
$K:\mathcal{D}\rightarrow\mathcal{C}^{T}$ defined by
$K(d)=(Gd,G\epsilon_{d})$. This functor is called the _comparison functor_ and
satisfies $U_{T}K=G$ and $KF=F_{T}$. We say the functor $G$ is _monadic_ if
$K$ is an equivalence of categories.

Now we recall Beck’s Theorem. The proof of this theorem and several equivalent
formulations of it can be found in Section VI. 7 of [39].

######  Theorem 2.11 (Beck’s Theorem).

Given an adjunction $F\dashv G:\mathcal{D}\leftrightarrows\mathcal{C}$, $G$ is
monadic if and only if the functor $G$ creates coequalizers for parallel pairs
$f,g:X\rightrightarrows Y$ for which $Gf,Gg$ has a split coequalizer.

Split coequalizers should not to be confused with reflexive coequalizers. A
pair $Gf,Gg$ has a _split_ coequalizer if there exists an object $C$ in
$\mathcal{C}$ along with morphisms $s:C\rightarrow GY$, $h:GY\rightarrow GX$
and $t:GY\rightarrow C$ such that $t$ is a coequalizer and
$ts=\mathrm{id}_{C}$, $(Gg)h=\mathrm{id}_{GY}$ and $(Gf)h=st$ hold. A
coequalizer of a parallel pair $f,g:X\rightrightarrows Y$ is called
_reflexive_ if there exists a morphism $h$ such that $hf=hg=\mathrm{id}_{X}$.
The importance of reflexive coequalizers within the theory of monads was
described in a theorem of Linton [38], which concerns the existence of
colimits in the category of $T$-modules. We will refer to reflexive
coequalizers and coreflexive equalisers by RCs and CEs, respectively.

If $\mathcal{C}$ is a monoidal category, then a monoid structures $(A,m,\eta)$
on an object $A$ in $\mathcal{C}$ gives rise to a monad structures on the
endofunctor $A\otimes-$ with $\mu=m\otimes-$. The Eilenberg-Moore category
$\mathcal{C}^{T}$ in this case becomes the category of left $A$-modules i.e.
objects $X$ with an action $r:A\otimes X\rightarrow X$. The free functor
$F_{T}$ sends an object $X$ to the free module generated by it $(A\otimes
X,m\otimes\mathrm{id}_{X})$ and the forgetful functor $U_{T}$ sends a module
$(X,r)$ to the object $X$.

The notation for monads is slightly confusing when compared to algebras. We
usually denote the category of (left) modules over an algebra $A$ by
${}_{A}{\mathcal{M}}$ and the category of (left) comodules over a coalgebra
$C$ by ${}^{C}{\mathcal{M}}$, while in the monadic case, the category of
modules over a monad $T$ is usually written as $\mathcal{C}^{T}$ and category
of comodules of a comonad $S$ as $\mathcal{C}_{S}$. Moreover, in many texts
including [39] $T$-modules are referred to as _$T$ -algebras_ to stay
consistent with the point of view that monads generalise algebraic theories
and $\mathcal{C}^{T}$ plays the role of the category of algebras over a theory
(see Section 4.3). Here we use the term $T$-modules to stay closer to
representation theoretic language and the example $T=A\otimes-$.

Finally, we recall the following well-known fact.

######  Theorem 2.12.

Given a pair of adjoint endofunctors $F\dashv
G:\mathcal{C}\leftrightarrows\mathcal{C}$, monad structures on $F$ correspond
to comonad structures on $G$.

This can be seen easily from the fact that $F$ and $G$ are duals in the
monoidal category of $\mathrm{End}(\mathcal{C})$ and this duality can be used
to take an algebra structure on one object to a coalgebra structure on the
other, see Proposition 2.4 of [44]. Explicitly, if $\eta$ and $\epsilon$
denote the unit and counit of $F\dashv G$ and $(F,\mu,\nu)$ is a monad
structure, then $(GG\epsilon)(GG\mu_{G})(G\eta_{FG})\eta_{G}$ and
$\epsilon\nu_{G}$ provide a comonad structure on $G$.

###  2.6 Ends and Coends

In this section we briefly recall the notion of ends and coends, with Chapter
IX of [39] serving as our main reference.

For categories $\mathcal{C}$ and $\mathcal{D}$ and bifunctors
$F,G:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathcal{D}$, a
_dinatural transformation_ $d:F\rightarrow G$ consists of a family of
morphisms $d_{X}:F(X,X)\rightarrow G(X,X)$ such that for any morphism
$f:X\rightarrow Y$, we have an equality of morphisms

$G(\mathrm{id}_{X},f)d_{X}F(f,\mathrm{id}_{X})=G(f,\mathrm{id}_{Y})d_{Y}F(\mathrm{id}_{Y},f):F(Y,X)\rightarrow
G(X,Y)$ (19)

An _end_ for a bifunctor
$F:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathcal{D}$ is a pair
$(E,e)$ consisting of an object $E$ of $\mathcal{D}$ and a dinatural
transformation $e:E\rightarrow F$, where we regard $E$ as a constant functor
$E:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathcal{D}$ sending
all objects in $\mathcal{C}^{\mathrm{op}}\times\mathcal{C}$ to $E$ and all
morphisms to $\mathrm{id}_{E}$, where $e$ is universal in the sense that for
any other such pair $(E^{\prime},e^{\prime})$, there exists a unique morphism
$f:E^{\prime}\rightarrow E$ satisfying $ef=e^{\prime}$. Note that for an end
$(E,e)$ of $F$, the equations (19) reduce to
$F(\mathrm{id}_{X},f)e_{X}=F(f,\mathrm{id}_{Y})e_{Y}$. Dually, we can define a
_coend_ of $F$ as a pair $(C,d)$ where $d:F\rightarrow C$ is a universal
dinatural transformation satisfying
$d_{X}F(f,\mathrm{id}_{X})=d_{Y}F(\mathrm{id}_{Y},f)$. The end and coend of
$F$ will be denoted by $\int_{X\in\mathcal{C}}F(X,X)$ and
$\int^{X\in\mathcal{C}}F(X,X)$, respectively.

Given any bifunctor
$F:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathcal{D}$, we
obtain a diagram in $\mathcal{D}$ containing morphisms $F(\mathrm{id}_{Y},f)$
and $F(f,\mathrm{id}_{X})$ corresponding to $f:X\rightarrow Y$ in
$\mathcal{C}$. An end $\int_{X\in\mathcal{C}}F(X,X)$ and a coend
$\int^{X\in\mathcal{C}}F(X,X)$ are precisely a limit and colimit for this
diagram, respectively. Hence, (co)ends for a bifunctor
$F:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\rightarrow\mathcal{D}$ exists if
$\mathcal{D}$ is (co)complete.

##  3 Hopf and Bimonads

In this section, we review the definitions and basic properties of bimonads
and Hopf monads from [14, 12]. After first reviewing the theory of bimonads,
we will look at the definition of Hopf monads on rigid monoidal categories,
defined in [14]. We will then review the notion of Hopf monads on general
monoidal categories and closed monoidal categories, which were defined in
[12].

###  3.1 Bimonads and Comonoidal Adjunctions

In this section, we first recall the theory of _bimonads_ or _comonoidal
monads_ from [14], which first appeared in [47] under the name of _opmonoidal
monads_ and independently in [51] under the name of _Hopf monads_. Since these
objects generalise braided bialgebras to arbitrary monoidal categories, we
choose to stay consistent with the terminology of [14, 12] and refer to them
as bimonads.

######  Definition 3.1.

A monad $(T,\mu,\eta)$ on a monoidal category $\mathcal{C}$ is said to be a
_bimonad_ or an _comonoidal monad_ if it also has a compatible comonoidal
structure $(T_{2},T_{0})$ satisfying

$\displaystyle T_{2}(X,Y)\mu_{X\otimes Y}=(\mu_{X}\otimes\mu_{Y})$
$\displaystyle T_{2}(T(X),T(Y))TT_{2}(X,Y)$ (20) $\displaystyle
T_{2}(X,Y)\eta_{X\otimes Y}=$ $\displaystyle\eta_{X}\otimes\eta_{Y}$ (21)
$\displaystyle T_{0}\mu_{\mathtt{1}}=$ $\displaystyle T_{0}(TT_{0})$ (22)
$\displaystyle T_{0}\eta_{\mathtt{1}}=$
$\displaystyle\mathrm{id}_{\mathtt{1}}$ (23)

for arbitrary objects $X$ and $Y$ in $\mathcal{C}$.

The conditions in the above definition are just stating that $\mu$ and $\eta$
are _comonoidal natural transformations_. A _bimonad morphism_ between two
bimonads is exactly a morphism between monads which is also a comonoidal
natural transformation. A bimonad structure on a monad $T$ is precisely the
structure needed to lift the monoidal structure on the base category
$\mathcal{C}$ to $\mathcal{C}^{T}$:

######  Theorem 3.2 (Theorem 7.1 [51]).

If $T$ is a monad on a monoidal category $\mathcal{C}$, then bimonad
structures on $T$ are in correspondence with liftings of the monoidal
structure of $\mathcal{C}$ onto $\mathcal{C}^{T}$ i.e. monoidal structures on
$\mathcal{C}^{T}$ such that $U_{T}$ is strict monoidal.

#### Sketch of proof.

If $T$ has a compatible comonoidal structure $(T_{2},T_{0})$, then we define
the monoidal structure on $\mathcal{C}^{T}$ as

$(M,r)\otimes(N,s):=\left(M\otimes N,\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
26.5746pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T(M\otimes
N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
30.8819pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{T_{2}(M,N)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 69.19368pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
44.88414pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 69.19368pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T(M)\otimes
T(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
137.469pt\raise 5.33333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{r\otimes s}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 155.66289pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
155.66289pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{M\otimes
N}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)$ (24)

with $\mathtt{1}_{\mathcal{C}^{T}}=(\mathtt{1},T_{0})$ as the unit object.
Conditions (20) and (21) ensure that (24) is a well-defined object in
$\mathcal{C}^{T}$. Conditions (22) and (23) ensure that $T_{0}$ is a well-
defined $T$-action on $\mathtt{1}$ so that $\mathtt{1}_{\mathcal{C}^{T}}$
becomes an object in $\mathcal{C}^{T}$.

In the converse direction, let us assume that $\mathcal{C}^{T}$ has a well-
defined monoidal structure $\overline{\otimes}$ such that $U_{T}$ is strict
monoidal, defined by $(M,r)\overline{\otimes}(N,s)=\left(M\otimes
N,{r\overline{\otimes}s}\right)$. Then consider the arbitrary free $T$-modules
$(T(M),\mu_{M})$ and $(T(N),\mu_{N})$ and define $T_{2}(N,M)$ as the composite

$\textstyle{T_{2}(M,N):T(M\otimes
N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T(\eta_{M}\otimes\eta_{N})}$$\textstyle{T(T(M)\otimes
T(N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{M}\overline{\otimes}\mu_{N}}$$\textstyle{T(M)\otimes
T(N)}$ (25)

It should be clear that due to the functoriality of $T$ and $\mu
T\eta=\mathrm{id}_{T}$ holding, this process gives the inverse of the one
describe above. For further details of this proof, we refer the reader to
Theorem 7.1 in [51]. $\diamond$

Now we reflect on the manner in which bimonads generalise braided bialgebras.
Recall from Section 2.5 that for any algebra $(B,m,\eta)$ in a braided
category $(\mathcal{C},\Psi)$ we obtain a monad
$(B\otimes-,m\otimes-,\eta\otimes-)$, which we will denote by $\mathbb{B}$. It
is well-known that the category of modules over a bialgebra lifts the monoidal
structure of the base category, via its coalgebra structure. From the
perspective of Theorem 3.2, the coalgebra structure of a bialgebra provides a
comonoidal structure on the functor $B\otimes-$ which turns $\mathbb{B}$ into
a bimonad. Explicitly, the comonoidal structure is defined by

$\mathbb{B}_{2}(M,N)=(\mathrm{id}_{B}\otimes\Psi_{B,M}\otimes\mathrm{id}_{N})(\Delta\otimes\mathrm{id}_{M\otimes
N}),\quad\mathbb{B}_{0}=\epsilon$ (26)

and the conditions in Definition 3.1 translate exactly to the bialgebra
axioms. In this way, we recover precisely the action of a bialgebra on the
tensor product of its modules from (24).

