#  Relative adjoint algebras

Martín Mombelli  Facultad de Matemática, Astronomía y Física
Universidad Nacional de Córdoba
CIEM – CONICET
Medina Allende s/n
(5000) Ciudad Universitaria, Córdoba, Argentina<EMAIL_ADDRESS><EMAIL_ADDRESS>
_URL:_ https://www.famaf.unc.edu.ar/$\sim$mombelli

###### Abstract.

Given a finite tensor category ${\mathcal{C}}$, an exact indecomposable
${\mathcal{C}}$-module category ${\mathcal{M}}$, and a tensor subcategory
${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$, we describe a way to
produce exact commutative algebras in the center $Z({\mathcal{C}})$, measuring
this inclusion. The construction of such algebras is done in an analogous way
as presented by Shimizu [20], but using instead the relative (co)end, a
categorical tool developed in [1] in the realm of representations of tensor
categories. We provide some explicit computations.

###### Key words and phrases:

tensor category; module category

###### 2010 Mathematics Subject Classification:

18D20, 18D10

## Introduction

Assume ${\mathcal{M}}$ is a module category over a finite tensor category
${\mathcal{C}}$. In [20], the author introduces the notion of adjoint algebra
${\mathcal{A}}_{\mathcal{M}}$ and the space of class functions
$\rm{CF}({\mathcal{M}})$ associated to ${\mathcal{M}}$, generalizing the
definitions given in [19]. This algebra is an interesting object associated to
${\mathcal{M}}$. In fact, this algebra is a complete invariant of
${\mathcal{M}}$.

If $\triangleright:{\mathcal{C}}\times{\mathcal{M}}\to{\mathcal{M}}$ is the
action of ${\mathcal{C}}$ on ${\mathcal{M}}$, then we can consider the action
functor

$\rho_{\mathcal{M}}:{\mathcal{C}}\to\operatorname{End}({\mathcal{M}}),$
$\rho_{\mathcal{M}}(X)(M)=X\triangleright
M,\,\,X\in{\mathcal{C}},M\in{\mathcal{M}}.$

The right adjoint of this functor is
$\rho_{\mathcal{M}}^{\rm{ra}}:\operatorname{End}({\mathcal{M}})\to{\mathcal{C}}$,
wich can be explicitly described as an end

$\rho_{\mathcal{M}}^{\rm{ra}}(F)=\int_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(M,F(M)),$

for any $F\in\operatorname{End}({\mathcal{M}})$, see [20, Thm. 3.4]. Here for
any $M\in{\mathcal{M}}$, $\underline{\operatorname{Hom}}(M,-)$ is the right
adjoint of the functor ${\mathcal{C}}\to{\mathcal{M}}$, $X\mapsto
X\triangleright M$. It is called the internal Hom of the module category
${\mathcal{M}}$. The adjoint algebra is defined as

${\mathcal{A}}_{\mathcal{M}}=\rho_{\mathcal{M}}^{\rm{ra}}(\mbox{\rm
Id\,}_{\mathcal{M}}).$

This object has structure of commutative algebra in the Drinfeld center
$Z({\mathcal{C}})$. A key ingredient of the construction is the end of the
functor

$\underline{\operatorname{Hom}}(-,-):{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{C}}.$

In this paper we investigate what kind of structure it is obtained, when we
replace the usual end by the relative end, a new categorical tool introduced
in [1].

Assume ${\mathcal{A}}$ is any category, and
$S:{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{A}}$ is a functor
equipped with natural isomorphisms

$\beta^{X}_{M,N}:S(M,X\triangleright N)\to S(X^{*}\triangleright M,N).$

We call this isomorphism a ${\mathcal{C}}$-prebalancing of $S$. Any additive
${\Bbbk}$-linear functor $S$ posses a $\mbox{\rm
vect\,}_{\Bbbk}$-prebalancing. However, in the general case, the prebalancing
is an extra structure of the functor $S$. The relative end of $S$ is an object
$E\in{\mathcal{A}}$ that comes with dinatural transformations
$\pi_{M}:E\xrightarrow{..}S(M,M)$ such that the equation

(0.1) $S(\mbox{ev}_{X}\triangleright\mbox{\rm id\,}_{M},\mbox{\rm
id\,}_{M})\pi_{M}=S(m_{X^{*},X,M},\mbox{\rm
id\,}_{M})\beta^{X}_{X\triangleright M,M}\pi_{X\triangleright M},$

is fulfilled, and it is universal among all objects in ${\mathcal{A}}$ with
dinatural transformations that satisfy (0.1). Unlike the case
${\mathcal{C}}=\mbox{\rm vect\,}_{\Bbbk}$, it may happen that a dinatural
transformation does not satisfy (0.1). The relative end of $S$ is denoted by
$\oint_{M\in{\mathcal{M}}}(S,\beta)$. When ${\mathcal{C}}=\mbox{\rm
vect\,}_{\Bbbk}$, the relative end coincides with the usual end. Relative
coends are defined similarly.

If ${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$ is a tensor
subcategory, then ${\mathcal{D}}$ acts on the left and on the right on
$\operatorname{End}({\mathcal{M}})$ making it a ${\mathcal{D}}$-bimodule
category. The action functor

$\rho_{\mathcal{M}}:{\mathcal{C}}\to\operatorname{End}({\mathcal{M}})$

lands in $Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$, the relative
center of $\operatorname{End}({\mathcal{M}})$. Thus we can consider the
functor

(0.2) $\rho:{\mathcal{C}}\to
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}})).$

Given any $F\in Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$, the
internal Hom of the module category ${\mathcal{M}}$ posses a canonical
${\mathcal{D}}$-prebalancing, see Lemma 6.3. Thus we can consider the relative
end

$\oint_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(-,F(-))\in{\mathcal{C}}.$

The adjoint algebra relative to ${\mathcal{D}}$ is then defined as

(0.3)
${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}:=\oint_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(-,-)\in{\mathcal{C}}.$

See Definition 6.6. The algebra ${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$
seems to measure very well the inclusion
${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$. The object
${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ is in fact a commutative
connected algebra in the Drinfeld center $Z({\mathcal{C}})$ such that
$Z({\mathcal{C}})_{{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}}$ is a rigid
monoidal category. This last condition is equivalent to
$Z({\mathcal{C}})_{{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}}$ being an
exact module category over $Z({\mathcal{C}})$, that is
${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ is an exact algebra. We called
this type of algebras étale, generalizing the notion of étale algebras in the
semisimple case. See [3, Section 3]. In our case, separability of the algebra
is replaced by the condition of exactness.

The key step to prove all these facts, is the computation of a right adjoint
of the functor $\rho$, given in (0.2). A right adjoint is described as

$\bar{\rho}:Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))\to{\mathcal{C}},$
$\bar{\rho}(F)=\oint_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(M,F(M)).$

In Theorem 6.4 we prove this fact, and we also prove that $\bar{\rho}$ is
exact, faithful and dominant. This implies that the adjunction
$(\rho,\bar{\rho})$ is monadic, and this fact implies that there is a monoidal
equivalence

$Z({\mathcal{C}})_{A_{{\mathcal{D}},{\mathcal{M}}}}\simeq
Z_{\mathcal{D}}({\mathcal{C}}^{*}_{\mathcal{M}}).$

Hence we can compute its Frobenius-Perron dimension as

$\mbox{\rm
FPdim\,}({\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}})=\frac{\mbox{\rm
FPdim\,}({\mathcal{C}})}{\mbox{\rm FPdim\,}({\mathcal{D}})}.$

These assertions are stated in Proposition 6.10.

The correspondence between tensor subcategories of
${\mathcal{C}}^{*}_{\mathcal{M}}$ and certain commutative algebras in the
center is not new, at least in the semisimple case. In [3, Thm. 4.10], the
authors described a bijective correspondence between tensor subcategories of a
fixed tensor category ${\mathcal{C}}$ and étale subalgebras of
${\mathcal{A}}_{\mathcal{C}}.$ This subalgebra is also described as
$I(\mathbf{1})$, where $I$ is the right adjoint of a certain functor. What we
present new in this paper is an explicit description of the functor $I$, and
the means to compute the algebra
${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$.

One of the advantages of the description presented in (0.3), is that it
provides a formula for its computation. In Section 7 we present explicit
calculations. We provide examples when ${\mathcal{C}}$ is an
equivariantization by a finite group, when ${\mathcal{C}}$ is the tensor
category of representations of a finite-dimensional Hopf algebra, and when
${\mathcal{C}}$ is the Deligne tensor product of two tensor categories.

### Acknowledgments

This work was partially supported by CONICET and Secyt (UNC), Argentina.

##  1\. Preliminaries

Throughout this paper, ${\Bbbk}$ will denote an algebraically closed field of
characteristic zero. All categories will be finite, in the sense of [7],
abelian ${\Bbbk}$-linear categories, and all functors will be additive
${\Bbbk}$-linear. If $F:{\mathcal{C}}\to{\mathcal{D}}$ is a functor, we shall
denote by $F^{\rm{la}}$, $F^{\rm{ra}}$ the left and right adjoint to $F$, if
they exist.

###  1.1. Monads

A monad over a category ${\mathcal{C}}$ is an algebra object in the monoidal
category $\operatorname{End}({\mathcal{C}})$. That is, a functor
$T:{\mathcal{C}}\to{\mathcal{C}}$ equipped with natural morphisms

$\mu:T\circ T\to T,\,\,\,\eta:\mbox{\rm Id\,}\to T,$

the multiplication and the unit of $T$, such that they satisfy associativity
and unity axioms. The category of $T$-modules, ${\mathcal{C}}^{T}$ is the
category whose objects are pairs $(M,s)$, where $M$ is an object of
${\mathcal{C}}$, $s:T(M)\to M$ is a morphism in ${\mathcal{C}}$ such that

$sT(s)=s\mu_{M},\quad s\eta_{M}=\mbox{\rm id\,}_{M}.$

The forgetful functor ${\mathcal{U}}:{\mathcal{C}}^{T}\to{\mathcal{C}}$ has
left adjoint given by the free functor
${\mathcal{L}}:{\mathcal{C}}\to{\mathcal{C}}^{T}$,
${\mathcal{L}}(M)=(T(M),\mu_{M}).$

If $(G:{\mathcal{D}}\to{\mathcal{C}},F:{\mathcal{C}}\to{\mathcal{D}})$ is an
adjunction, with unit and counit $e:G\circ F\to\mbox{\rm Id\,},$ $c:\mbox{\rm
Id\,}\to F\circ G$, then $T=F\circ G$ is a monad. The comparison functor

$\kappa:{\mathcal{D}}\to{\mathcal{C}}^{T},$

is defined by $\kappa(M)=(F(M),F(e_{M})),$ for any $M\in{\mathcal{D}}$. If the
comparison functor $\kappa$ is an equivalence of categories, the adjunction
$(G,F)$ is said to be monadic. Several results concerning monadicity of an
adjunction are given by Beck’s Theorem. See for example [15]. As a consequence
of Beck’s Monadicity Theorem, it follows that, if
$F:{\mathcal{C}}\to{\mathcal{D}}$ is faithful and exact, then $(G,F)$ is
monadic.

###  1.2. Finite tensor categories

A tensor category ${\mathcal{C}}$ is a monoidal rigid category, with simple
unit object $\mathbf{1}\in{\mathcal{C}}$, satisfying certain finitness
condition. We refer to [7] for more details on finite tensor categories.

We shall denote by ${\mathcal{C}}^{\rm{rev}}$, the tensor category whose
underlying abelian category is ${\mathcal{C}}$, with reverse monoidal product

${\otimes}^{\rm{rev}}:{\mathcal{C}}\times{\mathcal{C}}\to{\mathcal{C}},\quad
X{\otimes}^{\rm{rev}}Y=Y{\otimes}X,$

and associativity constraints

$a^{\rm{rev}}_{X,Y,Z}:(X{\otimes}^{\rm{rev}}Y){\otimes}^{\rm{rev}}Z\to
X{\otimes}^{\rm{rev}}(Y{\otimes}^{\rm{rev}}Z),$
$a^{\rm{rev}}_{X,Y,Z}:=a^{-1}_{Z,Y,X},$

for any $X,Y,Z\in{\mathcal{C}}$. Here $a_{X,Y,Z}:(X{\otimes}Y){\otimes}Z\to
X{\otimes}(Y{\otimes}Z)$ is the associativity of ${\mathcal{C}}$.

The next result is [2, Prop. 5.1]. It will be useful later.

######  Proposition 1.1.

Let ${\mathcal{C}},{\mathcal{D}}$ be tensor categories, and let
$F:{\mathcal{C}}\to{\mathcal{D}}$ be an exact tensor functor with left adjoint
$G:{\mathcal{D}}\to{\mathcal{C}}$. Let $T=FG$ be the monad associated to $F$.
Let $T=FG$ be the Hopf monad associated to the adjunction $(G,F)$. Then the
following assertions are equivalent:

  * (i)

The functor F is dominant;

  * (ii)

The unit $\eta$ of the monad T is a monomorphism;

  * (iii)

The monad $T$ is faithful;

  * (iv)

The left adjoint of F is faithful;

  * (v)

The right adjoint of F is faithful.∎

If $A\in{\mathcal{C}}$ is an algebra, then ${\mathcal{C}}_{A}$, respectively
${}_{A}{\mathcal{C}}_{A}$, denote the category of right $A$-modules in
${\mathcal{C}}$, respectively the category of $A$-bimodules in $A$. We shall
denote by ${\mathcal{U}}:{\mathcal{C}}_{A}\to{\mathcal{C}},$ the forgetful
functor, and a left adjoint
${\mathcal{L}}_{A}:{\mathcal{C}}\to{\mathcal{C}}_{A}$,
${\mathcal{L}}_{A}(V)=V{\otimes}A$, the free functor. A central algebra for
${\mathcal{C}}$ is an algebra in the Drinfeld center $Z({\mathcal{C}})$; that
is a pair $(A,\sigma)$, where $\sigma$ is a half-braiding for $A$. It is
commutative if $m\sigma=m$, where $m:A{\otimes}A\to A$ is the multiplication
of $A$.

The category ${}_{A}{\mathcal{C}}_{A}$ has a monoidal product ${\otimes}_{A}$.
For this category to be rigid, we need to ask for more conditions on the
algebra $A$. If $(A,\sigma)$ is a commutative central algebra, then there is a
faithful functor

${\mathcal{C}}_{A}\to{}_{A}{\mathcal{C}}_{A},$

described as follows. If $V\in{\mathcal{C}}_{A}$, then we can define a left
$A$-module structure using the half-braiding of $A$:

$A{\otimes}V\xrightarrow{\sigma_{V}}V{\otimes}A\to V.$

The category ${\mathcal{C}}_{A}$ inherits from ${}_{A}{\mathcal{C}}_{A}$ a
structure of monoidal category with unit object $A$, such that the free
functor ${\mathcal{L}}_{A}:{\mathcal{C}}\to{\mathcal{C}}_{A}$ is a (strong)
monoidal functor. Observe that both, the monoidal structure of
${\mathcal{C}}_{A}$ and the monoidal structure of ${\mathcal{L}}$ depend on
the half-braiding $\sigma$.

