${}_{B}H\otimes_{B}-$ on ${}_{B}{\mathcal{M}}$.

###### Proof.

Recall from Chapter 2 of [29] that $H$ is defined as the quotient of $B\langle
x\rangle$ by the ideal $\langle x.b-b.x-\mathsf{d}(b)\mid b\in B\rangle$. It
was already shown in [52] that when $\mathsf{d}$ satisfies the additional
compatibility condition with $\Delta$ and $\epsilon$, then $H$ obtains a
natural bialgebra structure with
$\Delta(x)=x\otimes_{\mathbb{K}}1+1\otimes_{\mathbb{K}}x$ and $\epsilon(x)=0$
extending the coalgebra structure defined on elements of $B$. Hence, the
natural map $f:B\rightarrow H$ defined by $b\mapsto b$ becomes a bialgebra
morphism.

Here we will define an action of $H$ on
$\mathrm{Hom}_{B}(B\otimes_{\mathbb{K}}M,N)$ such that the unit and counit in
(58) lift to ${}_{H}{\mathcal{M}}$. For a pair of $H$-modules $M$ and $N$, we
extend the $B$-action on $\mathrm{Hom}_{B}(B\otimes_{\mathbb{K}}M,N)$ to an
$H$-action by defining
$(x.f)(b\otimes_{\mathbb{K}}m)=x.f(b\otimes_{\mathbb{K}}m)-f(\mathsf{d}(b)\otimes_{\mathbb{K}}m)-f(b\otimes_{\mathbb{K}}x.m)$.
First note that this action is well-defined i.e.
$x.f\in\mathrm{Hom}_{B}(B\otimes_{\mathbb{K}}M,N)$:

$\displaystyle b.\big{(}(x.f)(b^{\prime}\otimes_{\mathbb{K}}m)\big{)}=$
$\displaystyle
b.x.f(b^{\prime}\otimes_{\mathbb{K}}m)-b.f(\mathsf{d}(b^{\prime})\otimes_{\mathbb{K}}m)-b.f(b^{\prime}\otimes_{\mathbb{K}}x.m)$
$\displaystyle=$ $\displaystyle
b.x.f(b^{\prime}\otimes_{\mathbb{K}}m)-f(b_{(1)}\mathsf{d}(b^{\prime})\otimes_{\mathbb{K}}b_{(2)}.m)-f(b_{(1)}b^{\prime}\otimes_{\mathbb{K}}b_{(2)}.x.m)$
$\displaystyle=$ $\displaystyle
x.f(b_{(1)}b^{\prime}\otimes_{\mathbb{K}}b_{(2)}.m)-\mathsf{d}(b).f(b^{\prime}\otimes_{\mathbb{K}}m)-f(b_{(1)}\mathsf{d}(b^{\prime})\otimes_{\mathbb{K}}b_{(2)}.m)$
$\displaystyle-f(b_{(1)}b^{\prime}\otimes_{\mathbb{K}}x.b_{(2)}.m)+f(b_{(1)}b^{\prime}\otimes_{\mathbb{K}}\mathsf{d}(b_{(2)}).m)$
$\displaystyle=$ $\displaystyle
x.f(b_{(1)}b^{\prime}\otimes_{\mathbb{K}}b_{(2)}.m)-f(\mathsf{d}(b_{(1)})b^{\prime}\otimes_{\mathbb{K}}b_{(2)}.m)$
$\displaystyle-f(b_{(1)}\mathsf{d}(b^{\prime})\otimes_{\mathbb{K}}b_{(2)}.m)-f(b_{(1)}b^{\prime}\otimes_{\mathbb{K}}x.b_{(2)}.m)$
$\displaystyle=$
$\displaystyle(x.f)(b_{(1)}b^{\prime}\otimes_{\mathbb{K}}b_{(2)}.m)$

Secondly, note that this gives rise to a well-defined action of $H$ since

$\displaystyle(x.b^{\prime}.f)(b\otimes_{\mathbb{K}}m)=$ $\displaystyle
x.(b^{\prime}.f)(b\otimes_{\mathbb{K}}m)-(b^{\prime}.f)(\mathsf{d}(b)\otimes_{\mathbb{K}}m)-(b^{\prime}.f)(b\otimes_{\mathbb{K}}x.m)$
$\displaystyle=$ $\displaystyle
x.f(bb^{\prime}\otimes_{\mathbb{K}}m)-f(\mathsf{d}(b)b^{\prime}\otimes_{\mathbb{K}}m)-f(bb^{\prime}\otimes_{\mathbb{K}}x.m)$
$\displaystyle(b^{\prime}.x.f)(b\otimes_{\mathbb{K}}m)=$
$\displaystyle(x.f)(bb^{\prime}\otimes_{\mathbb{K}}m)=x.f(bb^{\prime}\otimes_{\mathbb{K}}m)-f(\mathsf{d}(bb^{\prime})\otimes_{\mathbb{K}}m)-f(bb^{\prime}\otimes_{\mathbb{K}}x.m)$
$\displaystyle(\mathsf{d}(b^{\prime}).f)(b\otimes_{\mathbb{K}}m)=$
$\displaystyle f(b\mathsf{d}(b^{\prime})\otimes_{\mathbb{K}}m)$

Now we must check that $\mathrm{coev}^{M}_{N}$ and $\mathrm{ev}^{M}_{N}$
become $H$-module morphisms:

$\displaystyle[x.\mathrm{coev}^{M}_{N}(n)](b\otimes_{\mathbb{K}}m)=$
$\displaystyle
x.\big{(}\mathrm{coev}^{M}_{N}(n)(b\otimes_{\mathbb{K}}m)\big{)}-\mathrm{coev}^{M}_{N}(n)(\mathsf{d}(b)\otimes_{\mathbb{K}}m)$
$\displaystyle-\mathrm{coev}^{M}_{N}(n)(b\otimes_{\mathbb{K}}x.m)$
$\displaystyle=$ $\displaystyle
x.(b.n\otimes_{\mathbb{K}}m)-(\mathsf{d}(b).n\otimes_{\mathbb{K}}m)-(b.n\otimes_{\mathbb{K}}x.m)$
$\displaystyle=$
$\displaystyle(x.b.n\otimes_{\mathbb{K}}m)+(b.n\otimes_{\mathbb{K}}x.m)-(\mathsf{d}(b).n\otimes_{\mathbb{K}}m)-(b.n\otimes_{\mathbb{K}}x.m)$
$\displaystyle=$
$\displaystyle(b.x.n\otimes_{\mathbb{K}}m)=[\mathrm{coev}^{M}_{N}(x.n)](b\otimes_{\mathbb{K}}m)$

For $\mathrm{ev}^{M}_{N}$ the calculation follows in a trivial manner from
$\mathsf{d}(1)=0$ and its verification is left to the reader. ∎

Hence any suitable Ore extension $f:B\rightarrow B[x;\mathsf{d}]$ gives rise
to a left Hopf monad on ${}_{B}{\mathcal{M}}$. Additionally, note that this
induced left Hopf monad admits an augmentation precisely when $f$ admits a
retraction, which cannot happen unless $\mathsf{d}$ is an inner derivation.

###  4.5 Hopf Monads from Pivotal Pairs

In this section, we review our construction of Hopf monads from _pivotal
pairs_ in monoidal categories, which appeared in [27]. We will be assuming
that the base category $\mathcal{C}$ is closed and admits countable colimits.
Thereby, $\otimes$ commutes with colimits and the category of endofunctors
$\mathrm{End}(\mathcal{C})$ also has countable colimits. First we will recall
the notion of pivotal pairs and their categories of intertwined objects.

Notation: Throughout this section, we write $X$ instead of the morphism
$\mathrm{id}_{X}$ for brevity.

Recall from [27], that we call a pair of objects $(P,Q)$ a _pivotal pair_ in
$\mathcal{C}$ if there exist a quadruple of morphisms
$\mathrm{coev}:\mathtt{1}\rightarrow P\otimes Q$, $\mathrm{ev}:Q\otimes
P\rightarrow\mathtt{1}$ and $\underline{\mathrm{coev}}:\mathtt{1}\rightarrow
Q\otimes P$, $\underline{\mathrm{ev}}:P\otimes Q\rightarrow\mathtt{1}$, making
$Q$ a left and right dual of $P$, respectively. In fact, whenever we talk
about a pivotal pair, we always make an implicit choice for such morphisms.

Given a pivotal pair $(P,Q)$, we defined the category of $P$ and $Q$
_intertwined objects_ , denoted by $\mathcal{C}(P,Q)$, as the category whose
objects are pairs $(X,\sigma)$, where $X$ is an object of $\mathcal{C}$ and
$\sigma:X\otimes P\rightarrow P\otimes X$ an invertible morphism in
$\mathcal{C}$ such that

$\displaystyle(\mathrm{ev}\otimes X\otimes Q)(Q\otimes\sigma\otimes
Q)(Q\otimes X\otimes\mathrm{coev}):Q\otimes X\rightarrow X\otimes Q$ (59)
$\displaystyle(Q\otimes
X\otimes\underline{\mathrm{ev}})(Q\otimes\sigma^{-1}\otimes
Q)(\underline{\mathrm{coev}}\otimes X\otimes Q):X\otimes Q\rightarrow Q\otimes
X$ (60)

are inverses. Morphisms between objects $(X,\sigma)$, $(Y,\tau)$ of
$\mathcal{C}(P,Q)$ are morphisms $f:X\rightarrow Y$ in $\mathcal{C}$, which
satisfy $\tau(f\otimes P)=(P\otimes f)\sigma$. For an object $(X,\sigma)$ in
$\mathcal{C}(P,Q)$, we call $\sigma$ a $P$-_intertwining_ and denote the
induced morphisms (59) and (60), by $\overline{\sigma}$ and
$\overline{\sigma}^{-1}$, respectively, and call them _induced_
$Q$_-intertwinings_.

The category $\mathcal{C}(P,Q)$ obtains a natural monoidal structure by
defining $(X,\sigma)\otimes(Y,\tau)$ to be the pair $(X\otimes
Y,(\sigma\otimes Y)(X\otimes\tau))$ [Theorem 4.1 [27]]. It was already noted
in [27] that the category $\mathcal{C}(P,Q)$ can be viewed as the dual of a
strong monoidal functor in the sense of Section 2.3. The choice of a pivotal
pair $(P,Q)$ in a monoidal category $\mathcal{C}$ corresponds to the choice of
a strict monoidal functor from the monoidal category generated by a single
pivotal object, which we denote by $\mathrm{Piv}(1)$. The category
$\mathrm{Piv}(1)$ is the monoidal category generated by two objects $+$ and
$-$ and two pairs of duality morphisms making $-$ both the left and right dual
of $+$. It should be clear that given any pivotal pair in $\mathcal{C}$, we
have a strict monoidal functor
$\omega_{(P,Q)}:\mathrm{Piv}(1)\rightarrow\mathcal{C}$ which sends $+$ to $P$
and $-$ to $Q$ and the relevant duality morphisms to the duality morphisms in
$\mathcal{C}$.

######  Theorem 4.13.

If $(P,Q)$ is a pivotal pair in $\mathcal{C}$ and
$\omega:\mathrm{Piv}(1)\rightarrow\mathcal{C}$ its corresponding functor, then
there is a monoidal isomorphism
$\mathcal{C}(P,Q)\cong\left({}_{\mathrm{Piv}(1)}{\omega}_{\mathcal{C}}\right)^{\circ}$
which commutes with the forgetful functors from each category to
$\mathcal{C}$.

###### Proof.

In one direction, any object $(A,\sigma)$ of $\mathcal{C}(P,Q)$ has a natural
braiding
$\mathbf{\sigma}:A\otimes\mathrm{id}_{\omega}\rightarrow\mathrm{id}_{\omega}\otimes
A$ defined by $\mathbf{\sigma}_{P}=\sigma$ and
$\mathbf{\sigma}_{Q}=\overline{\sigma}^{-1}$. It follows by definition that
the duality morphisms between $P$ and $Q$ commute with these braidings and
that $(A,\mathbf{\sigma})$ belongs to
$\left({}_{\mathrm{Piv}(1)}{\omega}_{\mathcal{C}}\right)^{\circ}$. Conversely,
for any object $(A,\mathbf{\sigma})$ in
$\left({}_{\mathrm{Piv}(1)}{\omega}_{\mathcal{C}}\right)^{\circ}$ the pair
$(A,\mathbf{\sigma}_{P})$ is an object in $\mathcal{C}(P,Q)$, since
$\overline{\mathbf{\sigma}_{P}}^{-1}$ and $\overline{\mathbf{\sigma}_{P}}$
become inverses because of $\sigma$ commuting with the duality morphisms e.g.

$\displaystyle\overline{\mathbf{\sigma}_{P}}\
\overline{\mathbf{\sigma}_{P}}^{-1}=$ $\displaystyle(\mathrm{ev}\otimes
X\otimes Q)(Q\otimes\mathbf{\sigma}_{P}\otimes Q)(Q\otimes
X\otimes\mathrm{coev})(Q\otimes
X\otimes\underline{\mathrm{ev}})(Q\otimes\mathbf{\sigma}_{P}^{-1}\otimes Q)$
$\displaystyle(\underline{\mathrm{coev}}\otimes X\otimes Q)$ $\displaystyle=$
$\displaystyle(\mathrm{ev}\otimes X\otimes
Q)(Q\otimes\mathbf{\sigma}_{P}\otimes Q)(Q\otimes
X\otimes\mathrm{coev})\sigma_{Q}\sigma_{Q}^{-1}(Q\otimes
X\otimes\underline{\mathrm{ev}})(Q\otimes\mathbf{\sigma}_{P}^{-1}\otimes Q)$
$\displaystyle(\underline{\mathrm{coev}}\otimes X\otimes Q)$ $\displaystyle=$
$\displaystyle(\mathrm{ev}\otimes X\otimes
Q)(Q\otimes\mathbf{\sigma}_{P}\otimes Q)(\sigma_{Q}\otimes P\otimes
Q)(X\otimes Q\otimes\mathrm{coev})(X\otimes Q\otimes\underline{\mathrm{ev}})$
$\displaystyle(\sigma_{Q}^{-1}\otimes P\otimes
Q)(Q\otimes\mathbf{\sigma}_{P}^{-1}\otimes Q)(\underline{\mathrm{coev}}\otimes
X\otimes Q)$ $\displaystyle=$ $\displaystyle(X\otimes\mathrm{ev}\otimes
Q)(X\otimes Q\otimes\mathrm{coev})(X\otimes
Q\otimes\underline{\mathrm{ev}})(X\otimes\underline{\mathrm{coev}}\otimes
Q)=\mathrm{id}_{X\otimes Q}$

It should be clear that the described correspondence is a monoidal isomorphism
of categories preserving the underlying object of the pairs in each category.∎

Consequently, all the properties of the dual of a monoidal functor which we
discussed in Section 2.3 hold for $\mathcal{C}(P,Q)$. For example, by Theorem
2.3 we can conclude:

######  Corollary 4.14.

If $\mathcal{C}$ is a left (right) closed monoidal category, then
$\mathcal{C}(P,Q)$ has a unique left (right) closed monoidal structure lifting
that of $\mathcal{C}$, such that the forgetful functor $U$ becomes left
(right) closed.

By Corollary 4.14, we know that when $\mathcal{C}$ is closed we obtain a
$P$-intertwining on $[A,B]^{l(r)}$ corresponding to every pair of objects
$(A,\sigma_{A})$ and $(B,\sigma_{B})$ in $\mathcal{C}(P,Q)$. The precise
definition of the induced $P$-intertwinings on the inner-homs in
$\mathcal{C}(P,Q)$ can be found in Section 4 of [27].

Next we recall how we can construct a Hopf monad whose Eilenberg-Moore
category recovers $\mathcal{C}(P,Q)$ from Section 5 of [27]. We will call a
morphism $F(X)\rightarrow X$, for any endofunctor
$F:\mathcal{C}\rightarrow\mathcal{C}$, an _action_ of $F$ on an object $X$.
Although the functors in question will not carry any monad structures, we will
build a monad on the colimit of a diagram of these functors, so that these
_actions_ induce a genuine module structure over the resulting monad, thereby
justifying our terminology.

Observe that for a pair $(X,\sigma)$ in $\mathcal{C}(P,Q)$, we can view
$\sigma$ and $\sigma^{-1}$ as certain actions of the functors
$Q\otimes-\otimes P$ and $P\otimes-\otimes Q$ on $X$:

Moreover, for any pair $(X,\sigma)$ in $\mathcal{C}(P,Q)$, we can translate
the mentioned actions in terms of the induced $Q$-intertwinings, since
$(X\otimes\mathrm{ev})(\overline{\sigma}\otimes P)=(\mathrm{ev}\otimes
X)(Q\otimes\sigma)$ and $(\underline{\mathrm{ev}}\otimes
X)(P\otimes\overline{\sigma}^{-1})=(X\otimes\underline{\mathrm{ev}})(\sigma^{-1}\otimes
Q)$.

Conversely, when provided with two morphisms $\alpha:Q\otimes X\otimes
P\rightarrow X$ and $\beta:P\otimes X\otimes Q\rightarrow X$, we can recover
right and left $P$-intertwinings as set out below:

If we want the induced $P$-intertwinings of $\alpha$ and $\beta$ to be
inverses, we need the following equalities to hold:

$\displaystyle\mathrm{ev}\otimes X=\alpha(Q\otimes\beta\otimes P)(Q\otimes
P\otimes X\otimes\underline{\mathrm{coev}}):Q\otimes P\otimes X\rightarrow X$
(61) $\displaystyle
X\otimes\underline{\mathrm{ev}}=\beta(P\otimes\alpha\otimes
Q)(\mathrm{coev}\otimes X\otimes P\otimes Q):X\otimes P\otimes Q\rightarrow X$
(62)

Similarly, $\alpha$ and $\beta$ induce $Q$-intertwinings, (59) and (60) which
can be written as

In order for the induced $Q$-intertwinings to be inverses, we require the
following equalities to hold:

$\displaystyle\underline{\mathrm{ev}}\otimes X=\beta(P\otimes\alpha\otimes
Q)(P\otimes Q\otimes X\otimes\mathrm{coev}):P\otimes Q\otimes X\rightarrow X$
(63) $\displaystyle X\otimes\mathrm{ev}=\alpha(Q\otimes\beta\otimes
P)(\underline{\mathrm{coev}}\otimes X\otimes Q\otimes P):X\otimes Q\otimes
P\rightarrow X$ (64)

With this view of $P$-intertwinings in mind, we construct the left adjoint
functor to the forgetful functor $U:\mathcal{C}(P,Q)\rightarrow\mathcal{C}$.

Define the endofunctors $F_{+},F_{-}:\mathcal{C}\rightarrow\mathcal{C}$ by
$F_{+}(X)=Q\otimes X\otimes P$ and $F_{-}(X)=P\otimes X\otimes Q$. Let the
endofunctor $F^{\star}$ be defined as the coproduct

$F^{\star}=\coprod_{n\in\mathbb{N}_{0},(i_{1},i_{2},\dots,i_{n})\in\\{-,+\\}^{n}}F_{i_{1}}F_{i_{2}}\cdots
F_{i_{n}}$

where the term $F_{i_{1}}F_{i_{2}}\cdots F_{i_{n}}$ at $n=0$, is just the
identity functor $\mathrm{id}_{\mathcal{C}}$. For arbitrary $n\in\mathbb{N}$
and $(i_{1},i_{2},\dots,i_{n})\in\\{-,+\\}^{n}$, we denote
$F_{i_{1}}F_{i_{2}}\cdots F_{i_{n}}$ by $F_{i_{1},i_{2},\dots,i_{n}}$ and the
respective natural transformations $F_{i_{1},i_{2},\dots,i_{n}}\Rightarrow
F^{\star}$ by $\iota_{i_{1},i_{2},\dots,i_{n}}$. We denote the additional
natural transformation $\mathrm{id}\Rightarrow F^{\star}$ by $\iota_{0}$.
Hence, for any $F_{i_{1},i_{2},\dots,i_{n}}$ we have four parallel pairs:

(67) (70) (73) (76)

Consider the diagram in $\mathrm{End}(\mathcal{C})$ which the described
parallel pairs create. We denote the colimit of this diagram by $T$, the
unique natural transformation $F^{\star}\Rightarrow T$, by $\psi$, and the
compositions $\psi\iota_{i_{1},i_{2},\dots,i_{n}}$ and $\psi\iota_{0}$, by
$\psi_{i_{1},i_{2},\dots,i_{n}}$ and $\psi_{0}$, respectively.