As we recalled in Section 2.5, the theory of monads and adjunctions are
intertwined and it is natural to expect that adjunctions which giving rise to
bimonads should have a corresponding structure. These adjunctions appeared
first in Section 2.4 of [14]:

######  Definition 3.3.

An adjunction $F\dashv U:\mathcal{D}\leftrightarrows\mathcal{C}$ between
monoidal categories is called a _comonoidal adjunction_ if
$U:\mathcal{D}\rightarrow\mathcal{C}$ is strong monoidal.

######  Lemma 3.4 (Theorem 2.6 [14]).

If $F\dashv U:\mathcal{D}\leftrightarrows\mathcal{C}$ is an adjunction between
monoidal categories then TFAE

  1. (I)

$U$ is strong comonoidal.

  2. (II)

$F$ and $U$ are both comonoidal and the unit and counit of the adjunction are
comonoidal natural transformations.

Furthermore, if the above conditions hold then $T=UF$ becomes a bimonad.

#### Sketch of proof.

Given a comonoidal adjunction $F\dashv U$, we obtain a comonoidal structure on
$F$ as follows

$\textstyle{F(X\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}(X,Y)}$$\scriptstyle{F(\eta_{X}\otimes\eta_{Y})}$$\textstyle{F(X)\otimes
F(Y)}$$\textstyle{F(UF(X)\otimes
UF(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{FU_{2}^{-1}(F(X),F(Y))}$$\textstyle{FU(F(X)\otimes
F(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{F(X)\otimes
F(Y)}}$ (27)

and $F_{0}=\epsilon_{\mathtt{1}}F(U_{0}^{-1})$, where $\eta$ and $\epsilon$
denote the unit and counit of the adjunction. It then follows by definition
that $\eta$ and $\epsilon$ are comonoidal natural transformations.

In the converse direction if $F$ and $U$ are both comonoidal and the $\eta$
and $\epsilon$ respect these comonoidal structures, then $U$ is strong
comonoidal and the inverse of $U_{2}(X,Y)$ is given by

$\textstyle{U(X)\otimes
U(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{2}^{-1}(X,Y)}$$\scriptstyle{\eta_{U(X)\otimes
U(Y)}}$$\textstyle{U(X\otimes Y)}$$\textstyle{UF(U(X)\otimes
U(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{UF_{2}(U(X),U(Y))}$$\textstyle{U(FU(X)\otimes
FU(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U(\epsilon_{F(X)}\otimes\epsilon_{F(Y)})}$
(28)

Consequently, for a comonoidal adjunction $F\dashv U$ the monad $T=UF$ obtains
a comonoidal structure by the composition of functors and the conditions in
Definition 3.1 follow from $\eta$ and $\epsilon$ respecting this comonoidal
structure. $\diamond$

In a symmetric manner to Lemma 3.4, given a bimonad $T$ we obtain a comonoidal
adjunction, namely the free/forgetful adjunction $F_{T}\dashv
U_{T}:\mathcal{C}^{T}\leftrightarrows\mathcal{C}$. We have already seen in
Theorem 3.2, that $U_{T}$ is strict monoidal and carries a trivial strong
comonoidal structure. Note that when the monoidal structure on $\mathcal{C}$
is cartesian, any adjunction $F\dashv
U:\mathcal{D}\leftrightarrows\mathcal{C}$ is comonoidal, since $U$ preserves
all limits and is, thereby, a strong monoidal functor.

We should also note that in the circumstances of Lemma 3.4, the comparison
functor $K:\mathcal{D}\rightarrow\mathcal{C}^{T}$ obtains a natural strong
monoidal structure such that equalities $U_{T}K=U$ and $KF=F_{T}$ hold as
equalities of comonoidal functors. Explicitly $K_{2}(X,Y)=U_{2}$ and
$K_{0}=U_{0}$. For further details we refer the reader to the proof of Theorem
2.6 in [14].

Dual Notion: The appropriate dualisation of the notion of bimonads is that of
_bicomonads_ or _monoidal comonads_. While a bimonad consists of a monad and a
comonoidal structure, a bicomonad refers to a comonad with a compatible
monoidal structure, and these compatibility conditions are obtained by
reversing the morphisms in Definition 3.1. By symmetric arguments, one can
prove that the monoidal structure of the base category lifts to the category
of comodules over the bicomonad. Given a bialgebra $B$ in a braided category
$\mathcal{C}$, we also obtain a natural bicomonad structure on the endofunctor
$B\otimes-$ where the coalgebra structure of $B$ appears in the comonad
structure and the algebra structure in the monoidal structure. This is a
unique case, and in general we do not expect a single endofunctor to admit
both a bimonad and a bicomonad structure.

After reviewing the connection between comonoidal monads and their lifting
properties for their Eilenberg-Moore categories, there are two natural
questions which arise. 1) What happens if we consider a monoidal structure on
a monad? 2) Which monads lift the monoidal structure of the base category to
their Kleisli categories?

_Monoidal Monads:_ A monoidal monad is a monad $(T,\mu,\eta)$ with a monoidal
structure $(T_{2},T_{0})$ such that $\mu$ and $\eta$ become monoidal
morphisms. If suitable coequalizers exist, then the Eilenberg-Moore category
of a monoidal monad obtains a natural monoidal structure $\otimes_{T}$ defined
as the following coequalizer:

$\textstyle{T(T(X)\otimes
T(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T(r\otimes
s)}$$\scriptstyle{\mu_{M\otimes N}TT_{2}(M,N)}$$\textstyle{T(X\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X,r)\otimes_{T}(Y,s)}$
(29)

for a pair of $T$-modules $(X,r)$ and $(Y,s)$. In this setting, the free
module functor $F_{T}$ becomes strong monoidal. Consequently, the Kleisli
category which can be identified as the subcategory of free $T$-modules of the
monad lifts the monoidal structure of the base category. This monoidal
structure generalises the tensor product in the category of $R$-modules for a
commutative ring $R$. Lastly, the reader should note that $U_{T}F_{T}$ becomes
a monoidal comonad on $\mathcal{C}^{D}$. In a symmetric way, any comonoidal
comonad on $\mathcal{C}$ with suitable equalizers would give rise a comonoidal
monad on $\mathcal{C}^{T}$. We will see a refined form of this relation in
Section 6 for Hopf monads.

######  Remark 3.5.

The theory of monoidal monads is embedded in the theory of commutative monads
within the literature. A functor on a monoidal category can be equipped with
two natural transformations called a strength and a costrength, which were
introduced in [35]. If the base category is symmetric monoidal, we can ask for
these morphisms to be compatible via the symmetry and a monad equipped with
such compatible structures is called _commutative_. In fact, any monoidal
monad obtains a strength and a costrength in a natural way, and monoidal
monads which respect the symmetric structure are in bijection with commutative
monads [36].

###  3.2 Hopf Monads on Rigid Monoidal Categories

In this section we review the first definition of Hopf monads based on [14].
As we will see, the definition provided in [14] is only sensible when the base
category is rigid and was later generalised to arbitrary monoidal categories
in [12].

Bialgebras in a braided monoidal category are known to lift the monoidal
structure of the base category to their categories of modules. As we saw in
the previous section, bimonads generalise this aspect of the theory of
bialgebras. In the same direction, a bialgebra admitting an antipode and
becoming a Hopf algebra implies that the duality morphisms in the base
category lift to its category of modules. More concretely, let $B$ be a
braided bialgebra in $(\mathcal{C},\Psi)$, and ${}^{\vee}{X}$ denote the left
dual of an object $X$ in $\mathcal{C}$, with duality morphisms $\mathrm{ev}$
and $\mathrm{coev}$. If $B$ admits an antipode $S$ then for any $B$-action on
$X$, $\triangleright:B\otimes X\rightarrow X$, we obtain a natural $B$-action
on ${}^{\vee}{X}$ by
$(\mathrm{ev}\otimes\mathrm{id}_{{}^{\vee}{X}})(\mathrm{id}_{{}^{\vee}{X}}\otimes\triangleright(S\otimes\mathrm{id}_{X})\otimes\mathrm{id}_{{}^{\vee}{X}})(\Psi_{B,{}^{\vee}{X}}\otimes\mathrm{coev})$.
In particular, $\mathrm{ev}$ and $\mathrm{coev}$ respect these $B$-actions and
become duality morphisms in ${}_{B}\mathcal{C}$. When $B$ admits an invertible
antipode, we obtain a similar action on right dual objects using $S^{-1}$. The
first Hopf condition introduced in [14] ensures that the dualities in a rigid
monoidal category lift to the Eilenberg-Moore category of the bimonad. Let us
recall the definition of antipodes for bimonads from Section 3.3 of [14].

######  Definition 3.6.

A bimonad $T$ on a left rigid monoidal category $\mathcal{C}$ is called a
_left Hopf_ monad if there exists a natural transformation
$s^{l}_{X}:T({}^{\vee}{T(X)})\rightarrow{}^{\vee}{X}$ satisfying

$\displaystyle
T_{0}T(\mathrm{ev}_{X}({}^{\vee}{\eta_{X}}\otimes\mathrm{id}_{X}))=\mathrm{ev}_{T(X)}(s^{l}_{T(X)}T({}^{\vee}{\mu_{X}})\otimes\mathrm{id}_{T(X)})T_{2}({}^{\vee}{T(X)},X):T({}^{\vee}{T(X)}\otimes
X)\rightarrow\mathtt{1}$ $\displaystyle(\mu_{X}\otimes
s^{l}_{X})T_{2}(T(X),{}^{\vee}{T(X)})T(\mathrm{coev}_{T(X)})=(\eta_{X}\otimes\mathrm{id}_{{}^{\vee}{X}})\mathrm{coev}_{X}T_{0}:T(\mathtt{1})\rightarrow
T(X)\otimes{}^{\vee}{X}$

Symmetrically, a bimonad $T$ on a right rigid monoidal category is called a
_right Hopf_ monad if there exists a natural transformation
$s^{r}_{X}:T(T(X)^{\vee})\rightarrow X^{\vee}$ satisfying

$\displaystyle
T_{0}T(\underline{\mathrm{ev}}_{X}(\mathrm{id}_{X}\otimes\eta_{X}^{\vee}))=\underline{\mathrm{ev}}_{T(X)}(\mathrm{id}_{T(X)}\otimes
s^{r}_{T(X)}T(\mu_{X}^{\vee}))T_{2}(X,T(X)^{\vee}):T(X\otimes
T(X)^{\vee})\rightarrow\mathtt{1}$
$\displaystyle(\mathrm{id}_{X^{\vee}}\otimes\eta_{X})\underline{\mathrm{coev}}_{X}T_{0}=(s^{r}_{X}\otimes\mu_{X})T_{2}(T(X)^{\vee},T(X))T(\underline{\mathrm{coev}}_{T(X)}):T(\mathtt{1})\rightarrow
X^{\vee}\otimes T(X)$

We say a bimonad $T$ on a rigid category is a _Hopf_ monad if it is both left
Hopf and right Hopf. The natural transformation $s^{l}$ (resp. $s^{r}$) is
called a left (right) _antipode_ for the bimonad $T$.

This notion of antipode generalises the notion of antipode for Hopf algebras
in the following way: If $\mathcal{C}$ is a rigid braided monoidal category
and $B$ a Hopf algebra in it, then its corresponding bimonad $\mathbb{B}$,
obtains a left and right antipode:

$\displaystyle s^{l}_{X}$
$\displaystyle=(\mathrm{id}_{{}^{\vee}{X}}\otimes\mathrm{ev}_{B})(\mathrm{id}_{{}^{\vee}{X}\otimes{}^{\vee}{B}}\otimes
S)(\Psi_{B,{}^{\vee}{X}\otimes{}^{\vee}{B}})\longleftrightarrow
S=(s^{l}_{\mathtt{1}}\otimes\mathrm{id}_{B})(\mathrm{id}_{B}\otimes\Psi^{-1}_{B,{}^{\vee}{B}}\mathrm{coev}_{B})$
$\displaystyle s^{r}_{X}$
$\displaystyle=(\mathrm{id}_{X^{\vee}}\otimes\underline{\mathrm{ev}}_{B}(S^{-1}\otimes\mathrm{id}_{B^{\vee}}))(\Psi_{B,X^{\vee}}\otimes\mathrm{id}_{B^{\vee}})\longleftrightarrow
S^{-1}=(s^{r}_{\mathtt{1}}\otimes\mathrm{id}_{B})(\mathrm{id}_{B}\otimes\underline{\mathrm{coev}}_{B})$

where we have used the isomorphisms ${}^{\vee}{(B\otimes
X)}\cong{}^{\vee}{X}\otimes{}^{\vee}{B}$ and $(B\otimes X)^{\vee}\cong
X^{\vee}\otimes B^{\vee}$. The conditions in Definition 3.6 simply require $S$
to satisfy the usual antipode conditions. Note that for a braided bialgebra
$B$ in a rigid category, the bimonad $\mathbb{B}$ being only right Hopf
implies the existence of an _opantipode_ $S^{\prime}$ in the place of
$S^{-1}$. An opantipode is simply a map $S^{\prime}:B\rightarrow B$ satisfying
the basic properties which the inverse of an antipode would satisfy i.e.
$m(S^{\prime}\otimes\mathrm{id}_{B})\Psi^{-1}_{B,B}\Delta=\eta\epsilon=m(\mathrm{id}_{B}\otimes
S^{\prime})\Psi_{B,B}^{-1}\Delta$.