The next result is [2, Prop. 6.1], and it will be used later.

######  Proposition 1.2.

Let ${\mathcal{C}},{\mathcal{D}}$ be tensor categories, and let
$F:{\mathcal{C}}\to{\mathcal{D}}$ be an exact dominant tensor functor with
right adjoint $R:{\mathcal{D}}\to{\mathcal{C}}$. The following statements
hold.

  * 1.

The object $R(\mathbf{1})=A$ has a natural structure of central commutative
algebra in ${\mathcal{C}}$.

  * 2.

If $R$ is exact and faithful, then ${\mathcal{C}}_{A}$ is a tensor category,
and there exists a tensor equivalence $K:{\mathcal{D}}\to{\mathcal{C}}_{A}$
such that $K\circ F\simeq{\mathcal{L}}_{A}$ as tensor functors.∎

##  2\. Representations of tensor categories

A left _module_ category over ${\mathcal{C}}$ is a category ${\mathcal{M}}$
together with a bifunctor
$\rhd:{\mathcal{C}}\times{\mathcal{M}}\to{\mathcal{M}}$, exact in each
variable, endowed with natural associativity and unit isomorphisms

$m_{X,Y,M}:(X\otimes Y)\triangleright M\to X\triangleright(Y\triangleright
M),\ \ \ell_{M}:\mathbf{1}\triangleright M\to M.$

These isomorphisms are subject to the following conditions:

(2.1) $m_{X,Y,Z\triangleright M}\;m_{X\otimes Y,Z,M}=(\mbox{\rm
id\,}_{X}\triangleright m_{Y,Z,M})\;m_{X,Y\otimes
Z,M}(a_{X,Y,Z}\triangleright\mbox{\rm id\,}_{M}),$ (2.2) $(\mbox{\rm
id\,}_{X}\triangleright\ell_{M})m_{X,{\bf 1},M}=r_{X}\triangleright\mbox{\rm
id\,}_{M},$

for any $X,Y,Z\in{\mathcal{C}},M\in{\mathcal{M}}.$ Here $a$ is the
associativity constraint of ${\mathcal{C}}$. Sometimes we shall also say that
${\mathcal{M}}$ is a (left) ${\mathcal{C}}$-_module category_ or a (left)
representation of ${\mathcal{C}}$.

Let ${\mathcal{M}}$ and ${\mathcal{M}}^{\prime}$ be a pair of
${\mathcal{C}}$-modules. A _module functor_ is a pair $(F,c)$, where
$F:{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ is a functor equipped with natural
isomorphisms

$c_{X,M}:F(X\triangleright M)\to X\triangleright F(M),$

$X\in{\mathcal{C}}$, $M\in{\mathcal{M}}$, such that for any
$X,Y\in{\mathcal{C}}$, $M\in{\mathcal{M}}$:

(2.3) $\displaystyle(\mbox{\rm id\,}_{X}\triangleright
c_{Y,M})c_{X,Y\triangleright M}F(m_{X,Y,M})$
$\displaystyle=m_{X,Y,F(M)}\,c_{X\otimes Y,M}$ (2.4)
$\displaystyle\ell_{F(M)}\,c_{\mathbf{1},M}$ $\displaystyle=F(\ell_{M}).$

There is a composition of module functors: if ${\mathcal{M}}^{\prime\prime}$
is a ${\mathcal{C}}$-module category and
$(G,d):{\mathcal{M}}^{\prime}\to{\mathcal{M}}^{\prime\prime}$ is another
module functor then the composition

(2.5) $(G\circ F,e):{\mathcal{M}}\to{\mathcal{M}}^{\prime\prime},\qquad
e_{X,M}=d_{X,F(M)}\circ G(c_{X,M}),$

is also a module functor.

A natural module transformation between module functors $(F,c)$ and $(G,d)$ is
a natural transformation $\theta:F\to G$ such that

(2.6) $\displaystyle d_{X,M}\theta_{X\triangleright M}=(\mbox{\rm
id\,}_{X}\triangleright\theta_{M})c_{X,M},$

for any $X\in{\mathcal{C}}$, $M\in{\mathcal{M}}$. The vector space of natural
module transformations will be denoted by $\mbox{\rm Nat\,}_{\\!m}(F,G)$. Two
module functors $F,G$ are _equivalent_ if there exists a natural module
isomorphism $\theta:F\to G$. We denote by
$\operatorname{Fun}_{{\mathcal{C}}}({\mathcal{M}},{\mathcal{M}}^{\prime})$ the
category whose objects are module functors $(F,c)$ from ${\mathcal{M}}$ to
${\mathcal{M}}^{\prime}$ and arrows module natural transformations.

Two ${\mathcal{C}}$-module categories ${\mathcal{M}}$ and
${\mathcal{M}}^{\prime}$ are equivalent if there exist module functors
$F:{\mathcal{M}}\to{\mathcal{M}}^{\prime}$,
$G:{\mathcal{M}}^{\prime}\to{\mathcal{M}}$, and natural module isomorphisms
$\mbox{\rm Id\,}_{{\mathcal{M}}^{\prime}}\to F\circ G$, $\mbox{\rm
Id\,}_{{\mathcal{M}}}\to G\circ F$.

A module is indecomposable if it is not equivalent to a direct sum of two non
trivial modules. Recall from [7], that a module ${\mathcal{M}}$ is _exact_ if
for any projective object $P\in{\mathcal{C}}$ the object $P\triangleright M$
is projective in ${\mathcal{M}}$, for all $M\in{\mathcal{M}}$. If
${\mathcal{M}}$ is an exact indecomposable module category over
${\mathcal{C}}$, the dual category
${\mathcal{C}}^{*}_{\mathcal{M}}=\operatorname{End}_{\mathcal{C}}({\mathcal{M}})$
is a finite tensor category [7]. The tensor product is the composition of
module functors.

A _right module category_ over ${\mathcal{C}}$ is a finite category
${\mathcal{M}}$ equipped with an exact bifunctor
$\triangleleft:{\mathcal{M}}\times{\mathcal{C}}\to{\mathcal{M}}$ and natural
isomorphisms

$\widetilde{m}_{M,X,Y}:M\triangleleft(X{\otimes}Y)\to(M\triangleleft
X)\triangleleft Y,\quad r_{M}:M\triangleleft\mathbf{1}\to M$

such that

(2.7) $\widetilde{m}_{M\triangleleft
X,Y,Z}\;\widetilde{m}_{M,X,Y{\otimes}Z}(\mbox{\rm id\,}_{M}\triangleleft
a_{X,Y,Z})=(\widetilde{m}_{M,X,Y}\triangleleft\mbox{\rm
id\,}_{Z})\,\widetilde{m}_{M,X{\otimes}Y,Z},$ (2.8)
$(r_{M}\triangleleft\mbox{\rm
id\,}_{X})\widetilde{m}_{M,\mathbf{1},X}=\mbox{\rm id\,}_{M}\triangleleft
l_{X}.$

If ${\mathcal{M}},{\mathcal{M}}^{\prime}$ are right ${\mathcal{C}}$-modules, a
module functor from ${\mathcal{M}}$ to ${\mathcal{M}}^{\prime}$ is a pair
$(T,d)$ where $T:{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ is a functor and
$d_{M,X}:T(M\triangleleft X)\to T(M)\triangleleft X$ are natural isomorphisms
such that for any $X,Y\in{\mathcal{C}}$, $M\in{\mathcal{M}}$:

(2.9) $\displaystyle(d_{M,X}{\otimes}\mbox{\rm id\,}_{Y})d_{M\triangleleft
X,Y}T(m_{M,X,Y})$ $\displaystyle=m_{T(M),X,Y}\,d_{M,X{\otimes}Y},$ (2.10)
$\displaystyle r_{T(M)}\,d_{M,\mathbf{1}}$ $\displaystyle=T(r_{M}).$

We recall the next well-known result since it will be needed later. The proof
it is done by a straightforward calculation. Let
${\mathcal{M}},{\mathcal{M}}^{\prime}$ be left ${\mathcal{C}}$-module
categories. Assume that $F:{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ is a
functor with left adjoint
$F^{\rm{la}}:{\mathcal{M}}^{\prime}\to{\mathcal{M}}$, with associated natural
isomorphisms

$\alpha_{M,N}:\operatorname{Hom}_{{\mathcal{M}}}(F^{\rm{la}}(M),N)\to\operatorname{Hom}_{{\mathcal{M}}^{\prime}}(M,F(N)).$

######  Lemma 2.1.

Let $e:F^{\rm{la}}\circ F\to\mbox{\rm Id\,}$, $\eta:\mbox{\rm Id\,}\to F\circ
F^{\rm{la}}$ be the unit and counit of the adjunction. Also assume that
$(F,c):{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ is a module functor. Then
$F^{\rm{la}}$ has structure of module functor, with structure given by

$\widetilde{c}_{X,M}:F^{\rm{la}}(X\triangleright M)\to X\triangleright
F^{\rm{la}}(M),$ $\widetilde{c}_{X,M}=e_{X\triangleright
F^{\rm{la}}(M)}F^{\rm{la}}(c^{-1}_{X,F^{\rm{la}}(M)})F^{\rm{la}}(\mbox{\rm
id\,}_{X}\triangleright\eta_{M}),$

for any $X\in{\mathcal{C}}$, $M\in{\mathcal{M}}^{\prime}.$ Aslo, equation

(2.11) $\alpha^{-1}_{X\triangleright F(M),X\triangleright
M}(c^{-1}_{X,M})=(\mbox{\rm id\,}_{X}\triangleright
e_{M})\widetilde{c}_{X,F(M)}.$

is fulfilled, for any $X\in{\mathcal{C}},$ $M\in{\mathcal{M}}^{\prime}.$∎

###  2.1. Étale algebras in finite tensor categories

We propose the following definition of étale algebras in the non-semisimple
case, generalizing [3, Definition 3.1]. Assume that ${\mathcal{C}}$ is a
braided tensor category.

######  Definition 2.2.

An algebra $A\in{\mathcal{C}}$ is étale if:

  * $\bullet$

$A$ is commutative;

  * $\bullet$

the category ${\mathcal{C}}_{A}$ is an exact indecomposable left module
category over ${\mathcal{C}}$;

  * $\bullet$

$A$ is simple as a right $A$-module; that is $A\in{\mathcal{C}}_{A}$ is a
simple object.

It is connected if
$\dim_{\Bbbk}\operatorname{Hom}_{\mathcal{C}}(\mathbf{1},A)=1$.

######  Remark 2.3.

The exactness and indecomposability of ${\mathcal{C}}_{A}$ as a left
${\mathcal{C}}$-module category may be a superfluous condition. We expect that
$A$ being right simple implies that ${\mathcal{C}}_{A}$ is an exact and
indecomposable module category.

######  Remark 2.4.

Condition of ${\mathcal{C}}_{A}$ being exact and indecomposable module
category is equivalent to ${}_{A}{\mathcal{C}}_{A}$ being a rigid monoidal
category. Indeed, there is a monoidal equivalence
${}_{A}{\mathcal{C}}_{A}\simeq{\mathcal{C}}^{*}_{{\mathcal{C}}_{A}}$. One can
prove that this is also equivalent to ${\mathcal{C}}_{A}$ being a rigid
monoidal category, since rigidity of objects in ${\mathcal{C}}_{A}$ is
iherited by rigidity in ${}_{A}{\mathcal{C}}_{A}$.

There is a canonical correspondence between étale algebras and dominant tensor
functors with an exact faithful right adjoint. The proof of the next result
follows from Proposition 1.2. See also [3, Lemma 3.5].

######  Corollary 2.5.

If $F:Z({\mathcal{C}})\to{\mathcal{D}}$ is an exact dominant tensor functor
with an exact faithful right adjoint $R:{\mathcal{D}}\to Z({\mathcal{C}})$,
then $(A,\sigma)=R(\mathbf{1})$ is a connected étale algebra in the center
$Z({\mathcal{C}})$.∎

###  2.2. The internal Hom

Let ${\mathcal{C}}$ be a tensor category and ${\mathcal{M}}$ be a left
${\mathcal{C}}$-module category. For any pair of objects
$M,N\in{\mathcal{M}}$, the _internal Hom_ is an object
$\underline{\operatorname{Hom}}(M,N)\in{\mathcal{C}}$ representing the left
exact functor

$\operatorname{Hom}_{{\mathcal{M}}}(-\triangleright
M,N):{\mathcal{C}}^{\rm{op}}\to\mbox{\rm vect\,}_{\Bbbk}.$

This means that, there are natural isomorphisms, one the inverse of each
other,

(2.12)
$\begin{split}\phi^{X}_{M,N}:\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N))\to\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N),\\\ \psi^{X}_{M,N}:\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N)\to\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N)),\end{split}$

for all $M,N\in{\mathcal{M}}$, $X\in{\mathcal{C}}$. Sometimes we shall denote
the internal Hom of the module category ${\mathcal{M}}$ by
$\underline{\operatorname{Hom}}_{\mathcal{M}}$ to emphasize that it is related
to ${\mathcal{M}}$.

Naturality of $\psi$ implies that diagrams

(2.13) $\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\mapsto
f\circ\beta}$$\scriptstyle{\psi^{X}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\mapsto\underline{\operatorname{Hom}}(M,f)\circ\alpha}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,\widetilde{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{X}_{M,\widetilde{N}}}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,\widetilde{N})),}$
(2.14)
$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright\widetilde{M},N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\mapsto\beta(\mbox{\rm
id\,}_{X}\triangleright
g)}$$\scriptstyle{\psi^{X}_{\widetilde{M},N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(\widetilde{M},N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\mapsto\underline{\operatorname{Hom}}(g,N)\circ\alpha}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{X}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N)),}$
(2.15) $\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(Y\triangleright
M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\mapsto\beta(h\triangleright\mbox{\rm
id\,}_{M})}$$\scriptstyle{\psi^{Y}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(Y,\underline{\operatorname{Hom}}(M,N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\mapsto\alpha\circ
h}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{X}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N)),}$

commute, for any pair of morphism $f:N\to\widetilde{N}$, $g:M\to\widetilde{M}$
in ${\mathcal{M}}$, and any morphism $h:X\to Y$ in ${\mathcal{C}}$. Also,
naturality of $\phi$, implies that diagrams

(2.16)
$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(\widetilde{M},N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\mapsto\underline{\operatorname{Hom}}(g,N)\circ\beta}$$\scriptstyle{\phi^{X}_{\widetilde{M},N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright\widetilde{M},N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\mapsto\alpha(\mbox{\rm
id\,}_{X}\triangleright
g)}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{X}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N),}$ (2.17)
$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\mapsto\underline{\operatorname{Hom}}(M,f)\beta}$$\scriptstyle{\phi^{X}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\mapsto
f\alpha}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,\widetilde{N}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{X}_{M,\widetilde{N}}}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,\widetilde{N}),}$ (2.18)
$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(Y,\underline{\operatorname{Hom}}(M,N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\mapsto\beta\circ
h}$$\scriptstyle{\phi^{Y}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(Y\triangleright
M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\mapsto\alpha(h\triangleright\mbox{\rm
id\,}_{M})}$$\textstyle{\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{X}_{M,N}}$$\textstyle{\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N),}$

commutes for any $f:N\to\widetilde{N}$, $g:M\to\widetilde{M}$ morphisms in
${\mathcal{M}}$, and any morphism $h:X\to Y$ in ${\mathcal{C}}$. The next
result will be needed later.