Since $\otimes$ commutes with colimits, the family of morphisms
$\psi_{+,i_{1},i_{2},\dots,i_{n}}:Q\otimes F_{i_{1},i_{2},\dots,i_{n}}\otimes
P\rightarrow T$ induce a unique morphism $\alpha:Q\otimes T\otimes
P\rightarrow T$ such that
$\alpha(Q\otimes\psi_{i_{1},i_{2},\dots,i_{n}}\otimes
P)=\psi_{+,i_{1},i_{2},\dots,i_{n}}$. Similarly, the family of morphisms
$\psi_{-,i_{1},i_{2},\dots,i_{n}}:P\otimes F_{i_{1},i_{2},\dots,i_{n}}\otimes
Q\rightarrow T$ induce a morphism $\beta:P\otimes T\otimes Q\rightarrow T$
such that $\beta(P\otimes\psi_{i_{1},i_{2},\dots,i_{n}}\otimes
Q)=\psi_{-,i_{1},i_{2},\dots,i_{n}}$. As mentioned at the start of the
section, such actions $\alpha$ and $\beta$ provide us with the necessary
$P$-intertwinings.

Let us denote the natural transformation
$(P\otimes\alpha)(\mathrm{coev}\otimes T\otimes P):T\otimes P\Rightarrow
P\otimes T$ by $\sigma^{T}$. In Lemma 5.1 of [27], we demonstrated that indeed
for any object $X$ in $\mathcal{C}$, the pair $F(X):=(T(X),\sigma^{T}_{X})$
belongs to $\mathcal{C}(P,Q)$. Additionally, the assignment
$F:\mathcal{C}\rightarrow\mathcal{C}(P,Q)$ is functorial by construction where
with $F(f)=T(f)$ for morphisms $f$ of $\mathcal{C}$. Finally, in Theorem 5.2
of [27], we showed that $F$ is left adjoint to the relevant forgetful functor
$U:\mathcal{C}(P,Q)\rightarrow\mathcal{C}$.

The unit of the adjunction $F\dashv U$ has already appeared as
$\nu:=\psi_{0}:\mathrm{id}_{\mathcal{C}}\Rightarrow UF=T$. For the counit,
consider a pair $(X,\sigma)$ in $\mathcal{C}(P,Q)$ and denote its induced
actions $(\mathrm{ev}\otimes X)(Q\otimes\sigma)$ and
$(X\otimes\underline{\mathrm{ev}})(\sigma^{-1}\otimes Q)$ by
$\alpha_{\sigma}:F_{+}(X)\rightarrow X$ and
$\beta_{\sigma}:F_{-}(X)\rightarrow X$, respectively. One can then define
$\theta_{i_{1},i_{2},\dots,i_{n}}:F_{i_{1},i_{2},\dots,i_{n}}(X)\rightarrow
X$, for arbitrary $n\in\mathbb{N}$ and
$(i_{1},i_{2},\dots,i_{n})\in\\{-,+\\}^{n}$, by applying $\alpha_{\sigma}$ and
$\beta_{\sigma}$ iteratively so that
$\theta_{+,i_{1},i_{2},\dots,i_{n}}=\alpha_{\sigma}(Q\otimes\theta_{i_{1},i_{2},\dots,i_{n}}\otimes
P)$ and
$\theta_{-,i_{1},i_{2},\dots,i_{n}}=\beta_{\sigma}(P\otimes\theta_{i_{1},i_{2},\dots,i_{n}}\otimes
Q)$, where $\theta_{+}=\alpha_{\sigma}$ and $\theta_{-}=\beta_{\sigma}$.
Together with $\theta_{0}=\mathrm{id}_{X}$, we obtain a family of morphisms
from $F_{i_{1},i_{2},\dots,i_{n}}(X)$ to $X$, which must factorise through
$F^{\star}(X)$. In Theorem 5.2 of [27], we denoted the unique morphism
$F^{\star}(X)\rightarrow X$ by $\theta^{\star}$ and observed that the family
of morphisms described commute with the parallel pairs (67), (70), (73) and
(76), in a way that $\theta^{\star}$ must factorise through $T(X)$. Hence,
there exists a unique morphism $\theta_{(X,\sigma)}:T(X)\rightarrow X$ such
that
$\theta_{(X,\sigma)}\psi_{i_{1},i_{2},\dots,i_{n}}=\theta_{i_{1},i_{2},\dots,i_{n}}$.
It then follows that $\theta$ is a morphism between $(T(X),\sigma^{T}_{X})$
and $(X,\sigma)$ in $\mathcal{C}(P,Q)$, and
$(P\otimes\theta_{(X,\sigma)})\sigma^{T}_{X}=\sigma(\theta_{(X,\sigma)}\otimes
P)$ holds with $\theta$ becoming a natural transformation
$\theta:FU\Rightarrow\mathrm{id}_{\mathcal{C}(P,Q)}$ acting as the counit of
$F\dashv U$.

Now let us reflect on the the bimonad structure on the functor
$T:\mathcal{C}\rightarrow\mathcal{C}$. First, we observe the monad structure.
By definition, for any pair $F(X)=(T(X),\sigma^{T}_{X})$, the multiplication
of the monad $\theta_{F(X)}:TT(X)\rightarrow T(X)$ will be the unique morphism
such that

$\theta_{F(X)}(\psi_{i_{1},\dots,i_{n}})_{T(X)}F_{i_{1},\dots,i_{n}}\big{(}(\psi_{j_{1},\dots,j_{m}})_{X}\big{)}=(\psi_{i_{1},\dots,i_{n},j_{1},\dots,j_{m}})_{X}$

for arbitrary non-negative integers $n,m$ and
$i_{1},i_{2},\dots,i_{n},j_{1},j_{2},\dots,j_{m}\in\\{+,-\\}$. Meanwhile, the
unit of the monad was provided in the definition of $T$ as
$\psi_{0}:\mathrm{id}_{\mathcal{C}}\Rightarrow T$. The comonoidal structure on
$T$ arises directly from the monoidal structure of $\mathcal{C}(P,Q)$. Observe
that for pairs $(X,\sigma)$ and $(Y,\tau)$ the induced action
$\alpha_{\sigma\otimes\tau}$ on $A\otimes B$ is the composition
$(\alpha_{\sigma}\otimes\alpha_{\tau})(P\otimes
X\otimes\underline{\mathrm{coev}}\otimes Y\otimes Q)$. Consequently, we can
obtain the comonoidal structure of $T$, $T_{2}:T(-\otimes-)\rightarrow
T(-)\otimes T(-)$, by Theorem 3.2 as the unique morphism satisfying

$T_{2}\psi_{i_{1},\dots,i_{n}}=\big{(}\psi_{i_{1},\dots,i_{n}}\otimes\psi_{i_{1},\dots,i_{n}}\big{)}F_{i_{1},\dots,i_{n}}(-\otimes\mathrm{coev}_{i_{1},\dots,i_{n}}\otimes-)$

where $\mathrm{coev}_{i_{1},\dots,i_{n}}:\mathtt{1}\rightarrow
F_{-i_{n},\dots,-i_{1}}(\mathtt{1})$ are iteratively defined by
$\mathrm{coev}_{+}=\mathrm{coev}$ and
$\mathrm{coev}_{-}=\underline{\mathrm{coev}}$ and
$\mathrm{coev}_{\pm,i_{1},i_{2},\dots,i_{n}}=F_{-i_{n},\dots,-i_{1}}(\mathrm{coev}_{\pm})\mathrm{coev}_{i_{1},i_{2},\dots,i_{n}}$.
Note that the $P$-intertwining making $\mathtt{1}$ the unit of the monoidal
structure in $\mathcal{C}(P,Q)$ is simply the identity morphism
$\mathrm{id}_{P}$ and its induced actions are
$\alpha_{\mathrm{id}_{P}}=\mathrm{ev}$ and
$\beta_{\mathrm{id}_{P}}=\underline{\mathrm{ev}}$. Hence, we obtain
$T_{0}:T(\mathtt{1})\rightarrow\mathtt{1}$ as the unique morphism satisfying
$T_{0}\psi_{i_{1},i_{2},\dots,i_{n}}=\mathrm{ev}_{i_{1},i_{2},\dots,i_{n}}$,
where
$\mathrm{ev}_{i_{1},\dots,i_{n}}:F_{i_{1},\dots,i_{n}}(\mathtt{1})\rightarrow\mathtt{1}$
is defined iteratively by
$\mathrm{ev}_{\pm,i_{1},\dots,i_{n}}=\mathrm{ev}_{\pm}F_{\pm}(\mathrm{ev}_{i_{1},i_{2},\dots,i_{n}})$
with $\mathrm{ev}_{+}=\mathrm{ev}$ and
$\mathrm{ev}_{-}=\underline{\mathrm{ev}}$.

In Corollary 5.3 of [27], we noted that since $\mathcal{C}(P,Q)$ lifts the
closed monoidal structure of $\mathcal{C}$ via $U$ when $\mathcal{C}$ is
closed, then $T$ will be a Hopf monad under our base assumptions.
Equivalently, since $\mathcal{C}(P,Q)$ can be viewed as the dual of a strong
monoidal functor, this is an application of Theorem 2.3. However, this is the
only point where we needed $\mathcal{C}$ to be closed in [27]. The other
application of this condition was the fact that $-\otimes-$ would preserve
colimits in both entries as a consequence. Here we will show that $T$ will be
a Hopf monad under more relaxed conditions:

######  Theorem 4.15.

Let $\mathcal{C}$ be a monoidal category (not necessarily closed) where
$\otimes$ preserves colimits in both entries and $(P,Q)$ be a pivotal pair and
assume suitable colimits exist so that the bimonad $T$ can be constructed as
before. In this case, $T$ is a Hopf monad and the fusion operators of
Definition 3.10 are invertible.

###### Proof.

We will discuss this for the left fusion operator and leave the computations
for the right fusion operator to the reader. For a pair of arbitrary objects
$X$ and $Y$ in $\mathcal{C}$, we first note that because $\otimes$ preserves
colimits in each entry and, thereby, functors $F_{i_{1},\dots,i_{n}}$ also
preserve colimits, then $T(X\otimes T(Y))$ and $T(X)\otimes T(Y)$ both become
colimits with respect to the following families of morphisms:

$\displaystyle(\psi_{i_{1},\dots,i_{n}})_{X\otimes
T(Y)}F_{i_{1},\dots,i_{n}}(X\otimes(\psi_{j_{1},\dots,j_{m}})_{Y}):$
$\displaystyle F_{i_{1},\dots,i_{n}}(X\otimes
F_{j_{1},\dots,j_{m}}(Y))\rightarrow T(X\otimes T(Y))$
$\displaystyle(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{j_{1},\dots,j_{m}})_{Y}:$
$\displaystyle F_{i_{1},\dots,i_{n}}(X)\otimes
F_{j_{1},\dots,j_{m}}(Y)\rightarrow T(X)\otimes T(Y)$

We define the inverse $Q_{X,Y}$ of $H^{l}_{X,Y}$ as the unique morphism
satisfying

$\displaystyle
Q_{X,Y}\big{(}(\psi_{i_{1},i_{2},\dots,i_{n}})_{X}\otimes(\psi_{j_{1},j_{2},\dots,j_{m}})_{Y}\big{)}=$
$\displaystyle(\psi_{i_{1},\dots,i_{n}})_{X\otimes T(Y)}$ $\displaystyle
F_{i_{1},\dots,i_{n}}\big{(}X\otimes(\psi_{-i_{n},\dots,-i_{1},j_{1},\dots,j_{m}})_{Y}\big{)}$
$\displaystyle(F_{i_{1},\dots,i_{n}}(X)\otimes
F_{j_{1},\dots,j_{m}}(Y)\otimes\mathrm{coev}_{-i_{n},\dots,-i_{1}})$

Using this definition and the universal property of $T$ with respect to
iterations of (70) and (76), we now show that
$Q_{X,Y}H^{l}_{X,Y}=\mathrm{id}_{T(X\otimes T(Y))}$:

$\displaystyle Q$ ${}_{X,Y}H^{l}_{X,Y}(\psi_{i_{1},\dots,i_{n}})_{X\otimes
T(Y)}F_{i_{1},\dots,i_{n}}(X\otimes(\psi_{j_{1},\dots,j_{m}})_{Y})$
$\displaystyle=$ $\displaystyle
Q_{X,Y}(\mathrm{id}_{T(X)}\otimes\theta_{F(Y)})T_{2}(X,T(Y))(\psi_{i_{1},\dots,i_{n}})_{X\otimes
T(Y)}F_{i_{1},\dots,i_{n}}(X\otimes(\psi_{j_{1},\dots,j_{m}})_{Y})$
$\displaystyle=$ $\displaystyle
Q_{X,Y}(\mathrm{id}_{T(X)}\otimes\theta_{F(Y)})\big{(}(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{i_{1},\dots,i_{n}})_{T(Y)}\big{)}$
$\displaystyle
F_{i_{1},\dots,i_{n}}(X\otimes\mathrm{coev}_{i_{1},\dots,i_{n}}\otimes(\psi_{j_{1},\dots,j_{m}})_{Y})$
$\displaystyle=$ $\displaystyle
Q_{X,Y}\big{(}(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{i_{1},\dots,i_{n},j_{1},\dots,j_{m}})_{Y}\big{)}F_{i_{1},\dots,i_{n}}(X\otimes\mathrm{coev}_{i_{1},\dots,i_{n}}\otimes
F_{j_{1},\dots,j_{m}}(Y))$ $\displaystyle=$
$\displaystyle(\psi_{i_{1},\dots,i_{n}})_{X\otimes
T(Y)}F_{i_{1},\dots,i_{n}}\big{(}X\otimes(\psi_{-i_{n},\dots,-i_{1},i_{1},\dots,i_{n},j_{1},\dots,j_{m}})_{Y}\big{)}$
$\displaystyle(F_{i_{1},\dots,i_{n}}(X)\otimes
F_{i_{1},\dots,i_{n},j_{1},\dots,j_{m}}(Y)\otimes\mathrm{coev}_{-i_{n},\dots,-i_{1}})$
$\displaystyle
F_{i_{1},\dots,i_{n}}(X\otimes\mathrm{coev}_{i_{1},\dots,i_{n}}\otimes
F_{j_{1},\dots,j_{m}}(Y))$ $\displaystyle=$
$\displaystyle(\psi_{i_{1},\dots,i_{n}})_{X\otimes
T(Y)}F_{i_{1},\dots,i_{n}}\big{(}X\otimes(\psi_{j_{1},\dots,j_{m}})_{Y}\big{)}F_{i_{1},\dots,i_{n}}(X\otimes
F_{j_{1},\dots,j_{m}}(Y)\otimes\mathrm{ev}_{-i_{n},\dots,-i_{1}})$
$\displaystyle\big{(}F_{i_{1},\dots,i_{n}}(X\otimes
F_{j_{1},\dots,j_{m}}(Y))\otimes\mathrm{coev}_{-i_{n},\dots,-i_{1}}\big{)}$
$\displaystyle=$ $\displaystyle(\psi_{i_{1},\dots,i_{n}})_{X\otimes
T(Y)}F_{i_{1},\dots,i_{n}}\big{(}X\otimes(\psi_{j_{1},\dots,j_{m}})_{Y}\big{)}$

By the universal property of $T(X\otimes T(Y))$ we conclude that
$Q_{X,Y}H^{l}_{X,Y}=\mathrm{id}_{T(X\otimes T(Y))}$. By a similar argument we
see that $H^{l}_{X,Y}Q_{X,Y}=\mathrm{id}_{T(X)\otimes T(Y)}$:

$\displaystyle H$
${}^{l}_{X,Y}Q_{X,Y}\big{(}(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{j_{1},\dots,j_{m}})_{Y}\big{)}=(\mathrm{id}_{T(X)}\otimes\theta_{F(Y)})T_{2}(X,T(Y))(\psi_{i_{1},\dots,i_{n}})_{X\otimes
T(Y)}$ $\displaystyle
F_{i_{1},\dots,i_{n}}\big{(}X\otimes(\psi_{-i_{n},\dots,-i_{1},j_{1},\dots,j_{m}})_{Y}\big{)}(F_{i_{1},\dots,i_{n}}(X)\otimes
F_{j_{1},\dots,j_{m}}(Y)\otimes\mathrm{coev}_{-i_{n},\dots,-i_{1}})$
$\displaystyle=$
$\displaystyle(\mathrm{id}_{T(X)}\otimes\theta_{F(Y)})\big{(}(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{i_{1},\dots,i_{n}})_{T(Y)}\big{)}F_{i_{1},\dots,i_{n}}(X\otimes\mathrm{coev}_{i_{1},\dots,i_{n}}\otimes
T(Y))$ $\displaystyle
F_{i_{1},\dots,i_{n}}\big{(}X\otimes(\psi_{-i_{n},\dots,-i_{1},j_{1},\dots,j_{m}})_{Y}\big{)}(F_{i_{1},\dots,i_{n}}(X)\otimes
F_{j_{1},\dots,j_{m}}(Y)\otimes\mathrm{coev}_{-i_{n},\dots,-i_{1}})$
$\displaystyle=$
$\displaystyle\big{(}(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{i_{1},\dots,i_{n},-i_{n},\dots,-i_{1},j_{1},\dots,j_{m}})_{Y}\big{)}$
$\displaystyle
F_{i_{1},\dots,i_{n}}(X\otimes\mathrm{coev}_{i_{1},\dots,i_{n}}\otimes
F_{-i_{n},\dots,-i_{1},j_{1},\dots,j_{m}}(Y))$
$\displaystyle(F_{i_{1},\dots,i_{n}}(X)\otimes
F_{j_{1},\dots,j_{m}}(Y)\otimes\mathrm{coev}_{-i_{n},\dots,-i_{1}})$
$\displaystyle=$
$\displaystyle\big{(}(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{j_{1},\dots,j_{m}})_{Y}\big{)}\big{(}F_{i_{1},\dots,i_{n}}(X)\otimes\mathrm{ev}_{i_{1},\dots,i_{n}}\otimes
F_{j_{1},\dots,j_{m}}(Y)\big{)}$
$\displaystyle\big{(}F_{i_{1},\dots,i_{n}}(X\otimes\mathrm{coev}_{i_{1},\dots,i_{n}})\otimes
F_{j_{1},\dots,j_{m}}(Y)\big{)}=(\psi_{i_{1},\dots,i_{n}})_{X}\otimes(\psi_{j_{1},\dots,j_{m}})_{Y}$

Here we used the universal property of $T$ with respect to iterations of (67)
and (73). The inverse of the right fusion operator is the unique map
satisfying:

$\displaystyle(H^{r}_{X,Y})^{-1}\big{(}(\psi_{i_{1},i_{2},\dots,i_{n}})_{X}\otimes(\psi_{j_{1},i_{2},\dots,j_{m}})_{Y}\big{)}=(\psi_{i_{1},\dots,i_{n}})_{T(X)\otimes
Y}$ $\displaystyle
F_{j_{1},\dots,j_{m}}((\psi_{-j_{m},\dots,-j_{1},i_{1},\dots,i_{n}})_{X}\otimes
Y)(\mathrm{coev}_{-j_{m},\dots,-j_{1}}\otimes F_{i_{1},\dots,i_{n}}(X)\otimes
F_{j_{1},\dots,j_{m}}(Y))$

and the corresponding computation follows in a symmetric manner. ∎

Finally, we recall the criterion for when the constructed Hopf monad is
augmented from [27]. We say a pivotal pair $(P,Q)$ in $\mathcal{C}$ _lifts to_
$Z(C)$, if there exist braidings
$\lambda:P\otimes\mathrm{id}_{\mathcal{C}}\Rightarrow\mathrm{id}_{\mathcal{C}}\otimes
P$ and
$\chi:Q\otimes\mathrm{id}_{\mathcal{C}}\Rightarrow\mathrm{id}_{\mathcal{C}}\otimes
Q$, such that $(P,\lambda)$ and $(Q,\chi)$ are objects in $Z(\mathcal{C})$ and
$\mathrm{coev},\mathrm{ev},\underline{\mathrm{coev}}$ and
$\underline{\mathrm{ev}}$ become morphisms in $Z(\mathcal{C})$, making
$(P,\lambda)$ and $(Q,\chi)$ a pivotal pair in $Z(\mathcal{C})$.