The reader should note that because the definitions of Hopf monads with
antipodes require the base category to be left or right rigid, this theory
generalises the theory of _dualisable_ braided Hopf algebras and _finite-
dimensional_ Hopf algebras. For any Hopf algebra $B$ in a category
$\mathcal{C}$, the endofunctor $B\otimes-$ does not restrict to an endofunctor
on the (left or right) rigid subcategory of $\mathcal{C}$ unless $B$ itself is
(left or right) dualisable. In particular, in the case of
$\mathcal{C}=\mathsf{Vec}$, we need $H$ to be finite-dimensional to obtain an
endofunctor on $\mathsf{Vec}_{\mathrm{fd}}$.

Now we recall how the existence of left and right antipodes relates to the
lifting of left and right dualities to $\mathcal{C}^{T}$:

######  Theorem 3.7 (Theorem 3.8 [14]).

If $\mathcal{C}$ is left (right) rigid, and $T$ a bimonad on $\mathcal{C}$
then $T$ is left (right) Hopf if and only if $\mathcal{C}^{T}$ is left (right)
rigid.

#### Sketch of proof.

If $X$ is a $T$-module with action $r:T(X)\rightarrow X$, then the left and
right antipodes, if they exist, provide the following actions on
${}^{\vee}{X}$ and $X^{\vee}$, receptively:

$s^{l}_{X}T\left({}^{\vee}{r}\right):T({}^{\vee}{X})\rightarrow{}^{\vee}{X},\quad\quad
s^{r}_{X}T(r^{\vee}):T(X^{\vee})\rightarrow X^{\vee}$ (30)

Furthermore, with these actions the usual evaluation and coevaluation
morphisms lift to $\mathcal{C}^{T}$ as $T$-module morphisms. For further
details on why the above morphisms are $T$-actions we refer the reader to
Theorem 3.8 of [14].

In the converse direction, assume an arbitrary object $(A,r)$ in
$\mathcal{C}^{T}$ has a dual $({}^{\vee}{A},\rho_{(A,r)})$ with a pair of
duality morphisms $\mathrm{ev}$ and $\mathrm{coev}$. Since $U$ is strict
monoidal then ${}^{\vee}{A}$ is a left dual of $A$ in $\mathcal{C}$. With this
notation one can show that $\rho$ is natural and the left antipode is
recovered as
${}^{\vee}{\eta_{A}}\rho_{F_{T}(A)}:T({}^{\vee}{T(A)})\rightarrow{}^{\vee}{A}$.
See the proof of Theorem 3.8 of [14] for more details. $\diamond$

We should note that the actions described in Theorem 3.7 are unique for each
choice of duality morphisms. In particular, in Lemma 3.9 of [14] it is proved
that the described action on ${}^{\vee}{X}$ is the unique $T$-action on
${}^{\vee}{X}$ which makes both $\mathrm{coev}_{X}$ and $\mathrm{ev}_{X}$ into
$T$-module morphisms.

As in the case of comonoidal adjunctions, one can define _Hopf adjunctions_
between rigid categories. An adjunction $F\dashv
U:\mathcal{D}\rightarrow\mathcal{C}$ is said to be (left or right) Hopf, if
$U$ is strong monoidal and both categories are (left or right) rigid. By
Theorem 3.7, for a (left or right) Hopf monad $T$, the free/forgetful
adjunction $F_{T}\dashv U_{T}$ becomes such an adjunction. In the converse
direction, if $F\dashv U$ is a (left or right) Hopf adjunction, in Theorem
3.14 of [14] it is shown that $UF$ becomes a (left or right) Hopf monad with
the antipodes being obtained in a similar fashion to the proof of Theorem 3.7.
In the next section, we will encounter a more general notion of Hopf
adjunction.

Given any functor $F:\mathcal{C}\rightarrow\mathcal{D}$ between rigid monoidal
categories, we can form a pair of functors $F^{!}$ and ${}^{!}F$ by

$F^{!}(X)=F({}^{\vee}{X})^{\vee},\hskip
56.9055pt\prescript{!}{}{F(X)}={}^{\vee}{F(X^{\vee})}$ (31)

Observe that if $F$ is strong monoidal then ${}^{!}F\cong F\cong F^{!}$.
Furthermore, it is not hard to see that if $F\dashv U$, then $U^{!}\dashv
F^{!}$ and $\prescript{!}{}{U}\dashv\prescript{!}{}{F}$. With these
observations in mind, consider the adjunction $F_{T}\dashv U_{T}$ for a Hopf
monad $T$ on a rigid monoidal category. By Theorem 3.7, $\mathcal{C}^{T}$ is
rigid and thereby there exists an adjunction $U_{T}^{!}\dashv F_{T}^{!}$.
Since $U_{T}\cong U_{T}^{!}$ then $U_{T}$ admits a right adjoint. Furthermore,
since $T^{!}=U_{T}^{!}F_{T}^{!}$ and $F_{T}\dashv U_{T}\cong U_{T}^{!}\dashv
F_{T}^{!}$, we obtain an adjunction $T\dashv T^{!}$. An alternate description
of this adjunction comes directly from the antipodes:

######  Theorem 3.8 (Proposition 3.11 [14]).

If $T$ is a Hopf monad on a rigid monoidal category $\mathcal{C}$, with
antipodes $s^{l}$ and $s^{r}$ then
$s^{r}_{{}^{\vee}{T(X)}}T((s^{l}_{X})^{\vee})=\mathrm{id}_{T(X)}=s^{l}_{T(X)^{\vee}}T({}^{\vee}{(s^{r}_{X})})$.
In particular, $(s^{l}_{-})^{\vee}$ and $s^{r}_{{}^{\vee}{(-)}}$ provide the
unit and counit for the adjunction $T\dashv T^{!}$.

######  Corollary 3.9.

Any Hopf monad $T$ on a rigid category admits a right adjoint and hence
preserves colimits.

First note that when working over rigid monoidal abelian (or triangulated)
categories, Corollary 3.9 allows one to classify Hopf monads using variations
of the Eilenberg-Watts Theorem. The reader should also keep in mind the well-
known fact that the forgetful functor $U_{T}$ of a colimit preserving monad
$T$ creates (rather than just preserves) colimits. We also observe that a
symmetric argument to Theorem 3.8 can be made to prove that
$T\dashv\prescript{!}{}{T}$ via
$\prescript{!}{}{U_{T}}\dashv\prescript{!}{}{F_{T}}$ and since adjoints are
unique upto isomorphism, then $\prescript{!}{}{T}\cong T^{!}$.

_Dual Notion:_ Based on the adjunction $T\dashv T^{!}$ and Theorem 2.12 the
functor $T^{!}$ obtains a comonad structure. The right adjoint $T^{!}$ also
obtains a monoidal structure as in Equation (27). In particular, $T^{!}$ has a
Hopf comonad structure, with a dual notion of antipodes, and its category of
comodules is isomorphic to $\mathcal{C}^{T}$. By a dual argument, we can send
any Hopf comonad $T$ to a Hopf monad ${}^{!}T$. Consequently, we see that for
a rigid monoidal category $\mathcal{C}$, $T\leftrightarrow T^{!}$ provides a
correspondence between Hopf monads and Hopf comonads on $\mathcal{C}$. We
should also note that bicomonads correspond to _monoidal adjunctions_ where
the left adjoint functor is strong monoidal .

###  3.3 Hopf Monads on General Monoidal Categories

In [12], a new Hopf condition for bimonads was introduced which can be applied
to arbitrary monoidal categories instead of rigid ones. In this section, we
briefly review this definition and its properties.

######  Definition 3.10.

Given a bimonad $T$ on a monoidal category $\mathcal{C}$, one obtains a pair
of natural transformations called the left and right _fusion operators_ ,
denoted by $H^{l}$ and $H^{r}$, respectively:

(34) (37)

The bimonad is said to be _left (right) Hopf_ , if the left (right) fusion
operator is invertible.

Observe that for the bimonad $\mathbb{B}$ corresponding to a bialgebra $B$ in
a braided category, we always have isomorphisms
$\mathbb{B}(X\otimes\mathbb{B}(Y))\cong\mathbb{B}(X)\otimes\mathbb{B}(Y)$ and
$\mathbb{B}(\mathbb{B}(X)\otimes Y)\cong\mathbb{B}(X)\otimes\mathbb{B}(Y)$ via
the braiding. However, these are due to the form of the functor and most
importantly these isomorphisms do not provide inverses for the fusion
operators. Translating the conditions of Definition 3.10 for the bimonad
$\mathbb{B}=B\otimes-$, we see that the left and right fusion operators being
invertible imply the invertibility of maps
$\mathsf{H}_{1}=(\mathrm{id}_{B}\otimes
m)(\Delta\otimes\mathrm{id}_{B})=H^{l}_{\mathtt{1},\mathtt{1}}$ and
$\mathsf{H}_{2}=(m\otimes\mathrm{id}_{B})(\mathrm{id}_{B}\otimes\Psi_{B,B})(\Delta\otimes\mathrm{id}_{B})=H^{r}_{\mathtt{1},\mathtt{1}}$,
respectively. In turn, the invertibility of $\mathsf{H}_{1}$ and
$\mathsf{H}_{2}$ correspond to $B$ admitting an antipode $S$ and an opantipode
$S^{\prime}$, respectively:

$\displaystyle
S=(\epsilon\otimes\mathrm{id}_{B})\mathsf{H}_{1}^{-1}(\mathrm{id}_{B}\otimes\eta)\longleftrightarrow\mathsf{H}_{1}^{-1}=(\mathrm{id}_{B}\otimes
m)(\mathrm{id}_{B}\otimes
S\otimes\mathrm{id}_{B})(\Delta\otimes\mathrm{id}_{B})$ $\displaystyle
S^{\prime}=(\epsilon\otimes\mathrm{id}_{B})\mathsf{H}_{2}^{-1}(\eta\otimes\mathrm{id}_{B})\longleftrightarrow\mathsf{H}_{2}^{-1}=(\mathrm{id}_{B}\otimes
m)(\mathrm{id}_{B}\otimes
S^{\prime}\otimes\mathrm{id}_{B})(\Psi_{B,B}^{-1}\Delta\otimes\mathrm{id}_{B})\Psi_{B,B}^{-1}$

As far as the author is aware, this interpretation of the map $\mathsf{H}_{1}$
first appeared in [61], where it was shown that $\mathsf{H}_{1}$ satisfies the
_fusion equation_. Due to the above correspondence, the invertibility of
$\mathsf{H}_{1}$ and $\mathsf{H}_{2}$ becomes equivalent to the invertibility
of the left and right fusion operators of $\mathbb{B}$:

$\displaystyle(H^{l}_{M,N})^{-1}=(\mathrm{id}_{B\otimes M}\otimes
m(S\otimes\mathrm{id}_{B})\otimes\mathrm{id}_{N})(\mathrm{id}_{B}\otimes\Psi_{B,M}\otimes\mathrm{id}_{B\otimes
N})(\Delta\otimes\mathrm{id}_{M\otimes B\otimes N})$
$\displaystyle(H^{r}_{M,N})^{-1}=\big{(}(\mathrm{id}_{B}\otimes
m)(\mathrm{id}_{B}\otimes
S^{\prime}\otimes\mathrm{id}_{B})(\Psi_{B,B}^{-1}\Delta\otimes\mathrm{id}_{B})\otimes\mathrm{id}_{M\otimes
N}\big{)}(\Psi_{B\otimes M,B}^{-1}\otimes\mathrm{id}_{N})$

As in previous sections, we now recall the appropriate notion for a comonoidal
adjunction to be Hopf. For a comonoidal adjunction $F\dashv
U:\mathcal{D}\leftrightarrows\mathcal{C}$, we can define analogous _fusion
operators_ :

(40) (43)

where $X$ and $Y$ are objects of $\mathcal{C}$ and $\mathcal{D}$,
respectively, and $\epsilon$ is the counit of the adjunction. Such an
adjunction with an invertible left (right) fusion operator is called a left
(right) _Hopf_ adjunction. Observe that the fusion operators of the adjunction
are families of morphisms in $\mathcal{D}$, whereas the fusion operators of
the bimonad are morphisms of $\mathcal{C}$.