######  Lemma 2.6.

[11, Lemma 3] Let ${\mathcal{M}}$ be an exact module category over
${\mathcal{C}}$, and $F:{\mathcal{M}}\to{\mathcal{M}}$ be a
${\mathcal{C}}$-module functor with left adjoint
$F^{\rm{la}}:{\mathcal{M}}\to{\mathcal{M}}$. Then, there are natural
isomorphisms

$\xi^{F}_{M,N}:\underline{\operatorname{Hom}}(M,F(N))\to\underline{\operatorname{Hom}}(F^{\rm{la}}(M),N).$

∎

######  Remark 2.7.

Isomorphisms $\xi^{F}$ can be explicitly described using the proof of [11,
Lemma 3]. One can see that

(2.19)
$\xi^{F}_{M,N}=\psi^{Y}_{F^{\rm{la}}(M),N}\big{(}\alpha^{-1}_{Y\triangleright
M,N}(\phi^{Y}_{M,F(N)}(\mbox{\rm id\,}))\tilde{d}^{-1}_{Y,M}\big{)}.$

Here $Y=\underline{\operatorname{Hom}}(M,F(N))$, $\tilde{d}$ is the module
structure of the functor $F^{\rm{la}}$, and

$\alpha_{M,N}:\operatorname{Hom}_{\mathcal{M}}(F^{\rm{la}}(M),N)\to\operatorname{Hom}_{\mathcal{M}}(M,F(N))$

are the natural isomorphisms of the adjunction $(F^{\rm{la}},F).$

Next, we shall define a collection of morphisms that will be used continuosly.
If $X\in{\mathcal{C}}$, $M,N\in{\mathcal{M}}$, set

$\mbox{coev}^{\mathcal{M}}_{M}:\mathbf{1}\to\underline{\operatorname{Hom}}(M,M),\quad\mbox{ev}^{\mathcal{M}}_{M,N}:\underline{\operatorname{Hom}}(M,N)\triangleright
M\to N,$ $\mbox{coev}^{\mathcal{M}}_{M}=\psi^{\mathbf{1}}_{M,M}(\mbox{\rm
id\,}_{M}),\quad\mbox{ev}^{\mathcal{M}}_{M,N}=\phi^{\underline{\operatorname{Hom}}(M,N)}_{M,N}(\mbox{\rm
id\,}_{\underline{\operatorname{Hom}}(M,N)}).$

Define also $f_{M}=\mbox{ev}^{\mathcal{M}}_{M,M}(\mbox{\rm
id\,}_{\underline{\operatorname{Hom}}(M,M)}\triangleright\mbox{ev}^{\mathcal{M}}_{M,M})$,
and

(2.20)
$\mbox{comp}^{\mathcal{M}}_{M}:\underline{\operatorname{Hom}}(M,M){\otimes}\underline{\operatorname{Hom}}(M,M)\to\underline{\operatorname{Hom}}(M,M),$
$\mbox{comp}^{\mathcal{M}}_{M}=\psi^{\underline{\operatorname{Hom}}(M,M){\otimes}\underline{\operatorname{Hom}}(M,M)}_{M,M}(f_{M}).$

For any $M\in{\mathcal{M}}$ denote by $R_{M}:{\mathcal{C}}\to{\mathcal{M}}$
the ${\mathcal{C}}$-module functor $R_{M}(X)=X\triangleright M$. Its right
adjoint $R^{ra}_{M}:{\mathcal{M}}\to{\mathcal{C}}$ is then
$R^{\rm{ra}}_{M}(N)=\underline{\operatorname{Hom}}(M,N)$. The functor
$R^{\rm{ra}}_{M}$ is also a ${\mathcal{C}}$-module functor. Denote by

(2.21) $\mathfrak{a}_{X,M,N}:\underline{\operatorname{Hom}}(M,X\triangleright
N)\to X{\otimes}\underline{\operatorname{Hom}}(M,N)$

the left ${\mathcal{C}}$-module structure of $R^{\rm{ra}}_{M}$. Let
$\mathfrak{b}^{1}_{X,M,N}:\underline{\operatorname{Hom}}(X\triangleright
M,N)\to\underline{\operatorname{Hom}}(M,N){\otimes}X^{*}$ be the isomorphisms
induced by the natural isomorphisms

$\displaystyle\operatorname{Hom}_{\mathcal{C}}(Z,\underline{\operatorname{Hom}}(X\triangleright
M,N))\simeq\operatorname{Hom}_{\mathcal{M}}(Z\triangleright(X\triangleright
M),N)$
$\displaystyle\simeq\operatorname{Hom}_{\mathcal{M}}((Z{\otimes}X)\triangleright
M,N)\simeq\operatorname{Hom}_{\mathcal{C}}(Z\otimes
X,\underline{\operatorname{Hom}}(M,N))$
$\displaystyle\simeq\operatorname{Hom}_{\mathcal{C}}(Z,\underline{\operatorname{Hom}}(M,N)\otimes
X^{*}),$

for any $X,Z\in{\mathcal{C}}$, $M,N\in{\mathcal{M}}$. Define also

$\mathfrak{b}_{X,M,N}:\underline{\operatorname{Hom}}(X\triangleright
M,N){\otimes}X\to\underline{\operatorname{Hom}}(M,N)$

as the composition

$\mathfrak{b}_{X,M,N}=(\mbox{\rm
id\,}{\otimes}\mbox{ev}_{X})(\mathfrak{b}^{1}_{X,M,N}{\otimes}\mbox{\rm
id\,}_{X}).$

###  2.3. The canonical action of $\mbox{\rm vect\,}_{\Bbbk}$

Any abelian ${\Bbbk}$-linear category is a $\mbox{\rm vect\,}_{\Bbbk}$-module
category. If ${\mathcal{M}}$ is such a category, there exists a canonical
action of $\mbox{\rm vect\,}_{\Bbbk}$ on ${\mathcal{M}}$

$\blacktriangleright:\mbox{\rm
vect\,}_{\Bbbk}\times{\mathcal{M}}\to{\mathcal{M}},$

determined by

$\operatorname{Hom}_{\Bbbk}(V,\operatorname{Hom}_{\mathcal{M}}(A,B))\simeq\operatorname{Hom}_{\mathcal{M}}(V\blacktriangleright
A,B)$

for any $V\in\mbox{\rm vect\,}_{\Bbbk}$, $A,B\in{\mathcal{M}}$. Any additive
${\Bbbk}$-linear functor, results a $\mbox{\rm vect\,}_{\Bbbk}$-module
functor.

There exists a compatibility of the action of $\mbox{\rm vect\,}_{\Bbbk}$ and
the action of any tensor category.

######  Lemma 2.8.

If ${\mathcal{C}}$ is a tensor category and ${\mathcal{M}}$ is a left
${\mathcal{C}}$-module category with action given by
$\rhd:{\mathcal{C}}\times{\mathcal{M}}\to{\mathcal{M}}$, then there are
natural isomorphisms

$(V\blacktriangleright\mathbf{1})\rhd M\simeq V\blacktriangleright M,$

for any $V\in\mbox{\rm vect\,}_{\Bbbk}$, $X\in{\mathcal{C}}$,
$M\in{\mathcal{M}}$. ∎

##  3\. Bimodule categories and the relative center

Assume that ${\mathcal{C}},{\mathcal{D}},{\mathcal{E}}$ are tensor categories.
A $({\mathcal{C}},{\mathcal{D}})-$_bimodule category_ is a category
${\mathcal{M}}$ with left ${\mathcal{C}}$-module category structure
$\triangleright:{\mathcal{C}}\times{\mathcal{M}}\to{\mathcal{M}}$, and right
${\mathcal{D}}$-module category structure
$\triangleleft:{\mathcal{M}}\times{\mathcal{D}}\to{\mathcal{M}}$, equipped
with natural isomorphisms

$\\{\delta_{X,M,Y}:(X\triangleright M)\triangleleft Y\to
X\triangleright(M\triangleleft
Y),X\in{\mathcal{C}},Y\in{\mathcal{D}},M\in{\mathcal{M}}\\}$

satisfying certain axioms. For details the reader is referred to [12], [13].

Assume that ${\mathcal{M}}$ is a ${\mathcal{C}}$-bimodule category, with right
and left associativities given by

$m^{r}_{M,X,Y}:M\triangleleft(X{\otimes}Y)\to(M\triangleleft X)\triangleleft
Y,$ $m^{l}_{X,Y,M}:(X\otimes Y)\triangleright M\to
X\triangleright(Y\triangleright M).$

The  relative center [14] of ${\mathcal{M}}$ is the category
$Z_{\mathcal{C}}({\mathcal{M}})$ that consists of pairs $(M,\gamma)$, where
$M\in{\mathcal{M}}$ and $\gamma_{X}:M\triangleleft X\to X\triangleright M$ is
a family of natural isomorphism, called the half-braiding, such that

(3.1) $m^{l}_{X,Y,M}\gamma_{X{\otimes}Y}=(\mbox{\rm
id\,}_{X}\triangleright\gamma_{Y})\delta_{X,M,Y}(\gamma_{X}\triangleleft\mbox{\rm
id\,}_{Y})m^{r}_{M,X,Y},$

for any $X,Y\in{\mathcal{C}}$. If $(M,\gamma)$, $(M^{\prime},\gamma^{\prime})$
are two elements in $Z_{\mathcal{C}}({\mathcal{M}})$, a morphism between them
is a morphism $f:M\to M^{\prime}$ such that

(3.2) $(\mbox{\rm id\,}_{X}\triangleright
f)\gamma_{X}=\gamma^{\prime}_{X}(f\triangleleft\mbox{\rm id\,}_{X}),$

for any $X\in{\mathcal{C}}$. The next result is [10, Corollary B.5].

######  Proposition 3.1.

The forgetful functor
${\mathcal{U}}:Z_{\mathcal{C}}({\mathcal{M}})\to{\mathcal{M}},$
${\mathcal{U}}(M,\gamma)=M$, for any $(M,\gamma)\in
Z_{\mathcal{C}}({\mathcal{M}})$ is exact, and has left adjoint
$I:{\mathcal{M}}\to Z_{\mathcal{C}}({\mathcal{M}})$ given by

$I(M)=\int^{X\in{\mathcal{C}}}X\triangleright M\triangleleft{}^{*}X,$

for any $M\in{\mathcal{M}}$.∎

The relative center can be thought of as a 2-functor

${\mathcal{Z}}_{\mathcal{C}}:{}_{\mathcal{C}}{\mathcal{B}imod}\to Ab_{\Bbbk}$

Here ${}_{\mathcal{C}}{\mathcal{B}imod}$ is the 2-category of
${\mathcal{C}}$-bimodule categories, and $Ab_{\Bbbk}$ is the 2-category of
finite ${\Bbbk}$-linear abelian categories. If ${\mathcal{M}},{\mathcal{N}}$
are ${\mathcal{C}}$-bimodule categories, and
$(F,c,d):{\mathcal{M}}\to{\mathcal{N}}$ is a bimodule functor, then

${\mathcal{Z}}_{\mathcal{C}}(F):{\mathcal{Z}}_{\mathcal{C}}({\mathcal{M}})\to{\mathcal{Z}}_{\mathcal{C}}({\mathcal{N}})$

is the functor given as follows. If
$(M,\gamma)\in{\mathcal{Z}}_{\mathcal{C}}({\mathcal{M}}),$ then
${\mathcal{Z}}_{\mathcal{C}}(F)(M,\gamma)=(F(M),\widetilde{\gamma})$, where
$\widetilde{\gamma}_{X}:F(M)\triangleleft X\to X\triangleright F(M)$ is
defined as

(3.3) $\widetilde{\gamma}_{X}=c_{X,M}F(\gamma_{X})d^{-1}_{M,X},$

for any $X\in{\mathcal{C}}$. The next result seem to be well-known.

######  Proposition 3.2.

Let ${\mathcal{M}},{\mathcal{N}}$ be ${\mathcal{C}}$-bimodule categories, and
$(F,c,d):{\mathcal{M}}\to{\mathcal{N}}$ be a bimodule functor. The following
statements hold.

  * 1.

If $G:{\mathcal{N}}\to{\mathcal{M}}$ is a bimodule functor, right adjoint to
$F$, then ${\mathcal{Z}}_{\mathcal{C}}(G)$ is right adjoint to
${\mathcal{Z}}_{\mathcal{C}}(F)$.

  * 2.

If $F$ is exact (faithful) then ${\mathcal{Z}}_{\mathcal{C}}(F)$ is exact
(respect. faithful).∎

##  4\.  Adjoint algebras associated to module categories

Let ${\mathcal{C}}$ be a tensor category, and let ${\mathcal{M}}$ be an exact
indecomposable left ${\mathcal{C}}$-module category. In this Section, we shall
recall the definition of the adjoint algebra associated to ${\mathcal{M}}$, as
introduced by K. Shimizu [20].

The adjoint algebra of the module category ${\mathcal{M}}$ was defined as
follows. If

$\rho_{\mathcal{M}}:{\mathcal{C}}\to\mbox{\rm Rex\,}({\mathcal{M}}),$
$\rho_{\mathcal{M}}(X)(M)=X\triangleright
M,X\in{\mathcal{C}},M\in{\mathcal{M}},$

is the action functor, since it is a ${\mathcal{C}}$-bimodule functor, then
its right adjoint is also a ${\mathcal{C}}$-bimodule action. Whence we can
consider the functor

$Z_{\mathcal{C}}(\rho_{\mathcal{M}}^{\rm{ra}}):{\mathcal{C}}^{*}_{\mathcal{M}}=Z_{\mathcal{C}}(\mbox{\rm
Rex\,}({\mathcal{M}}))\to Z({\mathcal{C}}).$

Then
${\mathcal{A}}_{\mathcal{M}}:=Z_{\mathcal{C}}(\rho_{\mathcal{M}}^{\rm{ra}})(\mbox{\rm
Id\,}_{\mathcal{M}})\in Z({\mathcal{C}})$. The adjoint algebra of the tensor
category ${\mathcal{C}}$ is the algebra ${\mathcal{A}}_{\mathcal{C}}$ of the
regular module category ${\mathcal{C}}$.

In [19], [20] the author describes the right adjoint
$\rho_{\mathcal{M}}^{\rm{ra}}$ using the end of the internal hom; that is, it
was proven that

$\rho_{\mathcal{M}}^{\rm{ra}}:\mbox{\rm
Rex\,}({\mathcal{M}})\to{\mathcal{C}},$
$\rho_{\mathcal{M}}^{\rm{ra}}(F)=\int_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(M,F(M)),$

for any $F\in\mbox{\rm Rex\,}({\mathcal{M}})$. This description has several
advantages, one of them is that it is possible to describe the algebra
structure and the half braiding of ${\mathcal{A}}_{\mathcal{M}}$ in terms of
the dinatural transformations of the end.