######  Theorem 4.16 (Theorem 5.4 [27]).

The Hopf monad $T$ is augmented if and only if the pivotal pair $(P,Q)$ lifts
to $Z(\mathcal{C})$.

#### Sketch of proof.

($\Rightarrow$) Assuming $T$ is augmented with a Hopf monad morphism
$\epsilon:T\Rightarrow\mathrm{id}_{\mathcal{C}}$, we obtain a braiding on $P$
by $\lambda:=(\epsilon\psi_{-}\otimes
P)(P\otimes\mathrm{id}_{\mathcal{C}}\otimes\underline{\mathrm{coev}})$.
Additionally, $\lambda$ is invertible and
$\lambda^{-1}:=(P\otimes\epsilon\psi_{+})(\mathrm{coev}\otimes\mathrm{id}_{\mathcal{C}}\otimes
P)$ . In a symmetric manner, one can introduce $\chi:=(\epsilon\psi_{+}\otimes
Q)(Q\otimes\mathrm{id}_{\mathcal{C}}\otimes\mathrm{coev})$ with
$\chi^{-1}:=(Q\otimes\epsilon\psi_{-})(\underline{\mathrm{coev}}\otimes\mathrm{id}_{\mathcal{C}}\otimes
Q)$ as its inverse and show that
$\mathrm{coev},\mathrm{ev},\underline{\mathrm{coev}}$ and
$\underline{\mathrm{ev}}$ become morphisms in $Z(\mathcal{C})$ and commute
with the braidings of $\mathtt{1}$, $P\otimes Q$ and $Q\otimes P$.

($\Leftarrow$) Assuming there exist braidings
$\lambda:P\otimes\mathrm{id}_{\mathcal{C}}\Rightarrow\mathrm{id}_{\mathcal{C}}\otimes
P$ and
$\chi:Q\otimes\mathrm{id}_{\mathcal{C}}\Rightarrow\mathrm{id}_{\mathcal{C}}\otimes
Q$ making $(P,\lambda)$ and $(Q,\chi)$ objects in $Z(\mathcal{C})$, such that
$\mathrm{coev},\mathrm{ev},\underline{\mathrm{coev}}$ and
$\underline{\mathrm{ev}}$ are morphisms in $Z(\mathcal{C})$, we can
iteratively define the natural transformations $\epsilon_{i_{1},\dots
i_{n}}:F_{i_{1},\dots i_{n}}\Rightarrow\mathrm{id}_{\mathcal{C}}$ by
$\epsilon_{+,i_{1},\dots
i_{n}}=(\mathrm{id}_{\mathcal{C}}\otimes\mathrm{ev})(\chi\otimes
P)F_{+}(\epsilon_{i_{1},\dots i_{n}})$ and $\epsilon_{-,i_{1},\dots
i_{n}}=(\mathrm{id}_{\mathcal{C}}\otimes\underline{\mathrm{ev}})(\lambda\otimes
Q)F_{-}(\epsilon_{i_{1},\dots i_{n}})$ where
$\epsilon_{0}=\mathrm{id}_{\mathcal{C}}$. Since $\mathrm{ev}$ and
$\underline{\mathrm{ev}}$ commute with the braidings, then
$\epsilon_{+}=(\mathrm{ev}\otimes\mathrm{id}_{\mathcal{C}})(Q\otimes\lambda^{-1})$
and
$\epsilon_{-}=(\underline{\mathrm{ev}}\otimes\mathrm{id}_{\mathcal{C}})(P\otimes\chi^{-1})$.
It is straightforward to check that $\epsilon_{i_{1},\dots i_{n}}$ commute
with the parallel pairs (67), (70), (73) and (76), and therefore induce a
unique morphism $\epsilon:T\rightarrow\mathrm{id}_{\mathcal{C}}$. and from the
universal properties of $TT$ and $T(-\otimes-)$, we can conclude that
$\epsilon$ is a bimonad morphism. $\diamond$

In Examples 5.6 of [27], we looked at the case when
$\mathcal{C}=\mathsf{Vec}$. In this case, pivotal pairs in $\mathsf{Vec}$
correspond to invertible matrices and given any such matrix, we obtain an
involutive Hopf algebra $H$ such that $T\cong H\otimes_{\mathbb{K}}-$. On the
other hand, as we will see in Theorem 4.19, additive Hopf monads on
${}_{A}{\mathcal{M}}_{A}$ which admit a right adjoint correspond to Hopf
algebroids over $A$, in the sense of Schauenburg [56]. In Theorem 4.3 of [26],
we constructed the relevant Hopf algebroids which arise in this way by
considering a pivotal pair $(P,Q)$ in ${}_{A}{\mathcal{M}}_{A}$.

###  4.6 Bialgebroids and Hopf Algebroids

In this section we will review the theory of bialgebroids and Schauenburg’s
notion of Hopf algebroids as examples of bimonads and Hopf monads on the
category of bimodules ${}_{A}{\mathcal{M}}_{A}$. The notation used for
describing Hopf algebroids varies quite a bit depending on the reference, but
here we adapt the notation from Chapter 5 of [9]. Throughout this section $A$
denotes a fixed $\mathbb{K}$-algebra and $\otimes$ denotes $\otimes_{A}$,
unless it appears with a different subscript.

For an algebra $A$, the _opposite algebra_ $A^{\mathrm{op}}$ is the algebra
structure defined on $A$ by $(\overline{a})(\overline{b})=\overline{ba}$,
where we denote elements of the opposite algebra with a line above i.e $a,b\in
A$ and $\overline{a},\overline{b}\in A^{\mathrm{op}}$. It is a well-known fact
that $A$-bimodules correspond to left
$A\otimes_{\mathbb{K}}A^{\mathrm{op}}$-modules, where
$A^{e}=A\otimes_{\mathbb{K}}A^{\mathrm{op}}$ is called the _enveloping
algebra_ of $A$. More concretely, there exists an isomorphism of categories,
between the category of $A$-bimodules $\prescript{}{A}{\mathcal{M}}_{A}$ and
that of left $A^{e}$-modules $\prescript{}{A^{e}}{\mathcal{M}}$. Hence, we use
${}_{A^{e}}{\mathcal{M}}$ and ${}_{A}{\mathcal{M}}_{A}$ interchangeably. We
will denote elements of $A^{e}=A\otimes_{\mathbb{K}}A^{\mathrm{op}}$ by
$a\overline{b}$ where $a\in A$ and $\overline{b}\in A^{\mathrm{op}}$. The
reader should note, that other authors often prefer the notation
$a\overline{b}=\mathsf{s}(a)\mathsf{t}(b)$, where the algebra morphisms
$A\hookrightarrow A^{e}\rightarrow H$ and $A^{\mathrm{op}}\hookrightarrow
A^{e}\rightarrow H$ are denoted by $\mathsf{s}$ and $\mathsf{t}$ and called
_source_ and _target_ , respectively.

For an $A^{e}$-bimodule $B$ we denote the functor
$\prescript{}{A^{e}}{B}\otimes_{A^{e}}-$ by
$B\boxtimes-:{}_{A}{\mathcal{M}}_{A}\rightarrow{}_{A}{\mathcal{M}}_{A}$. This
functor absorbs the bimodule structure of its input via its right
$A^{e}$-action and produces a new bimodule actions via its left
$A^{e}$-action. Explicitly, for an $A$-bimodule $M$ we have

$\displaystyle B\boxtimes
M=B\otimes_{\mathbb{K}}M/\\{(br\overline{s})\otimes_{\mathbb{K}}m$
$\displaystyle-b\otimes_{\mathbb{K}}(rms)\mid m\in M,\ r,s\in A,\ b\in B\\}$
$\displaystyle r(b\boxtimes m)s=(r\overline{s}b)\boxtimes m$
$\displaystyle\quad\forall m\in M,\ \forall r,s\in A,\ \forall b\in B$

Note that any $A^{e}$-bimodule $B$ can be considered as an $A$-bimodule either
by its right or left $A^{e}$-action, and we denote the latter $A$-bimodule by
$|B$. We continue to adapt the notation of [9] and recall the following
definitions from Chapter 5.

######  Definition 4.17.

Let $A$ be an algebra and $B$ an $A^{e}$-bimodule.

  1. (I)

An $A^{e}$-_ring_ structure on $B$ consists of a $\mathbb{K}$-algebra
structure $(\mu,1_{B})$ on $B$ with an algebra homomorphism
$\eta:A^{e}\rightarrow B$, such that the $A^{e}$-bimodule structure on $B$ is
induced by the algebra homomorphism i.e.
$\mu(\eta\otimes_{\mathbb{K}}\mathrm{id}_{B})$ coincides with the left action
of $A^{e}$ and $\mu(\mathrm{id}_{B}\otimes_{\mathbb{K}}\eta)$ with the right
action of $A^{e}$.

  2. (II)

An $A|A$-_coring_ structure on $B$ consists of bimodule maps
$\Delta:|B\rightarrow|B\otimes|B$ and $\epsilon:|B\rightarrow A$ satisfying

$\displaystyle b_{(1)}\otimes(b_{(2)})_{(1)}\otimes(b_{(2)})_{(2)}=$
$\displaystyle(b_{(1)})_{(1)}\otimes(b_{(1)})_{(2)}\otimes b_{(2)}$ (77)
$\displaystyle\epsilon(b_{(1)})b_{(2)}=b$
$\displaystyle=\overline{\epsilon(b_{(2)})}b_{(1)}$ (78)
$\displaystyle\Delta(br\overline{s})=$ $\displaystyle b_{(1)}r\otimes
b_{(2)}\overline{s}$ (79) $\displaystyle\epsilon(br)=$
$\displaystyle\epsilon(b\overline{r})$ (80)

for any $b\in B$ and $r,s\in A$, where $\Delta(b)=b_{(1)}\otimes b_{(2)}$ is
denoted by Sweedler’s notation as in the case of Hopf algebras. Conditions
(77) and (78) are equivalent to $(|B,\Delta,\epsilon)$ being a comonoid in the
category of $A$-bimodules.

  3. (III)

A _left_ $A$-_bialgebroid_ structure on $B$ consists of an $A^{e}$-ring
structure $(\mu,\eta)$ and an $A|A$-coring structure $(\Delta,\epsilon)$ on
$B$ satisfying

$\displaystyle(bb^{\prime})_{(1)}\otimes(bb^{\prime})_{(2)}$
$\displaystyle=b_{(1)}b_{(1)}^{\prime}\otimes b_{(2)}b_{(2)}^{\prime}$ (81)
$\displaystyle\Delta(1_{B})$ $\displaystyle=1_{B}\otimes 1_{B}$ (82)
$\displaystyle\epsilon(1_{B})$ $\displaystyle=1_{A}$ (83)
$\displaystyle\epsilon(bb^{\prime})=\epsilon$
$\displaystyle(b\epsilon(b^{\prime}))=\epsilon\left(b\overline{\epsilon(b^{\prime})}\right)$
(84)

for any $b,b^{\prime}\in B$, where $1_{B}=\eta(1_{A^{e}})$.

First note that an $A^{e}$-ring structure as defined above is equivalent
picking $A^{e}$-bimodule maps $\mu_{A^{e}}:B\otimes_{A^{e}}B\rightarrow B$ and
$\eta_{A^{e}}:A^{e}\rightarrow B$, which provide $B$ with the structure of a
monoid in the monoidal category of $A^{e}$-bimodules. Secondly, we can deduce
from the definition of an $A|A$-coring $B$ that the image of $\Delta$ lands in

$B\times_{A}B:=\left\\{\sum_{i}b_{i}\otimes
b^{\prime}_{i}\in|B\otimes|B\middle|\sum_{i}b_{i}\overline{a}\otimes
b^{\prime}_{i}=\sum_{i}b_{i}\otimes b^{\prime}_{i}a,\ a\in
A\right\\}\subset|B\otimes|B$

Bialgebroids are often defined with reference to $B\times_{A}B$, the _Takeuchi
$\times$-product_ [63], and therefore called $\times$-bialgebras. The
equivalence of the above definition and the more popular variation is present
in both [9, 11].

It is easy to check that $A^{e}$-ring structures on $B$ correspond to monad
structures on the endofunctor $B\boxtimes-$ (Proposition 5.2 [9]) since there
is a monoidal embedding of ${}_{A^{e}}\mathcal{M}_{A^{e}}$ into
$\mathrm{End}({}_{A^{e}}{\mathcal{M}})$. In particular, by restriction of
scalars along $\eta$, any $B$-module is equipped with an $A$-bimodule
structure and there exists a forgetful functor
$U:\prescript{}{B}{\mathcal{M}}\rightarrow{}_{A}{\mathcal{M}}_{A}$ with its
free left adjoint functor also denoted by $B\boxtimes-$. In this way we
consider left actions $B\otimes_{\mathbb{K}}M\rightarrow M$ of $B$ on
$A$-bimodules $M$ which factor through an $A$-bimodule map $B\boxtimes
M\rightarrow M$.

Similarly, $A|A$-coring structures on $B$ correspond to comonoidal structures
on the functor $B\boxtimes-$ (Proposition 5.5 [9]). In our notation if
$T=B\boxtimes-$, then $T_{2}(M,N)=\Delta_{M,N}$ is defined as the map

$\textstyle{\Delta_{M,N}:B\boxtimes(M\otimes N)\longrightarrow(B\boxtimes
M)\otimes(B\boxtimes N)}$$\textstyle{\hskip 52.63777ptb\boxtimes(m\otimes
n)\longmapsto(b_{(1)}\boxtimes m)\otimes(b_{(2)}\boxtimes n)}$ (85)

which is well-defined and a bimodule map for any pair of $A$-bimodules $M$ and
$N$ by condition (79). Additionally, by (80) the map $\epsilon$ factorizes
through a map $T_{0}:B\boxtimes A\rightarrow A$. Furthermore, conditions (77)
and (78) assure that (1) and (2) hold, respectively.

Given the above correspondences, the conditions in Definition 4.17 (III)
become equivalent to the conditions in Definition 3.1 and bimonad structures
on $B\boxtimes-$ correspond precisely to left $A$-bialgebroid structures on
$B$ (Theorem 5.9 [9]). From this point of view the category of $B$-modules
lifts the monoidal structure of ${}_{A}{\mathcal{M}}_{A}$: If
$(M,\triangleright_{M})$ and $(N,\triangleright_{N})$ are $B$-modules, the
$B$-action on $M\otimes N$ given by (24) is defined by the composition
$(\triangleright_{M}\otimes\triangleright_{N})\Delta_{M,N}$.

We must point out that the theory described above is not symmetric. A _right_
$A$-bialgebroid structure on $B$ arises when we ask the category of right
$B$-modules to be monoidal so that the forgetful functor
$\mathcal{M}_{B}\rightarrow{}_{A}{\mathcal{M}}_{A}$ becomes strong monoidal.
In other words, right $A$-bialgebroid structures are those which make the
functor $-\boxtimes B=-\otimes_{A^{e}}B_{A^{e}}$ a bimonad. From here onwards
an $A$-bialgebroid structure always refers to a left $A$-bialgebroid
structure.

There have been several variations of the Hopf condition for bialgebroids to
mimic the Hopf condition for bialgebras. The choice which interests us is the
condition which makes the corresponding bimonad of a bialgebroid $B$ into a
Hopf monad. This is the case for Schauenburg’s Hopf algebroids which were
introduced in [56].

######  Definition 4.18.

A _Schauenburg Hopf algebroid_ or $\times$-_Hopf algebra_ structure on $B$
consists of an $A$-bialgebroid structure as above, such that the induced maps

(90)

are invertible, where we define the tensor products
$\otimes_{A^{\mathrm{op}}}$, $\odot$ and $\diamond$ as follows:

$\displaystyle B\otimes_{A^{\mathrm{op}}}B$
$\displaystyle=B\otimes_{\mathbb{K}}B/\\{b\overline{s}\otimes_{\mathbb{K}}b^{\prime}-b\otimes_{\mathbb{K}}\overline{s}b^{\prime}\mid
b,b^{\prime}\in B,\overline{s}\in A^{\mathrm{op}}\\}$ $\displaystyle B\odot B$
$\displaystyle=B\otimes_{\mathbb{K}}B/\\{br\otimes_{\mathbb{K}}b^{\prime}-b\otimes_{\mathbb{K}}rb^{\prime}\mid
b,b^{\prime}\in B,r\in A\\}$ $\displaystyle B\diamond B$
$\displaystyle=B\otimes_{\mathbb{K}}B/\\{\overline{s}b\otimes_{\mathbb{K}}b^{\prime}-b\otimes_{\mathbb{K}}sb^{\prime}\mid
b,b^{\prime}\in B,s\in A\\}$

Once one writes down the fusion operators for the bimonad $T=B\boxtimes-$, it
is easy to see that the maps $\beta$ and $\vartheta$ being invertible are
equivalent to the bimonad $T=B\boxtimes-$ being left and right Hopf,
respectively. Consequently, if $B$ is a Schauenburg Hopf algebroid and
$\beta,\vartheta$ are invertible, we usually denote $\beta^{-1}(b\diamond
1)=b_{(+)}\otimes_{A^{\mathrm{op}}}b_{(-)}$ and $\vartheta^{-1}(1\diamond
b)=b_{[+]}\odot b_{[-]}$ and thereby observe that the closed structure of
${}_{A}{\mathcal{M}}_{A}$ is lifted to $\prescript{}{B}{\mathcal{M}}$ via the
following $B$-actions:

$\textstyle{B\boxtimes\mathrm{Hom}_{A}(M,N)\rightarrow\mathrm{Hom}_{A}(M,N)}$$\textstyle{B\boxtimes\prescript{}{A}{\mathrm{Hom}}(M,N)\rightarrow\prescript{}{A}{\mathrm{Hom}}(M,N)}$$\textstyle{b\boxtimes
f\mapsto(m\mapsto b_{(+)}f(b_{(-)}m))}$$\textstyle{b\boxtimes
g\mapsto(m\mapsto b_{[+]}g(b_{[-]}m))}$ (91)

for any pair of $B$-bimodules $M$ and $N$.

Finally, we refer the reader to Chapter 5 of [9] and [11] for further details
on these facts. We conclude by presenting the following observation. The
Eilenberg-Watts theorem [66] tells us that any additive left adjoint functor
$F:{}_{A^{e}}{\mathcal{M}}\rightarrow{}_{A^{e}}{\mathcal{M}}$ is isomorphic to
a functor $\prescript{}{A^{e}}{B}\otimes_{A^{e}}-$, where $B$ is an
$A^{e}$-bimodule. Using this result one can classify additive left adjoint
bimonads and Hopf monads on ${}_{A}{\mathcal{M}}_{A}$:

######  Theorem 4.19 ([62]).

For an algebra $A$ and an abelian monoidal category $\mathcal{C}$, if
$G:\mathcal{C}\rightarrow{}_{A}{\mathcal{M}}_{A}$ is an additive functor with
a left adjoint $F$, such that
$GF:{}_{A}{\mathcal{M}}_{A}\rightarrow{}_{A}{\mathcal{M}}_{A}$ has a right
adjoint , then $G$ is (closed) strong monoidal if and only if $\mathcal{C}$ is
equivalent to $\prescript{}{B}{\mathcal{M}}$ for a left (Hopf) bialgebroid
$B$.

This result was translated into the language of Hopf monads in [12] and simply
states that colimit-preserving (Hopf monad) bimonads on
${}_{A}{\mathcal{M}}_{A}$ correspond to (Schauenburg Hopf algebroids) left
bialgebroids over $A$.

######  Remark 4.20.

We should warn the reader that the notion of a Hopf algebroid over an algebra
$A$ described here is not the same as the notion of a _weak Hopf algebra_.
However, any weak Hopf algebra structure defined over a field $\mathbb{K}$ is
equivalent to a Hopf algebroid structure over a separable Frobenius
$\mathbb{K}$-algebra $A$. The details of this correspondence are explained in
Chapter 6 of [9].

##  5 Combining Hopf monads and Bimonads

In this section, we review two ways in which we can combine Hopf monads to
obtain new Hopf monads: Bosonisation or cross product, and distributive laws.