For any bimonad $T=UF$, we have equalities

$H^{l}_{X,Y}=U_{2}(F(X),F(Y))U(\overline{H}^{l}_{X,F(Y)}),\hskip
28.45274ptH^{r}_{X,Y}=U_{2}(F(X),F(Y))U(\overline{H}^{r}_{F(X),Y})$

Since $U$ is strong monoidal and $U_{2}$ is invertible, then $H^{l}$ (resp.
$H^{r}$) being invertible follows from $\overline{H}^{l}$ (resp.
$\overline{H}^{r}$) being invertible. Hence, a left (right) Hopf adjunction in
this sense gives rise to a left (right) Hopf monad. In the converse direction,
a Hopf monad gives rise to such an adjunction as well:

######  Theorem 3.11 (Theorem 2.15 [12]).

A bimonad $T$ is a (left or right) Hopf monad, if and only if the adjunction
$F_{T}\dashv U_{T}:\mathcal{C}^{T}\leftrightarrows\mathcal{C}$ is a (left or
right) Hopf adjunction.

#### Sketch of proof.

We have already discussed one direction of the proof. For the second part,
assume that $T$ is either left or right Hopf. We will simply provide the
inverses to the fusion operators of $F_{T}\dashv
U_{T}:\mathcal{C}^{T}\leftrightarrows\mathcal{C}$. Let $X$ and $(M,r)$ be
objects of $\mathcal{C}$ and $\mathcal{C}^{T}$, respectively, and denote
$\overline{H}^{l}_{X,(M,r)}=f$ and $\overline{H}^{r}_{(M,r),X}=g$. Depending
on whether $H^{r}_{M,X}$ and $H^{r}_{M,X}$ are invertible, we obtain inverses
for $f$ and $g$ by

Since the above morphisms are in $\mathcal{C}^{T}$, one does not only need to
check that the above expressions provide the inverses for the fusion
operators, but also that they are $T$-module maps. We refer the reader to
Lemmas 2.18 and 2.19 in [12] for a detailed proof of these facts. $\diamond$

In [12], the authors also present the notion of left (right) _pre-Hopf_ monads
which are bimonads with invertible $H^{l}_{\mathtt{1},-}$ (resp.
$H^{r}_{-,\mathtt{1}}$). Several of the results proved in [12] only require
this weaker condition rather than the full Hopf condition. We will briefly
discuss when a bimonad being pre-Hopf is equivalent to it being Hopf in
Section 6. An example of a pre-Hopf monad which is not Hopf is provided in
Example 2.8 of [12].

###  3.4 Hopf Monads on Closed Monoidal Categories

In this section, we will look at the case where the base category
$\mathcal{C}$ is monoidal closed and describe the relation between the fusion
operators and the lifting of the closed structure of $\mathcal{C}$ to
$\mathcal{C}^{T}$. To do this, we need to recall the theory of liftings from
Section 3.5 of [12]. In [12] a new set of binary antipodes were also
introduced, in addition to the ordinary antipodes of [14]. Without delving too
much into the details of proofs we will present the definitions of these
notions.

First, let us recall how an ordinary Hopf algebra $H$ with an invertible
antipode lifts the closed structure of $\mathsf{Vec}$ to its category of
modules. Given any pair of $H$-modules $(V,\triangleright)$ and
$(W,\triangleright^{\prime})$, we obtain two $H$-actions $\triangleright^{l}$
and $\triangleright^{r}$ on $\mathrm{Hom}_{\mathbb{K}}(V,W)$ defined by
$h\triangleright^{l}f=h_{(1)}\triangleright^{\prime}f(S(h_{(2)})\triangleright-)$
and
$h\triangleright^{r}f=h_{(1)}\triangleright^{\prime}f(S^{-1}(h_{(2)})\triangleright-)$
for $f\in\mathrm{Hom}_{\mathbb{K}}(V,W)$. In this way, the unit and counit of
$\mathrm{Hom}_{\mathbb{K}}(V,-)\dashv-\otimes_{\mathbb{K}}V\cong
V\otimes_{\mathbb{K}}-$ become $H$-module morphisms so that endofunctors
$(\mathrm{Hom}_{\mathbb{K}}(V,-),\triangleright^{l})$ and
$(\mathrm{Hom}_{\mathbb{K}}(V,-),\triangleright^{r})$ on ${}_{H}{\mathcal{M}}$
become right adjoint to $-\otimes_{\mathbb{K}}(V,\triangleright)$ and
$(V,\triangleright)\otimes_{\mathbb{K}}-$, respectively. A similar statement
can be made for braided Hopf algebras in arbitrary closed braided categories.

Now we recall how adjunctions on a base category can lift to the category of
modules over a monad. In full generality, given monads $(T,\mu,\eta)$ and
$(T^{\prime},\mu^{\prime},\eta^{\prime})$ on categories $\mathcal{C}$ and
$\mathcal{C}^{\prime}$, respectively, a _lift_ of functor
$L:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ along $(T,T^{\prime})$ is a
functor $\overline{L}:\mathcal{C}^{T}\rightarrow\mathcal{C}^{\prime
T^{\prime}}$ such that $U_{T^{\prime}}\overline{L}=LU_{T}$ holds. It is a
well-known fact that lifts $\overline{L}$ correspond to natural
transformations $\theta:T^{\prime}L\rightarrow LT$ which satisfy
$\theta\mu^{\prime}_{L}=G(\mu)\theta_{T}T^{\prime}(\theta)$ and
$\theta\eta^{\prime}_{L}=L(\eta)$. Such a natural transformation $\theta$ is
referred to as a _lifting datum_ and defines $\overline{L}$ by
$\overline{L}(M,r)=(L(M),(Lr)\theta_{M})$, where $(M,r)$ is an object of
$\mathcal{C}^{T}$. In the converse direction, if $\overline{L}$ exists and
sends any free module $(T(M),\mu_{M})$ to a $T^{\prime}$-module
$(LT(M),\rho_{M})$, we obtain $\theta=\rho(T^{\prime}L\eta)$.

If in the above scenario, $L$ admits a right adjoint
$R:\mathcal{C}^{\prime}\rightarrow\mathcal{C}$, then liftings $\overline{R}$
of $R$ which are right adjoint to $\overline{L}$ together with a unit and
counit which lift the unit and counit of the original adjunction $L\dashv R$,
are in bijection with special lifting data $\xi:TR\rightarrow RT^{\prime}$
which satisfy additional compatibility conditions (Equations (3a)-(3f) [12]).
In Theorem 3.13 of [12], it is proved that such a $\xi$ exists if and only if
the lifting datum $\theta$ of $L$ is invertible. If so, then $\xi$ is uniquely
determined by the expression $\xi=RT^{\prime}(e)R(\theta^{-1}_{R})h_{TR}$,
where $h$ and $e$ denote the unit and counit of $L\dashv R$.

The case which concerns us is when $\mathcal{C}=\mathcal{C}^{\prime}$ and
$T=T^{\prime}$ is a bimonad on $\mathcal{C}$. In this situation, any
$T$-module $(M,r)$ provides a lifting $\overline{L}=-\otimes(M,r)$ of
$L=-\otimes M$. In particular, the left fusion operator of $F_{T}\dashv U_{T}$
provides a lifting datum $\overline{H}_{-,(M,r)}:F_{T}(-\otimes
M)\longrightarrow F_{T}(-)\otimes M$. If $\mathcal{C}$ is left closed and the
adjunction in consideration is that of $L=-\otimes M$ and $R=[M,-]^{l}$, then
the mentioned theory shows that the left fusion operator is invertible if and
only if $\overline{L}=-\otimes(M,r)$ has a right adjoint $\overline{R}$
lifting $[M,-]^{l}$. We denote $\overline{R}$ by $[(M,r),-]^{l}$ and observe
that

$[(M,r),(N,t)]^{l}=\left([M,N]^{l},\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.18741pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T[M,N]^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
27.3852pt\raise 6.86389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.49722pt\hbox{$\scriptstyle{\xi^{(M,r)}_{N}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 51.87787pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
51.87787pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{[M,T(N)]^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
107.47641pt\raise 7.19444pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.44444pt\hbox{$\scriptstyle{[M,t]^{l}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 133.72095pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
133.72095pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{[M,N]^{l}}$}}}}}}}\ignorespaces}}}}\ignorespaces\right)$
(44)

where $\xi^{(M,r)}_{N}$ itself decomposes as

$\textstyle{T[M,N]^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{(M,r)}_{N}}$$\scriptstyle{\mathrm{coev}^{M}_{T[M,N]^{l}}}$$\textstyle{[M,T(N)]^{l}}$$\textstyle{[M,T[M,N]^{l}\otimes
M]^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left[M,\left(\overline{H}_{N,(M,r)}\right)^{-1}\right]^{l}}$$\textstyle{[M,T([M,N]^{l}\otimes
M)]^{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[M,T(\mathrm{ev}^{M}_{N})]^{l}}$

Hence, over closed monoidal categories the invertibility of the fusion
operators for a bimonad become equivalent to the lifting of the closed
structures of the base category to the category of modules over the monad.

######  Theorem 3.12 (Theorem 3.6 [12]).

If $T$ is a bimonad on a left (resp. right) closed monoidal category
$\mathcal{C}$, then $T$ is left (resp. right) Hopf if and only if the category
$\mathcal{C}^{T}$ is left (resp. right) closed and $U_{T}$ preserves the
closed structure.

#### Sketch of proof.

The only subtlety which we have not addressed is that $U_{T}$ preserving the
closed structure of $\mathcal{C}^{T}$, in the sense described in Section 2.1,
is truly equivalent to the adjunctions $-\otimes M\dashv[M,-]^{l}$ lifting to
$\mathcal{C}^{T}$. If the latter holds then it is clear that $\mathcal{C}^{T}$
becomes left closed and $U_{T}$ preserves this structure with
$U_{T}^{l}=\mathrm{id}$. In the other direction, $U_{T}$ preserving the closed
structure provides an isomorphism of bifunctors
$U_{T}^{l}:U_{T}[,]_{\mathcal{C}^{T}}^{l}\rightarrow[U_{T},U_{T}]^{l}$ and
thereby we can define a left closed structure on $\mathcal{C}^{T}$ with
$[,]_{\mathcal{C}^{T}}^{l}=[U_{T},U_{T}]^{l}$. For more details on this
correspondence, see Lemma 3.15 and more generally Section 3 of [12].
$\diamond$

In the particular case where $\mathcal{C}$ is closed, Section 3.3 of [12]
provides an equivalent formulation of the Hopf condition for bimonads. First
note that $\xi$ is natural in term $(M,r)$ as well i.e. $\xi$ can be viewed as
a natural transformation between bifunctors
$\xi:T[U_{T},\mathrm{id}_{\mathcal{C}}]^{l}\rightarrow[U_{T},T]^{l}$. Using
the adjunction $F_{T}\dashv U_{T}$, we obtain a bijection between ${\rm
Mor}_{\mathcal{C}^{T}}(T[U_{T},N]^{l},[U_{T},T(N)]^{l})$ and ${\rm
Mor}_{\mathcal{C}}(T[T,N]^{l},[\mathrm{id}_{\mathcal{C}},T(N)]^{l})$ for any
object $N$ in $\mathcal{C}$. The corresponding morphism to $\xi_{N}$ under
this bijection is
$[\eta_{-},N]^{l}\xi^{F_{T}(-)}_{N}:T[T(-),N]^{l}\rightarrow[-,T(N)]^{l}$ and
is denoted by $\overline{s}^{l}_{-,N}$. In this way,
$\xi^{(M,r)}_{N}=\overline{s}^{l}_{M,N}T[r,N]^{l}$. The family of morphisms
$\overline{s}^{l}$ are called the _left binary antipodes_ and satisfy certain
compatibility conditions (Equations (1a) and (1b) [12]), which can be written
without reference to the fusion operators. A notion of right binary antipode
$\overline{s}^{r}_{M,N}:T[T(M),N]^{r}\rightarrow[M,T(N)]^{r}$ can also be
defined in a symmetric manner.