Assume that
$\pi^{\mathcal{M}}_{M}:{\mathcal{A}}_{\mathcal{M}}\xrightarrow{..}\underline{\operatorname{Hom}}(M,M)$
are the corresponding dinatural transformations. The half braiding of
${\mathcal{A}}_{\mathcal{M}}$ is the unique isomorphism
$\sigma^{\mathcal{M}}_{X}:{\mathcal{A}}_{\mathcal{M}}{\otimes}X\to
X{\otimes}{\mathcal{A}}_{\mathcal{M}}$ such that the diagram

(4.1) $\textstyle{{\mathcal{A}}_{\mathcal{M}}\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\mathcal{M}}_{X\triangleright
M}\otimes\mbox{\rm
id\,}_{X}}$$\scriptstyle{\sigma^{{\mathcal{M}}}_{X}}$$\textstyle{\underline{\operatorname{Hom}}(X\triangleright
M,X\triangleright M)\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{b}_{X,M,X\triangleright
M}}$$\textstyle{\underline{\operatorname{Hom}}(M,X\triangleright
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{a}_{X,M,M}}$$\textstyle{X\otimes{\mathcal{A}}_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mbox{\rm
id\,}_{X}\otimes\pi^{\mathcal{M}}_{M}}$$\textstyle{X{\otimes}\underline{\operatorname{Hom}}(M,M)}$

is commutative. Recall from Section 2.2 the definition of the morphisms
$\mathfrak{b},\mathfrak{a}$. It was explained in [20, Subection 4.2] that the
algebra structure of ${\mathcal{A}}_{\mathcal{M}}$ is given as follows. The
product and the unit of ${\mathcal{A}}_{\mathcal{M}}$ are

$m_{\mathcal{M}}:{\mathcal{A}}_{\mathcal{M}}{\otimes}{\mathcal{A}}_{\mathcal{M}}\to{\mathcal{A}}_{\mathcal{M}},\quad
u_{\mathcal{M}}:\mathbf{1}\to{\mathcal{A}}_{\mathcal{M}},$

defined to be the unique morphisms such that they satisfy

(4.2) $\begin{split}\pi^{\mathcal{M}}_{M}\circ
m_{\mathcal{M}}=\mbox{comp}^{\mathcal{M}}_{M}\circ(\pi^{\mathcal{M}}_{M}{\otimes}\pi^{\mathcal{M}}_{M}),\\\
\pi^{\mathcal{M}}_{M}\circ
u_{\mathcal{M}}=\mbox{coev}^{\mathcal{M}}_{M},\end{split}$

for any $M\in{\mathcal{M}}$. For the definition of $\mbox{coev}^{\mathcal{M}}$
and $\mbox{comp}^{\mathcal{M}}$ see Section 2.2.

##  5\.  (Co)ends for module categories

In this Section we recall the definition of relative (co)ends. This is a
useful tool in the theory of representations of tensor categories,
generalizing the usual (co)ends, that we developed in [1].

Let ${\mathcal{C}}$ be a tensor category and ${\mathcal{M}}$ be a left
${\mathcal{C}}$-module category. Let $m$ be the associativity constraint of
${\mathcal{M}}$. Assume that ${\mathcal{A}}$ is a category together with a
functor $S:{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{A}}$,
equipped with natural isomorphisms

$\beta^{X}_{M,N}:S(M,X\triangleright N)\to S(X^{*}\triangleright M,N),$

for any $X\in{\mathcal{C}},M,N\in{\mathcal{M}}$. Isomorphisms $\beta$ are
called a prebalancing of the functor $S$. Sometimes we shall refer to $\beta$
as a ${\mathcal{C}}$-prebalancing, to emphasize the fact that $\beta$ is
related with the action of ${\mathcal{C}}$.

######  Definition 5.1.

The relative end of the pair $(S,\beta)$ is an object $E\in{\mathcal{A}}$
equipped with dinatural transformations $\pi_{M}:E\xrightarrow{..}S(M,M)$ such
that

(5.1) $S(\mbox{ev}_{X}\triangleright\mbox{\rm id\,}_{M},\mbox{\rm
id\,}_{M})\pi_{M}=S(m_{X^{*},X,M},\mbox{\rm
id\,}_{M})\beta^{X}_{X\triangleright M,M}\pi_{X\triangleright M},$

for any $X\in{\mathcal{C}},M\in{\mathcal{M}}$, and it is universal among those
objects with this property. That is, if $E^{\prime}\in{\mathcal{A}}$ is
another object, with dinatural transformations
$\pi^{\prime}_{M}:\widetilde{E}\xrightarrow{..}S(M,M)$, such that they verify
(5.1), then there exists a unique morphism $h:\widetilde{E}\to E$ such that
$\pi^{\prime}_{M}=\pi_{M}\circ h$. The relative coend is defined similarly. We
refer to [1] for its definition.

We will denote the relative end as $\oint_{M\in{\mathcal{M}}}(S,\beta)$, or
simply as $\oint_{M\in{\mathcal{M}}}S$, when the prebalancing $\beta$ is
understood from the context. In the next result we collect some facts about
the relative end that will be needed later. The reader is referred to [1,
Prop. 3.3], [1, Prop. 4.2].

######  Theorem 5.2.

Assume ${\mathcal{M}},{\mathcal{N}}$ are left ${\mathcal{C}}$-module
categories, and
$S,\widetilde{S}:{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{A}}$
are functors equipped with ${\mathcal{C}}$-prebalancings

$\beta^{X}_{M,N}:S(M,X\triangleright N)\to S(X^{*}\triangleright M,N),$
$\widetilde{\beta}^{X}_{M,N}:\widetilde{S}(M,X\triangleright
N)\to\widetilde{S}(X^{*}\triangleright M,N),$

$X\in{\mathcal{C}},M,N\in{\mathcal{M}}$. The following assertions holds

  * (i)

Assume that the module ends
$\oint_{M\in{\mathcal{M}}}(S,\beta),\oint_{M\in{\mathcal{M}}}(\widetilde{S},\widetilde{\beta})$
exist and have dinatural transformations $\pi,\widetilde{\pi}$, respectively.
If $\gamma:S\to\widetilde{S}$ is a natural transformation such that

(5.2) $\widetilde{\beta}^{X}_{M,N}\gamma_{(M,X\triangleright
N)}=\gamma_{(X^{*}\triangleright M,N)}\beta^{X}_{M,N},$

then there exists a unique map
$\widehat{\gamma}:\oint_{M\in{\mathcal{M}}}(S,\beta)\to\oint_{M\in{\mathcal{M}}}(\widetilde{S},\widetilde{\beta})$
such that $\widetilde{\pi}_{M}\widehat{\gamma}=\gamma_{(M,M)}\pi_{M}$ for any
$M\in{\mathcal{M}}$. If $\gamma$ is a natural isomorphism, then
$\widehat{\gamma}$ is an isomorphism.

  * (ii)

If the end $\oint_{M\in{\mathcal{M}}}(S,\beta)$ exists, then for any object
$U\in{\mathcal{A}}$, the end
$\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{A}}(U,S(-,-))$ exists,
and there is an isomorphism

$\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{A}}(U,S(-,-))\simeq\operatorname{Hom}_{\mathcal{A}}(U,\oint_{M\in{\mathcal{M}}}(S,\beta)).$

Moreover, if
$\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{A}}(U,S(-,-))$ exists
for any $U\in{\mathcal{A}}$, then the end $\oint_{M\in{\mathcal{M}}}(S,\beta)$
exists.

  * (iii)

For any pair of ${\mathcal{C}}$-module functors
$(F,c),(G,d):{\mathcal{M}}\to{\mathcal{N}},$ the functor

$\operatorname{Hom}_{{\mathcal{N}}}(F(-),G(-)):{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to\mbox{\rm
vect\,}_{\Bbbk}$

has a canonical ${\mathcal{C}}$-pre-balancing given by

(5.3)
$\beta^{X}_{M,N}:\operatorname{Hom}_{{\mathcal{N}}}(F(M),G(X\triangleright
N))\to\operatorname{Hom}_{{\mathcal{N}}}(F(X^{*}\triangleright M),G(N))$
$\beta^{X}_{M,N}(\alpha)=(ev_{X}\triangleright\mbox{\rm
id\,}_{G(N)})m^{-1}_{X^{*},X,G(N)}(\mbox{\rm id\,}_{X^{*}}\triangleright
d_{X,N}\alpha)c_{X^{*},M},$

for any $X\in{\mathcal{C}},M,N\in{\mathcal{M}}$. There is an isomorphism

$\mbox{\rm
Nat\,}_{\\!m}(F,G)\simeq\oint_{M\in{\mathcal{M}}}(\operatorname{Hom}_{{\mathcal{N}}}(F(-),G(-)),\beta).$

∎

######  Remark 5.3.

If the functor $S$ has ${\mathcal{C}}$-prebalancing given by $\beta$, then,
according to [1], for any $U\in{\mathcal{A}}$, the prebalancing for the
functor $\operatorname{Hom}_{\mathcal{A}}(U,S(-,-))$ is defined as

$\beta^{U}_{X,M,N}:\operatorname{Hom}_{\mathcal{A}}(U,S(M,X\triangleright
N))\to\operatorname{Hom}_{\mathcal{A}}(U,S(X^{*}\triangleright M,N)),$
$\beta^{U}_{X,M,N}(f)=\beta^{X}_{M,N}\circ f.$

######  Remark 5.4.

According to [1, Prop. 4.2], the dinatural transformations associated with the
end given in Theorem 5.2 (iii) are

$\pi_{M}:\mbox{\rm
Nat\,}_{\\!m}(F,G)\to\operatorname{Hom}_{{\mathcal{N}}}(F(M),G(M)),\,\pi_{M}(\alpha)=\alpha_{M}.$

Let ${\mathcal{D}}\subseteq{\mathcal{C}}$ be a tensor subcategory of
${\mathcal{C}}$. If ${\mathcal{M}}$ is a left ${\mathcal{C}}$-module category,
we can consider the restricted ${\mathcal{D}}$-module category $\mbox{\rm
Res\,}^{\mathcal{C}}_{\mathcal{D}}{\mathcal{M}}$. We can consider both
relative ends

$\pi_{M}:\oint_{M\in{\mathcal{M}}}(S,\beta)\to S(M,M),$
$\lambda_{M}:\oint_{M\in\mbox{\rm
Res\,}^{\mathcal{C}}_{\mathcal{D}}{\mathcal{M}}}(S,\beta)\to S(M,M).$

The next result is a straightforward consequence of the definition of the
relative ends.

######  Proposition 5.5.

[1, Prop. 3.12] Let
$S:{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{A}}$ be a functor
equipped with a prebalancing $\beta^{X}_{M,N}:S(M,X\triangleright N)\to
S(X^{*}\triangleright M,N)$. There exists a monomorphism in ${\mathcal{A}}$

$\iota:\oint_{M\in{\mathcal{M}}}(S,\beta)\hookrightarrow\oint_{M\in\mbox{\rm
Res\,}^{\mathcal{C}}_{\mathcal{D}}{\mathcal{M}}}(S,\beta),$

such that $\lambda_{M}\circ\iota=\pi_{M}$. ∎

The next result says that, the relative end coincides with the usual end in
the case ${\mathcal{C}}=\mbox{\rm vect\,}_{\Bbbk}$. Recall from Section 2.3
that, any category has a canonical module structure over $\mbox{\rm
vect\,}_{\Bbbk}$ that we denote by $\blacktriangleright$.

######  Proposition 5.6.

[1, Prop. 3.14] Let ${\mathcal{M}},{\mathcal{B}}$ be categories together with
a functor $S:{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{B}}$. The
assertions following holds.

  * (i)

The functor $S$ posses a canonical $\mbox{\rm vect\,}_{\Bbbk}$-prebalancing

$\gamma^{V}_{M,N}:S(M,X\blacktriangleright N)\to S(X^{*}\blacktriangleright
M,N),$

  * (ii)

Any dinatural transformation $\pi_{M}:E\to S(M,M)$ satisfies equation (5.1)
for the canonical action of $\mbox{\rm vect\,}_{\Bbbk}$ on ${\mathcal{M}}$.
That is

(5.4) $S(\mbox{ev}_{V}\blacktriangleright\mbox{\rm id\,}_{M},\mbox{\rm
id\,}_{M})\pi_{M}=S(m_{V^{*},V,M},\mbox{\rm
id\,}_{M})\gamma^{V}_{V\blacktriangleright M,M}\pi_{V\blacktriangleright M},$

for any $V\in\mbox{\rm vect\,}_{\Bbbk}$, $M\in{\mathcal{M}}$.

∎

##  6\. Relative adjoint algebras

For the rest of this Section ${\mathcal{C}}$ will be a tensor category and
${\mathcal{M}}$ be an indecomposable exact left ${\mathcal{C}}$-module
category, with structure functor given by
$\rho:{\mathcal{C}}\to\operatorname{End}({\mathcal{M}})$.

For each tensor subcategory
${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$ we shall construct a
connected étale algebra ${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ in the
center $Z({\mathcal{C}})$. When ${\mathcal{D}}=\mbox{\rm vect\,}_{\Bbbk}$ is
the tensor subcategory of ${\mathcal{C}}^{*}_{\mathcal{M}}$ generated by the
unit object, the algebra ${\mathcal{A}}_{\mbox{\rm
vect\,}_{\Bbbk},{\mathcal{M}}}$ coincides with Shimizu’s adjoint algebra
${\mathcal{A}}_{\mathcal{M}}$.

###  6.1. Computing adjoint of the action functor

Take ${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$ a tensor
subcategory.

We shall denote by $\operatorname{End}({\mathcal{M}})$ the category of right
exact functors $F:{\mathcal{M}}\to{\mathcal{M}}$. Then, the category
$\operatorname{End}({\mathcal{M}})$ has an obvious structure of
${\mathcal{D}}$-bimodule category. Hence, we can consider the relative center
$Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$. Let us explicitly
describe objects in this category. An object in
$Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ is a pair $(F,\gamma)$,
where $F:{\mathcal{M}}\to{\mathcal{M}}$ is a right exact functor,
$\gamma_{(D,d)}:F\circ D\to D\circ F$ is a family of natural isomorphisms, for
any $(D,d)\in{\mathcal{D}}$, such that

(6.1) $(\mbox{\rm
id\,}_{(D,d)}\circ\gamma_{(D^{\prime},d^{\prime})})(\gamma_{(D,d)}\circ\mbox{\rm
id\,}_{(D^{\prime},d^{\prime})})=\gamma_{(D,d)\circ(D^{\prime},d^{\prime})},$

for any $(D,d),(D^{\prime},d^{\prime})\in{\mathcal{D}}$. Since ${\mathcal{M}}$
is a left module category over ${\mathcal{C}}^{*}_{\mathcal{M}}$, in
particular, it is a ${\mathcal{D}}$-module category. Turns out that
$Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))={\mathcal{D}}^{*}_{\mathcal{M}}.$

######  Lemma 6.1.

The following assertions hold.

  * 1.

The category $Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ has
structure of ${\mathcal{C}}$-bimodule category.

  * 2.