###  5.1 Bosonisation and Cross Products

In this section we review the theory of cross products or bosonisation for
Hopf monads which describes how we can compose a Hopf monad with a secondary
Hopf monad on its module category. This in turn generalises Radford’s
biproduct construction and Majid’s bosonisation in the theory of ordinary Hopf
algebras.

Assume $T$ is a monad on $\mathcal{C}$ and $P$ is a monad on
$\mathcal{C}^{T}$. Then we can consider the composition of the following
adjunctions

$\textstyle{\left(\mathcal{C}^{T}\right)^{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{P}}$$\textstyle{\mathcal{C}^{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{P}}$$\scriptstyle{U_{T}}$$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{T}}$

In this situation, the endofunctor $U_{T}PF_{T}$ is called the _cross product_
of $T$ by $P$ and is denoted by $P\rtimes T$. If $\eta$ and $\epsilon$ denote
the unit and counit of the adjunction $F_{T}\dashv U_{T}$ and $\eta^{\prime}$
and $\mu^{\prime}$ denote the unit and multiplication providing the monad
structure on $P$, then the monad structure $(P\rtimes T,q,\nu)$ is defined as
follows

$q=(U_{T}\mu^{\prime}_{F_{T}})(U_{T}P\epsilon_{PF_{T}}):U_{T}PF_{T}U_{T}PF_{T}\rightarrow
U_{T}PF_{T},\quad\nu=(U_{T}\eta^{\prime}_{F_{T}})\eta$ (92)

Observe that although $P\rtimes T$ is the corresponding monad of the
adjunction $F_{P}F_{T}\dashv U_{T}U_{P}$, which is the composition of two
monadic adjunctions, $F_{P}F_{T}\dashv U_{T}U_{P}$ itself is not necessarily
monadic. In other words the comparison functor
$K:\left(\mathcal{C}^{T}\right)^{P}\rightarrow\mathcal{C}^{P\rtimes T}$ might
not be an equivalence.

######  Theorem 5.1 (Proposition 5.1 [4]).

If $T$ is a monad on $\mathcal{C}$ and $P$ a monad on $\mathcal{C}^{T}$ which
preserves reflexive coequalizers then the adjunction $F_{P}F_{T}\dashv
U_{T}U_{P}$ is monadic. Moreover, the comparison functor
$K:\left(\mathcal{C}^{T}\right)^{P}\rightarrow\mathcal{C}^{P\rtimes T}$ is an
isomorphism of categories.

If $\mathcal{C}$ is monoidal and $T$ and $P$ are both bimonads then $U_{T}$
and $U_{P}$ are both strong monoidal and thereby so is their composition
$U_{T}U_{P}$. Therefore the cross product of two bimonads also becomes a
bimonad, where $P\rtimes T$ obtains a canonical comonoidal structure as the
composition of comonoidal functors. In fact one can show that the composition
of Hopf monads also satisfies the Hopf condition:

######  Theorem 5.2 (Proposition 4.4 [12]).

The cross product of two (left or right) Hopf monads is also a (left or right)
Hopf monad.

#### Sketch of proof.

Let us assume that $T$ and $P$ are both left Hopf monads and denote the units
and counits of adjunctions $F_{T}\dashv U_{T}$ and $F_{P}\dashv U_{P}$ by
$\eta$, $\epsilon$ and $\eta^{\prime}$, $\epsilon^{\prime}$, respectively. We
observe that the left fusion operator of the adjunction $F_{P}F_{T}\dashv
U_{T}U_{P}$ appears in the left edge of the commuting rectangle below:

$\textstyle{F_{P}F_{T}(X\otimes
U_{T}U_{P}(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F_{P})_{2}(F_{T}(X),F_{T}U_{T}U_{P}(Y))(F_{T})_{2}(X,U_{T}U_{P}(Y))}$$\scriptstyle{F_{P}(F_{T})_{2}(X,U_{T}U_{P}(Y))}$$\textstyle{F_{P}(F_{T}(X)\otimes
F_{T}U_{T}U_{P}(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{P}\left(\mathrm{id}_{F_{T}(X)}\otimes\epsilon_{U_{P}(Y)}\right)}$$\textstyle{F_{P}F_{T}(X)\otimes
F_{P}F_{T}U_{T}U_{P}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id}_{F_{P}F_{T}(X)}\otimes\epsilon^{\prime}_{Y}F_{P}\epsilon_{U_{P}(Y)})}$$\textstyle{F_{P}(F_{T}(X)\otimes
U_{P}(Y))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F_{P})_{2}(F_{T}(X),U_{P}(Y))}$$\textstyle{F_{P}F_{T}(X)\otimes
Y}$$\textstyle{F_{P}F_{T}(X)\otimes
F_{P}U_{P}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{F_{P}F_{T}(X)}\otimes\epsilon^{\prime}_{Y}}$
(93)

Since the above diagram commutes, the left fusion operator of the composed
adjunction can be written as the composition of the two fusion operators of
$T$ and $P$. Hence, if $T$ and $P$ both have invertible left fusion operators,
then so does $P\rtimes T$. A symmetric argument can be applied to the right
fusion operator. $\diamond$

This notion of cross product for Hopf monads generalises Radford’s biproduct
and Majid’s bosonisation for ordinary Hopf algebras. Given a bialgebra
$(H,m_{H},1_{H},\Delta_{H},\epsilon_{H})$ and an $H$-module algebra $A$ i.e.
an algebra $((A,\triangleright),m_{A},1_{A})$ in the category of left
$H$-modules ${}_{H}{\mathcal{M}}$, we can form a new algebra on the vector
space $A\otimes_{\mathbb{K}}H$ called the _cross product algebra_ $A\rtimes
H$. This structure is exactly what we obtain by composing monads
$T=H\otimes_{\mathbb{K}}-$ on $\mathsf{Vec}$ and $P=A\otimes_{\mathbb{K}}-$ on
${}_{H}{\mathcal{M}}$, where we obtain a monad structure on $P\rtimes
T=A\otimes_{\mathbb{K}}H\otimes_{\mathbb{K}}-$ and an algebra structure on
$A\otimes_{\mathbb{K}}H$ with $1_{A}\otimes_{\mathbb{K}}1_{H}$ as its unit and
its multiplication defined by

$\displaystyle(m_{A}\otimes\mathrm{id}_{H})(\mathrm{id}_{A}\otimes\triangleright\otimes
m_{H})(\mathrm{id}_{A\otimes
H}\otimes\Psi_{H,A}\otimes\mathrm{id}_{H})(\mathrm{id}_{A}\otimes\Delta\otimes\mathrm{id}_{A\otimes
H})$

where $\triangleright:H\otimes_{\mathbb{K}}A\rightarrow A$ denotes the
$H$-module action on $A$ and $\Psi$ the symmetry on $\mathsf{Vec}$.

Now let us assume $(H,m_{H},1_{H},\Delta_{H},\epsilon_{H},S_{H})$ is a Hopf
algebra and that the category of $H$-modules is braided (in this case $H$ has
a quasitriangular structure, see Section 7.3). In this case, one can pick an
$H$-module Hopf algebra
$((A,\triangleright),m_{A},1_{A},\Delta_{A},\epsilon_{A},S_{A})$ i.e. a Hopf
algebra in the braided category of $H$-modules, and as previously discussed
$P=A\otimes_{\mathbb{K}}-$ will have a natural left Hopf monad structure. By
Theorem 5.2 the bimonad $P\rtimes
T=A\otimes_{\mathbb{K}}H\otimes_{\mathbb{K}}-$ will also be left Hopf. Hence,
we obtain a new Hopf algebra $A\mathrel{\rtimes\cdot}H$ where the coalgebra
and antipode, now use the braiding $\Psi^{\prime}$ of the category
${}_{H}{\mathcal{M}}$:

$\displaystyle\Delta_{A\mathrel{\rtimes\cdot}H}:=$
$\displaystyle(\mathrm{id}_{A}\otimes_{\mathbb{K}}\Psi^{\prime}_{A,H}\otimes_{\mathbb{K}}\mathrm{id}_{H})(\Delta_{A}\otimes_{\mathbb{K}}\Delta_{H})$
$\displaystyle S_{A\mathrel{\rtimes\cdot}H}:=$
$\displaystyle(\triangleright\otimes_{\mathbb{K}}\mathrm{id}_{H})(\mathrm{id}_{H}\otimes_{\mathbb{K}}\Psi_{H,A})(\Delta_{H}\otimes_{\mathbb{K}}\mathrm{id}_{A})\Psi^{\prime}_{A,H}(S_{A}\otimes_{\mathbb{K}}S_{H})$

This procedure is called _bosonisation_ and was introduced by Majid in [45].

Even if the category of $H$-modules is not necessarily braided, but $(A,\tau)$
has a central Hopf algebra structure in ${}_{H}{\mathcal{M}}$ or equivalently
$A$ is a braided Hopf algebra in ${}^{H}_{H}\mathcal{YD}$, then we obtain a
natural left Hopf monad $P=A\otimes_{\tau}-$ on ${}_{H}{\mathcal{M}}$. By
Theorem 5.2 we again obtain a Hopf algebra structure
$A\mathrel{\rtimes\cdot}H$ on $A\otimes H$, where the coalgebra and antipode
are defined in the same way but the braiding $\Psi^{\prime}_{A,H}$ is replaced
with the braiding $\tau_{H}$. This construction is referred to as _Radford’s
biproduct_ and first appeared in [53] without any reference to braided Hopf
algebras. Its interpretation in terms of braided Hopf algebras appeared in the
Appendix of [42]. While from the monadic point of view, the constructions look
exactly the same, in bosonisation the braidings arise from the braiding on
${}_{H}{\mathcal{M}}$ via a quasitriangular structure on $H$, while in
Radford’s biproduct the braiding $\tau$ can be written in terms of the
$H$-comodule structure on $A$ making $A$ a Yetter-Drinfeld module and the
formulas take a very different look in this way. We refer the reader to
Section 9.4 of [46] for further details on these constructions.

######  Example 5.3 (Cross product for Hopf algebroids).

Consider a Hopf algebroid $H$ over a base algebra $A$ and its corresponding
Hopf monad $T=H\boxtimes-$ on ${}_{A}{\mathcal{M}}_{A}$. By the described
theory of cross products, we can define an analogous procedure for Hopf
algebroids if we are given a braided Hopf algebra $B$ in the center of
${}_{H}{\mathcal{M}}$. This provides us with a Hopf monad $B\otimes_{A}-$ on
${}_{H}{\mathcal{M}}$ and, by Theorem 5.2, we can compose these monads to
obtain a new Hopf monad on ${}_{A}{\mathcal{M}}_{A}$, which itself will
correspond to a new Hopf algebroid over $A$, by Theorem 4.19. Let us briefly
describe the new Hopf algebroid structure on $B\otimes_{A}H$.

First, we recall from Proposition 4.4 [56] that the lax left dual
$\mathcal{Z}_{l,\mathrm{lax}}({}_{H}{\mathcal{M}})$ can be identified with the
category of left _Yetter-Drinfeld_ modules over $H$, which are defined in a
completely analogous way to classical Hopf algebras. First recall that over a
bialgebroid $H$, a left $H$-_comodule structure_ means a left $A$-module $B$
morphism $\delta:{}_{A}B\rightarrow|H\otimes_{A}B$ which atisfies
$\Delta(b_{(-1)})\otimes_{A}b_{(0)}=b_{(-1)}\otimes_{A}\delta(b_{(0)})$ and
$\epsilon_{H}(b_{(-1)}).b_{(0)}=b$, where we denote
$\delta(b)=b_{(-1)}\otimes_{A}b_{(0)}$. The left $A$-module $B$ obtains a
natural $A$-bimodule structure with its right $A$-action defined by
$ma=\epsilon_{H}(h_{(-1)}a).h_{(0)}$ so that $\delta$ factors through

$H\times_{A}B=\\{\sum_{i}h_{i}\otimes_{A}b_{i}\in|H\otimes_{A}B\mid\sum_{i}h_{i}\overline{a}\otimes_{A}b_{i}=\sum_{i}h_{i}\otimes_{A}b_{i}a\
\text{for}\ \forall a\in A\\}$

A Yetter-Drinfeld module over $H$ is an $A$-bimodule $B$ with a left
$H$-action $\triangleright$ and a compatible left $H$-coaction $\delta$
satisfying $h_{(1)}.b_{(-1)}\otimes_{A}h_{(2)}\triangleright
b_{(0)}=(h_{(1)}\triangleright
b)_{(-1)}.h_{(2)}\otimes_{A}(h_{(1)}\triangleright b)_{(0)}$. The natural
braiding obtained for such a Yetter-Drinfeld module is defined by
$b\otimes_{A}m\mapsto b_{(-1)}\triangleright^{\prime}m\otimes_{A}b_{(0)}$
where $m\in M$, for arbitrary $(M,\triangleright^{\prime})$ in
${}_{H}{\mathcal{M}}$. We refer the reader to Section 4 of [56] for more
details.

Assume $(B,\triangleright,\delta)$ is equipped with a braided Hopf algebra
structure in $\mathcal{Z}_{l,\mathrm{lax}}({}_{H}{\mathcal{M}})$ via
$(\Delta_{B},\epsilon_{B},S_{B})$. Note that these maps are all $A$-bimodule
maps and respect the relevant $H$-actions and $H$-coactions. Additionally, the
codomain of $\epsilon_{B}$ is the trivial bimodule $A$ which acts as the unit
of ${}_{A}{\mathcal{M}}_{A}$ and $\mathcal{Z}_{l,lax}({}_{H}{\mathcal{M}})$.
By Theorem 5.2, we obtain a Hopf algebroid structure on
$B\otimes_{A}{}_{\mathsf{s}}H$ where ${}_{\mathsf{s}}H$ denotes the left
$A$-module structure of $|H$ and the $A^{e}$-bimodule structure on
$B\otimes_{A}{}_{\mathsf{s}}H$ is defined by
$a_{1}\overline{a_{2}}(b\otimes_{A}h)a_{3}\overline{a_{4}}=a_{1}b\otimes_{A}\overline{a_{2}}ha_{3}\overline{a_{4}}$.
Furthermore, the element $1_{B}\otimes_{A}1_{H}$ acts as the unit and the
other structural morphisms can be obtained as follows:

$\displaystyle(b\otimes_{A}h).(b^{\prime}\otimes_{A}h^{\prime})=$
$\displaystyle b.(h_{(1)}\triangleright
b^{\prime})\otimes_{A}h_{(2)}.h^{\prime}$
$\displaystyle(b\otimes_{A}h)_{(1)}\otimes_{A}(b\otimes_{A}h)_{(2)}=$
$\displaystyle(b_{(1)}\otimes_{A}b_{(2)(-1)}.h_{(1)})\otimes_{A}(b_{(2)(0)}\otimes_{A}h_{(2)})$
$\displaystyle\epsilon(b\otimes_{A}h)=$
$\displaystyle\epsilon_{B}(b).\epsilon_{H}(h)$

Similarly, the inverse of the left canonical map is obtained by observing
(93):

$\displaystyle(b\otimes_{A}h)_{(+)}\otimes_{A^{\mathrm{op}}}(b\otimes_{A}h)_{(-)}=(b_{(1)}\otimes_{A}b_{(2)(-1)}.h_{(+)})\otimes_{A^{\mathrm{op}}}(S_{B}(b_{(2)(0)})\otimes_{A}h_{(-)})$

With any result regarding Hopf algebroids, one needs to check that all the
maps and compositions defined behave well with regard to the various $A$ and
$A^{\mathrm{op}}$ actions. But in our case, since we are simply writing out
the relevant morphisms after regarding the structures as Hopf monads, Theorem
5.2 guarantees that the morphisms will be well-defined with respect to the
relevant actions. We will denote the obtained Hopf algebroid by
$B\mathrel{\rtimes\cdot}H$.

###  5.2 Distributive Laws

In this section we review the notion of distributive laws and how one can
compose two (Hopf) bimonads with a distributive law between them. This
construction can be viewed as a generalisation of the tensor product of two
braided Hopf algebras.

Distributive laws were introduced by Beck in [5] and determine when a monad
(or an adjunction, as seen in Section 3.4) can lift to the Eilenberg-Moore
category of another monad. We recall the basic theory of distributive laws
without providing proofs and refer the reader to [67] for detailed references
and historical notes on the topic. Throughout this section $(T,\mu,\eta)$ and
$(S,\nu,\iota)$ will denote a pair of monads on $\mathcal{C}$.

A _distributive law_ or _entwining_ from a monad $T$ to a monad $S$ is a
natural transformation $\lambda:TS\rightarrow ST$ such that the following
diagrams commute

$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{T}}$$\scriptstyle{T\iota}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{ST}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{ST}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{S}}$$\scriptstyle{S\eta}$
$\textstyle{TTS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\lambda}$$\scriptstyle{\mu_{S}}$$\textstyle{TST\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{T}}$$\textstyle{STT\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\mu}$$\textstyle{TSS\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{S}}$$\scriptstyle{T\nu}$$\textstyle{STS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S\lambda}$$\textstyle{SST\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{T}}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{ST}$$\textstyle{TS\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{ST}$

Such a distributive law $\lambda:TS\rightarrow ST$ from a monad $T$ to a monad
$S$, allows $S$ to lift to $\mathcal{C}^{T}$ as
$(\hat{S},\hat{\nu},\hat{\iota})$, where
$\hat{S}:\mathcal{C}^{T}\rightarrow\mathcal{C}^{T}$, $\hat{\nu}$ and
$\hat{\iota}$ are defined as

$\hat{S}(M,r)=(S(M),(Sr)\lambda_{M}:TS(M)\rightarrow ST(M)\rightarrow
S(M)),\quad\hat{\nu}_{(M,r)}=\nu_{M},\quad\hat{\iota}_{(M,r)}=\iota_{M}$

for any $T$-module $(M,r)$, such that $U_{T}\hat{S}=SU_{T}$. The objects of
$\left(\mathcal{C}^{T}\right)^{\hat{S}}$ have an equivalent description as
triples $(M,r,\rho)$ where $M$ is an object of $\mathcal{C}$,
$r:T(M)\rightarrow M$ is a $T$-action and $\rho:S(M)\rightarrow M$ is a
$S$-action satisfying

$\textstyle{TS(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\rho}$$\scriptstyle{\lambda_{M}}$$\textstyle{ST(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Sr}$$\textstyle{T(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{M}$$\textstyle{S(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$

Note that the theory is not symmetric and that we cannot necessarily lift $T$
to $\mathcal{C}^{S}$ via $\lambda$. For this we require another distributive
law $\xi:ST\rightarrow TS$. In particular, if $\lambda$ is invertible then we
can lift $T$ via $\xi=\lambda^{-1}$ and in this case
$\left(\mathcal{C}^{T}\right)^{\hat{S}}\cong\left(\mathcal{C}^{S}\right)^{\hat{T}}$
since for any triple $(M,r,\rho)$ the conditions
$\rho(Sr)\lambda_{M}=r(T\rho)$ and $\rho(Sr)=r(T\rho)\xi_{M}$ become
equivalent by $\xi=\lambda^{-1}$.

Given a distributive law $\lambda:TS\rightarrow ST$, the endofunctor $ST$
obtains a natural monad structure via multiplication
$\nu(SS\mu)(S\lambda_{T})$ and unit $\iota_{T}\eta$. We will follow the
notation of [12] and denote this monad by $S\circ_{\lambda}T$. One can
directly show that the monads $\hat{S}\rtimes T$ and $S\circ_{\lambda}T$ are
isomorphic and that $\mathcal{C}^{\hat{S}\rtimes
T}\cong\mathcal{C}^{S\circ_{\lambda}T}$. Furthermore, when $T$ and $S$ are
bimonads the following observation can be made:

######  Lemma 5.4.

If $T,S$ and $\lambda$ are as above and $T$ and $S$ carry bimonad structures
such that $\lambda:TS\rightarrow ST$ is a comonoidal natural transformation,
then $S\circ_{\lambda}T$ also becomes a bimonad.