In the case where $T$ corresponds to an ordinary Hopf algebra $H$, for an
arbitrary $H$-module $(M,\triangleright)$ the binary antipode and
$\xi^{(M,\triangleright)}_{N}$ take the following forms:

$\displaystyle\overline{s}^{l}_{M,N}(h\otimes_{\mathbb{K}}g)=$
$\displaystyle\big{(}m\mapsto
h_{(1)}\otimes_{\mathbb{K}}g(S(h_{(2)})\otimes_{\mathbb{K}}m)\big{)}\in\mathrm{Hom}_{\mathbb{K}}(M,H\otimes_{\mathbb{K}}N)$
$\displaystyle\xi^{(M,\triangleright)}_{N}(h\otimes_{\mathbb{K}}f)=$
$\displaystyle\big{(}m\mapsto
h_{(1)}\otimes_{\mathbb{K}}f(S(h_{(2)})\triangleright
m)\big{)}\in\mathrm{Hom}_{\mathbb{K}}(M,H\otimes_{\mathbb{K}}N)$

where $h,h^{\prime}\in H$, $f\in\mathrm{Hom}_{\mathbb{K}}(M,N)$ and
$g\in\mathrm{Hom}_{\mathbb{K}}(H\otimes_{\mathbb{K}}M,N)$.

Since $\overline{s}^{l}$ was determined as the unique map corresponding to
$\xi$ under the adjunction, it should be clear that for a closed monoidal
category, the existence of the binary antipodes is equivalent to the fusion
operators being invertible. For a detailed proof of this correspondence we
refer the reader to Theorem 3.6 of [12]. Here we will simply recall how the
fusion operators and binary antipodes can be described in terms of each other
from Proposition 3.9 of [12]:

$\displaystyle(H_{M,N}^{l})^{-1}=$ $\displaystyle
T(M\otimes\mu_{N})\mathrm{ev}^{T(N)}_{T(M\otimes
TT(N))}\left(s^{l}_{T(N),M\otimes
TT(N)}T\big{(}\mathrm{coev}_{M}^{TT(N)}\big{)}\otimes\mathrm{id}_{T(N)}\right)$
(45) $\displaystyle(H_{M,N}^{r})^{-1}=$ $\displaystyle T(\mu_{M}\otimes
N)\underline{\mathrm{ev}}^{T(M)}_{T(TT(M)\otimes
N)}\left(\mathrm{id}_{T(M)}\otimes s^{r}_{T(M),TT(M)\otimes
N}T\big{(}\underline{\mathrm{coev}}_{N}^{TT(M)}\big{)}\right)$ (46)
$\displaystyle s^{l}_{M,N}=$
$\displaystyle\left[M,\mathrm{ev}^{T(M)}_{N}\right]^{l}\left[\eta_{M},(H^{l}_{M,[T(M),N]^{l}})^{-1}\right]^{l}\mathrm{coev}_{T[T(M),N]^{l}}^{T(M)}$
(47) $\displaystyle s^{r}_{M,N}=$
$\displaystyle\left[M,\underline{\mathrm{ev}}^{T(M)}_{N}\right]^{r}\left[\eta_{M},(H^{r}_{M,[T(M),N]^{l}})^{-1}\right]^{r}\underline{\mathrm{coev}}_{T[T(M),N]^{r}}^{T(M)}$
(48)

The particular benefit of considering the binary antipodes is that the
$T$-actions on the inner-homs are simplified. In particular, the $T$-action on
$[M,N]^{l}$ for $T$-modules $(M,r)$ and $(N,t)$, described in (44), becomes
$[M,t]^{l}\overline{s}^{l}_{M,N}T[r,N]^{l}$.

Finally, we should comment on why the notions of Hopf monads on rigid
categories from Definition 3.6 and those defined for general monoidal
categories in Definition 3.10 agree when the base monoidal category is rigid:

######  Theorem 3.13.

If $\mathcal{C}$ is a left (right) rigid category and $T$ a bimonad on it,
then $T$ is a left (right) Hopf monad as in Definition 3.6 if and only if it
is a left (right) Hopf monad as in Definition 3.10.

#### Sketch of proof.

This statement follows simply by Theorems 3.7 and 3.12 and the fact that any
left (right) rigid category is left (right) closed with
$[M,N]^{l}=N\otimes{}^{\vee}{M}$ (resp. $[M,N]^{r}=M^{\vee}\otimes N$).
Alternatively, one can prove Theorem 3.13 by directly showing how the binary
and ordinary antipodes are related:

$\displaystyle\overline{s}^{l}_{M,N}$ $\displaystyle=(T(N)\otimes
s^{l}_{M})T_{2}(N,{}^{\vee}{T(M)}),$ $\displaystyle s^{l}_{X}$
$\displaystyle=(T_{0}\otimes{}^{\vee}{X})\overline{s}^{l}_{X,\mathtt{1}}$ (49)
$\displaystyle\overline{s}^{r}_{M,N}$ $\displaystyle=(s^{r}_{M}\otimes
T(N))T_{2}(T(M)^{\vee},N),$ $\displaystyle s^{r}_{X}$
$\displaystyle=(X^{\vee}\otimes T_{0})\overline{s}^{r}_{X,\mathtt{1}}$ (50)

The relation between these antipodes is presented in Remark 3.11 of [12] with
some minor mistakes, which we have corrected here. $\diamond$

_Cartesian Case:_ Adjunctions between cartesian closed categories $L\dashv
R:\mathcal{D}\leftrightarrows\mathcal{C}$ where the right adjoint $R$ is also
cartesian closed had been considered long before the appearance of Hopf
monads. It was also well-known that $L$ being closed becomes equivalent to the
invertibility of the fusion operator (40) (see Lemma A1.5.8 of [32]). The
latter condition in the cartesian setting is often referred to as the
_Frobenius reciprocity law_ for $L\dashv R$ in the literature.

##  4 Examples

In this section we present various examples of Hopf monads. In [9] several
structures such as ordinary Hopf algebras, Hopf algebroids and bimonoids in
duoidal categories are described as examples of bimonads. Here we review some
of these examples and provide some novel examples of our own.

###  4.1 Central Hopf Algebras as Hopf monads

In this section we review the theory of augmented Hopf monads from [12] and
recall a criteria for telling when a Hopf monad arises from a braided Hopf
algebra.

So far we have described how any bialgebra (Hopf algebra) $B$ in a braided
category $\mathcal{C}$ gives rise to a bimonad (left Hopf monad) on
$\mathcal{C}$. If the category is monoidal but does not admit a braiding, then
a bialgebra $(B,\tau)$ in the lax center of the category
$Z_{l,\mathrm{lax}}(\mathcal{C})$ would also produce a bimonad structure on
the endofunctor $B\otimes-$ in the same way. The underlying monad structure
will still be defined by the algebra structure of $B$, while for defining the
comonoidal structure of $B\otimes-$ we simply replace the use of the braiding
of the category $\Psi_{B,-}$ by the braiding $\tau_{-}$ which $(B,\tau)$ comes
equipped with. We follow the notation of [12] and call $(B,\tau)$ a _central
(Hopf) bialgebra_ and denote its corresponding (Hopf) bimonad by
$B\otimes_{\tau}-$. Note that any braided bialgebra $B$ in a braided category
$(\mathcal{C},\Psi)$ can be viewed as a central bialgebra $(B,\Psi_{B,-})$.

In [12] the authors provided a characterisation for Hopf monads which arise
from central Hopf algebras. An arbitrary bimonad $T$ on a monoidal category is
called _augmented_ if there exists a bimonad morphism
$\epsilon:T\rightarrow\mathrm{id}_{C}$. Given a central bialgebra $(B,\tau)$,
the bimonad $B\otimes_{\tau}-$ automatically becomes augmented via the counit
map $\epsilon\otimes\mathrm{id}_{\mathcal{C}}:T=B\otimes-\rightarrow-$.
Without acknowledging this augmentation morphism, the counit is hidden in the
comonoidal structure, as a morphism from $T(\mathtt{1})$ to $\mathtt{1}$.

######  Theorem 4.1 (Theorem 5.7 [12]).

Let $\mathcal{C}$ be a monoidal category. There is an equivalence of
categories between the category of central Hopf algebras in $\mathcal{C}$ and
that of augmented left Hopf monads on $\mathcal{C}$.

#### Sketch of proof.

We have already mentioned one direction of the argument. For the converse
direction, the difficulty is showing that for an augmented left Hopf monad
$T$, $T(\mathtt{1})$ becomes a central Hopf algebra and that its corresponding
left Hopf monad is isomorphic to $T$. First observe that given an augmentation
$\epsilon:T\rightarrow\mathrm{id}_{\mathcal{C}}$ we can reconstruct the
following natural transformations:

$\displaystyle u^{\epsilon}$
$\displaystyle=(T(\mathtt{1})\otimes\epsilon)T_{2}(\mathtt{1}\otimes\mathrm{id}_{\mathcal{C}}):T\rightarrow
T(\mathtt{1})\otimes\mathrm{id}_{\mathcal{C}}$ (51) $\displaystyle
v^{\epsilon}$ $\displaystyle=(\epsilon\otimes
T(\mathtt{1}))T_{2}(\mathrm{id}_{\mathcal{C}}\otimes\mathtt{1}):T\rightarrow\mathrm{id}_{\mathcal{C}}\otimes
T(\mathtt{1})$ (52)

In Lemma 5.16 of [12], it is shown that when $T$ is left Hopf, then
$u^{\epsilon}$ is invertible and its inverse is given by
$T(\epsilon)(H^{l}_{\mathtt{1},-})^{-1}(T(\mathtt{1})\otimes\eta)$. In Lemma
5.15 [12], it is demonstrated that
$\tau:=v^{\epsilon}(u^{\epsilon})^{-1}:T(\mathtt{1})\otimes\mathrm{id}_{\mathcal{C}}\rightarrow\mathrm{id}_{\mathcal{C}}\otimes
T(\mathtt{1})$ provides a lax braiding for $T(\mathtt{1})$ and moreover that
$u^{\epsilon}:T\rightarrow T(\mathtt{1})\otimes_{\tau}-$ is an isomorphism of
bimonads. One can thereby define the central Hopf algebra structure on
$T(\mathtt{1})$ via this isomorphism. $\diamond$

First observe that a symmetric argument can prove a similar statement for
augmented right Hopf monads and bialgebras in the lax right center
$Z_{r,\mathrm{lax}}(\mathcal{C})$ which admit opantipodes. Secondly, we note
that the use of Lemma 5.16 from [12] is essential and without knowing that the
left fusion operator is invertible, we can not provide an inverse for
$u^{\epsilon}$. In [12], augmented bimonads $(T,\epsilon)$ for which
$u^{\epsilon}$ is invertible are called _left regular_. By the same argument
as above, we can describe an equivalence between central bialgebras in
$\mathcal{C}$ and left regular augmented bimonads on $\mathcal{C}$ [Theorem
5.17 [12]]. We should also note that for the corresponding bimonad
$B\otimes_{\tau}-$ of a central bialgebra $(B,\tau)$, we have
$u^{\epsilon}=\mathrm{id}_{B}\otimes\mathrm{id}_{\mathcal{C}}$ and
$v^{\epsilon}=\tau$.

Observe that for an arbitrary central bialgebra $(B,\tau)$ in a monoidal
category $\mathcal{C}$, the module category $\mathcal{C}^{B\otimes_{\tau}-}$
can not be viewed as the module category over a bialgebra in $\mathcal{C}$
itself. Hence, although these bimonads correspond to braided bialgebras in
some braided category, namely $Z_{l,\mathrm{lax}}(\mathcal{C})$, they are the
first genuine examples of bimonads on a base category $\mathcal{C}$ that go
beyond the study of bialgebras within $\mathcal{C}$ itself. In the next
sections we will discuss examples of Hopf monads which can not be augmented,
but first we can look at an application of this theory which was described in
[16].