For any $X\in{\mathcal{C}}$, the functor $\rho(X)$ belongs to
$Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$. Thus we can consider the
functor $\rho:{\mathcal{C}}\to
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$. Turns out that this
functor is monoidal, exact and faithful.

###### Proof.

1\. Take $X\in{\mathcal{C}}$, $(F,\gamma)\in
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$, and define

$X\triangleright(F,\gamma)=(X\triangleright
F,{}^{X}\gamma),\,(F,\gamma)\triangleleft X=(F\triangleleft X,\gamma^{X}),$

where

$X\triangleright F,\;F\triangleleft X:{\mathcal{M}}\to{\mathcal{M}},$
$(X\triangleright F)(M)=X\triangleright F(M),\,(F\triangleleft
X)(M)=F(X\triangleright M),$

for any $M\in{\mathcal{M}}$. For any $(D,d)\in{\mathcal{D}}$ define the
natural isomorphism

${}^{X}\gamma_{(D,d)}:(X\triangleright F)\circ D\to D\circ(X\triangleright
F),$ $\big{(}{}^{X}\gamma_{(D,d)}\big{)}_{M}=d^{-1}_{X,F(M)}(\mbox{\rm
id\,}_{X}\triangleright(\gamma_{D})_{M}),$

for any $M\in{\mathcal{M}}$. Also define, for any $(D,d)\in{\mathcal{D}}$

$\gamma^{X}_{(D,d)}:(X\triangleright F)\circ D\to D\circ(X\triangleright F),$
$\big{(}\gamma^{X}_{(D,d)}\big{)}_{M}=(\gamma_{(D,d)})_{X\triangleright
M}F(d^{-1}_{X,M}),$

for any $M\in{\mathcal{M}}$.

2\. It follows from item (1), since $\rho(X)=X\triangleright\mbox{\rm Id\,}$.
Exactness of the functor $\rho$ follows from the exactness of the bifunctor
$\triangleright$ in the first variable. There is a commutative diagram of
functors

(6.2)
$\textstyle{{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{X\mapsto
X\triangleright-}$$\textstyle{Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{U}}}$$\textstyle{\operatorname{End}({\mathcal{M}}),}$

where
${\mathcal{U}}:Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))\to\operatorname{End}({\mathcal{M}})$
is the forgetful functor. Since the action functor
${\mathcal{C}}\to\operatorname{End}({\mathcal{M}})$ is faithful, it follows
that $\rho$ is faithful.

∎

######  Remark 6.2.

For future reference, for any $X\in{\mathcal{C}}$, we denote the functor
$L^{X}=\rho(X)\in\operatorname{End}({\mathcal{M}}).$ For any
$M\in{\mathcal{M}}$, $L^{X}(M)=X\triangleright M.$ It follows from part (2) of
the above Lemma that $(L^{X},l^{X})\in
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$, where

$l^{X}_{D}:L^{X}\circ D\to D\circ L^{X},$
$\big{(}l^{X}_{D}\big{)}_{M}=d^{-1}_{X,M},$

for any $(D,d)\in{\mathcal{D}}$.

The adjoint algebra associated to ${\mathcal{M}}$, introduced in [20], was
defined using the right adjoint of the action functor
$\rho:{\mathcal{C}}\to\operatorname{End}({\mathcal{M}})$. Turns out that the
right adjoint is given by

$\operatorname{End}({\mathcal{M}})\to{\mathcal{C}},\,F\mapsto\int_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(M,F(M)).$

In loc. cit. the author defines

${\mathcal{A}}_{\mathcal{M}}=\int_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(M,M)$

We intend to generalize this construction using the functor
$\rho:{\mathcal{C}}\to Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$
instead, and replace the usual end with the relative end. Let us show first
that the bifunctor $\underline{\operatorname{Hom}}(-,F(-))$ posses a
prebalancing.

######  Lemma 6.3.

For any $(F,\gamma)\in Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ let
us denote by $S^{F}$ the functor given by

$S^{F}:{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{C}},(M,N)\mapsto\underline{\operatorname{Hom}}(M,F(N)).$

It has a canonical ${\mathcal{D}}$-prebalancing $\beta^{D}_{M,N}$ as follows.
For any $D\in{\mathcal{D}}$, $\beta$ is the composition

$S^{F}(M,D(N))=\underline{\operatorname{Hom}}(M,F(D(N)))\xrightarrow{\underline{\operatorname{Hom}}(M,(\gamma_{D})_{N})}\underline{\operatorname{Hom}}(M,D(F(N)))\xrightarrow{}$
$\xrightarrow{\xi^{D}_{M,F(N)}}\underline{\operatorname{Hom}}(D^{\rm{la}}(M),F(N))=S^{F}(D^{\rm{la}}(M),N)$

Here, isomorphism $\xi^{D}$ is the one described in Lemma 2.6. ∎

The next result is a generalization of a result of K. Shimizu. When applied to
${\mathcal{D}}=\mbox{\rm vect\,}_{\Bbbk}$, it recovers exactly [20, Thm. 3.4].
However, the proof of this result is more involved as the one presented in
loc.cit., mainly due to the presence of prebalancings.

######  Theorem 6.4.

Let ${\mathcal{M}}$ be an exact indecomposable ${\mathcal{C}}$-module
category. The following assertions hold.

  * 1.

The functor
$\bar{\rho}:Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))\to{\mathcal{C}}$

(6.3)
$\bar{\rho}(F)=\oint_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(M,F(M))=\oint_{M\in{\mathcal{M}}}(S^{F},\beta),$

is well-defined, and it is a right adjoint of the functor
$\rho:{\mathcal{C}}\to Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$.

  * 2.

The functor $\bar{\rho}$ is exact and faithful.

  * 3.

The functor $\rho:{\mathcal{C}}\to
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ is dominant.

###### Proof.

1\. Let us prove first that the right adjoint to $\rho$ is given by formula
(6.3) on objects. Let be $X\in{\mathcal{C}}$, $(F,\gamma)\in
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$. We have the functor

$\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright-,F(-)):{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to
vect_{\Bbbk},$

with ${\mathcal{D}}$-prebalancing given by

(6.4)
$\displaystyle\begin{split}&\beta^{X}_{D,M,N}:\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,F(D(N)))\to\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
D^{\rm{la}}(M),F(N)),\\\
&\beta^{X}_{D,M,N}(f)=(ev_{D})_{F(N)}D^{\rm{la}}((\gamma_{D})_{N}f)\tilde{d}^{-1}_{X,M},\end{split}$

for any $(D,d)\in{\mathcal{D}}$. Here $ev_{D}:D^{\rm{la}}\circ D\to\mbox{\rm
Id\,}$ is the evaluation of the adjunction $(D^{\rm{la}},D)$, and
$\tilde{d}_{X,M}:D^{\rm{la}}(X\triangleright M)\to X\triangleright
D^{\rm{la}}(M)$ is the ${\mathcal{C}}$-module structure of the functor
$D^{\rm{la}}$. The formula for this prebalancing is taken from (5.3). We have
natural isomorphisms

(6.5) $\displaystyle\operatorname{Hom}_{\mathcal{C}}(X,\rho^{\rm{ra}}(F))$
$\displaystyle\simeq\mbox{\rm
Nat\,}_{m}(\rho(X),F)\simeq\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{M}}(X\triangleright
M,F(M))$

The first isomorphism is just the definition of the right adjoint, and the
second isomorphism follows from Theorem 5.2 (iii).

For any $X\in{\mathcal{C}}$, we also have the functor

$\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(-,F(-))):{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to
vect_{\Bbbk}.$

This functor has a ${\mathcal{D}}$-prebalancing given

$b^{X}_{D,M,N}:\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,F(D(N))))\to\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(D^{\rm{la}}(M),F(N))),$
$b^{X}_{D,M,N}(f)=\beta^{D}_{M,N}f.$

Here $\beta^{D}_{M,N}$ is the prebalancing of the functor $S^{F}$ given in
Lemma 6.3. Recall the natural isomorphisms presented in (2.12)

$\phi^{X}_{M,N}:\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N))\to\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N).$

######  Claim 6.1.

For any $X\in{\mathcal{C}}$ we have

$b^{X}_{D,M,N}\phi^{X}_{M,F(D(N))}=\phi^{X}_{D^{\rm{la}}(M),F(N)}\beta^{X}_{D,M,N}.\qed$

The proof of this claim is given in the Apendix in Proposition 8.1. Using this
claim, it follows from Theorem 5.2 (i) that there exist an isomorphism

$\displaystyle\widehat{\phi}^{X}:\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{C}}(X,\underline{\operatorname{Hom}}(M,F(M)))\to\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{M}}(X\triangleright
M,F(M))$

such that the diagram

(6.6)
$\textstyle{\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{C}}(X,\underline{\operatorname{Hom}}(M,F(M)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{X,F}_{M}}$$\scriptstyle{\widehat{\phi}^{X}}$$\textstyle{\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{M}}(X\triangleright
M,F(M))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\pi}^{X,F}_{M}}$$\textstyle{\operatorname{Hom}_{\mathcal{C}}(X,\underline{\operatorname{Hom}}(M,F(M)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{X}_{M,N}}$$\textstyle{\operatorname{Hom}_{\mathcal{M}}(X\triangleright
M,F(M))}$

is commutative. Here $\pi^{X,F}_{M}$ and $\widetilde{\pi}^{X,F}_{M}$ are the
corresponding dinatural transformations. Hence, using isomorphisms (6.5)

$\displaystyle\operatorname{Hom}_{\mathcal{C}}(X,\rho^{\rm{ra}}(F))$
$\displaystyle\simeq\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{C}}(X,\underline{\operatorname{Hom}}(M,F(M)))$
$\displaystyle\simeq\operatorname{Hom}_{\mathcal{C}}(X,\bar{\rho}(F)).$

The second isomorphism follows from Theorem 5.2 (ii). This proves that the
corresponding relative end exists and it is a right adjoint of the functor
$\rho$.

Let us see how to define $\bar{\rho}$ on morphisms. Note that, under the
identification

$\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{C}}(X,\underline{\operatorname{Hom}}(M,F(M)))\simeq\operatorname{Hom}_{\mathcal{C}}(X,\bar{\rho}(F)),$

if
$\pi^{F}_{M}:\oint_{M\in{\mathcal{M}}}\underline{\operatorname{Hom}}(M,F(M))\to\underline{\operatorname{Hom}}(M,F(M))$
is the associated dinatural transformation, then

(6.7) $\pi^{X,F}_{M}(g)=\pi^{F}_{M}\circ g,$

for any $g\in\operatorname{Hom}_{\mathcal{C}}(X,\bar{\rho}(F)).$ This follows
from the proof of Theorem 5.2 (ii). Let $f:(F,\gamma)\to(G,\delta)$ be a
morphism in $Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$. This map
induces a natural morphism $\widetilde{f}:S^{F}\to S^{G}$,
$\widetilde{f}_{M,N}=\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{M},f_{N})$, for any $M,N\in{\mathcal{M}}$. Using the naturality of
$\xi$ (see Lemma 2.6), one can prove that

$\beta^{D}_{M,N}\widetilde{f}_{M,D(N)}=\widetilde{f}_{D^{\rm{la}}(M),N}\beta^{D}_{M,N},$

for any $D\in{\mathcal{D}}$, $M,N\in{\mathcal{M}}$. This implies, using
Theorem 5.2 (i) that, there exists a morphism
$\bar{\rho}(f):\oint_{M\in{\mathcal{M}}}(S^{F},\beta)\to\oint_{M\in{\mathcal{M}}}(S^{G},\beta)$
such that

(6.8) $\displaystyle\widetilde{f}_{M,M}\pi^{F}_{M}=\pi^{G}_{M}\bar{\rho}(f),$

for any $M\in{\mathcal{M}}.$ This finishes the proof that the functor
$\bar{\rho}$ is well-defined on morphisms.

2\. Next, we shall prove that $\bar{\rho}$ is exact. For this, we shall prove
that $\bar{\rho}$ preserves kernels and cokernels. Let $(F,\gamma),(G,\delta)$
be objects in $Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$, and
$f:(F,\gamma)\to(G,\delta)$ be a morphism. Let $k:(K,\lambda)\to(F,\gamma)$ be
a kernel of $f$. So, we have an exact sequence

$0\to(K,\lambda)\xrightarrow{k}(F,\gamma)\xrightarrow{f}(G,\delta).$

We shall prove that $\ker(\bar{\rho}(f))=\bar{\rho}(k).$ Let
$h:Q\to\bar{\rho}(F)$ be a morphism in ${\mathcal{C}}$ such that

$\bar{\rho}(f)\circ h=0.$

For any $M\in{\mathcal{M}}$, the map $\pi^{F}_{M}\circ
h\in\operatorname{Hom}_{\mathcal{C}}(Q,\underline{\operatorname{Hom}}(M,F(M)))$.
Using the natural isomorphism (2.12), set

$\alpha_{M}=\phi^{Q}_{M,F(M)}(\pi^{F}_{M}h):Q\triangleright M\to F(M).$

For any $X\in{\mathcal{C}}$, define $L^{X}:{\mathcal{M}}\to{\mathcal{M}}$,
$L^{X}(M)=X\triangleright M$. Using Lemma 6.1 (2), one can see that $L^{X}\in
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$. See also Remark 6.2.

######  Claim 6.2.

$\alpha:L^{Q}\to F$ is a morphism in
$Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ such that
$f\circ\alpha=0$.

###### Proof of claim.

It follows from commutativity of diagram (6.6) and equation (6.7) that

$\alpha_{M}=\phi^{Q}_{M,F(M)}(\pi^{Q,F}_{M}(h))=\widetilde{\pi}^{Q,F}_{M}(\widehat{\phi}^{Q}(h)).$

Using identification

$\mbox{\rm
Nat\,}_{\\!m}(Q\triangleright-,F)\simeq\oint_{M\in{\mathcal{M}}}\operatorname{Hom}_{\mathcal{M}}(Q\triangleright
M,F(M)),$

and Remark 5.4, it follows that $\alpha$ indeed is a natural module
transformation. For any $M\in{\mathcal{M}}$

$\displaystyle f_{M}\alpha_{M}$
$\displaystyle=f_{M}\phi^{Q}_{M,F(M)}(\pi^{F}_{M}h)=\phi^{Q}_{M,G(M)}(\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{M},f_{M})\pi^{F}_{M}h)$
$\displaystyle=\phi^{Q}_{M,G(M)}(\pi^{G}_{M}\bar{\rho}(f)h)=0.$

The second equality is commutativity of diagram (2.17), the third equality
follows from (6.8). This finishes the proof of the Claim. ∎

Since $k:(K,\lambda)\to(F,\gamma)$ is the kernel of $f$, there exists a
morphism $\mu:L^{Q}\to K$ in
$Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ such that

(6.9) $k\circ\mu=\alpha.$

The morphism

$\psi^{Q}_{M,K(M)}(\mu_{M}):Q\to\underline{\operatorname{Hom}}(M,K(M)),$

is a dinatural transformation. This fact can be checked using diagrams (2.13),
(2.14).