#### Sketch of proof.

The endofunctor $ST$ has a natural comonoidal structure with
$(S_{2})_{T\otimes T}S(T_{2})$ and $S_{0}S(T_{0})$. Now observe that for
$S\circ_{\lambda}T$, conditions (21) and (23) automatically follow from $S$
and $T$ being bimonads. Conditions (20) and (22) then follow from $\lambda$
being a comonoidal natural transformation or equivalently satisfying
$(\lambda\otimes\lambda)(T_{2})_{S\otimes S}T(S_{2})=(S_{2})_{T\otimes
T}ST_{2}\lambda_{-\otimes-}$ and
$T_{0}T(S_{0})=S_{0}S(T_{0})\lambda_{\mathtt{1}}$. $\diamond$

Instead of verifying the bimonad conditions on $S\circ_{\lambda}T$, one can
simply observe that $\lambda$ being a comonoidal natural transformation
implies that $(S_{2},S_{0})$ lift to a well-defined comonoidal structure on
$\hat{S}$. Consequently, $\hat{S}$ becomes a bimonad and by the results of
Section 5.1, so does $\hat{S}\rtimes T=S\circ_{\lambda}T$.

######  Theorem 5.5 (Corollary 4.7 [12]).

If $T$ and $S$ are Hopf monads on a monoidal category $\mathcal{C}$ and
$\lambda:TS\rightarrow ST$ a comonoidal distributive law from $T$ to $S$, then
the lifted monad $\hat{S}$, on $\mathcal{C}^{T}$, has a Hopf monad structure
and the composition $S\circ_{\lambda}T$ defines a Hopf monad on $\mathcal{C}$.

#### Sketch of proof.

By the observations made above its easy to see that the image of the left
(right) fusion operator of $\hat{S}$ under $U_{T}$ becomes the left (right)
fusion operator of $S$. Since $U_{T}$ is conservative, then $\hat{S}$ is left
(right) Hopf if and only if $S$ is left (right) Hopf. Consequently, if $S$ is
left (right) Hopf then by Theorem 5.2, $\hat{S}\rtimes T=S\circ_{\lambda}T$
becomes left (right) Hopf. $\diamond$

Composing Hopf monads via distributive laws can be viewed as a generalisation
of tensoring Hopf algebras: If $(\mathcal{C},\Psi)$ is a braided category and
$B$ and $C$ are two braided Hopf algebras in $\mathcal{C}$, then it is well-
known that $C\otimes B$ obtains a natural braided Hopf algebra structure by:

$\displaystyle m_{C\otimes B}:=(m_{C}\otimes m_{B}$
$\displaystyle)(\mathrm{id}_{C}\otimes\Psi_{B,C}\otimes\mathrm{id}_{B}),\quad\Delta_{C\otimes
B}:=(\mathrm{id}_{C}\otimes\Psi_{C,B}\otimes\mathrm{id}_{B})(\Delta_{C}\otimes\Delta_{B})$
$\displaystyle\eta_{C\otimes B}$
$\displaystyle:=\eta_{C}\otimes\eta_{B},\quad\epsilon_{C\otimes
B}:=\epsilon_{C}\otimes\epsilon_{B},\quad S_{C\otimes B}:=S_{C}\otimes S_{B}$

Viewing $B$ and $C$ as left Hopf monads via $T=B\otimes-$ and $S=C\otimes-$,
we see that $\lambda=\Psi_{B,C}\otimes-:TS\rightarrow ST$ defines a
distributive law between the two monads. Moreover, $\lambda$ is comonoidal.
All the necessary conditions simply follow because the structural morphisms of
$B$ and $C$ respect the braiding of the category. It is then easy to see that
the Hopf monad structure on $S\circ_{\lambda}T=C\otimes B\otimes-$ agrees with
the described Hopf algebra structure on $C\otimes B$.

##  6 Central Coalgebras and Hopf Monads

In this section we review the correspondence between Hopf monads and central
cocommutative coalgebras, as presented in Section 6 of [12]. In summary, every
(pre-)Hopf monad $T$ on a category $\mathcal{C}$ induces a central
cocommutative coalgebra in $\mathcal{C}^{T}$. In the converse direction, for
any central cocommutative coalgebra $(C,\tau)$ in a category $\mathcal{D}$,
under suitable exactness conditions, ${}^{C}\mathcal{D}$ becomes monoidal and
the adjunction $U_{C\otimes-}\dashv F_{C\otimes-}$ a comonoidal adjunction,
inducing a Hopf monad on ${}^{C}\mathcal{D}$. Moreover, under additional
assumptions discussed in Theorem 6.6 these procedures are inverses to each
other upto isomorphism.

###  6.1 From Central Coalgebras to Hopf Adjunctions

In this section we will recall how suitable central coalgebras $C$ in a
category $\mathcal{D}$ induce a Hopf monads on their category of comodules
${}^{C}\mathcal{D}$.

Assume $(C,\Delta,\epsilon)$ is a coalgebra in a monoidal category
$\mathcal{D}$. The corresponding free/forgetful functors for the comonad
$C\otimes-$, provide an adjunction $V\dashv
R:\mathcal{D}\leftrightarrows\prescript{C}{}{\mathcal{D}}$, where the
forgetful functor $V$ is left adjoint to the free functor $R$ defined by
$R(M)=(C\otimes M,\Delta\otimes\mathrm{id}_{M})$. Hence, we obtain a monad
$T=RV$ on ${}^{C}\mathcal{D}$.

We say $(C,\tau)$ is a _(lax) central coalgebra_ if
$\tau:C\otimes-\rightarrow-\otimes C$ is a (lax) braiding such that $(C,\tau)$
is a coalgebra in the (lax) center of $\mathcal{D}$ and thereby satisfies
$(\tau\otimes\mathrm{id}_{C})(\mathrm{id}_{C}\otimes\tau)(\Delta\otimes\mathrm{id}_{\mathcal{D}})=(\mathrm{id}_{\mathcal{D}}\otimes\Delta)\tau$
and
$(\mathrm{id}_{\mathcal{D}}\otimes\epsilon)\tau=\epsilon\otimes\mathrm{id}_{\mathcal{D}}$.
Furthermore, we say $C$ is a _cocommutative (lax) central coalgebra_ or (lax)
_CCC_ if $\tau_{C}\Delta=\Delta$.

In [12], the authors discuss the conditions under which a CCC structure on $C$
provides a monoidal structure on the category of $C$-comodules. A lax CCC is
called _cotensorable_ if for each pair of comodules $(M,\delta)$ and
$(N,\delta^{\prime})$, the pair

$\textstyle{M\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{M}\otimes\delta^{\prime}}$$\scriptstyle{\tau_{M}\delta\otimes\mathrm{id}_{N}}$$\textstyle{M\otimes
C\otimes N}$ (94)

admits an equalizer and $C\otimes-$ preserves them. We denote the equalizer of
this pair by $M\otimes_{C}N$. Note that such pairs are coreflexive via
$\mathrm{id}_{M}\otimes\epsilon\otimes\mathrm{id}_{N}$ and thereby if
$\mathcal{D}$ admits CEs and $\otimes$ preserves them, then any lax CCC is
cotensorable. In particular, this theory generalises the theory of _cotensors_
for ordinary coalgebras in $\mathsf{Vec}$ where $\tau$ is given by the
symmetry of the category, see Section 10 of [19].

######  Theorem 6.1 (Theorem 6.4 [12]).

If $C$ denotes a lax cotensorable CCC as above:

  1. (I)

The equalizers $M\otimes_{C}N$ have a natural $C$-coaction and the bifunctor
$\otimes_{C}$ defines a monoidal structure on ${}^{C}\mathcal{D}$.

  2. (II)

The free functor $R:\mathcal{D}\rightarrow{}^{C}\mathcal{D}$ is strong
monoidal and $V\dashv R$ is a comonoidal adjunction.

  3. (III)

The adjunction $V\dashv R$ is in fact left Hopf. If $\tau$ is invertible, then
the adjunction is also right Hopf.

#### Sketch of proof.

For part (I), we observe that for a pair $(M,\delta)$ and
$(N,\delta^{\prime})$, the object $M\otimes_{C}N$ carries a natural
$C$-comodule structure $t$, where $t$ is defined as the unique map satisfying

$\textstyle{M\otimes_{C}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{t}$$\textstyle{M\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\otimes\mathrm{id}_{M\otimes
N}}$$\textstyle{C\otimes M\otimes N}$$\textstyle{C\otimes
M\otimes_{C}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{C}\otimes\pi}$

where $\pi:M\otimes_{C}N\rightarrow M\otimes N$ is the equalizer of the pair
(94). Note that $t$ is determined uniquely because $\mathrm{id}_{C}\otimes\pi$
is an equalizer for the coreflexive pair
$\mathrm{id}_{C}\otimes\tau_{M}\delta\otimes\mathrm{id}_{N}$ ,
$\mathrm{id}_{C\otimes M}\otimes\delta^{\prime}$. It is then straightforward
to show that $t$ defines a coaction.

For (II), first recall that the free functor $R$ takes an object $M$ in
$\mathcal{D}$ to the free comodule $(C\otimes
M,\Delta\otimes\mathrm{id}_{M})$. Additionally, observe that the coproduct
$\Delta$ forms a split equalizer for the pair $\Delta\otimes\mathrm{id}_{C}$
and $\mathrm{id}_{C}\otimes\Delta$, with sections
$\mathrm{id}_{C}\otimes\epsilon$ and $\mathrm{id}_{C\otimes
C}\otimes\epsilon$. Hence, if we denote the equalizer of (94) for comodules
$R(M)$ and $R(N)$ by $\pi^{\prime}$, we obtain a natural isomorphism between
two equalizers and a commutative diagram:

|  |
---|---|---
$\textstyle{(C\otimes M)\otimes_{C}(C\otimes
N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\cong}$$\textstyle{C\otimes
M\otimes C\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{C\otimes
M}\otimes\Delta\otimes\mathrm{id}_{N}}$$\scriptstyle{\tau_{C\otimes
M}(\Delta\otimes\mathrm{id}_{M})\otimes\mathrm{id}_{C\otimes
N}}$$\textstyle{C\otimes M\otimes C\otimes C\otimes N}$$\textstyle{C\otimes
M\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt(\mathrm{id}_{C}\otimes\tau_{M})(\Delta\otimes\mathrm{id}_{M})\otimes\mathrm{id}_{N}}$

Notice that $(C\otimes M)\otimes_{C}(C\otimes N)=R(M)\otimes_{C}R(N)$ and
$C\otimes M\otimes N=R(M\otimes N)$ and thereby we obtain a natural strong
monoidal structure on $R$. Of course one must also check whether the
isomorphism respects the coactions, but this also follows easily.

Now we demonstrate that $V\dashv R$ is left Hopf. Recall from Lemma 3.4 that a
strong monoidal structure on $R$ provides a comonoidal structure on its left
adjoint $V$. In the diagram below, the upper edges form the comonoidal
structure of $V$ as described by (27):

$\textstyle{M\otimes_{C}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\otimes_{C}\delta^{\prime}}$$\scriptstyle{\pi}$$\textstyle{(C\otimes
M)\otimes_{C}(C\otimes
N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\pi^{\prime}}$$\textstyle{C\otimes
M\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon\otimes\mathrm{id}_{M\otimes
N}}$$\scriptstyle{\tau_{C\otimes
M}(\Delta\otimes\mathrm{id}_{M})\otimes\mathrm{id}_{N}}$$\textstyle{M\otimes
N}$$\textstyle{M\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta\otimes\delta^{\prime}}$$\textstyle{C\otimes
M\otimes C\otimes
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\epsilon\otimes\mathrm{id}_{M}\otimes\epsilon\otimes\mathrm{id}_{N}}$

Since the diagram commutes, we can alternatively identify the comonoidal
structure with the composition of the lower edges of the diagram. In
particular, $V_{2}((M,\delta),(N,\delta^{\prime}))=\pi$ where
$V(M\otimes_{C}N,t)=M\otimes_{C}N$ and $V(M,\delta)\otimes
V(N,\delta^{\prime})=M\otimes N$. Therefore, the left fusion operator of the
adjunction $V\dashv R$ decomposes as

$\overline{H}^{l}_{(M,\delta),d}=(\mathrm{id}_{M}\otimes\epsilon\otimes\mathrm{id}_{d})\pi:V((M,\delta)\otimes_{C}R(d))\rightarrow
V((M,\delta))\otimes d$

for an object $d$ in $\mathcal{D}$ and a $C$-comodule $(M,\delta)$. By a
similar argument to the one above we can show that
$\tau_{M}\delta\otimes\mathrm{id}_{d}:V((M,\delta))\otimes d\rightarrow
M\otimes C\otimes d$ becomes a split equalizer for the same parallel pair
which $\pi:M\otimes_{C}R(d)\rightarrow M\otimes C\otimes d$ is an equalizer
of. Thereby there exists a natural isomorphism such that the following diagram
commutes:

$\textstyle{M\otimes_{C}R(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\cong}$$\textstyle{M\otimes
C\otimes
d\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id}_{M}\otimes\epsilon\otimes\mathrm{id}_{d})}$$\scriptstyle{\tau_{M}\delta\otimes\mathrm{id}_{C\otimes
d}}$$\scriptstyle{\mathrm{id}_{M}\otimes\Delta\otimes\mathrm{id}_{d}}$$\textstyle{M\otimes
C\otimes C\otimes d}$$\textstyle{M\otimes
d\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{M}\delta\otimes\mathrm{id}_{d}}$

Hence, the left fusion operator is invertible. When $\tau$ is invertible a
symmetric argument can be applied to show that the right fusion operator is
invertible. $\diamond$

The above result can provide us with many additional examples of Hopf monads.
However, unlike Section 4 where we were looking at describing Hopf monads over
a fixed base category $\mathcal{C}$, a suitable CCC in a category
$\mathcal{D}$ provides us with a Hopf monad on ${}^{C}\mathcal{D}$ rather than
the category $\mathcal{D}$ which we started with.

######  Example 6.2.

For any cocommutative coalgebra $(C,\Delta,\epsilon)$ in $\mathsf{Vec}$, the
category of left $C$-comodules ${}^{C}{\mathcal{M}}$ is monoidal via the
cotensor product of comodules (Section 10 of [19]) and by Theorem 6.1 the
free/forgetful adjunction
$\mathrm{forg.}\dashv(C\otimes_{\mathbb{K}}-,\Delta\otimes_{\mathbb{K}}-):\mathsf{Vec}\leftrightarrows{}^{C}{\mathcal{M}}$
is Hopf. In particular the monad $T$ on ${}^{C}{\mathcal{M}}$ which sends
every $C$-comodule $(M,\delta)$ to the free comodule
$(C\otimes_{\mathbb{K}}M,\Delta\otimes_{\mathbb{K}}\mathrm{id}_{M})$ is Hopf.
The multiplication and unit of the monad $T$ are given by
$(\mathrm{id}_{C}\otimes_{\mathbb{K}}\epsilon\otimes_{\mathbb{K}}\mathrm{id}_{{}^{C}{\mathcal{M}}})$
and
$\delta:(M,\delta)\rightarrow(C\otimes_{\mathbb{K}}M,\Delta\otimes_{\mathbb{K}}\mathrm{id}_{M})$,
respectively. The comonoidal structure on $T$ is defined using
$T((M,\delta))\otimes_{C}T((N,\delta^{\prime}))\cong(C\otimes_{\mathbb{K}}M\otimes_{\mathbb{K}}N,\Delta\otimes_{\mathbb{K}}\mathrm{id}_{M\otimes_{\mathbb{K}}N})$:

$\textstyle{T_{2}((M,\delta),(N,\delta^{\prime})):C\otimes_{\mathbb{K}}(M\otimes_{C}N)\rightarrow
C\otimes_{\mathbb{K}}M\otimes_{\mathbb{K}}N}$$\textstyle{\hskip
99.58464ptc\otimes_{\mathbb{K}}m\otimes_{C}n\longmapsto
c\otimes_{\mathbb{K}}m\otimes_{\mathbb{K}}n}$

and
$T_{0}=\mathrm{id}_{C}\otimes_{\mathbb{K}}\epsilon:C\otimes_{\mathbb{K}}C\rightarrow
C$.

###  6.2 Induced Central Coalgebra of a Hopf Adjunction

In this section, we review the construction of central coalgebras from Hopf
adjunctions based on Section 6.2 of [12]. It is well-known that any braided
Hopf algebra $B$ in $\mathcal{C}$ becomes an object of
${}^{B}_{B}\mathcal{YD}$ via its _left adjoint coaction_
$(m\otimes\mathrm{id}_{B})(\mathrm{id}_{B}\otimes\Psi_{B,B})(\Delta\otimes
S)\Delta$. In particular, this coaction respects the original coalgebra
structure on $B$ and thereby $(B,\Delta,\epsilon)$ becomes a natural coalgebra
object in $Z({}_{B}\mathcal{C})$. Below we will describe how any Hopf monad
$T$ induces a coalgebra object in $Z(\mathcal{C}^{T})$.

Given a comonoidal adjunction $F\dashv
U:\mathcal{D}\leftrightarrows\mathcal{C}$, the object $\hat{C}:=F(\mathtt{1})$
obtains a coalgebra structure via $F_{2}(\mathtt{1},\mathtt{1})$ and $F_{0}$,
and is referred to as the _induced coalgebra_ of the adjunction. Hence, we
obtain three distinct comonads $\hat{T}=FU$, $\hat{C}\otimes-$, and
$-\otimes\hat{C}$ on $\mathcal{D}$. Observe that the fusion operators of the
adjunctions provide natural transformations between two of the three comonads
as presented:

$\textstyle{\overline{H}^{l}_{\mathtt{1},X}:}$$\textstyle{\hat{T}(X)=F(\mathtt{1}\otimes
U(X))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id}_{F(\mathtt{1})}\otimes\epsilon_{X})F_{2}(\mathtt{1},U(X))}$$\textstyle{\hat{C}\otimes
X=F(\mathtt{1})\otimes X}$
$\textstyle{\overline{H}^{r}_{X,\mathtt{1}}:}$$\textstyle{\hat{T}(X)=F(U(X)\otimes\mathtt{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\epsilon_{X}\otimes\mathrm{id}_{F(\mathtt{1})})F_{2}(U(X),\mathtt{1})}$$\textstyle{X\otimes\hat{C}=X\otimes
F(\mathtt{1})}$

In Lemma 6.5 of [12] it is demonstrated that $\overline{H}^{l}_{\mathtt{1},-}$
and $\overline{H}^{r}_{-,\mathtt{1}}$ are comonad morphisms.

######  Theorem 6.3 (Corollary 6.7 [12]).

With notation as above, if $F\dashv U$ is a left pre-Hopf adjunction, then

$\tau_{X}:=\overline{H}^{r}_{X,\mathtt{1}}\left(\overline{H}^{l}_{\mathtt{1},X}\right)^{-1}:\hat{C}\otimes
X\rightarrow X\otimes\hat{C}$ (95)

defines a lax braiding in $\mathcal{D}$ satisfying
$F_{2}(X,\mathtt{1})=\tau_{F(X)}F_{2}(\mathtt{1},X)$. In particular,
$(C,\tau)$ becomes a lax CCC in $\mathcal{D}$. Additionally, If the adjunction
is pre-Hopf, then $\tau$ is invertible.

The proof of the above result is simply checking that the necessary conditions
hold and we refer the reader to Section 6.2 of [12]. Note that
$\overline{H}^{l}_{\mathtt{1},X}$ being invertible tells us that
$\hat{T}=FU\cong\hat{C}\otimes-$ are isomorphic comonads. The above result
also shows that given a pre-Hopf adjunction
$\hat{C}\otimes-\cong-\otimes\hat{C}$ as comonads. Additionally, note that for
the free/forgetful adjunction induced by a braided Hopf algebra $B$ it follows
from the description of the fusion operators and their inverses in Section 3.3
that we can recover the natural braiding on $B$, induced by its adjoint
coaction, as $\tau$ in Theorem 6.3.

In [12], the lax CCC $\hat{C}$ of a pre-Hopf adjunction $F\dashv U$ is called
its _induced_ CCC. As seen in the last section, CCCs can induce comonoidal
adjunctions and the next step is to understand the connection between this
adjunction induced by $\hat{C}$ and the original adjunction $F\dashv U$. We
will first recall the conditions which make the induced CCC cotensorable and
the comparison functor between $\mathcal{C}$ and
$\mathcal{D}_{\hat{T}}\cong\mathcal{D}^{\hat{C}}$ strong monoidal.