It is well-known that for any given Hopf algebra $B$ in a braided category
$(\mathcal{D},\Psi)$, the center of its category of modules
$Z_{l,\mathrm{lax}}({}_{B}\mathcal{D})$ has an equivalent formulation as the
category of _Yetter-Drinfeld_ modules ${}^{B}_{B}\mathcal{YD}$ over $B$. The
category ${}^{B}_{B}\mathcal{YD}$ consists of triples
$(M,\triangleright,\delta)$ where $M$ is an object of $\mathcal{D}$,
$\triangleright:B\otimes M\rightarrow M$ is a $B$-action and
$\delta:M\rightarrow B\otimes M$ a $B$-coaction satisfying the following
compatibility condition:

$(m\otimes\triangleright)(\mathrm{id}_{H}\otimes\Psi_{H,H}\otimes\mathrm{id}_{M}\\!)(\Delta\otimes\delta)\\!=\\!(m\otimes\mathrm{id}_{M}\\!)(\mathrm{id}_{H}\otimes\Psi_{M,H})(\delta\triangleright\otimes\mathrm{id}_{M}\\!)(\mathrm{id}_{H}\otimes\Psi_{H,M})(\Delta\otimes\mathrm{id}_{M}\\!)$
(53)

The equivalence between $Z_{l,\mathrm{lax}}({}_{B}\mathcal{D})$ and
${}^{B}_{B}\mathcal{YD}$ is given by sending a Yetter-Drinfeld module
$(M,\triangleright,\delta)$ to $(M,\triangleright)$ with the lax braiding
$\tau_{(N,r)}=(r\otimes\mathrm{id}_{M})(\mathrm{id}_{H}\otimes\Psi_{M,N})(\delta\otimes\mathrm{id}_{N})$.
In the converse direction, any $B$-module $(M,\triangleright)$ with a lax
braiding $\tau$ is sent to
$(M,\triangleright,\tau_{H}(\mathrm{id}_{M}\otimes\eta))$. The resulting
braiding on the category ${}^{B}_{B}\mathcal{YD}$ is given by
$(\triangleright_{N}\otimes\mathrm{id}_{M})(\mathrm{id}_{H}\otimes\Psi_{M,N})(\delta_{M}\otimes\mathrm{id}_{N})$
for a pair of objects $(M,\triangleright_{M},\delta_{M})$ and
$(N,\triangleright_{N},\delta_{N})$. Note that when the antipode of $B$ is
invertible then $Z_{l,\mathrm{lax}}({}_{B}\mathcal{D})=Z({}_{B}\mathcal{D})$.
We refer the reader to [6] for additional details on Yetter-Drinfeld modules
in braided categories.

If we start with a monoidal category $\mathcal{C}$ and pick a braided Hopf
algebra $B$ in $Z_{l,\mathrm{lax}}(\mathcal{C})$, it is an interesting
question whether we can express the category
${}^{B}_{B}\mathcal{YD}\left(Z_{l,\mathrm{lax}}(\mathcal{C})\right)\equiv
Z_{l,\mathrm{lax}}\left({}_{B}Z_{l,\mathrm{lax}}(\mathcal{C})\right)$ as the
center of a category of modules over a bialgebra-like structure in
$\mathcal{C}$ itself. This gap is precisely filled by the theory of Hopf
monads:

######  Theorem 4.2.

[Proposition 2.13 [16]] If $(B,\tau)$ is a braided Hopf algebra in the center
of $\mathcal{C}$, then there exists a braided monoidal isomorphism
${}^{(B,\tau)}_{(B,\tau)}\mathcal{YD}\left(Z_{l,\mathrm{lax}}(\mathcal{C})\right)\cong
Z_{l,\mathrm{lax}}\left(\mathcal{C}^{B\otimes_{\tau}-}\right)$.

Note that ${}_{(B,\tau)}Z_{l,\mathrm{lax}}(\mathcal{C})$ and
$\mathcal{C}^{B\otimes_{\tau}-}$ will generally not be the equivalent, while
the above result tells us that their centres are.

###  4.2 Hopf Monads on $\mathbb{N}_{0}$

In this section we will classify Hopf monads on the monoidal category
$(\mathbb{N}_{0},+,0)$. The category $\mathbb{N}_{0}$ has natural numbers
(including $0$) as objects and morphisms are given by the natural order $\leq$
on $\mathbb{N}_{0}$. Explicitly, for two numbers $m,n\in\mathbb{N}_{0}$, there
exists a unique arrow $\mathrm{min}(m,n)\rightarrow\mathrm{max}(m,n)$. Recall
that more generally, any poset $(P,\leq)$ can be viewed as a category in the
same way. The category $\mathbb{N}_{0}$ also obtains a symmetric monoidal
structure via addition $+$ and $0$ acting as the monoidal unit.

It is well-known that monads $(T,\mu,\eta)$ on poset categories $(P,\leq)$
correspond to _closure operators_ on $(P,\leq)$ i.e. an order-preserving map
$T:P\rightarrow P$ such that $p\leq T(p)$ and $T(p)=T^{2}(p)$ for any $p\in P$
(Example 5.1.7 [54]).

For the particular case of $\mathbb{N}_{0}$, closure operators correspond to
infinite subsets of $\mathbb{N}_{0}$. Given a closure operators $T$, since
$n\leq T(n)$, the set $S_{T}:=\\{T(n)\mid n\in\mathbb{N}_{0}\\}$ defines an
infinite subset of $\mathbb{N}_{0}$. Additionally, for any pair of numbers $m$
and $k$ which satisfy $m\leq m+k\leq T(m)$, we have that $T(m)\leq T(m+k)\leq
T(m)$ and thereby $T(m+k)=T(m)$. Hence $T(n)$ for any $n\in\mathbb{N}_{0}$ is
defined solely by $m_{n}:=\mathrm{min}\\{m\in S_{T}\mid n\leq m\\}$: By
definition $T(n)\leq T(m_{n})=m_{n}$ and since $T(n)\in\\{m\in S_{T}\mid n\leq
m\\}$, then $T(n)=m_{n}$. In the converse direction, we obtain a closure
operator $T$ for any infinite subset $S$ of $\mathbb{N}_{0}$ defined by
$T(n)=\mathrm{min}\\{m\in S\mid n\leq m\\}$.

######  Lemma 4.3.

Bimonads on $(\mathbb{N}_{0},+,0)$ correspond to non-trivial (not $\\{0\\}$)
submonoids of $(\mathbb{N}_{0},+)$.

###### Proof.

Assume $(T,\mu,\eta)$ is defined by an infinite subset
$S\subseteq\mathbb{N}_{0}$ and has a compatible comonoidal structure
$(T_{2},T_{0})$. Firstly note that the existence of $T_{0}$ implies that
$0\leq T(0)\leq 0$, and thereby $T(0)=0\in S$. Secondly, $T_{2}$ implies that
$T(m+n)\leq T(m)+T(n)$ for all pairs $m,n\in\mathbb{N}_{0}$. In particular,
for $m^{\prime},n^{\prime}\in S$ we have that $m^{\prime}+n^{\prime}\leq
T(m^{\prime}+n^{\prime})\leq m^{\prime}+n^{\prime}$. Thereby $S$ must be
closed under addition. This condition is in fact necessary and sufficient
since for any $m,n\in\mathbb{N}_{0}$ we have $m+n\leq T(m)+T(n)$ and thereby
$T(m+n)\leq T(T(m)+T(n))=T(m)+T(n)$. The compatibility conditions of
Definition 3.1 do not need to be checked, since there exists a unique morphism
between any two objects in a poset category. ∎

######  Theorem 4.4.

Hopf monads on $(\mathbb{N}_{0},+,0)$ are in bijection with submonoids
$\langle n\rangle$ of $(\mathbb{N}_{0},+)$ which are generated by a single
non-zero element $n\in\mathbb{N}_{0}$.

###### Proof.

It is easy to show that any Hopf monad $T$ on $(\mathbb{N}_{0},+,0)$ is
uniquely determined by $T(1)$. The left fusion operator being invertible
implies that $T(T(n)+m)=T(n)+T(m)$. For any $n\in S_{T}$,
$T(n+1)=\mathrm{min}\\{m\in S_{T}\mid n\lneq m\\}$. By the Hopf condition
$T(n+1)=n+T(1)$. Consequently, $S_{T}=\\{0,T(1),2T(1),\ldots\\}=\langle
T(1)\rangle$.

In the converse direction, assume that $S_{T}$ has the mentioned form and for
any $m,n\in\mathbb{N}_{0}$, there exist $k,l\in\mathbb{N}_{0}$ such that
$T(m)=kT(1)$ and $T(n)=lT(1)$. In this case, $T(kT(1)+n)=k^{\prime}T(1)$ where
$k^{\prime}$ is minimal so that $kT(1)+n\leq k^{\prime}T(1)$. Consequently,
$l=k^{\prime}-k$ and $T(T(n)+m)=T(n)+T(m)$ holds. ∎

Another interpretation of the above results comes from looking at the
Eilenberg-Moore category. Any $T$-module $n$ for a closure operator must
satisfy $T(n)=n$. Therefore $\mathcal{C}^{T}$ is precisely the set $S_{T}$
defined above. From this perspective Lemma 4.3 is a consequence of Theorem
3.2. Also note that the Hopf monads from Theorem 4.4 are only augmented if
$T(1)=1$ and $S_{T}=\mathbb{N}_{0}$.

###  4.3 Hopf Monads on $\mathsf{Set}$

In this section we will classify finitary Hopf monads on the category
$(\mathsf{Set},\times,1)$. It is well-known that _finitary_ monads
$(T,\mu,\eta)$, meaning monads where $T$ preserves filtered-colimits, on
$\mathsf{Set}$ correspond to _Lawvere theories_ or _algebraic theories_. The
latter were introduced in [37] and we will refer to them as _theories_ here.
We also base our notation on the modern treatment of this topic in [30] which
focuses on the correspondence between theories and monads. A classification of
theories whose categories of models are cartesian closed has already appeared
in [31]. Here, we will extend this work and show that any theory whose
category of models lifts the cartesian closed structure of $\mathsf{Set}$ must
correspond to the theory of $G$-sets for a group $G$.

First, we will recall the basics of algebraic theories from Section 2 of [30].
Let $\aleph_{0}$ denote the skeleton of the category of full subcategory of
finite sets. Hence, $\aleph_{0}$ has the natural numbers as its objects
(including $0$ instead of $\emptyset$) and morphisms $\alpha:m\rightarrow n$
correspond to morphisms $\alpha:\\{1,\ldots m\\}\rightarrow\\{1,\ldots n\\}$.
A Lawvere theory $\mathbb{T}$ consists of a small category with finite
products and a strict product-preserving identity-on-objects functor
$t:\aleph_{0}^{\mathrm{op}}\rightarrow\mathbb{T}$ where every object is a
power of $t(1)$. We will abuse notation and denote the images of
$t(n)=t(1)\times\cdots\times t(1)$ by $n$.

Models over the theory $\mathbb{T}$ are limit-preserving functors $\mathbb{T}$
into $\mathsf{Set}$. In terms of classical universal algebra language, any
model $M:\mathbb{T}\rightarrow\mathsf{Set}$ corresponds to the choice of a set
$M(1)$ and maps $M(m)=M(1)^{m}\rightarrow M(1)^{n}=M(n)$ corresponding to
morphisms in $\mathbb{T}$, which compose in a functorial manner. To ease
notation, we will discuss $\mathbb{T}$-models by referring to $M(1)$ and the
operations acting on $f_{M(1)}=M(f)$ instead of the functor $M$. The inclusion
of maps $\alpha:\\{1,\ldots m\\}\rightarrow\\{1,\ldots n\\}$ via
$t:\aleph_{0}^{\mathrm{op}}\rightarrow\mathbb{T}$ gives rise to elementary
operations $\overline{\alpha}:X^{n}\rightarrow X^{m}$, where
$\overline{\alpha}(x_{1},\ldots,x_{n})=(x_{\alpha(1)},\ldots,x_{\alpha(m)})$
for any $\mathbb{T}$-model on the set $X$. In particular, all the diagonal
maps $\Delta^{n}:X\rightarrow X^{n}$ sending $x\mapsto(x,\ldots,x)$ act on
each model.

Throughout this section we assume that $\mathbb{T}$ is a _non-degenerate_
theory, meaning that not all its models are trivial (have a singleton set as
their underlying set).