######  Claim 6.3.

The dinatural transformation $\psi^{Q}_{M,K(M)}(\mu_{M})$ satisfies (5.1) with
respect to the prebalancing of the functor $S^{K}$. That is, the following
equation is fulfilled

$\displaystyle\underline{\operatorname{Hom}}((\mbox{ev}_{D})_{M},$
$\displaystyle\mbox{\rm id\,}_{K(M)})\psi^{Q}_{M,K(M)}(\mu_{M})=$
$\displaystyle=\xi^{D}_{D(M),K(M)}\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{D(M)},(\lambda_{D})_{M})\psi^{Q}_{D(M),K(D(M))}(\mu_{D(M)}),$

for any $D\in{\mathcal{D}}$, $M\in{\mathcal{M}}$. Here
$\mbox{ev}_{D}:D^{\rm{la}}\circ D\to\mbox{\rm Id\,}$ is the evaluation of the
adjunction $(D^{\rm{la}},D)$.∎

The proof of this claim is given in the Apendix, in Proposition 8.2. It
follows from Claim 6.3, and the universal property of the relative end, that,
there exists a morphism $\widehat{h}:Q\to\bar{\rho}(K)$ such that

(6.10) $\pi^{K}_{M}\circ\widehat{h}=\psi^{Q}_{M,K(M)}(\mu_{M}),$

for any $M\in{\mathcal{M}}$. We shall prove that

(6.11) $\bar{\rho}(k)\widehat{h}=h.$

This will end the proof that $\ker(\bar{\rho}(f))=\bar{\rho}(k).$ Indeed,
applying $\pi^{F}_{M}$ to the left hand side of (6.11) we get

$\displaystyle\pi^{F}_{M}\bar{\rho}(k)\widehat{h}$
$\displaystyle=\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{M},k_{M})\pi^{K}_{M}\widehat{h}$
$\displaystyle=\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{M},k_{M})\psi^{Q}_{M,K(M)}(\mu_{M})$
$\displaystyle=\psi^{Q}_{M,F(M)}(k_{M}\mu_{M})=\psi^{Q}_{M,F(M)}(\alpha_{M})$
$\displaystyle=\pi^{F}_{M}h$

The first equality follows from (6.8), the second equality follows from
(6.10), the third one follows from (2.13), the fourth equality is (6.9) and
the last one is the definition of $\alpha$. Equation (6.11) then follows from
the universal property of the dinatural transformation $\pi^{F}.$

This finishes the proof that $\ker(\bar{\rho}(f))=\bar{\rho}(k).$ The proof
that $\bar{\rho}$ preserves cokernels is done similarly. Hence $\bar{\rho}$ is
exact.

Let us prove that $\bar{\rho}$ is faithfull. Since it is exact, it is enough
to prove that reflects zero objects. Take a non-zero object $(F,\gamma)\in
Z_{\mathcal{D}}(\operatorname{End}(({\mathcal{M}}))$. Again, lets keep in mind
the commutative diagram

(6.12)
$\textstyle{{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{X\mapsto
X\triangleright-}$$\textstyle{Z_{\mathcal{D}}(\operatorname{End}(({\mathcal{M}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{U}}}$$\textstyle{\operatorname{End}(({\mathcal{M}}),}$

where ${\mathcal{U}}:Z_{\mathcal{D}}(\mbox{\rm
Rex\,}({\mathcal{M}}))\to\operatorname{End}(({\mathcal{M}})$ is the forgetful
functor. Since ${\mathcal{M}}$ is an exact indecomposable as a
${\mathcal{C}}$-module category, it follows from [5, Prop. 2.6 (ii)] that the
functor ${\mathcal{C}}\to\mbox{\rm Rex\,}({\mathcal{M}})$, $X\mapsto
X\triangleright-$ is dominant. Whence, there exists an object
$Y\in{\mathcal{C}}$ such that

$\mbox{\rm Nat\,}(Y\triangleright-,{\mathcal{U}}(F,\gamma))\neq 0.$

Thus

$0\neq\mbox{\rm
Nat\,}(Y\triangleright-,{\mathcal{U}}(F,\gamma))\simeq\operatorname{Hom}_{Z_{\mathcal{D}}(\operatorname{End}(({\mathcal{M}}))}(I(Y\triangleright-),(F,\gamma)).$

Here $I:\mbox{\rm Rex\,}({\mathcal{M}})\to Z_{\mathcal{D}}(\mbox{\rm
Rex\,}({\mathcal{M}}))$ is the left adjoint of the forgetful functor, and it
is given by

$I(F)=\int^{X\in{\mathcal{C}}}X\triangleright F\triangleleft{}^{*}X.$

The functor $I$ was presented in Proposition 3.1. Let us denote

$Z=\int^{X\in{\mathcal{C}}}X{\otimes}Y{\otimes}{}^{*}X.$

Then, $I(Y\triangleright-)\simeq\rho(Z)$. Thus

$0\neq\operatorname{Hom}_{Z_{\mathcal{D}}(\mbox{\rm
Rex\,}({\mathcal{M}}))}(\rho(Z),(F,\gamma))\simeq\operatorname{Hom}_{\mathcal{C}}(Z,\bar{\rho}(F,\gamma)),$

which implies that $\bar{\rho}(F,\gamma)\neq 0$.

3\. It follows from Proposition 1.1 since $\rho$ is faithful. ∎

###  6.2. Étale algebras corresponding to tensor subcategories of
${\mathcal{C}}^{*}_{\mathcal{M}}$

In this Section we shall construct étale algebras out of tensor subcategories.
This construction was already presented in [3, Thm. 4.10]. Let
${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$ be a tensor
subcategory. Observe that, since
$Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ is a
${\mathcal{C}}$-bimodule category (see Lemma 6.1), then we can consider the
relative center
$Z_{\mathcal{C}}(Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}})))=Z_{\mathcal{D}}({\mathcal{C}}^{*}_{\mathcal{M}}).$

######  Proposition 6.5.

The following assertions hold.

  * 1.

The functor $\rho:{\mathcal{C}}\to
Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ is a
${\mathcal{C}}$-bimodule functor.

  * 2.

Using (1) we can consider the functor $Z(\rho):Z({\mathcal{C}})\to
Z_{\mathcal{D}}({\mathcal{C}}^{*}_{\mathcal{M}}).$ This tensor functor is
dominant.

###### Proof.

The proof of (1) follows by a straightforward computation. It follows from
Proposition 3.2 that $Z(\bar{\rho})$ is a right adjoint to $Z(\rho)$, and this
functor is exact and faithful. It follows from Proposition 1.1 that $Z(\rho)$
is dominant. ∎

Now we can give the main definition of the paper.

######  Definition 6.6.

Let ${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$ be a tensor
subcategory. The object

${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}=Z(\bar{\rho})(\mbox{\rm
Id\,})=\oint_{M\in{\mathcal{M}}}(\underline{\operatorname{Hom}}(-,-),\beta)\in
Z({\mathcal{C}})$

is called the adjoint algebra of ${\mathcal{M}}$ relative to ${\mathcal{D}}$.
The morphisms
$\pi_{M}:{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}\to\underline{\operatorname{Hom}}(M,M)$
are the dinatural transformations associated to this relative end.

######  Remark 6.7.

Since the relative end coincides with the usual end when
${\mathcal{D}}=\mbox{\rm vect\,}_{\Bbbk}$, then ${\mathcal{A}}_{\mbox{\rm
vect\,}_{\Bbbk},{\mathcal{M}}}={\mathcal{A}}_{{\mathcal{M}}}$ is the adjoint
algebra associated to ${\mathcal{M}}$, as introduced by Shimizu [20].

The proof of the next result follows mutatis mutandis from the proof presented
in [20].

######  Proposition 6.8.

The object ${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ has a half-braiding,
multiplication and unit

$\sigma_{X}:{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}{\otimes}X\to
X{\otimes}{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}},$
$m:{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}{\otimes}{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}\to{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}},\quad
u:\mathbf{1}\to{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}},$

determined by

(6.13) $\textstyle{{\mathcal{A}}_{\mathcal{M}}\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\mathcal{M}}_{X\triangleright
M}\otimes\mbox{\rm
id\,}_{X}}$$\scriptstyle{\sigma^{{\mathcal{M}}}_{X}}$$\textstyle{\underline{\operatorname{Hom}}(X\triangleright
M,X\triangleright M)\otimes
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{b}_{X,M,X\triangleright
M}}$$\textstyle{\underline{\operatorname{Hom}}(M,X\triangleright
M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{a}_{X,M,M}}$$\textstyle{X\otimes{\mathcal{A}}_{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mbox{\rm
id\,}_{X}\otimes\pi^{\mathcal{M}}_{M}}$$\textstyle{X{\otimes}\underline{\operatorname{Hom}}(M,M)}$
(6.14) $\pi_{M}\circ
m=\mbox{comp}^{\mathcal{M}}_{M,M}(\pi_{M}{\otimes}\pi_{M}),\,\,\pi_{M}\circ
u=\mbox{coev}^{\mathcal{M}}_{M},$

for any $X\in{\mathcal{C}}$, $M\in{\mathcal{M}}$. Moreover,
${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ is a commutative algebra, that
is $m\circ\sigma_{{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}}=m.$ ∎

######  Remark 6.9.

Equations (6.13), (6.14) are the same as equations (4.1) and (4.2), describing
the product and the half-braiding in terms of the dinatural transformations.

######  Proposition 6.10.

Le ${\mathcal{M}}$ be an exact indecomposable module category over
${\mathcal{C}}$, and let
${\mathcal{D}}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$ be a tensor
subcategory. The following statements hold:

  * 1.

The object ${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ is a connected étale
algebra in the center ${\mathcal{Z}}({\mathcal{C}})$ such that there is a
monoidal equivalence

$Z({\mathcal{C}})_{A_{{\mathcal{D}},{\mathcal{M}}}}\simeq
Z_{\mathcal{D}}({\mathcal{C}}^{*}_{\mathcal{M}}).$

  * 2.

If
${\mathcal{D}}_{1}\subseteq{\mathcal{D}}_{2}\subseteq{\mathcal{C}}^{*}_{\mathcal{M}}$
are tensor subcategories, then there is an algebra inclusion

${\mathcal{A}}_{{\mathcal{D}}_{2},{\mathcal{M}}}\subseteq{\mathcal{A}}_{{\mathcal{D}}_{1},{\mathcal{M}}}.$

  * 3.

$\mbox{\rm
FPdim\,}_{Z({\mathcal{C}})}({\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}})=\frac{\mbox{\rm
FPdim\,}({\mathcal{C}})}{\mbox{\rm FPdim\,}({\mathcal{D}})}.$

###### Proof.

1\. Consider the adjoint functors $(Z(\rho),Z(\bar{\rho}))$. It follows from
Proposition 6.5 that they satisfy conditions of Corollary 2.5. Whence, the
algebra $Z(\bar{\rho})(\mbox{\rm
Id\,})={\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ is étale. It follows from
Proposition 1.2 that there is a tensor equivalence
$Z({\mathcal{C}})_{{\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}}\simeq
Z_{\mathcal{D}}({\mathcal{C}}^{*}_{\mathcal{M}}).$

2\. It follows from Proposition 5.5.

3\. Since $\mbox{\rm
FPdim\,}(Z_{\mathcal{D}}({\mathcal{C}}^{*}_{\mathcal{M}}))=\mbox{\rm
FPdim\,}({\mathcal{D}})\mbox{\rm FPdim\,}({\mathcal{C}}),$ the formula for the
dimension of ${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ follows from [6,
Lemma 6.2.4].

∎

##  7\. Some concrete computations

For the rest of this Section ${\mathcal{C}}$ will denote a tensor category. We
shall give some explicit computations of the relative adjoint
${\mathcal{A}}_{{\mathcal{D}},{\mathcal{M}}}$ for diverse
${\mathcal{C}}$-module categories ${\mathcal{M}}$ and tensor subcategories
${\mathcal{D}}\hookrightarrow{\mathcal{C}}^{*}_{\mathcal{M}}$.

To compute examples of such objects, let us describe, more explicitly, what
equation (5.1) looks like in this particular case.

In this case, the relative end is applied to the functor

$\underline{\operatorname{Hom}}(-,-):{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to{\mathcal{C}}.$

If ${\mathcal{D}}\hookrightarrow{\mathcal{C}}^{*}_{\mathcal{M}}$ is a tensor
subcategory, then ${\mathcal{D}}$ acts on ${\mathcal{M}}$ by evaluation. The
functor $\underline{\operatorname{Hom}}(-,-)$ has a
${\mathcal{D}}$-prebalancing

$\xi^{D}_{M,N}:\underline{\operatorname{Hom}}(M,D(N))\to\underline{\operatorname{Hom}}(D^{\rm{la}}(M),N),$

described in Lemma 2.6. If we denote

$\mbox{ev}^{D}:D^{\rm{la}}\circ D\to\mbox{\rm Id\,},$

the counit if the adjunction $(D^{\rm{la}},D)$, then, in this case, a
dinatural transformation $\pi_{M}:E\to\underline{\operatorname{Hom}}(M,M)$
satisfies equation (5.1) if and only if

(7.1) $\underline{\operatorname{Hom}}((\mbox{ev}^{D})_{M},\mbox{\rm
id\,}_{M})\pi_{M}=\xi^{D}_{D(M),M}\pi_{D(M)},$

for any $M\in{\mathcal{M}}$.

###  7.1. Case $\operatorname{Rep}(G/N)\hookrightarrow({\mathcal{C}}\rtimes
G)^{*}_{{\mathcal{C}}}$

Let $G$ be a finite group acting on ${\mathcal{C}}$ by tensor autoequivalences
$(F_{g},\xi^{g}):{\mathcal{C}}\to{\mathcal{C}}$. Here, for any $g\in G$

$\xi^{g}_{X,Y}:F_{g}(X){\otimes}F_{g}(Y)\to F_{g}(X{\otimes}Y)$

are the natural isomorphisms giving the monoidal functor structure on $F_{g}$.
We can consider the $G$-crossed product tensor category ${\mathcal{C}}\rtimes
G$.

As an abelian category ${\mathcal{C}}\rtimes G=\bigoplus_{g\in
G}{\mathcal{C}}_{g}$, where ${\mathcal{C}}_{g}={\mathcal{C}}$. An object in
${\mathcal{C}}_{g}$ will be denoted by $[X,g]$, where $X\in{\mathcal{C}}$.
Similarly, a morphism in ${\mathcal{C}}\rtimes G$ will be denoted by

$[f,g]:[X,g]\to[Y,g]$

where $f:X\to Y$ is a morphism in ${\mathcal{C}}$. The tensor product of
${\mathcal{C}}\rtimes G$ is determined by

$[X,g]\otimes[Y,h]:=[X\otimes F_{g}(Y),gh],\ \ \ X,Y\in{\mathcal{C}},\ \
g,h\in G,$

and the unit object is $[\mathbf{1},1]$. See [24] for details on the
associativity constraint and a proof of the pentagon identity.

The category ${\mathcal{C}}\rtimes G$ acts on ${\mathcal{C}}$ as follows. If
$[X,g]\in{\mathcal{C}}\rtimes G$ and $Y\in{\mathcal{C}}$ then

(7.2) $[X,g]\triangleright Y=X{\otimes}F_{g}(Y).$

It is well-known that the dual category $({\mathcal{C}}\rtimes
G)^{*}_{{\mathcal{C}}}$ is tensor equivalent to the $G$-equivariantization
${\mathcal{C}}^{G}$.