######  Theorem 6.4 (Proposition 6.9 [12]).

Let $F\dashv U$ be a left pre-Hopf adjunction as above. If the adjunction is
comonadic and satisfies

  1. (I)

For any pair of objects $X,Y$ in $\mathcal{C}$, the fork below forms an
equalizer:

$\textstyle{F(X\otimes
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{2}(X,Y)}$$\textstyle{F(X)\otimes
F(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{F(X)}\otimes
F_{2}(\mathtt{1},Y)}$$\scriptstyle{F_{2}(X,\mathtt{1})\otimes\mathrm{id}_{F(Y)}}$$\textstyle{F(X)\otimes\hat{C}\otimes
F(Y)}$ (96)

  2. (II)

The above equalizers are preserved by $F(\mathtt{1})\otimes-$

Then the induced coalgebra $(\hat{C},\tau)$ is cotensorable and the comparison
functor between $\mathcal{C}$ and $\mathcal{D}^{\hat{C}}$ is strong monoidal.
In particular, the adjunction is left Hopf.

#### Sketch of proof.

Note that the comparison functor $K:\mathcal{C}\rightarrow{}^{C}\mathcal{D}$
is defined by sending any object $X$ in $\mathcal{C}$ to
$(F(X),F_{2}(\mathtt{1},X))$. Since $K$ is an equivalence and
$F_{2}(X,\mathtt{1})=\tau_{F(X)}F_{2}(\mathtt{1},X)$, then the conditions for
$\hat{C}$ being cotensorable become equivalent to the conditions above.
Additionally, $K$ becomes a strong monoidal equivalence by (96). Finally, we
observe that by Theorem 6.1 the free/forgetful adjunction for the comonad
$\hat{C}\otimes-$ is left Hopf, and since $U\dashv F$ is comonadic then it is
also left Hopf. $\diamond$

While the induced coalgebra in the theory of Hopf algebras is quite well-
known, we will now look at the induced coalgebra for another examples of Hopf
monads:

######  Example 6.5.

Let $H$ be a bialgebroid over a base algebra $A$ as in Section 4.6. Recall
that the free/forgetful adjunction induced on ${}_{A}{\mathcal{M}}_{A}$ is a
comonoidal adjunction with a bimonad structure on the functor $H\boxtimes-$.
The induced CCC for this adjunction will be a natural coalgebra structure on
$H\boxtimes A$ which is the quotient of $H$ by the left ideal generated by
$\\{a-\overline{a}\mid a\in A\\}$. The vector space $H\boxtimes A$ has an
$A$-bimodule structure by the left $A^{e}$-action on $H$. It follows by the
above theory that $(\Delta,\epsilon)$ descend to a well-defined coalgebra
structure on $H\boxtimes A$. This can also be seen directly from (79) and (80)
and the fact that the image of $\Delta$ falls in the Takeuchi product. If $H$
is left Hopf, then by Theorem 6.3 the coalgebra $H\boxtimes A$ obtains a lax
braiding defined by

$\textstyle{(H\boxtimes A)\otimes_{A}M\rightarrow M\otimes_{A}(H\boxtimes
A)}$$\textstyle{(h\boxtimes a)\otimes_{A}M\longmapsto
h_{(+)(1)}ah_{(-)}\triangleright m\otimes_{A}(h_{(+)}\boxtimes 1)}$ (97)

where $(M,\triangleright)$ is an arbitrary $H$-module. It is a difficult task
to check that this map is indeed well-defined and a bimodule morphism, however
we get this result for free from Theorem 6.4.

###  6.3 Equivalence of Hopf Monads and Central Coalgebras

In this section, we review the equivalence between Hopf monads on a category
$\mathcal{C}$, Hopf adjunctions $F\dashv
U:\mathcal{D}\leftrightarrows\mathcal{C}$ and lax cotensorable CCCs in
$\mathcal{D}$ from Section 6.6 of [12]. We then provide a simplification of
this result for tensor categories.

######  Theorem 6.6 (Theorem 6.14 [12]).

If $\mathcal{C}$ and $\mathcal{D}$ are monoidal categories, then TFAE:

  1. (I)

A conservative Hopf monad $T$ on $\mathcal{C}$, where $\mathcal{C}$ admits RCs
and CEs, the monoidal product $\otimes$ preserves CEs and $T$ preserves RCs
and CEs.

  2. (II)

A Hopf adjunction $F\dashv U:\mathcal{D}\leftrightarrows\mathcal{C}$, where
both $\mathcal{C}$ and $\mathcal{D}$ admit RCs and CEs, both $F$ and $U$ are
conservative, $U$ preserves RCs and $F$ preserves CEs.

  3. (III)

A CCC $(C,\tau)$ in $\mathcal{D}$, where $\mathcal{D}$ admits RCs and CEs, the
monoidal product $\otimes$ preserves CEs, the functor $C\otimes-$ is
conservative and preserves RCs.

Note that when saying these structures are equivalent, we mean upto
isomorphism and the processes of sending an adjunction to a CCC and a CCC to
its corresponding adjunction are not strict inverses. Additionally, the
emphasis on the existence and preservation of RCs has to do with the
comonadicity of the adjunction corresponding to a CCC, see Remark 6.1 in [12].
For more details and the proof of this equivalence we refer the reader to to
Section 6.6 of [12].

Recall that _tensor categories_ , see [24], are rigid and abelian and thereby
admit equalizers and coequalizers. Moreover, in rigid monoidal categories the
tensor product preserves finite limits and colimits in both entries
[Proposition 4.2.1 [24]]. Additionally, _tensor functors_ , as considered in
[24], are defined as being faithful exact strong monoidal functors between two
tensor categories. Such functors are automatically conservative since abelian
categories are balanced and any functor with a balanced domain which is
faithful automatically becomes conservative. Finally, recall by Corollary 3.9,
that any Hopf monad on a rigid monoidal category is colimit-preserving. Hence,
we can re-write Theorem 6.6 for the setting of tensor categories:

######  Theorem 6.7.

If $\mathcal{C},\mathcal{D}$ are two tensor categories over a field
$\mathbb{K}$, then TFAE:

  1. (I)

A conservative Hopf monad $T$ on $\mathcal{C}$, where $\mathcal{C}$ and $T$
preserves CEs.

  2. (II)

A tensor functor $U:\mathcal{D}\rightarrow\mathcal{C}$ with a conservative
left adjoint $F$ which preserves CEs.

  3. (III)

A CCC $(C,\tau)$ in $\mathcal{D}$.

A dual statement to the above theorem can be made with regards to Hopf
comonads and central cocommutative algebras. This dual statement is in part
presented Section 6.1 of [13].

##  7 Classical Hopf-algebraic results for Hopf monads

In this section we review some classical results in the theory of Hopf
algebras which have been generalised to the monadic setting.

###  7.1 Radford’s Biproduct Theorem

Radford’s biproduct Theorem which we alluded to in Section 5.1 states the
following stronger result: Given a pair of Hopf algebras $H_{1}$ and $H_{2}$
and a Hopf algebra map $\pi:H_{2}\twoheadrightarrow H_{1}$ which splits by
$\iota:H_{1}\hookrightarrow H_{2}$, we can find a braided Hopf algebra $B$ in
${}^{H_{1}}_{H_{1}}\mathcal{YD}$, such that $H_{2}\cong
B\mathrel{\rtimes\cdot}H_{1}$. In Section 4.4 of [12], the authors discuss a
more general version of this result which we will review here under the name
of _cross quotients_. The more general question is understanding when a Hopf
monad can be obtained as a cross product from another Hopf monad.

First observe that for a pair of monads $(T,\mu,\eta)$ and
$(Q,\mu^{\prime},\eta^{\prime})$ on a category $\mathcal{C}$, there exists a
natural correspondence between monad morphisms and functors between
$\mathcal{C}^{T}$ and $\mathcal{C}^{Q}$ which commute with the forgetful
functors: Any monad morphism $\phi:T\rightarrow Q$ defines a functor
$\phi^{*}:\mathcal{C}^{Q}\rightarrow\mathcal{C}^{T}$ by
$\phi^{*}(M,r)=(M,r\phi)$ which satisfies $U_{T}\phi^{*}=U_{Q}$. Conversely,
from any functor $F:\mathcal{C}^{Q}\rightarrow\mathcal{C}^{T}$ satisfying
$U_{T}F=U_{Q}$, we can recover a monad morphism $F^{*}:T\rightarrow Q$ defined
by $F^{*}_{X}=\rho_{(Q(X),\mu^{\prime}_{X})}T\eta^{\prime}_{X}$ where $\rho$
is the natural transformation given by $F((M,r))=(M,\rho_{(M,r)})$. For
further details on this correspondence we refer the reader to Lemma 1.7 of
[14].

In [12] a monad morphism $\phi:T\rightarrow Q$ is called _cross quotientable_
if the functor $\phi^{*}$ is monadic. Such a functor, in turn, defines a monad
on $\mathcal{C}^{T}$ which is called the cross quotient of $\phi$ and denoted
by $Q\div T$. In this setting, it is easy to check that $\phi^{*}$ is monadic
if and only if it has a left adjoint and if so $(\mathcal{C}^{T})^{Q\div T}$
becomes isomorphic to $\mathcal{C}^{Q}$, Lemma 4.9 of [12]. Moreover, it is
shown that the left adjoint $G$ of $\phi^{*}$ exists precisely if for any
$T$-module $(M,r)$, the pair
$\mu^{\prime}_{M}Q(\phi_{M}),Q(r):F_{Q}T(M)\rightrightarrows F_{Q}(M)$ admits
a coequalizer $G(M,r)$ in $\mathcal{C}^{Q}$.

######  Lemma 7.1 (Proposition 4.11 (a) [12]).

If $\phi:T\rightarrow Q$ is a cross quotientable bimonad morphism, then $Q\div
T$ is a bimonad.

###### Proof.

This statement follows from the fact that $\phi^{*}$ becomes strong monoidal
when $\phi$ is a bimonad morphism. In this case the trivial comonoidal
structure $\phi^{*}_{2}\big{(}(M,r),(N,t)\big{)}=\mathrm{id}_{M\otimes N}$ is
well-defined since $(r\otimes t)Q_{2}(M,N)\phi_{M\otimes N}=(r\otimes
t)(\phi_{M}\otimes\phi_{N})T_{2}(M,N)$. ∎

At this point we should note that the operations ‘cross quotient’ and ‘cross
product’ are inverses, Proposition 4.11 [12]. Fixing a (bi)monad $T$ on
$\mathcal{C}$, we see that for every (bi)monad $Q$ equipped with a cross
quotientable (bi)monad morphism $\phi:T\rightarrow Q$, we have an isomorphism
$Q\cong(Q\div T)\rtimes T$. In the converse direction, for any (bi)monad $P$
on $\mathcal{C}^{T}$, the unit $\eta^{\prime\prime}$ of $P$ provides a cross
quotientable (bi)monad morphism $U_{T}\eta^{\prime\prime}_{F_{T}}$ from $T$ to
$Q=P\rtimes T$.

######  Theorem 7.2 (Proposition 4.13 [12]).

Let $T$ and $Q$ be left (right) Hopf monads on a monoidal category
$\mathcal{C}$. Assume $\mathcal{C}$ has RCs, $T$ and $Q$ both preserve RCs and
$\otimes$ preserves RCs in the left (right) entry. In this case, any bimonad
morphism $\phi:T\rightarrow Q$ is cross quotientable and $Q\div T$ is left
(right) Hopf.

#### Sketch of proof.

We previously mentioned that $\phi$ is cross quotientable if and only if
reflexive pairs $\mu^{\prime}_{M}Q(\phi_{M}),Q(r):F_{Q}T(M)\rightrightarrows
F_{Q}(M)$ corresponding to $T$-modules $(M,r)$ admit coequalizers in
$\mathcal{C}^{Q}$ [Lemma 4.9 of [12]]. Hence, under the assumptions made above
$Q\div T$ exists and is a bimonad by Lemma 7.1. Let $\overline{H}^{l}$ denote
the left fusion operator of the adjunction corresponding to $Q\div T$. Next we
observe that Diagram (93) commuting also shows that if the left fusion
operators of $T$ and $(Q\div T)\rtimes T\cong Q$ are invertible, then
$\overline{H}^{l}_{F_{T},-}$ is invertible ($P=Q\div T$).

The final step is to view any $T$-module $(M,r)$ as the coequalizer of its
corresponding reflexive pair $F_{T}(r),\mu_{M}:F_{T}T(M)\rightrightarrows
T(M)$ in $\mathcal{C}^{T}$ with $F_{T}(\eta_{M})$ and $r:T(M)\rightarrow M$.
By Lemma 4.2 of [12], $\otimes$ also preserves RCs in the left component in
$\mathcal{C}^{T}$ and $\mathcal{C}^{Q}$. Since $F_{P}$ is left adjoint and
preserves coequalizers, then the functors $F_{P}(-\otimes U_{P}(X))$ and
$F_{P}(-)\otimes X$ both preserve RCs for any $X\in\mathcal{C}^{Q}$. Therefore
if $\overline{H}^{l}_{F_{T}(M),X}$ is invertible, then so is
$\overline{H}^{l}_{(M,r),X}$. We refer the reader to Lemma 4.8 of [12] for
additional details on this argument. $\diamond$

Given a pair of ordinary Hopf algebras $H_{1}$ and $H_{2}$, their
corresponding left Hopf monads $Q=H_{2}\otimes_{\mathbb{K}}-$ and
$T=H_{1}\otimes_{\mathbb{K}}-$ on $\mathsf{Vec}$ satisfy the conditions in
Theorem 7.2. Hence we can consider the cross quotient of
$\phi=\iota\otimes_{\mathbb{K}}-$ for any Hopf algebra map
$\iota:H_{1}\rightarrow H_{1}$. In particular, the resulting adjunction given
by restriction and extension of scalars between ${}_{H_{2}}{\mathcal{M}}$ and
${}_{H_{1}}{\mathcal{M}}$ will be left Hopf. We already mentioned this fact in
Section 4.4. The corresponding Hopf monad $Q\div T$ is given by the functor
${}_{H_{1}}{H_{2}}\otimes_{H_{1}}-$ on ${}_{H_{1}}{\mathcal{M}}$, where we
consider the natural $H_{1}$-bimodule structure which $\iota$ induces on
$H_{2}$.

######  Theorem 7.3 (Corollary 5.12 [12]).

If $T$, $Q$ and $\mathcal{C}$ satisfy the conditions of Theorem 7.2 and the
bimonad morphism $\phi:T\rightarrow Q$ admits a retraction $\iota:Q\rightarrow
T$ then there exists a central Hopf algebra $(B,\tau)$ in $\mathcal{C}^{T}$
such that $Q\div T\cong B\otimes_{\tau}-$.

#### Sketch of proof.

The correspondence between bimonad morphisms $T\rightarrow Q$ and bimonads
$Q\div T$ is functorial (Remark 4.12 of [12]). Therefore, if view the
retraction of $\phi$ as a morphism between bimonad maps $T\rightarrow Q$ and
$T\rightarrow T$, it will be sent to an augmentation map $Q\div T\rightarrow
T\div T=\mathrm{id}_{\mathcal{C}^{T}}$ under this correspondence. Hence,
$Q\div T$ is augmented and by Theorem 4.1 it corresponds to a central Hopf
algebra in $\mathcal{C}^{T}$. $\diamond$

The setting of Radford’s biproduct Theorem for a pair of Hopf algebras $H_{1}$
and $H_{2}$ is precisely the reduction of the above result for monads
$Q=H_{2}\otimes_{\mathbb{K}}-$ and $T=H_{1}\otimes_{\mathbb{K}}-$ on
$\mathsf{Vec}$, where the Hopf algebra map $\iota:H_{1}\hookrightarrow H_{2}$
admits a retraction $\pi:H_{1}\twoheadrightarrow H_{2}$.

######  Example 7.4 (Radford’s biproduct Theorem for Hopf Algebroids).

Note that for an arbitrary $\mathbb{K}$-algebra $A$, the tensor product of
${}_{A}{\mathcal{M}}_{A}$ preserves coequalizers in both entries.
Additionally, the corresponding monads for bialgebroids were those which
preserved colimits (Theorem 4.19). Hence, we can apply Theorem 7.3 to the
setting of Hopf algebroids: If $H_{1}$ and $H_{2}$ are a pair of Hopf
algebroids over a base algebra $A$ and there exist a pair of bialgebroid
morphisms $\pi:H_{2}\twoheadrightarrow H_{1}$ and $\iota:H_{1}\hookrightarrow
H_{2}$ such that $\pi\iota=\mathrm{id}_{H_{1}}$, then there exists a Hopf
algebra $(B,\triangleright,\delta)$ in $\mathcal{Z}({}_{H_{1}}{\mathcal{M}})$
such that $H_{2}\cong B\mathrel{\rtimes\cdot}H_{1}$. As in Radford’s result,
$B$ can be identified as the subspace of elements of the form $h-\iota\pi(h)$,
for $h\in H$.

###  7.2 Hopf Modules

An important result in the theory of ordinary Hopf algebras concerns Hopf
modules and is often referred to as the Fundamental Theorem of Hopf Algebras.
This result describes an equivalence between the category of vector spaces and
the category of Hopf modules and has been generalised to the case of braided
Hopf algebras and braided categories in [6] under the additional requirement
that idempotents split in the base category. In this section we review the
analogous statement for Hopf monads.

A Hopf module for a Hopf algebra $H$ consists of an $H$-module with a
compatible $H$-comodule structure. Therefore, the first important step for
defining Hopf modules over Hopf monads is picking a notion of comodules over
the monad. Hopf modules over Hopf monads were first defined in [14], where
‘$T$-comodules’ were defined as comodules with respect to the induced
coalgebra of the monad $(T(\mathtt{1}),T_{2}(\mathtt{1},\mathtt{1}),T_{0})$.
Using this definition, a version of the Hopf module theorem was proved in [14]
for right Hopf monads on right rigid categories. This result was then extended
to Hopf monads on monoidal categories in [12]. More recently, the result of
[12] was interpreted as an example of Galois entwining structures in [50]. We
will discuss this viewpoint at the end of this section.

Note that for any bimonad $T$ on a monoidal category $\mathcal{C}$, the image
of the induced coalgebra $\hat{C}$ of Section 6.2 under $U_{T}$ becomes a
coalgebra $(T(\mathtt{1}),T_{2}(\mathtt{1},\mathtt{1}),T_{0})$ in
$\mathcal{C}$. With this notation, a triple $(M,r,\delta)$ is called a _(left)
Hopf module_ , if $r:T(M)\rightarrow M$ is a $T$-action and
$\delta:M\rightarrow T(\mathtt{1})\otimes M$ is a $T(\mathtt{1})$-coaction
satisfying

$\textstyle{T(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T\delta}$$\scriptstyle{r}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{T(\mathtt{1})\otimes
M}$$\textstyle{T(T(\mathtt{1})\otimes
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H^{r}_{\mathtt{1},X}}$$\textstyle{T(\mathtt{1})\otimes
T(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}_{T(\mathtt{1})}\otimes
r}$

Morphisms of Hopf modules are defined accordingly as morphisms in
$\mathcal{C}$ which commute with the relevant actions and coactions. We will
denote the category of (left) Hopf modules by $\mathcal{H}^{l}(T)$. Note that
for the corresponding Hopf monad of a braided Hopf algebra $B$, we recover the
usual notion of Hopf modules which are triples $(M,\triangleright,\delta)$
where $\triangleright$ is a $B$-action and $\delta$ a $B$-coaction satisfying
$\delta\triangleright=(m\otimes\triangleright)(\mathrm{id}_{H}\otimes\Psi_{B,B}\otimes\mathrm{id}_{M})(\Delta\otimes\delta)$.

Let us briefly recall the notion of a _mixed distributive law_ or _entwining_
from monad $(T,\mu,\eta)$ to a comonad $(G,\Delta,\epsilon)$ from [67] (the
axioms first appeared for entwinings between ordinary algebras and coalgebras
in [18]). Such an entwining is a natural transformation $\lambda:TG\rightarrow
GT$ satisfying analogous conditions to the distributive laws in Section 5.2.
Given such a $\lambda$, we can define a comonad $\hat{G}$ on $\mathcal{C}^{T}$
such that $U_{T}\hat{G}=GU_{T}$. Explicitly, the functor $\hat{G}$ is defined
by $\hat{G}(M,r)=(G(M),(Gr)\lambda_{M}:TG(M)\rightarrow GT(M)\rightarrow
G(M))$ for any $T$-module $(M,r)$. The compatibility conditions on $\lambda$
then ensure that $(\hat{G},\Delta,\epsilon)$ defines a comonad on
$\mathcal{C}^{T}$. In a symmetric fashion, $\lambda$ lifts $T$ to
$\mathcal{C}_{G}$, defining a monad
$\hat{T}(M,\delta)=(T(M),\lambda_{M}T\delta:T(M)\rightarrow TG(M)\rightarrow
GT(M))$, where $(M,\delta)$ is a $G$-comodule in $\mathcal{C}_{G}$. Finally,
we obtain a natural isomorphism
$\left(\mathcal{C}^{T}\right)_{\hat{G}}\cong\left(\mathcal{C}_{G}\right)^{\hat{T}}$.