Every theory $\mathbb{T}$ described above gives rise to a finitary monad $T$
and vice-versa. In particular, the category of $T$-modules is equivalent to
the category of $\mathbb{T}$-models. If $U:\mathbb{T}{\rm-
Mod}\leftrightarrows\mathsf{Set}$ denotes the forgetful functor sending a
model $M$ to its underlying set $U(M)=M(1)$, then its left adjoint $F$ is
defined as

$F(X)=\int^{n\in\aleph_{0}^{\mathrm{op}}}\mathbb{T}(n,1)\times X^{n}$ (54)

and $(T=UF,\mu,\eta)$ denotes the corresponding monad to the theory. The
equivalences between $\mathbb{T}$-models and $T$-modules commute with the
forgetful functors to $\mathsf{Set}$ so we can work with either notion
interchangeably. Starting with a finitary monad, its Kleisli category can be
viewed as a Lawvere theory and recovers $T$ by the described procedure, upto
isomorphism. See Section 4 of [30] for more details.

We will call the elements of $\mathbb{T}(n,1)$, $n$-ary operations. Note that
$F(X)$ becomes a quotient of the coproduct $\amalg_{n\in\mathbb{N}_{0},\
\rho\in\mathbb{T}(n,1)}X^{n}$. Hence an arbitrary element in $F(X)$ can be
written as the image of an element $(\rho;x_{1},\ldots,x_{n})$ where $x_{i}\in
X$ and $\rho$ is an $n$-ary operation and if $p=q\overline{\alpha}$ in
$\mathbb{T}$ for a morphism $\alpha$ in $\aleph_{0}$, then the terms
$(p;x_{1},\ldots,x_{n})$ and $(q;x_{\alpha(1)},\ldots,x_{\alpha(m)})$ are
identified in $F(X)$. If we denote the set with $m$ elements by $m$ in
$\mathsf{Set}$, then its easy to check that $F(m)\cong\mathbb{T}(m,1)$ becomes
the set of all possible $n$-ary operations: Any element in
$\amalg_{n\in\mathbb{N}_{0},\ \rho\in\mathbb{T}(n,1)}m^{n}$ can be written in
the form $(\rho;\overline{\alpha}(1,\ldots n))$ for some morphism
$\alpha\in\aleph_{0}$, which thereby gets identified with
$(\rho\overline{\alpha};1,\ldots m)$ in $F(m)$, where $\rho\overline{\alpha}$
is an $m$-ary operation. Hence, we will write elements
$(\rho;\alpha(1),\ldots,\alpha(n))$ of $F(m)$ as $m$-ary
$\rho\overline{\alpha}\in\mathbb{T}(m,1)$ instead.

It is also well-known that if $A$ and $B$ are $\mathbb{T}$-models, then
$U(A)\times U(B)$ obtains a natural $\mathbb{T}$-model structure define by

$\rho((a_{1},b_{1}),\ldots,(a_{n},b_{n}))=\big{(}\rho(a_{1},\ldots
a_{n}),\rho(b_{1},\ldots b_{n})\big{)}$

for an arbitrary $n$-ary operation $\rho$. By Theorem 3.2, the corresponding
monad $T$ of the theory $\mathbb{T}$ becomes a bimonad. In particular, the
comonoidal structure $F_{2}$ on $F$ is given by:

$\big{(}\rho;(a_{(1)},b_{(1)}),\ldots,(a_{(n)},b_{(n)})\big{)}\mapsto\big{(}(\rho;a_{1},\ldots
a_{n}),(\rho;b_{1},\ldots b_{n})\big{)}$

Now, we look at when the corresponding bimonad of a theory is Hopf. First
recall from Proposition 2.8 that $Z(\mathsf{Set})=\mathsf{Set}$, and thereby
the only possible augmented Hopf monads on $\mathsf{Set}$ arise from Hopf
algebras in $\mathsf{Set}$ i.e. groups $G$. The corresponding Hopf monad $T$
of group $G$, will be of the form $G\times-$ and preserves all colimits. In
particular, the corresponding algebraic theory $\mathbb{T}$ to this monad is
the category generated by $\mathbb{T}(1,1)=G$ and all other morphisms from
$\aleph_{0}^{\mathrm{op}}$. The category of models over the theory are
$G$-sets. Here, we will demonstrate that any algebraic theory whose
corresponding bimonad is Hopf must be of this form.

Assume that $\mathbb{T}$ is a theory whose corresponding monad $T$ becomes
Hopf. Hence, the category of $\mathbb{T}$-models becomes cartesian closed and
for any $T$-algebra $A$, the following morphism
($\overline{H}^{l}_{\mathtt{1},A}$) is bijective:

$\textstyle{\Psi_{A}:FU(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(1)\times
A}$$\textstyle{(\rho;a_{1},\ldots
a_{|\rho|})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{(}\rho^{\Delta},\rho_{A}(a_{1},\ldots
a_{|\rho|})\big{)}}$ (55)

where $|\rho|$ denotes the arity of $\rho$ and
$\rho^{\Delta}=\rho\Delta^{|\rho|}$ is the unary operation associated to
$\rho$.

Recall that a unary operation $u\in F(1)$ is called a pseudo-constant if
$u(x)=u(y)$ in all $\mathbb{T}$-models.

######  Theorem 4.5 (Theorem 1.2 [31]).

If the category of $\mathbb{T}$-models is cartesian closed, then $\mathbb{T}$
has no pseudo-constants.

Note that if a theory has no pseudo constants, it also has no _constants_ i.e.
operations of arity 0. if $\lambda\in T(0,1)$, then we would obtain a pseudo-
constant operation $\lambda\overline{\alpha}$, where $\alpha$ is the unique
morphism $\emptyset\rightarrow 1$.

######  Lemma 4.6.

For any unary $u$ operation in $\mathbb{T}$ and $\mathbb{T}$-model $A$, the
map $u_{A}$ is injective.

###### Proof.

Since $\mathbb{T}$ is non-degenerate we can pick an arbitrary
$\mathbb{T}$-model $A$ with at least two elements $a\neq b\in U(A)$. If $u$ is
a unary operation in $\mathbb{T}$ and $u_{A}(a)=u_{A}(b)$, then
$\Psi_{A}((u;a))=\Psi_{A}((u;b))$. Since $\Psi_{A}$ is a bijection, it means
that $(u;a)=(u;b)$ in $FU(A)$.

Now consider any $\mathbb{T}$-model $B$ and arbitrary elements $c,d\in U(B)$.
We can define a map $f:U(A)\rightarrow U(B)$ such that $f(a)=c$ and
$f(b)=(d)$. Consequently, $F(f):FU(A)\rightarrow FU(B)$ is an algebra morphism
and, by definition, $F(f)((u;a))=(u;c)$ and $F(f)((u;b))=(u;d)$ hold and
$(u;c)=(u;d)$ in $FU(B)$. Since the $\mathbb{T}$-model structure of $B$ is
given by an action $UFU(B)\rightarrow U(B)$, then $u_{B}(c)=u_{B}(d)$ in $B$.
Since, $B$, $c$ and $d$ were chosen arbitrarily, we conclude that $u$ is
pseudo-constant in all $\mathbb{T}$-models, which contradicts Theorem 4.5. ∎

######  Lemma 4.7.

Any unary operation in $\mathbb{T}$ is invertible and its inverse is another
unary operation of the theory.

###### Proof.

Let $u$ be a unary operation in $\mathbb{T}$. First note that $\Psi_{F(1)}$ is
a bijection and consider $\Psi_{F(1)}^{-1}((u,\mathrm{id}))$. There must exist
an operation $\rho$ of arity $n$ and a family of unaries
$u_{1},\ldots,u_{n}\in F(1)$, such that $\rho^{\Delta}=u$ and
$\rho(u_{1},\ldots,u_{n})\Delta^{n}=\mathrm{id}$, so that
$\Psi_{F(1)}((\rho;u_{1},\ldots,u_{n}))=(u,\mathrm{id})$.

Now consider a non-trivial $\mathbb{T}$-model $A$ and $a\in A$. By definition
$\Psi_{A}((u;a))=(u,u_{A}(a))$. However, we also know that
$\Psi_{A}((\rho;b_{1},\ldots b_{n}))=(u,u_{A}(a))$, where
$b_{i}:=(u_{i})_{A}u_{A}(a)$. Since $\Psi_{A}$ is a bijection, we conclude
that $(u;a)=(\rho;b_{1},\ldots b_{n})$ in $FU(A)$.

If for all $i$, we have that $b_{i}\neq a$, then we can consider any pair
$x\neq y\in B$ in any non-trivial $\mathbb{T}$-model $B$ and conclude that
$u_{B}(x)=u_{B}(y)$: This is because there exist a pair of functions
$f_{1},f_{2}:U(A)\rightarrow U(B)$ which map $b_{i}$ to the same collection of
elements but where $f_{1}$ maps $a$ to $x$ and $f_{2}$ maps $a$ to $y$. Hence,
by considering the images of the algebra maps $F(f_{1})$ and $F(f_{2})$ we see
that

$(u;x)=(\rho;f_{1}(b_{1}),\ldots,f_{1}(b_{n}))=(\rho;f_{2}(b_{1}),\ldots,f_{2}(b_{n}))=(u;y)\in
FU(B)$

Since the $\mathbb{T}$-model structure of $B$ is defined by a map from $FU(B)$
to $U(B)$, then $u_{B}(x)=u_{B}(y)$, which contradicts Lemma 4.6. Hence, there
must be an $1\leq m\leq n$ for which $a=(u_{m})_{A}u_{A}(a)$.

Now consider the case where $A=F(1)$ and $a=\mathrm{id}$. From the argument
above, there must exist an $m$ such that
$\mathrm{id}=(u_{m})_{F(1)}u_{F(1)}(\mathrm{id})=u_{m}u$. Consequently, we
obtain the following equality in $F(1)$:

$(u_{m})_{F(1)}(uu_{m})=u_{m}uu_{m}=u_{m}=(u_{m})_{F(1)}(\mathrm{id})$

But by Lemma 4.6, $(u_{m})_{F(1)}$ is injective and thereby,
$uu_{m}=\mathrm{id}$ as well. Hence, $u$ is invertible and
$u_{m}u=\mathrm{id}=uu_{m}$. Before concluding the proof, we note as a
consequence $(\rho;u_{1},\ldots,u_{n})=(u;u_{m})$ holds in $FUF(1)$. ∎

It follows immediately from Lemma 4.7, that the set of unary operations in
$\mathbb{T}$ form a group and as a consequence any $\mathbb{T}$-model is a
$F(1)$-set. Now we show that the theory $\mathbb{T}$ is fully generated by
unary operations.

######  Theorem 4.8.

For any operation $\rho$ in $\mathbb{T}$ of arity $n$, there exists an $1\leq
m\leq n$ and a unary operation $u$ such that $\rho=u\sigma_{m}$, where
$\sigma_{m}(1,\ldots n)=m$.

###### Proof.

Let $A$ be an arbitrary $\mathbb{T}$-model and $\rho$ an $n$-ary operation in
$A$. Since $\Psi_{A}$ is bijective, we have that

$(\rho;a_{1},\ldots,a_{n})=\big{(}\rho^{\Delta};(\rho^{\Delta}_{A})^{-1}\rho_{A}(a_{1},\ldots,a_{n})\big{)}\in
FU(A)$

If $b:=(\rho^{\Delta}_{A})^{-1}\rho(a_{1},\ldots,a_{n})\neq a_{i}$ for all
$1\leq i\leq n$, then we can consider maps $U(A)\rightarrow U(B)$ for
arbitrary models $B$, which send $b$ to various elements of $A$, but send
$a_{i}$ to a fixed set of elements. Similar to the argument in the proof of
Lemma 4.7, this implies that $\rho^{\Delta}$ must be a pseudo-constant which
contradicts Theorem 4.5. Therefore,
$(\rho^{\Delta}_{A})^{-1}\rho(a_{1},\ldots,a_{n})=a_{i}$ for some $i$.

Now let $A=F(n)$ and let $a_{i}=\sigma_{i}\in F(n)$ where
$\sigma_{i}(x_{1},\ldots,x_{n})=x_{i}$. By the above argument, we see that for
some $i$ the following equality holds

$(\rho;\sigma_{1},\ldots,\sigma_{n})=\big{(}\rho^{\Delta};\sigma_{i}\big{)}\in
FUF(n)$

Looking at the image of the two elements under $\Psi_{F(n)}$, we observe that
$\rho(x_{1},\ldots,x_{n})=\rho^{\Delta}(x_{i})$. ∎

By Theorem 4.8 and the construction of $F$ as a coend, we observe that
$F(X)\cong F(1)\times X$ for any set $X$. In particular, the monad structure
on $T$ is given by the group structure on $F(1)$.

######  Theorem 4.9.

If $T$ is a finitary Hopf monad on $\mathsf{Set}$, then there exists a group
$G$ such that $T\cong G\times-$. In particular, $T$ must be augmented.