######  Lemma 7.1.

For any normal subgroup $N\triangleleft G$, there is an inclusion of tensor
categories

$\operatorname{Rep}(G/N)\hookrightarrow({\mathcal{C}}\rtimes
G)^{*}_{{\mathcal{C}}}.$

###### Proof.

Since the category of finite-dimensional vector spaces $\mbox{\rm
vect\,}_{\Bbbk}$ is included in ${\mathcal{C}}$ via: $V\mapsto
V\blacktriangleright\mathbf{1}$. Here $\blacktriangleright:\mbox{\rm
vect\,}_{\Bbbk}\times{\mathcal{C}}\to{\mathcal{C}}$ is the canonical action of
$\mbox{\rm vect\,}_{\Bbbk}$ on any finite abelian ${\Bbbk}$-linear category
${\mathcal{C}}$. See Section 2.3. Hence
$\operatorname{Rep}(G)\hookrightarrow{\mathcal{C}}^{G}$. Since the canonical
projection $G\twoheadrightarrow G/N$ induces an inclusion
$\operatorname{Rep}(G/N)\hookrightarrow\operatorname{Rep}(G)$, and
${\mathcal{C}}^{G}\simeq({\mathcal{C}}\rtimes G)^{*}_{{\mathcal{C}}}$, the
inclusion $\operatorname{Rep}(G/N)\hookrightarrow({\mathcal{C}}\rtimes
G)^{*}_{{\mathcal{C}}}$ follows.

To be more explicit, let us describe this inclusion. If
$V\in\operatorname{Rep}(G/N)$, then $V\in\operatorname{Rep}(G)$ with action
given by

$g\cdot v=\overline{g}\cdot v.$

This implies that, for any $g\in G$, there is an isomorphism

$s_{g}:V\blacktriangleright\mathbf{1}\to V\blacktriangleright
F_{g}(\mathbf{1})\simeq F_{g}(V\blacktriangleright\mathbf{1}),$

given by the action of $g$ on $V$. For any $V\in\operatorname{Rep}(G/N)$ let
us denote by $L_{V}:{\mathcal{C}}\to{\mathcal{C}}$ the ${\mathcal{C}}\rtimes
G$-module functor described as follows. As a functor

$L_{V}(Y)=Y{\otimes}(V\blacktriangleright\mathbf{1}),$

For any $Y\in{\mathcal{C}}$. The module structure of the functor $L_{V}$ is
given by the composition

$L_{V}([Y,g]\triangleright
X)=Y{\otimes}F_{g}(X){\otimes}(V\blacktriangleright\mathbf{1})\xrightarrow{\mbox{\rm
id\,}_{Y{\otimes}F_{g}(X)}{\otimes}s_{g}}Y{\otimes}F_{g}(X){\otimes}F_{g}(V\blacktriangleright\mathbf{1})\to$
$\xrightarrow{\mbox{\rm
id\,}_{Y}{\otimes}\xi^{g}_{X,V\blacktriangleright\mathbf{1}}}Y{\otimes}F_{g}(X{\otimes}V\blacktriangleright\mathbf{1})=[Y,g]\triangleright
L_{V}(X),$

for any $X\in{\mathcal{C}}$, $[Y,g]\in{\mathcal{C}}\rtimes G$. ∎

We aim to compute the relative adjoint
${\mathcal{A}}_{\operatorname{Rep}(G/N),{\mathcal{C}}}$ associated with the
inclusion $\operatorname{Rep}(G/N)\hookrightarrow({\mathcal{C}}\rtimes
G)^{*}_{{\mathcal{C}}}$, as an object in ${\mathcal{C}}\rtimes G.$ For any
$g\in G$ we shall denote

${\mathcal{A}}_{g}=\int_{X\in{\mathcal{C}}}X{\otimes}F_{g}(X)^{*}\in{\mathcal{C}},$

with the corresponding dinatural transformations

$\pi^{g}_{X}:{\mathcal{A}}_{g}\to X{\otimes}F_{g}(X)^{*}.$

If $N\triangleleft G$ is a normal subgroup, we shall denote

${\mathcal{A}}(N,{\mathcal{C}})=\oplus_{n\in
N}[{\mathcal{A}}_{n},n]\in{\mathcal{C}}\rtimes G.$

######  Proposition 7.2.

There exists an isomorphisms of objects

${\mathcal{A}}_{\operatorname{Rep}(G/N),{\mathcal{C}}}\simeq{\mathcal{A}}(N,{\mathcal{C}}).$

###### Proof.

Let us, explicitly describe, all the necessary ingredients to understand
equation (5.1) in this particular example.

First of all, one can easily check that the internal hom of ${\mathcal{C}}$ as
a ${\mathcal{C}}\rtimes G$-module category is

(7.3) $\underline{\operatorname{Hom}}(X,Y)=\oplus_{g\in
G}[Y{\otimes}F_{g}(X)^{*},g],$

for any $X,Y\in{\mathcal{C}}$. For any $V\in\operatorname{Rep}(G/N)$ let us
denote by

$\beta^{V}_{M,N}:\underline{\operatorname{Hom}}(M,N{\otimes}V\blacktriangleright\mathbf{1})\to\underline{\operatorname{Hom}}(M{\otimes}V^{*}\blacktriangleright\mathbf{1},N)$

the $\operatorname{Rep}(G/N)$-prebalancing of the functor

$\underline{\operatorname{Hom}}(-,-):{\mathcal{C}}^{\rm{op}}\times{\mathcal{C}}\to{\mathcal{C}}\rtimes
G.$

Here, we are using that the left adjoint to $L_{V}$ is $L_{V^{*}}$. Observe
that, using (7.3), the prebalancing $\beta^{V}_{M,N}=\oplus_{g\in
G}\beta^{V,g}_{M,N}$, where

$\beta^{V,g}_{M,N}:[N{\otimes}V\blacktriangleright\mathbf{1}{\otimes}F_{g}(M)^{*},g]\to[N{\otimes}F_{g}(M{\otimes}V^{*}\blacktriangleright\mathbf{1})^{*},g].$

According to Lemma 6.3 and Lemma 2.6, one can compute $\beta^{V,g}_{M,N}$ as
the composition:

$N{\otimes}V\blacktriangleright\mathbf{1}{\otimes}F_{g}(M)^{*}\xrightarrow{\gamma^{V,g}_{M,N}}N{\otimes}F_{g}(M{\otimes}V^{*}\blacktriangleright\mathbf{1})^{*}\xrightarrow{t_{g}}N{\otimes}F_{g}(M{\otimes}V^{*}\blacktriangleright\mathbf{1})^{*}$

Here $\gamma^{V,g}_{M,N}$ is the morphism presented in Proposition 5.6
associated with the additivity of the functor
$\underline{\operatorname{Hom}}(-,-)$, and $t_{g}=\mbox{\rm
id\,}_{N}{\otimes}F_{g}(\mbox{\rm id\,}_{M}{\otimes}s_{g})$, where $s_{g}$ is
the action of $g$ on $V^{*}$.

Let $E$ be an object in ${\mathcal{C}}\rtimes G$ equipped with dinatural
transformations

$\lambda_{X}=\oplus_{g\in G}\lambda^{g}_{X}:E\to\oplus_{g\in
G}[X{\otimes}F_{g}(X)^{*},g].$

The dinatural maps $\lambda$ satisfies equation (7.1) if and only if
$\lambda^{g}_{X}=0$ for any $g\notin N$. ∎

###  7.2. Case ${\mathcal{D}}\hookrightarrow{\mathcal{C}}$ is a tensor
subcategory

If ${\mathcal{M}}={\mathcal{C}}$ with the regular action, then
${\mathcal{C}}^{*}_{\mathcal{M}}={\mathcal{C}}^{\rm{rev}}$. Hence any tensor
subcategory ${\mathcal{D}}\subseteq{\mathcal{C}}$ produces a tensor
subcategory
${\mathcal{D}}^{\rm{rev}}\subseteq{\mathcal{C}}^{*}_{\mathcal{C}}$. In this
particular case the funtor $S$ is

(7.4) $S:{\mathcal{C}}^{\rm{op}}\times{\mathcal{C}}\to{\mathcal{C}},$
$S(X,Y)=Y{\otimes}X^{*}.$

The action of ${\mathcal{D}}^{\rm{rev}}$ on ${\mathcal{C}}$ is given by

$D\triangleright X=X{\otimes}D,$

for any $D\in{\mathcal{D}}$, $X\in{\mathcal{C}}$. The prebalancing of the
functor $S$ is given by

(7.5) $\beta^{D}_{M,N}:(N{\otimes}D){\otimes}M^{*}\to
N{\otimes}(M{\otimes}{}^{*}D)^{*},$ $\beta^{D}_{M,N}=(\mbox{\rm
id\,}_{N}{\otimes}\phi^{-1}_{M,{}^{*}D})a_{N,D,M^{*}}.$

Here $a$ is the associativity constraint, and $\phi_{X,Y}:(X{\otimes}Y)^{*}\to
Y^{*}{\otimes}X^{*}$ is the canonical isomorphism. Note that the right dual in
${\mathcal{D}}^{\rm{rev}}$ is the left dual in ${\mathcal{D}}$. A dinatural
transformation $\pi$ satisfies equation (5.1) if and only if

$\pi_{X\triangleright M}=(\beta^{X}_{X\triangleright
M,M})^{-1}S(m^{-1}_{X^{*},X,M},\mbox{\rm
id\,}_{M})S(\mbox{ev}_{X}\triangleright\mbox{\rm id\,}_{M},\mbox{\rm
id\,}_{M})\pi_{M}$

for any $X\in{\mathcal{D}}$, $M\in{\mathcal{C}}$. And, in this particular
example it satisfies (5.1) if and only if

(7.6)
$\displaystyle\begin{split}\pi_{M{\otimes}X}&=a^{-1}_{M,X,(M{\otimes}X)^{*}}(\mbox{\rm
id\,}_{M}{\otimes}\phi_{M{\otimes}X,{}^{*}X}((\mbox{\rm
id\,}_{M}{\otimes}ev_{X})a_{M,X,{}^{*}X})^{*})\pi_{M}\end{split}$

######  Remark 7.3.

Equation (7.6) implies that the dinatural transformation $\pi$ depends only on
objects that belong to a “quotient” ${\mathcal{C}}/{\mathcal{D}}$. This
observation will be used in subsection 7.3, in the example when
${\mathcal{C}}$ equals the Deligne tensor product of two tensor categories
${\mathcal{D}}\boxtimes{\mathcal{B}}$.

Note that this is equivalent to

$a_{M,X,(M{\otimes}X)^{*}}\pi_{M{\otimes}X}=(\mbox{\rm
id\,}_{M}{\otimes}\phi_{M{\otimes}X,{}^{*}X}((\mbox{\rm
id\,}_{M}{\otimes}ev_{X})a_{M,X,{}^{*}X})^{*})\pi_{M}.$

Applying to both sides of this equality $\mbox{\rm
id\,}_{M{\otimes}X}{\otimes}\phi_{M,X}$ one gets

(7.7) $\displaystyle(\mbox{\rm
id\,}_{M{\otimes}X}{\otimes}\phi_{M,X})a_{M,X,(M{\otimes}X)^{*}}\pi_{M{\otimes}X}=(\mbox{\rm
id\,}_{M}{\otimes}coev_{X}{\otimes}\mbox{\rm id\,}_{M^{*}})\pi_{M}$

We then produce an étale algebra

${\mathcal{A}}({\mathcal{D}}):={\mathcal{A}}_{{\mathcal{D}}^{\rm{rev}},{\mathcal{C}}}=\oint_{X\in\operatorname{Rep}(H)}(S,\beta).$

This object is a subalgebra of the usual adjoint algebra
${\mathcal{A}}_{\mathcal{C}}$. Not to be confused with the adjoint algebra
${\mathcal{A}}_{\mathcal{D}}$ of the tensor category ${\mathcal{D}}$.

In this Section we shall focus our attention to these particular subalgebras
measuring the inclusion ${\mathcal{D}}\hookrightarrow{\mathcal{C}}.$ First,
let us show that there is some sort of exact sequence. This will be more clear
when computing some particular examples.

For any $X\in{\mathcal{C}}$, $Y\in{\mathcal{D}}$ let
$\lambda_{Y}:{\mathcal{A}}_{\mathcal{D}}\to Y{\otimes}Y^{*}$,
$\pi_{X}:{\mathcal{A}}_{\mathcal{C}}\to X{\otimes}X^{*}$ be the dinatural
transformations associated with these ends. Also,
$\alpha_{X}:{\mathcal{A}}({\mathcal{D}})\to X{\otimes}X^{*}$ denote the
dinatural transformations that satisfy (5.1).

######  Proposition 7.4.

Let ${\mathcal{D}}\subseteq{\mathcal{C}}$ be a tensor subcategory. There are
algebra morphisms

${\mathcal{A}}({\mathcal{D}})\xrightarrow{\iota}{\mathcal{A}}_{\mathcal{C}}\xrightarrow{q}{\mathcal{A}}_{\mathcal{D}},$

such that $q\circ\iota=u\circ\alpha_{\mathbf{1}}$.

###### Proof.

Since ${\mathcal{D}}$ is a subcategory of ${\mathcal{C}}$, by the universal
property of the end, there exists a map

$q:{\mathcal{A}}_{\mathcal{C}}\to{\mathcal{A}}_{\mathcal{D}}$

such that $\lambda_{Y}q=\pi_{Y}$ for any $Y\in{\mathcal{D}}$. By the
definition of the product of the adjoint algebras given in Section 4, it
follows that $q$ is an algebra map. It follows from Proposition 5.5 that,
there exists a map
$\iota:{\mathcal{A}}({\mathcal{D}})\to{\mathcal{A}}_{\mathcal{C}}$ such that
$\pi_{X}\iota=\alpha_{X}$ for any $X\in{\mathcal{C}}$. Using Proposition 6.8
it follows that $\iota$ is an algebra map.

Let $Y\in{\mathcal{D}}$. Then, it follows from the definition of the unit of
${\mathcal{A}}_{\mathcal{D}}$ that

$\lambda_{Y}\circ
u\circ\alpha_{\mathbf{1}}=\mbox{coev}_{Y}\circ\alpha_{\mathbf{1}}.$

Also

$\lambda_{Y}\circ q\circ\iota=\pi_{Y}\circ\iota=\alpha_{Y}.$

Since $\alpha$ satisfies (7.7), it follows that
$\alpha_{Y}=\mbox{coev}_{Y}\circ\alpha_{\mathbf{1}},$ for any
$Y\in{\mathcal{D}}$. Whence $q\circ\iota=u\circ\alpha_{\mathbf{1}}$. ∎

###  7.3. Case ${\mathcal{C}}={\mathcal{D}}\boxtimes{\mathcal{B}}$

Let ${\mathcal{D}}$, ${\mathcal{B}}$ be tensor categories. The Deligne tensor
product ${\mathcal{C}}={\mathcal{D}}\boxtimes{\mathcal{B}}$ has structure of
tensor category. The tensor category ${\mathcal{D}}$ can be seen as a tensor
subcategory of ${\mathcal{C}}$ in a canonical way

${\mathcal{D}}\hookrightarrow{\mathcal{D}}\boxtimes{\mathcal{B}},$ $X\mapsto
X\boxtimes\mathbf{1}.$

In the next Proposition we shall make use of the canonical equivalence

$Z({\mathcal{D}}\boxtimes{\mathcal{B}})\simeq Z({\mathcal{D}})\boxtimes
Z({\mathcal{B}}).$

Thus, identifying $Z({\mathcal{B}})$ with the tensor subcategory
$\mathbf{1}\boxtimes Z({\mathcal{B}})\hookrightarrow Z({\mathcal{D}})\boxtimes
Z({\mathcal{B}})=Z({\mathcal{C}}).$

######  Proposition 7.5.

There exists an isomorphism of algebras
${\mathcal{A}}({\mathcal{D}})\simeq{\mathcal{A}}_{\mathcal{B}}$.