In the definition of left Hopf modules, the right fusion operator
$H^{r}_{\mathtt{1},X}$ is acting as a distributive law. Hence,
$\mathcal{H}^{l}(T)=\left(\mathcal{C}^{T}\right)_{\hat{G}}$ where
$\hat{G}:\mathcal{C}^{T}\rightarrow\mathcal{C}^{T}$ is the comonad obtained by
lifting $T(\mathtt{1})\otimes-$ via $H^{r}_{\mathtt{1},X}$. We have already
encountered $\hat{G}$ in Section 6.2 with the notation $\hat{C}\otimes-$ and
know it to be isomorphic to the comonad $\hat{T}=FU$ when
$H^{l}_{\mathtt{1},X}$ is invertible.

The Fundamental Theorem of braided Hopf algebras describes an equivalence
between the base category and the category of Hopf modules. This equivalence
is given by the free Hopf module functor and the functor sending Hopf modules
to their coinvariant parts. For ordinary Hopf algebras the coinvariant part of
an $H$-comodule is the subspace of elements which satisfy
$\delta(h)=1\otimes_{\mathbb{K}}h$. Now we recall the analogous notions for
Hopf monads from [12].

For any left Hopf module $(M,r,\delta)$ over a bimonad $T$ the equalizer, if
it exists, of the coreflexive pair
$\delta,\eta_{\mathtt{1}}\otimes\mathrm{id}_{M}:M\rightarrow
T(\mathtt{1})\otimes M$ is called its _coinvariant part_ and we denote it by
$M^{T(\mathtt{1})}$. Furthermore, $T$ is said to preserve coinvariant parts of
(left) Hopf modules if $T$ preserves this class of equalizers. There also
exists a natural functor $H_{T}:\mathcal{C}\rightarrow\mathcal{H}^{l}(T)$
defined by $H_{T}(X)=(T(X),\mu_{X},T_{2}(\mathtt{1},X))$ which sends an object
to the _free Hopf module_ generated by it. As mentioned
$\mathcal{H}^{l}(T)=\left(\mathcal{C}^{T}\right)_{\hat{G}}$ and the free
functor $H_{T}$ satisfies $U^{\hat{G}}H_{T}=F_{T}$, where $F_{T}\dashv
U_{T}:\mathcal{C}\leftrightarrows\mathcal{C}^{T}$ and $U^{\hat{G}}\dashv
F^{\hat{G}}:\mathcal{H}^{l}(T)\leftrightarrows\mathcal{C}^{T}$ denote the
free/forgetful adjunctions for the monad $T$ and the comonad $\hat{G}$,
respectively.

For a general monad $T$, the question of when
$F^{\hat{T}}F_{T}:\mathcal{C}\rightarrow\left(\mathcal{C}^{T}\right)_{\hat{T}}$
induces an equivalence of categories is answered in [25]. Theorem 6.11 of [12]
combines this with the observation that $\hat{G}\cong\hat{T}$ for a left pre-
Hopf monad to obtain an analogous statement to the Fundamental Theorem:

######  Theorem 7.5 (Theorem 6.11 [12]).

Assuming $T$ is a left pre-Hopf monad, TFAE:

  1. (I)

The functor $H_{T}$ defines an equivalence of categories.

  2. (II)

The functor $T$ is conservative, left Hopf modules admit coinvariant parts and
$T$ preserves them.

#### Sketch of proof.

As in the classical setting, if Hopf modules admit coinvariants, the functor
$E:\mathcal{H}^{l}(T)\rightarrow\mathcal{C}$ which is defined on objects by
$E(M,r,\delta)=M^{T(\mathtt{1})}$ and extended naturally to morphisms, becomes
right adjoint to $H_{T}$. Furthermore, as demonstrated in [25], the counit of
the adjunction is an isomorphism if and only if $T$ preserves coinvariants.
When the counit of the adjunction is an isomorphism then the right adjoint $E$
is full and faithful and it is part of an equivalence if and only if it is
essentially surjective. It is an easy exercise to check that the latter
condition is true if and only if the left adjoint $H_{T}$ is conservative.
Since $T=U_{T}F_{T}=U_{T}U^{\hat{G}}H_{T}$, where $U_{T}$, $U^{\hat{G}}$ are
both conservative functors, then $H_{T}$ is conservative if and only if $T$
is. $\diamond$

First note that the corresponding Hopf monad of an ordinary Hopf algebra $H$
on $\mathsf{Vec}$ naturally satisfies the conditions of Theorem 7.5 and we
recover the classical Fundamental Theorem of Hopf algebras in this case.
Secondly, note that a symmetric theory of right Hopf modules can also be
defined over a Hopf monad, with $H^{l}_{\mathtt{1},-}$ acting as a
distributive law between $T$ and $-\otimes T(\mathtt{1})$. In the theory of
ordinary Hopf algebras and braided Hopf algebras, the notions of left and
right Hopf modules become equivalent via the braiding. However, this is not
the case for Hopf monads. In Example 5.6 of [50], a right pre-Hopf monad is
provided which does not satisfy the left Hopf module Theorem.

######  Example 7.6 (Fundamental Theorem of Hopf algebras for Hopf
algebroids).

Let $H$ be a bialgebroid over an algebra $A$ and let
$(\underline{\Delta},\underline{\epsilon})$ denote the coalgebra structure on
$\underline{H}:=H\boxtimes A$, which we described in Example 6.5. A left Hopf
module for the bimonad $H\boxtimes-$ on ${}_{A}{\mathcal{M}}_{A}$ will
consists of an $H$-module $(M,\triangleright)$ with an
$\underline{H}$-coaction $\delta:M\rightarrow\underline{H}\otimes_{A}M$
satisfying

$(h\triangleright m)_{(-1)}\otimes_{A}(h\triangleright
m)_{(0)}=\pi({h_{(1)}.m_{(-1)}})\otimes_{A}(h_{(2)}\triangleright
m_{(0)})\in\underline{H}\otimes_{A}M$

where we denote the natural projection $H\rightarrow H\boxtimes A$ by $\pi$
and $\delta(m)=m_{(-1)}\otimes_{A}m_{(0)}$ for $m\in M$. If $H$ is a Hopf
algebroid and $H$ is faithfully flat as a right $A^{e}$-module then
$H\boxtimes-$ becomes conservative and preserves coinvariant parts and by
Theorem 7.5 we obtain an equivalence between ${}_{A}{\mathcal{M}}_{A}$ and the
category of Hopf modules defined above. In particular, the equivalence is
given by the functor which sends any $A$-bimodule $M$ to the corresponding
Hopf module $H\boxtimes M$ with the relevant free $H$-action and
$\underline{H}$-coaction $h\boxtimes
m\rightarrow\pi(h_{(1)})\otimes_{A}(h_{(2)}\boxtimes m)$. Note that this
result differs from the other version of the Fundamental Theorem for Hopf
algebroids which appears in Section 4.3.4 of [7]. The latter result describes
an equivalence of categories between another notion of Hopf modules and
${}_{A}{\mathcal{M}}$.

The above description and proof of the Hopf module Theorem can be generalised
to the category mixed modules $\left(\mathcal{C}^{T}\right)_{\hat{G}}$
obtained from any distributive law $\lambda:TG\rightarrow GT$ between an
arbitrary monad $T$ and comonad $G$ on a category $\mathcal{C}$. This theory
is fully described in [48] and elaborated on in [49, 50]. In this framework, a
natural transformation $g:\mathrm{id}_{\mathcal{C}}\rightarrow G$ is called a
_group-like morphism_ if $g$ is a comonad morphism i.e. $\Delta g=g_{G}g$ and
$\epsilon g=\mathrm{id}_{\mathrm{id}_{\mathcal{C}}}$. If such a morphism
exists, one can construct a functor
$H^{g}:\mathcal{C}\rightarrow\left(\mathcal{C}^{T}\right)_{\hat{G}}$ defined
by $H^{g}(M)=(T(M),\mu_{M},\lambda Tg_{M})$ satisfying $U^{\hat{G}}K=F_{T}$.
Moreover, the functor $T$ obtains two left $G$-comodule structure
$\lambda(Tg)$ and $g_{T}$, where we view $G$ as a comonoid in the monoidal
category $\mathrm{End}(\mathcal{C})$. In the case of bimonads discussed above,
$\lambda=H^{r}_{\mathtt{1},-}$, $G=T(\mathtt{1})\otimes-$ and
$g=\eta_{\mathtt{1}}\otimes-$.

If the pairs $\lambda_{M}(Tg_{M})$ and $g_{T(M)}$ admit equalizers denoted by
$i^{g}:T^{g}(M)\rightarrow T(M)$, then $T^{g}$ becomes functorial and $i^{g}$
a monad morphism. Moreover, $H^{g}$ becomes monadic and $T^{g}$ the monad
generated by $H^{g}$. In the bimonad case, $T^{g}$ would exist if all free
$T$-modules admit coinvariants. The monad morphism $i^{g}$ naturally induces a
functor between $(i^{g})^{*}:\mathcal{C}^{T}\rightarrow\mathcal{C}^{T^{g}}$,
which under suitable conditions [Section 3.7 [48]] admits a left adjoint and
induces a comonad on $\mathcal{C}^{T}$. Whenever this comonad is isomorphic to
$\hat{G}$, and $\mathcal{C}^{T^{g}}$ is equivalent to
$\left(\mathcal{C}^{T}\right)_{\hat{G}}$, then the quadruple $(T,G,\lambda,g)$
are called a _Galois entwining_. In Proposition 4.6 of [50], it is shown for a
bimonad $T$ the quadruple
$(T,T(\mathtt{1})\otimes-,H^{r}_{\mathtt{1},-},\eta_{\mathtt{1}}\otimes-)$
becomes a Galois entwining if and only if $H^{l}_{\mathtt{1},-}$ is
invertible. For further details on Galois entwining and their connection to
bimonads we refer the reader to [48] and Section 4 of [50].

###  7.3 Quasitriangular Structures

Quasitriangular structures on ordinary Hopf algebras were originally
introduced by Drinfeld in [22, 23] and correspond to braidings on the category
of modules of the Hopf algebra. For braided Hopf algebras, a diagrammatic
generalisation was presented in [46]. In this section, we review the
corresponding notion for bimonads, which was introduced in [14].

An R-matrix or quasitriangular structure on an ordinary bialgebra $H$ is an
invertible element of $R\in H\otimes_{\mathbb{K}}H$ (where we
$H\otimes_{\mathbb{K}}H$ has an algebra structure $\bullet$ using the tensor
product of algebras) satisfying

$\displaystyle\Psi^{\mathsf{Vec}}_{H,H}\Delta(h)\bullet R=R\bullet\Delta(h)$
$\displaystyle(\Delta\otimes_{\mathbb{K}}\mathrm{id}_{H})R=R_{13}\bullet
R_{23}$
$\displaystyle(\mathrm{id}_{H}\otimes_{\mathbb{K}}\Delta)R=R_{13}\bullet
R_{12}$

where $R_{ij}:H^{\otimes 3}\rightarrow H^{\otimes 3}$ are the appearances of
$R$ to the $i$ and $j$-th components of $H^{\otimes 3}$ i.e.
$R_{23}=1_{H}\otimes_{\mathbb{K}}R$. To generalise this theory to braided
bialgebras, $B$, Majid [43] interpreted the R-matrix as a morphism
$R:\mathbb{K}\rightarrow H\otimes_{\mathbb{K}}H$ satisfying analogous
conditions (Fig. 9.13 of [46]).

A _quasitringular_ structure or _R-matrix_ on a bimonad $T$ is a natural
transformation $R_{X\otimes Y}:X\otimes Y\rightarrow T(Y)\otimes T(X)$
satisfying

$\displaystyle(\mu_{Y}\otimes\mu_{X})R_{T(X),T(Y)}T_{2}(X,Y)=(\mu_{Y}\otimes\mu_{X})T_{2}(T(Y),T(X))T(R_{X,Y})$
(98) $\displaystyle(\mathrm{id}_{T(Z)}\otimes$ $\displaystyle
T_{2}(X,Y))R_{X\otimes Y,Z}=(\mu_{Z}\otimes\mathrm{id}_{T(X)\otimes
T(Y)})(R_{X,T(Z)}\otimes\mathrm{id}_{T(Y)})(\mathrm{id}_{X}\otimes R_{Y,Z})$
(99) $\displaystyle(T_{2}(Y,Z)$
$\displaystyle\otimes\mathrm{id}_{T(X)})R_{X,Y\otimes
Z}=(\mathrm{id}_{T(Y)\otimes T(Z)}\otimes\mu_{X})(\mathrm{id}_{T(Y)}\otimes
R_{T(X),Z})(R_{X,Y}\otimes\mathrm{id}_{Z})$ (100)

for any triple of objects $X,Y,Z$ in $\mathcal{C}$.

######  Theorem 7.7 (Theorem 8.5 [14]).

If $T$ is a bimonad on a monoidal category $\mathcal{C}$, there is a bijection
between R-matrices on $T$ and lax braidings on $\mathcal{C}^{T}$.

#### Sketch of proof.

This bijection is constructed as follows: If $R$ is an R-matrix on $T$ then
$\tau_{(M,r),(N,s)}=(s\otimes r)R_{M,N}$, where $(M,r)$ and $(N,s)$ are
$T$-modules, defines a braiding on $\mathcal{C}^{T}$. The first condition (98)
ensures that $(s\otimes r)R_{M,N}$ is a well-defined $T$-module map, while the
other two conditions (99) and (100) dictate the braiding axioms. In the other
direction, if $\tau$ is a braiding on $\mathcal{C}^{T}$ then
$R_{M,N}=\tau_{(TX,\mu_{X}),(TY,\mu_{Y})}(\eta_{X}\otimes\eta_{Y})$ defines an
R-matrix on $T$. $\diamond$

Note that the R-matrix in the theory of ordinary bialgebras is expected to be
an invertible element of $H\otimes_{\mathbb{K}}H$. For bimonads, a similar
notion of invertiblilty is discussed in [14], where a natural transformation
$R_{X\otimes Y}:X\otimes Y\rightarrow T(Y)\otimes T(X)$ is called _convolution
invertible_ if there exists another natural transformation
$R^{-1}_{X,Y}:Y\otimes X\rightarrow T(X)\otimes T(Y)$ satisfying

$\eta_{X}\otimes\eta_{Y}=(\mu_{X}\otimes\mu_{Y})R^{-1}_{T(X),T(Y)}R_{X,Y},\quad\eta_{Y}\otimes\eta_{X}=(\mu_{Y}\otimes\mu_{X})R_{T(X),T(Y)}R^{-1}_{X,Y}$
(101)

Whenever $R^{-1}$ in the above sense exists, then the lax braiding $\tau$
defined in Theorem 7.7 obtains an inverse defined by by
$\tau^{-1}_{(M,r),(N,s)}=(r\otimes s)R^{-1}_{N,M}:M\otimes N\rightarrow
N\otimes M$.

If $\mathcal{C}=\mathsf{Vec}$ and $T=H\otimes_{\mathbb{K}}-$ is given by a
bialgebra $H$, then the existence of any R-matrix
$R_{V,W}:V\otimes_{\mathbb{K}}W\rightarrow
H\otimes_{\mathbb{K}}W\otimes_{\mathbb{K}}H\otimes_{\mathbb{K}}V$ for the
bimonad $T$ induces a classical R-matrix on $H$ defined by
$R_{\mathtt{1},\mathtt{1}}(1)\in H\otimes_{\mathbb{K}}H$. It is then easy to
check that the natural family of morphisms $R_{V,W}$ are solely determined by
the choice of $R_{\mathtt{1},\mathtt{1}}(1)$. In the opposite direction, we
can encode any classical R-matrix
$R=\sum_{i}h_{i}\otimes_{\mathbb{K}}h^{\prime}_{i}$ as an R-matrix on the
bimonad $H\otimes_{\mathbb{K}}-$ as follows
$R_{X\otimes_{\mathbb{K}}Y}(x\otimes
y)=\sum_{i}h^{\prime}_{i}\otimes_{\mathbb{K}}y\otimes_{\mathbb{K}}h_{i}\otimes_{\mathbb{K}}x$.

In the more general setting where $\mathcal{C}$ is an arbitrary braided
category and $T=\mathbb{B}$ corresponds to a braided bialgebra $B$ in
$\mathcal{C}$, R-matrices on $T$ do not necessarily correspond to a morphism
$\overline{R}:\mathtt{1}\rightarrow B\otimes B$ satisfying the conditions in
Fig. 9.13 of [46]. In particular, the latter R-matrices only define a braiding
on a particular subcategory of $B$-modules (called _commutative modules_). In
Section 8.6 of [15] it is shown that when $\mathcal{C}$ is rigid and admits a
coend $C=\int^{c\in\mathcal{C}}c\otimes c^{\vee}$, then R-matrices on
$\mathbb{B}$ are in bijection with morphisms $C\otimes C\rightarrow B\otimes
B$ satisfying analogous conditions to Fig. 9.13 of [46]. The case where
$\mathcal{C}$ is $\mathsf{Vec}_{\mathrm{fd}}$ is particular since
$C=\mathbb{K}$ and, therefore, the classical notion of R-matrices are
recovered.

Another key aspect of the theory of R-matrices and Hopf algebras is their
relation with the Yang-Baxter equation. An analogous version of the Yang-
Baxter equation can be shown to hold for Hopf monads as well, Corollary 8.7 in
[14]. Other concepts relating to braided categories and Hopf algebras such as
_Drinfeld elements_ for a Hopf monad and analogous notions of _ribbon_ and
_sovereign_ structures on Hopf monads have all been defined in [14]. We should
also note that the notion of quasitriangular structures on bialgebroids first
appeared in [21].

###  7.4 Tannaka Reconstruction for Hopf monads

In this section we review the construction of [60] which produces Hopf monads
from suitable strong monoidal functors. This theory generalises our
construction of Hopf monads from pivotal pairs in Section 4.5 and recovers the
usual Tannaka-Krein reconstruction for braided Hopf algebras [41] when the
base category is braided. Let us first recall what we mean by Tannaka-Krein
theory for Hopf algebras.

Let $\mathcal{D}$ be a small monoidal category and
$\omega:\mathcal{D}\rightarrow\mathcal{C}$ be a strict monoidal functor such
that $\omega$ factors through the inclusion functor
$\mathrm{inc.}:\mathcal{C}_{\mathrm{rig}}\rightarrow\mathcal{C}$, where
$\mathcal{C}_{\mathrm{rig}}$ denotes the subcategory of rigid objects in
$\mathcal{C}$. We consider the functor
$\omega\otimes\omega^{\vee}:\mathcal{D}\times\mathcal{D}^{\mathrm{op}}\rightarrow\mathcal{C}$
and denote its coend (see Section 2.6), if it exists, by

$H_{\omega}:=\int^{a\in\mathcal{D}}\omega(a)\otimes\omega(a)^{\vee}$ (102)

Recall that the coend is the colimit of the diagram consisting of objects
$\omega(a)\otimes\omega(b)^{\vee}$ and morphisms
$\omega(f)\otimes\mathrm{id}_{\omega(b)^{\vee}}$ and
$\mathrm{id}_{\omega(a)}\otimes\omega(f)^{\vee}$ corresponding to objects
$a,b\in\mathcal{D}$ and morphisms $f:a\rightarrow b$ in $\mathcal{D}$,
respectively.

######  Theorem 7.8.

If $(\mathcal{C},\Psi)$ is a braided monoidal category, and the mentioned
coend exists, it comes equipped with the structure of a braided bialgebra,
such that
$\left({}_{\mathcal{D}}{\omega}_{\mathcal{C}}\right)^{\circ}_{r,\mathrm{lax}}$
is monoidal equivalent to the the category of left $H_{\omega}$-modules,
${}_{H_{\omega}}{\mathcal{C}}$. Additionally, if $\mathcal{D}$ is rigid then
$H_{\omega}$ admits a bijective antipode making it a braided Hopf algebra
object in $\mathcal{C}$.