###  4.4 Galois and Ore Extensions of Bialgebras

In this section, we will assume $B$ and $H$ are a pair of bialgebras and
$f:B\rightarrow H$ a bialgebra injection and discuss when the induced
adjunction ${}_{H}{\mathcal{M}}\leftrightarrows{}_{B}{\mathcal{M}}$ is left
(pre-) Hopf. We will first show that the pre-Hopf condition corresponds to $H$
being a Galois extension of $B$ in the sense of [59] and then provide a more
general Galois condition (57) for the bimonad to be Hopf. Finally, we show
that any suitable Ore extension of a bialgebra provides a left Hopf monad in
this way.

Any algebra morphism $f$ gives rise to a pair of adjoint functors,
$H\otimes_{B}-\dashv U:{}_{H}{\mathcal{M}}\leftrightarrows{}_{B}{\mathcal{M}}$
where $U$ denotes the restriction of scalars via $f$ and $H$ has a natural
$B$-bimodule structure via $f$. If $f$ is a bialgebra morphism then $U$
becomes naturally strong monoidal since for any pair of $H$-modules $M$ and
$N$, $B$ acts on $U(M\otimes_{\mathbb{K}}N)$ by
$f(b)_{(1)}.m\otimes_{\mathbb{K}}f(b)_{(2)}.n$ while the $B$-action on
$U(M)\otimes_{\mathbb{K}}U(N)$ is given by
$f(b_{(1)}).m\otimes_{\mathbb{K}}f(b_{(2)}).n$. Consequently, if $f$ is a
bialgebra morphism then $H\otimes_{B}-\dashv U$ is a comonoidal adjunction.
Alternatively, one can look directly at the induced monad on
${}_{B}{\mathcal{M}}$ which is given by $T={}_{B}H\otimes_{B}-$ where we
consider the natural $B$-bimodule structure induced by $f$. The comonoidal
structure $T_{2}(M,N)$ on the monad is given by
$h\otimes_{B}(m\otimes_{\mathbb{K}}n)\mapsto(h_{(1)}\otimes_{B}m)\otimes_{\mathbb{K}}(h_{(2)}\otimes_{B}m)$
which is well-defined because $f$ is bialgebra map. Hence, we have a commuting
diagram of strong monoidal functors:

$\textstyle{{}_{H}{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forg.}}$$\scriptstyle{U}$$\textstyle{{}_{B}{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forg.}}$$\textstyle{\mathsf{Vec}}$

If $B$ and $H$ are both Hopf algebras then $f$ automatically commutes with the
antipodes since the antipodes, when they exist, are uniquely determined by the
bialgebra structures. In this case, the forgetful functors in the above
diagram both become left closed and $U$ also becomes left closed since $f$
respects the antipode and, thereby, $U$ respects the actions on the left
inner-homs.

When $B$ and $H$ are not Hopf, we can still consider the problem of when the
bimonad $T={}_{B}H\otimes_{B}-$ is left Hopf. This can be done in terms of the
fusion operator but also in terms of the closed structure. While
${}_{B}{\mathcal{M}}$ is does not lift the closed structure of $\mathsf{Vec}$,
it is still a closed monoidal category and $T={}_{B}H\otimes_{B}-$ being Hopf
would mean that ${}_{H}{\mathcal{M}}$ is lifting the inner-homs of
${}_{B}{\mathcal{M}}$.

Now assume $f:B\rightarrow H$ is a bialgebra injection and consider the subset
$I=\\{f(b)-\epsilon_{B}(b).1_{H}\mid b\in B\\}$ in $H$. It is straightforward
to check that this subset is a coideal i.e. satisfies $\Delta_{H}(I)\subset
I\otimes_{\mathbb{K}}H+H\otimes_{\mathbb{K}}I$. Hence, we obtain a quotient
coalgebra structure on $C:=H/I$ and we denote the coalgebra projection
$H\twoheadrightarrow C$ by $\pi$. Note that $H$ becomes a left $C$-comodule
via $\delta:=(\pi\otimes_{\mathbb{K}}\mathrm{id}_{H})\Delta$. Moreover, $H$
becomes a monoid in ${}^{C}{\mathcal{M}}$ (a $C$-_comodule algebra_) with this
coaction. With this structure, we can identify $f(B)$ with the space of
$C$-coinvariants ${}^{C}H$ i.e. elements $h$ satisfying
$\delta(h)=\pi(1_{H})\otimes_{\mathbb{K}}h$:

$\displaystyle\delta(h)=\pi(1_{H})\otimes_{\mathbb{K}}h\Rightarrow\pi(h_{(1)})\epsilon(h_{(2)})=\pi(1_{H})\epsilon(h)\Rightarrow\pi(h-\epsilon(h).1_{H})=0\Rightarrow
h\in f(B)$

Note that for a general comodule algebra the space of coinvariants are formed
by elements $h$ satisfying
$\delta(h^{\prime}h)=\pi(h^{\prime}_{(1)})\otimes_{\mathbb{K}}h^{\prime}_{(2)}h$
for any $h^{\prime}\in H$ but this is equivalent to
$\delta(h)=\pi(1_{H})\otimes_{\mathbb{K}}h$ in our case. In this setting, we
call $H$ a $C$-_Galois extension_ of $B$ if the map $\beta$ is a bijection:

$\textstyle{\beta:H\otimes_{B}H\longrightarrow
C\otimes_{\mathbb{K}}H}$$\textstyle{\hskip
14.22636pth\otimes_{B}h^{\prime}\longmapsto\pi(h_{(1)})\otimes_{\mathbb{K}}h_{(2)}h^{\prime}}$
(56)

See Definition 1.5 of [59] for comparison. The notion of $C$-Galois extension
also appeared earlier in [18, 17] for entwined algebras and coalgebras under
the name of _principal_ $C$-_bundles_. With this notation, we can prove the
following result:

######  Theorem 4.10.

If $f:B\rightarrow H$ is a bialgebra injection as above, then the bimonad
$T={}_{B}H\otimes_{B}-$ on ${}_{B}{\mathcal{M}}$ is left pre-Hopf if and only
if $H$ is a $C$-Galois extension of $B$.

###### Proof.

First assume that $T={}_{B}H\otimes_{B}-$ is left pre-Hopf. Now consider
$\mathbb{K}$ with its natural $B$-action induced by $\epsilon$ and $B$ with
its trivial left $B$-action. Note that
$T(\mathbb{K})=H\otimes_{B}\mathbb{K}\cong C$ and $T(B)=H\otimes_{B}B\cong H$
as left $B$-modules and $T(\mathbb{K}\otimes_{\mathbb{K}}T(B))\cong
H\otimes_{B}H$. By these isomorphisms we recover $\beta$ as
$H^{l}_{\mathbb{K},B}$ and if $H^{l}_{\mathbb{K},B}$ is invertible then so is
$\beta$.

In the converse direction, we assume $H$ is a $C$-Galois extension of $B$ and
denote $\beta(c\otimes_{\mathbb{K}}1)=c_{(+)}\otimes_{B}c_{(-)}$. Then we can
define the inverse of $H^{l}_{\mathbb{K},M}$ for any $B$-module $M$ by
$c\otimes_{\mathbb{K}}(h\otimes_{B}m)\mapsto
c_{(+)}\otimes_{B}(c_{(-)}.h\otimes_{B}m)$. It is easy to check that this map
is indeed well-defined and provides an inverse for $H^{l}_{\mathbb{K},M}$. ∎

For the induced bimonad of $f:B\rightarrow H$ to be left Hopf we need a more
generalised version of $\beta$:

######  Theorem 4.11.

If $f:B\rightarrow H$ is a bialgebra injection as above, then the bimonad
$T={}_{B}H\otimes_{B}-$ on ${}_{B}{\mathcal{M}}$ is left Hopf if and only if
the following map is a bijection:

$\textstyle{\Gamma:H\otimes_{B}(B\otimes_{\mathbb{K}}H)\longrightarrow
H\otimes_{\mathbb{K}}H}$$\textstyle{\hskip
14.22636pth\otimes_{B}(b\otimes_{\mathbb{K}}h^{\prime})\longmapsto
h_{(1)}f(b)\otimes_{\mathbb{K}}h_{(2)}h^{\prime}}$ (57)

Here we consider $B\otimes_{\mathbb{K}}H$ as the tensor product of two left
$B$-modules.

###### Proof.

As in the proof of Theorem 4.10, we observe that
$T(B\otimes_{\mathbb{K}}T(B))\cong H\otimes_{B}(B\otimes_{\mathbb{K}}H)$ and
recover $\Gamma$ from $H^{l}_{B,B}$. Hence, if $T$ is left Hopf then $\Gamma$
becomes invertible. In the converse direction, assume $\Gamma$ is invertible
and denote
$\Gamma(h\otimes_{\mathbb{K}}1)=h_{(+)}\otimes_{B}(h_{(+-)}\otimes_{\mathbb{K}}h_{(-)})$.
Note that the component $h_{(+-)}$ belongs to $B$. With this notation we can
define the inverse of $H^{l}_{M,N}$ for two arbitrary $B$-modules $M$ and $N$
by

$\displaystyle(H\otimes_{B}M)\otimes_{\mathbb{K}}(H\otimes_{B}N)$
$\displaystyle\longrightarrow
H\otimes_{B}(M\otimes_{\mathbb{K}}(H\otimes_{B}N))$
$\displaystyle(h\otimes_{B}m)\otimes_{\mathbb{K}}(h^{\prime}\otimes_{B}n)$
$\displaystyle\longmapsto
h_{(+)}\otimes_{B}(h_{(+-)}.m\otimes_{\mathbb{K}}(h_{(-)}h^{\prime}\otimes_{B}m))$

It follows in a straightforward manner that this map is well-defined and
provides an inverse for $H^{l}_{M,N}$. ∎

The invertibility of $\Gamma$ is a difficult problem to check. Here we will
provide an example of such an extension by directly checking that $H$ lifts
the left inner-homs of ${}_{B}{\mathcal{M}}$.

For any bialgebra, the category ${}_{B}{\mathcal{M}}$ is left closed with
$[M,N]^{l}=\mathrm{Hom}_{B}(B\otimes_{\mathbb{K}}M,N)$, where
$\mathrm{Hom}_{B}$ denotes the space of left $B$-module morphisms and
$\mathrm{Hom}_{B}(B\otimes_{\mathbb{K}}M,N)$ has a left $B$-action defined by
$(b.f)(b^{\prime}\otimes_{\mathbb{K}}m)=f(b^{\prime}.b\otimes_{\mathbb{K}}m)$.
The unit and counit of $-\otimes_{\mathbb{K}}M\dashv[M,-]^{l}$ are defined by

$\textstyle{\mathrm{coev}^{M}_{N}:N\rightarrow[M,N\otimes_{\mathbb{K}}M]^{l}}$$\textstyle{\quad
n\mapsto(b\otimes_{\mathbb{K}}m\mapsto b.n\otimes_{\mathbb{K}}m)}$
$\textstyle{\mathrm{ev}^{M}_{N}:[M,N]^{l}\otimes_{\mathbb{K}}M\rightarrow
N}$$\textstyle{\quad f\otimes_{\mathbb{K}}m\mapsto
f(1_{B}\otimes_{\mathbb{K}}m)}$ (58)

This closed structure was observed in [56] at the level of bialgebroids. By
Theorem 3.12, the bimonad $T$ being Hopf is equivalent to the functor $U$
lifting the adjunction above to ${}_{H}{\mathcal{M}}$. Hence, whenever $H$ has
a well-defined action on $[M,N]^{l}$ so that $\mathrm{coev}^{M}_{N}$ and
$\mathrm{ev}^{M}_{N}$ become $H$-module morphisms then $T$ becomes a left Hopf
monad. We will present one family of such examples here:

######  Theorem 4.12.

Let $B$ be a bialgebra and $\mathsf{d}:B\rightarrow B$ a derivation on $B$
satisfying
$\Delta(\mathsf{d}(b))=\mathsf{d}(b_{(1)})\otimes_{\mathbb{K}}b_{(2)}+b_{(1)}\otimes_{\mathbb{K}}\mathsf{d}(b_{(2)})$
and $\epsilon(\mathsf{d})=0$. If $H=B[x;\mathsf{d}]$ denotes the _Ore
extension_ of $B$, then $H$ is a bialgebra and the natural algebra map
$f:B\rightarrow H$ is a bialgebra map and induces a left Hopf monad