Tannaka-Krein reconstruction over $\mathsf{Vec}$ and symmetric monoidal
categories was first formally treated in [55] and later for algebraic groups
in [20]. We refer the reader to [34] for a detailed comparison with Tannaka-
Krein duality for compact topological groups. The formulation of braided Hopf
algebras and braided Tannaka-Krein reconstruction as we have presented here
appears in [41]. The detailed proof of this result can be found in Chapter 9
of [46], where the coend is is described in terms of natural transformations
between certain functors.

We will not go into the details of the proof of Theorem 7.8, but only present
the structural morphisms of the resulting Hopf algebra in the case where the
functor $\omega$ is strict monoidal and additionally
$\omega(x)^{\vee}=\omega(x^{\vee})$ for all $x\in\mathcal{D}$. If
$\mu_{x}:\omega(x)\otimes\omega(x)^{\vee}\rightarrow H_{\omega}$ denote the
unique natural morphisms, making $H_{\omega}$ the colimit of the diagram, then
the Hopf algebra structure $(m,\eta,\Delta,\epsilon,S)$ on $H_{\omega}$ is
defined via the unique morphisms satisfying:

$\displaystyle m:H_{\omega}\otimes H_{\omega}\rightarrow H_{\omega}\ ;$
$\displaystyle\quad m(\mu_{x}\otimes\mu_{y})=\mu_{x\otimes
y}\big{(}\mathrm{id}_{\omega(x)}\otimes\Psi_{\omega(x)^{\vee},\omega(y)}\otimes\mathrm{id}_{\omega(y)^{\vee}}\big{)}$
$\displaystyle\eta:\mathtt{1}\rightarrow H_{\omega}\ ;$
$\displaystyle\quad\eta=\mu_{\mathtt{1}}$
$\displaystyle\Delta:H_{\omega}\rightarrow H_{\omega}\otimes H_{\omega}\ ;$
$\displaystyle\quad\Delta\mu_{x}=(\mu_{x}\otimes\mu_{x})\big{(}\mathrm{id}_{\omega(x)}\otimes\mathrm{coev}_{\omega(x)}\otimes\mathrm{id}_{\omega(x)^{\vee}}\big{)}$
$\displaystyle\epsilon:H_{\omega}\rightarrow\mathtt{1}\ ;$
$\displaystyle\quad\epsilon\mu_{x}=\mathrm{ev}_{\omega(x)}$ $\displaystyle
S:H_{\omega}\rightarrow H_{\omega}\ ;$ $\displaystyle\quad
S\mu_{x}=\mu_{x^{\vee}}(\mathrm{ev}_{\omega(x)}\otimes\mathrm{id}_{\omega(x^{\vee})\otimes\omega(x^{\vee\vee})})(\Psi_{\omega(x^{\vee})\otimes\omega(x^{\vee\vee}),\omega(x^{\vee})})$
$\displaystyle\quad(\mathrm{id}_{\omega(x)\otimes\omega(x)^{\vee}}\otimes\omega(\mathrm{ev}_{x})^{\vee})$

where $x,y\in\mathcal{D}$. In [60], a generalisation of Theorem 7.8 is
described where $\mathcal{C}$ is a monoidal category, not necessarily braided,
with suitable colimits. The output in this setting is then a Hopf monad rather
than a braided Hopf algebra. We will briefly review this theory with reference
to our construction in Section 4.5.

Assume $\mathcal{C}$ is a monoidal category where $\otimes$ preserves colimits
in both entries. A _construction data_ over $\mathcal{C}$ consists of a strong
monoidal functor $\omega:\mathcal{D}\rightarrow\mathcal{C}$ where
$\mathcal{D}$ is a small monoidal category and $\omega$ factors through the
inclusion functor
$\mathrm{inc.}:\mathcal{C}_{\mathrm{rig}}\rightarrow\mathcal{C}$.
Additionally, we assume that the following coends exist for every object $X$
in $\mathcal{C}$:

$T_{\omega}(X):=\int^{d\in\mathcal{D}}\omega(d)\otimes
X\otimes\omega(d)^{\vee}$ (103)

Now we review the main result of [60]:

######  Theorem 7.9 (Theorem 3.3 [60]).

Let $\omega:\mathcal{D}\rightarrow\mathcal{C}$ be a functor factorising
through $\mathrm{inc.}:\mathcal{C}_{\mathrm{rig}}\rightarrow\mathcal{C}$ such
that the coends $T_{\omega}(X)$ in (103) exist. In this case:

  1. (I)

$T_{\omega}$ is functorial and comonoidal.

  2. (II)

If $\mathcal{D}$ is monoidal and $\omega$ is strong monoidal then $T_{\omega}$
also has a monad structure. Moreover, $\mathcal{C}^{T_{\omega}}$ becomes
isomorphic to the lax right dual of $\omega$.

  3. (III)

If $\mathcal{D}$ is left (right) rigid then $T_{\omega}$ is right (left) Hopf.

The proofs of these results are rather lengthy and elaborate and we refer the
reader to Section 3 of [60] for the full proofs. However, we will briefly
review how the Hopf monad structure on $T_{\omega}$ is defined with reference
to our construction from Section 4.5.

Let us denote the universal morphisms $\omega(d)\otimes
X\otimes\omega(d)^{\vee}\rightarrow T_{\omega}(X)$ by $\psi_{d,X}$. Recall
from Section 4.5 that any pivotal pair $(P,Q)$ in a monoidal category
$\mathcal{C}$ corresponds to a strict monoidal functor
$\omega:\mathrm{Piv}(1)\rightarrow\mathcal{C}$. By definition the image of
$\omega$ falls in $\mathcal{C}_{\mathrm{rig}}$ and the coends $T_{\omega}(X)$
become precisely the colimits $T(X)$ described in Section 4.5 in this case:
Objects of $\mathrm{Piv}(1)$ are tensor products of several objects of the
form $+$ and $-$ and their images under $\omega$ took the form
$P_{i_{1}}\otimes\cdots\otimes P_{i_{n}}$, corresponding to lists
$i_{j}\in\\{-,+\\}$. Hence, we wrote the universal morphisms
$\psi_{d,X}:\omega(d)\otimes X\otimes\omega(d)^{\vee}\rightarrow
T_{\omega}(X)$ in the form $(\psi_{i_{1},\ldots,i_{n}})_{X}$. Similarly, it
should be clear that by definition $T_{\omega}$ is the colimit of a diagram in
$\mathrm{End}(\mathcal{C})$ with functors
$F_{d}:=\omega(d)\otimes-\otimes\omega(d)^{\vee}$ and natural transformations
$\omega(f)\otimes-\otimes\omega(d^{\prime})$ and
$\omega(d)\otimes-\otimes\omega(f)^{\vee}$ corresponding to morphisms
$f:d\rightarrow d^{\prime}$ in $\mathcal{D}$. Note that
$\omega(\mathtt{1})\otimes-\otimes\omega(\mathtt{1})^{\vee}=F_{0}=\mathrm{id}_{\mathcal{C}}$
in Section 4.5 since $\omega$ was strict monoidal. Since all morphisms in
$\mathrm{Piv}(1)$ were generated by the quadruple of duality morphisms we only
needed to look at four families of parallel pairs to construct $T_{\omega}$ in
Section 4.5.

With the above notation $T_{\omega}$ obtains a natural comonoidal structure
$(T_{2},T_{0})$ where $T_{2}(X,Y)$ and $T_{0}$ are the unique morphisms
satisfying

$\displaystyle T_{2}(X,Y)\psi_{d,X\otimes Y}=(\psi_{d,X}\otimes\psi_{d,Y})$
$\displaystyle(\mathrm{id}_{\omega(d)\otimes
X}\otimes\underline{\mathrm{coev}}_{\omega(d)}\otimes\mathrm{id}_{Y\otimes\omega(d)^{\vee}})$
(104) $\displaystyle T_{0}\psi_{d,\mathtt{1}}$
$\displaystyle=\underline{\mathrm{ev}}_{\omega(d)}$ (105)

for all objects $d\in\mathcal{D}$. In the example of
$\omega:\mathrm{Piv}(1)\rightarrow\mathcal{C}$ the evaluation and coevaluation
maps for arbitrary objects $d\in\mathrm{Piv}(1)$ took the forms
$\mathrm{coev}_{i_{1},\ldots,i_{n}}$ and $\mathrm{ev}_{i_{1},\ldots,i_{n}}$
and gave rise to the same comonoidal structure described above.

If we further assume that $\omega$ is strong monoidal, then $T$ obtains a
monad structure $(\mu,\eta)$ where $\mu_{X}$ and $\eta_{X}$ are the unique
morphisms satisfying

$\displaystyle\mu_{X}\psi_{d,T(X)}(\mathrm{id}_{\omega(d)}\otimes\psi_{d^{\prime},X}\otimes\mathrm{id}_{\omega(d)^{\vee}})$
$\displaystyle=\psi_{d\otimes
d^{\prime},X}(\omega_{2}(d,d^{\prime})\otimes\mathrm{id}_{X}\otimes(\omega_{2}(d,d^{\prime})^{-1})^{\vee})$
(106) $\displaystyle\eta_{X}$
$\displaystyle=\psi_{\mathtt{1},X}(\omega_{0}\otimes
X\otimes(\omega_{0}^{-1})^{\vee})$ (107)

for any pair of objects $d,d^{\prime}\in\mathcal{D}$. In Section 4.5, (106)
translated to the universal property of $\theta$ and (107) simplified since
$\omega$ was strict monoidal and thereby
$\omega_{0}=\mathrm{id}_{\mathtt{1}}$.

The isomorphism
$\mathcal{C}^{T_{\omega}}\cong\left({}_{\mathcal{D}}{\omega}_{\mathcal{C}}\right)^{\circ}_{r,\mathrm{lax}}$
is defined as follows: For any $T$-module $(X,r:T(X)\rightarrow X)$ the object
$X$ obtains a lax right braiding
$(r\psi_{d,X}\otimes\mathrm{id}_{\omega(d)})(\mathrm{id}_{\omega(d)\otimes
X}\otimes\underline{\mathrm{coev}}_{\omega(d)})$. In the converse direction
for any object $(X,\tau)$ in
$\left({}_{\mathcal{D}}{\omega}_{\mathcal{C}}\right)^{\circ}_{r,\mathrm{lax}}$,
we obtain a $T$-action $r$ on $X$ as the unique morphism satisfying
$r\psi_{d,X}=(\mathrm{id}_{X}\otimes\underline{\mathrm{ev}}_{\omega(d)})(\tau_{d}\otimes\mathrm{id}_{\omega(d)^{\vee}})$.
In Section 4.5, we observed this isomorphism by showing that each object
$(X,\sigma)$ of $\mathcal{C}(P,Q)$ obtains a $T$-action via $\alpha_{\sigma}$
and $\beta_{\sigma}$ and vice versa.

If $\mathcal{C}$ is left or right closed then part (III) of Theorem 7.9
follows directly from the isomorphism of
$\mathcal{C}^{T}\cong\left({}_{\mathcal{D}}{\omega}_{\mathcal{C}}\right)^{\circ}_{r,\mathrm{lax}}$
and a symmetric version of Theorem 2.3. However, when $\mathcal{C}$ is not
closed, the statement is still shown to hold in [60] and the inverses of the
fusion operators of $T$ are constructed as in the proof of Theorem 4.15 but
with the additional use of the natural isomorphisms
$\zeta:\omega(-^{\vee})\rightarrow\omega(-)^{\vee}$. For our choice of
$\omega:\mathrm{Piv}(1)\rightarrow\mathcal{C}$, $\zeta$ were simply the
identity morphisms. We refer the reader to Section 3.8 of [60] for this
description since the details of this construction go beyond the scope of this
work.

In [60] it is also noted that if $\mathcal{C}$ admits a braiding then the
coend $T_{\omega}$ can be written as $-\otimes B$ for some braided Hopf
algebra $B$. As seen in Section 4.5, if $\omega$ factors through the center of
$\mathcal{C}$ then one can produce an augmentation on $T_{\omega}$ by using
the braidings $\omega(d)\otimes X\otimes\omega(d)^{\vee}\rightarrow
X\otimes\omega(d)\otimes\omega(d)^{\vee}$ as in Theorem 4.16.

Note that when applying Tannaka-Krein reconstruction for ordinary Hopf
algebras ($\mathcal{C}=\mathsf{Vec}$) we obtain an embedding of $\mathcal{D}$
into the category of comodules of the reconstructed Hopf algebra $H_{\omega}$.
In the monadic setting, the category of comodules over $T_{\omega}$ are
defined as the left lax dual of the forgetful functor $U_{T_{\omega}}$ and
this embedding follows trivially. We will briefly discuss the notion of
comodules in Section 8. In the classical setting, we can also use a
‘Recognition Theorem’ to tell if this embedding is an equivalence when
restricted to the rigid subcategory of comodule [Theorem 3 [34]]. As far as
the author is aware, such a result does not exist in the monadic setting for a
general base category $\mathcal{C}$.

Duals and Cenetralisers of Hopf monads: Before the work of Shimizu in [60]
which we discussed above, Bruguières and Virelizier [15] studied the question
of when the dual of the forgetful functor
$U_{T}:\mathcal{C}^{T}\rightarrow\mathcal{C}$ of a Hopf monad $T$ could be
identified as modules over another monad $Z_{T}$. In the Tannaka-Krein
terminology, we would obtain $Z_{T}$ as $T_{U_{T}}$ if suitable coends exist.
However, the work in [15] is presented from a slightly different point of
view.

First note that for an ordinary finite dimensional Hopf algebras $H$, the dual
of its category of finite dimensional modules can be identified with the
category of modules over the dual Hopf algebra $H^{*}$. Hence, $Z_{T}$ is
generalising the dual Hopf algebra construction and since the latter structure
on $H^{*}$ only makes sense when $H$ is finite dimensional, generalising this
theory to Hopf monads only makes sense if we assume $\mathcal{C}$ to be rigid.
In this setting, an endofunctor $T$ on a monoidal category $\mathcal{C}$ was
called _centralizable_ in [15] if for any object $X$ in $\mathcal{C}$, there
exists an object $Z_{T}(X)$ with a universal natural transformation
$\phi^{X}_{-}:X\otimes-\rightarrow T(-)\otimes Z_{T}(X)$, such that for any
other such pair $(\overline{Z(X)},\overline{\phi^{X}})$ there exists a
morphism $f:\overline{Z(X)}\rightarrow Z_{T}(X)$ satisfying
$\phi^{X}=\overline{\phi^{X}}f$. If $Z_{T}$ exists, $\phi^{X}_{Y}:X\otimes
Y\rightarrow T(Y)\otimes Z_{T}(X)$ becomes a natural transformation in both
entries [Lemma 5.2 [15]] and the pair $(Z_{T},\phi)$ is called a _centralizer_
for $T$.

In principle, the universal property described means that
$Z_{T}(X)=\int^{Y\in\mathcal{C}}T(Y)\otimes X\otimes Y^{\vee}$ and given the
adjunction $F_{T}\dashv U_{T}$ we obtain an equality

$Z_{T}(X)=\int^{Y\in\mathcal{C}}T(Y)\otimes X\otimes
Y^{\vee}=\int^{(M,r)\in\mathcal{C}^{T}}U_{T}(M,r)\otimes X\otimes
U_{T}(M,r)^{\vee}=T_{U_{T}}$

See Lemma 3.9 of [15] for more details. Thereby, Theorem 5.6 of [15] follows
as a consequence of Theorem 7.9 and
$\mathcal{C}^{Z_{T}}\cong\left({}_{\mathcal{C}^{T}}{(U_{T})}_{\mathcal{C}}\right)^{\circ}$.

Finally, we should note that $Z_{T}$ was described in [15] as the monad whose
module category recovers the _relative centre_ to $T$, which was denoted by
$\mathcal{Z}_{T}(\mathcal{C})$, rather than
$\left({}_{\mathcal{C}^{T}}{(U_{T})}_{\mathcal{C}}\right)^{\circ}$. The
relative centre is defined to have pairs $(M,\delta)$ as objects, where $M$ is
an object of $\mathcal{C}$ and
$\delta:M\otimes\mathrm{id}_{\mathcal{C}}\rightarrow T\otimes M$ a natural
transformation satisfying $(T_{2}(Y,Z)\otimes\mathrm{id}_{M})\delta_{Y\otimes
Z}=(\mathrm{id}_{T(Y)}\otimes\delta_{Z})(\delta_{Y}\otimes\mathrm{id}_{Z})$
and $\mathrm{id}_{M}=(T_{0}\otimes\mathrm{id}_{M})\delta_{\mathtt{1}}$. Using
the adjunction $F_{T}\dashv U_{T}$, we obtain a natural isomorphism between
$\mathcal{Z}_{T}(\mathcal{C})$ and
$\left({}_{\mathcal{C}^{T}}{(U_{T})}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$:
If $\eta$ and $\epsilon$ denote the unit and counit of $F_{T}\dashv U_{T}$,
then we can define a pair of inverse functors
$L:\mathcal{Z}_{T}(\mathcal{C})\rightarrow\left({}_{\mathcal{C}^{T}}{(U_{T})}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$
and
$K:\left({}_{\mathcal{C}^{T}}{(U_{T})}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}\rightarrow\mathcal{Z}_{T}(\mathcal{C})$
by

$\displaystyle L(M,\delta):=$
$\displaystyle\big{(}M,(U_{T}\epsilon_{(N,r)}\otimes\mathrm{id}_{M})\delta_{N}:M\otimes
U_{T}(N,r)\rightarrow U_{T}(N,r)\otimes M\big{)}$ $\displaystyle K(M,\tau):=$
$\displaystyle\big{(}M,\tau_{F_{T}(N)}(\mathrm{id}_{M}\otimes\eta_{N}):M\otimes
N\rightarrow U_{T}F_{T}(N)\otimes M\big{)}$

Note that since $\mathcal{C}$ was assumed to be rigid for constructing
$Z_{T}$, there was no difference between the dual and the lax dual of $U_{T}$,
as noted in Theorem 2.1.

##  8 Other aspects of Hopf monads

In this section, we briefly mention some other aspects of the theory of Hopf
monads which we have not covered.

Comodules over bimonads: For an ordinary bialgebra $H$, it is well-known that
the category of $H$-comodules ${}^{H}{\mathcal{M}}$ becomes isomorphic to the
dual $\left({}_{{}_{H}{\mathcal{M}}}{U}_{\mathsf{Vec}}\right)^{\circ}$ of the
forgetful functor $U:{}_{H}{\mathcal{M}}\rightarrow\mathsf{Vec}$. The
correspondence between $H$-coactions on $M$ and braidings $M\otimes
U\rightarrow U\otimes M$ follows exactly as in the correspondence between
${}^{H}_{H}\mathcal{YD}$ and $Z_{l,\mathrm{lax}}({}_{H}{\mathcal{M}})$. An
interesting question is finding an appropriate notion of comodules over an
arbitrary bimonad.

If we consider the bimonads arising from braided bialgebras $B$, we recover
the notion of $B$-comodules by looking at $T(\mathtt{1})$-comodules. Therefore
in the main references on Hopf monads [14, 12] comodules over $T(\mathtt{1})$
are called $T$-comodules, as in Section 7.2. However, given a bialgebroid $B$
over an algebra $A$, there exists a well-define notion of $B$-comodules such
that the category of these objects again become isomorphic to the lax left
dual of the forgetful functor
${}_{B}{\mathcal{M}}\rightarrow{}_{A}{\mathcal{M}}_{A}$ [58]. Therefore, given
a bimonad $T$ on a category $\mathcal{C}$, it is more natural to define the
category of $T$_-comodules_ as the category
$\left({}_{\mathcal{C}^{T}}{(U_{T})}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}$.
This is the point of view which is taken in [60]. However as pointed out in
Remark 2.13 of [60], this definition is not appropriate for braided bialgebras
in arbitrary braided categories.

Since the forgetful functor
$\overline{U}:\left({}_{\mathcal{C}^{T}}{(U_{T})}_{\mathcal{C}}\right)^{\circ}_{l,\mathrm{lax}}\rightarrow\mathcal{C}$