###### Proof.

To simplify calculations, we shall assume that both tensor categories are
strict. Let $\pi_{B}:{\mathcal{A}}_{\mathcal{B}}\to B{\otimes}B^{*}$,
$B\in{\mathcal{B}}$, be the dinatural transformations associated with
${\mathcal{A}}_{\mathcal{B}}$. For any $B\in{\mathcal{B}}$,
$X\in{\mathcal{D}}$ define

$\widehat{\pi}_{X\boxtimes B}=(\mbox{\rm id\,}_{\mathbf{1}\boxtimes
B}{\otimes}\phi_{X\boxtimes B,{}^{*}X\boxtimes\mathbf{1}}(\mbox{\rm
id\,}_{\mathbf{1}\boxtimes B}{\otimes}ev_{X})^{*})\pi_{B}.$

Note that this definition of $\widehat{\pi}$ is given by using (7.6), with
trivial associativities. One can prove that $\widehat{\pi}_{X\boxtimes
B}:{\mathcal{A}}_{\mathcal{B}}\to X\boxtimes B{\otimes}(X\boxtimes B)^{*}$
defines a dinatural transformation lifting the dinatural transformation $\pi$
and it satisfies equation (7.6). Let be
$E\in{\mathcal{D}}\boxtimes{\mathcal{B}}$ an object, and $\lambda_{A}:E\to
A{\otimes}A^{*}$, $A\in{\mathcal{C}}$, be a dinatural transformation such that
it satisfies (7.6). It follows that $\lambda$ is determined by its restriction
on elements of the form $\mathbf{1}\boxtimes B$, $B\in{\mathcal{B}}$, thus
defining a dinatural transformation

$\lambda_{\mathbf{1}\boxtimes B}:E\to B{\otimes}B^{*}.$

Hence, there exists a map $h:E\to{\mathcal{A}}_{\mathcal{B}}$ such that
$\pi_{B}\circ h=\lambda_{\mathbf{1}\boxtimes B}$. Since both, $\widehat{\pi}$
and $\lambda$ are determined by its values in elements of the form
$\mathbf{1}\boxtimes B$, then $\widehat{\pi}\circ h=\lambda$, proving that
${\mathcal{A}}({\mathcal{D}})\simeq{\mathcal{A}}_{\mathcal{B}}$. ∎

######  Question 7.1.

If
${\mathcal{D}}\to{\mathcal{C}}\to{\mathcal{B}}\boxtimes\operatorname{End}({\mathcal{M}})$
is an exact sequence of tensor categories relative to a module category
${\mathcal{M}}$, see [5], is there a relation between
${\mathcal{A}}({\mathcal{D}})$ and ${\mathcal{A}}_{\mathcal{B}}$ ?

###  7.4. Case when ${\mathcal{C}}=\operatorname{Rep}(H)$, $H$ a Hopf algebra

Observe first that
${\mathcal{C}}^{\rm{rev}}=\operatorname{Rep}(H^{\rm{cop}}).$ Tensor
subcategories of categories of representations of Hopf algebras are in
correspondence with Hopf algebra quotients. Let $\pi:H\to Q$ be a Hopf algebra
projection. In this case, define ${\mathcal{D}}=\operatorname{Rep}(Q)$. The
inclusion
$\operatorname{Rep}(Q)^{\rm{rev}}\hookrightarrow\operatorname{Rep}(H)^{\rm{rev}}$
is given via $\pi$.

######  Proposition 7.6.

There exists an isomorphism

${\mathcal{A}}({\mathcal{D}})\simeq\\{h\in
H:h\mbox{${}_{\left(1\right)}$}{\otimes}\pi(h\mbox{${}_{\left(2\right)}$})=h{\otimes}1\\}=H^{co\,\pi}.$

###### Proof.

For any $M\in\operatorname{Rep}(H)$ let us define

$\lambda_{M}:H^{co\,\pi}\to M{\otimes_{{\Bbbk}}}M^{*},$
$\lambda_{M}(h)=\sum_{i}h_{i}{\otimes_{{\Bbbk}}}h^{i},$

where $\sum_{i}h^{i}(m)\,h_{i}=h\cdot m$, for any
$h\in(H^{\rm{cop}})^{co\,\pi}$, $m\in M$.

######  Claim 7.1.

$\lambda$ is a dinatural transformation, and it satisfy (7.7).

###### Proof of Claim.

Dinaturality of $\lambda$ follows easily. Let $M\in\operatorname{Rep}(H)$,
$X\in\operatorname{Rep}(Q)$, $m\in M$, $y\in X$, then the right hand side of
(7.7) evaluated in $h\in H^{co\,\pi}$ equals

$\displaystyle(\mbox{\rm id\,}_{M}{\otimes}coev_{X}{\otimes}\mbox{\rm
id\,}_{M^{*}})\lambda_{M}(h)=\sum_{i,j}h_{i}{\otimes}x_{j}{\otimes}x^{j}{\otimes}h^{i}.$

Here $\\{x_{i}\\},\\{x^{i}\\}$ are dual basis if $X$ and $X^{*}$. When
evaluating the right most tensorands in $y{\otimes}m$ of the above equality it
gives

$\sum_{i,j}h_{i}{\otimes}x_{j}{\otimes}x^{j}(y)h^{i}(m)=h\cdot m{\otimes}y.$

The left hand side of (7.7) evaluated in $h\in H^{co\,\pi}$ equals

$\lambda_{M{\otimes}X}(h)=\sum_{i}\tilde{h}_{i}{\otimes_{{\Bbbk}}}\tilde{h}^{i}.$

Evaluating the right tensorand in $m{\otimes}y$ it gives

$\sum_{i}\tilde{h}_{i}{\otimes_{{\Bbbk}}}\tilde{h}^{i}(m{\otimes}y)=h\cdot(m{\otimes}y)=h\mbox{${}_{\left(1\right)}$}\cdot
m{\otimes}\pi(h\mbox{${}_{\left(2\right)}$})\cdot y.$

Since $h\in H^{co\,\pi}$ it follows that both sides are equal. ∎

Now, let be $E\in\operatorname{Rep}(H)$ an object equipped with a dinatural
transformation $\alpha_{M}:E\to M{\otimes_{{\Bbbk}}}M^{*}$ such that it
satisfies (7.7). If we set $\alpha_{H}(e)=\sum_{i}e_{i}{\otimes}e^{i}$, define

$\beta:E\to H,\quad\beta(e)=\sum_{i}e^{i}(1)e_{i}.$

Since $\alpha$ satisfies (7.7), it follows that $\beta(E)\subseteq
H^{co\,\pi}$, and dinaturality of $\alpha$ implies that
$\alpha=\lambda\circ\beta$, proving that ${\mathcal{A}}({\mathcal{D}})\simeq
H^{co\,\pi}$. ∎

##  8\. Apendix

Let ${\mathcal{M}}$ be an exact indecomposable left ${\mathcal{C}}$-module
category, and let ${\mathcal{D}}$ be a tensor subcategory of
${\mathcal{C}}^{*}_{\mathcal{M}}$. For any $X\in{\mathcal{C}}$ and
$(F,\gamma)\in Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$ the functor

$\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright-,F(-)):{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to
vect_{\Bbbk},$

has ${\mathcal{D}}$-prebalancing given by

(8.1)
$\displaystyle\begin{split}&\beta^{X}_{D,M,N}:\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,F(D(N)))\to\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
D^{\rm{la}}(M),F(N)),\\\
&\beta^{X}_{D,M,N}(f)=(ev_{D})_{F(N)}D^{\rm{la}}((\gamma_{D})_{N}f)\tilde{d}^{-1}_{X,M},\end{split}$

for any $(D,d)\in{\mathcal{D}}$. Here $ev_{D}:D^{\rm{la}}\circ D\to\mbox{\rm
Id\,}$ is the evaluation of the adjunction $(D^{\rm{la}},D)$, and
$\tilde{d}_{X,M}:D^{\rm{la}}(X\triangleright M)\to X\triangleright
D^{\rm{la}}(M)$ is the ${\mathcal{C}}$-module structure of the functor
$D^{\rm{la}}$. Also, the functor

$\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(-,F(-))):{\mathcal{M}}^{\rm{op}}\times{\mathcal{M}}\to
vect_{\Bbbk}$

has a ${\mathcal{D}}$-prebalancing given by

$b^{X}_{D,M,N}:\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,F(D(N))))\to\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(D^{\rm{la}}(M),F(N))),$
$b^{X}_{D,M,N}(f)=\beta^{D}_{M,N}f.$

Here $\beta^{D}_{M,N}$ is the ${\mathcal{D}}$-prebalancing presented in Lemma
6.3. Explicitly

$\beta^{D}_{M,N}=\xi^{D}_{M,F(N)}\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{M},(\gamma_{D})_{N}),$

where $\xi^{D}$ is the natural isomorphism presented in Lemma 2.6. The next
result is the proof of Claim 6.1.

######  Proposition 8.1.

Let

$\phi^{X}_{M,N}:\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,N))\to\operatorname{Hom}_{{\mathcal{M}}}(X\triangleright
M,N),$

be the natural isomorphisms presented in (2.12). We have that

(8.2)
$\beta^{X}_{D,M,N}\phi^{X}_{M,F(D(N))}=\phi^{X}_{D^{\rm{la}}(M),F(N)}b^{X}_{D,M,N},$

for any $X\in{\mathcal{C}}$, $M,N\in{\mathcal{M}}$.

###### Proof.

Take
$f\in\operatorname{Hom}_{{\mathcal{C}}}(X,\underline{\operatorname{Hom}}(M,F(D(N))))$.
Then, applying $\psi^{X}_{D^{\rm{la}}(M),F(N)}$ on both sides of (8.2) and
evaluating in $f$, one obtains that (8.2) is equivalent to

(8.3)
$\psi^{X}_{D^{\rm{la}}(M),F(N)}\big{(}\beta^{X}_{D,M,N}(\phi^{X}_{M,F(D(N))}(f))\big{)}=b^{X}_{D,M,N}(f).$

The right hand side of (8.3) is equal to

$\displaystyle=\xi^{D}_{M,F(N)}\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{M},(\gamma_{D})_{N})f$
$\displaystyle=\psi^{Y}_{D^{\rm{la}}(M),F(N)}\big{(}\alpha^{-1}_{Y\triangleright
M,F(N)}(\phi^{Y}_{M,D(F(N))}(\mbox{\rm
id\,}))\tilde{d}^{-1}_{Y,M}\big{)}\underline{\operatorname{Hom}}(\mbox{\rm
id\,}_{M},(\gamma_{D})_{F(N)})f$
$\displaystyle=\psi^{Y}_{D^{\rm{la}}(M),F(N)}\big{(}\alpha^{-1}_{Y\triangleright
M,F(N)}(\phi^{Y}_{M,D(F(N))}(\mbox{\rm id\,}))\tilde{d}^{-1}_{Y,M}$
$\displaystyle(\underline{\operatorname{Hom}}(\mbox{\rm
id\,},(\gamma_{D})_{F(N)})f)\triangleright\mbox{\rm
id\,}_{D^{\rm{la}}(M)}\big{)}$
$\displaystyle=\psi^{X}_{D^{\rm{la}}(M),F(N)}\big{(}\alpha^{-1}_{Y\triangleright
M,F(N)}(\phi^{Y}_{M,D(F(N))}(\mbox{\rm
id\,}))D^{\rm{la}}(\underline{\operatorname{Hom}}(\mbox{\rm
id\,},(\gamma_{D})_{F(N)})f)\triangleright\mbox{\rm id\,})$
$\displaystyle\tilde{d}^{-1}_{X,M}\big{)}$
$\displaystyle=\psi^{X}_{D^{\rm{la}}(M),F(N)}\big{(}\alpha^{-1}_{X\triangleright
M,F(N)}(\phi^{Y}_{M,D(F(N))}(\mbox{\rm
id\,})(\underline{\operatorname{Hom}}(\mbox{\rm
id\,},(\gamma_{D})_{F(N)})f)\triangleright\mbox{\rm id\,})$
$\displaystyle\tilde{d}^{-1}_{X,M}\big{)}$
$\displaystyle=\psi^{X}_{D^{\rm{la}}(M),F(N)}\big{(}\alpha^{-1}_{X\triangleright
M,F(N)}(\phi^{X}_{M,D(F(N))}(\underline{\operatorname{Hom}}(\mbox{\rm
id\,},(\gamma_{D})_{F(N)})f))\tilde{d}^{-1}_{X,M}\big{)}$

The second equality follows from the definition of $\xi^{D}$ given in 2.19.
Here $Y=\underline{\operatorname{Hom}}(M,D(F(N)))$, and $\tilde{d}$ is the
module structure of the functor $D^{\rm{la}}$. The third equality follows from
(2.15). Here we have used isomorphisms $\alpha$, see Remark 2.19. The fourth
equality follows from the naturality of $\tilde{d}$, the fifth one follows
from the naturality of $\alpha$, the sixth equality follows from (2.18). Using
the naturality of $\alpha$, one can see that the last expression is equal to
the left hand side of (8.3). ∎

######  Proposition 8.2.

Let $Q\in{\mathcal{C}}$, and $(\mu,\lambda):L^{Q}\to K$ be a natural
transformation in $Z_{\mathcal{D}}(\operatorname{End}({\mathcal{M}}))$. Then,
for any $(D,d)\in{\mathcal{D}}$ we have

(8.4)
$\displaystyle\begin{split}\underline{\operatorname{Hom}}((\mbox{ev}_{D})_{M},&\mbox{\rm
id\,}_{K(M)})\psi^{Q}_{M,K(M)}(\mu_{M})=\\\
&=\xi^{D}_{D(M),K(M)}\underline{\operatorname{Hom}}(\mbox{\rm
id\,},(\lambda_{D})_{M})\psi^{Q}_{D(M),K(D(M))}(\mu_{D(M)}),\end{split}$

###### Proof.

It follows from (2.14) that, the left hand side of (8.4) is equal to

$\psi^{Q}_{D^{\rm{la}}(D(M)),K(M)}(\mu_{M}(\mbox{\rm
id\,}_{Q}\triangleright(\mbox{ev}_{D})_{M})).$

It follows from (2.13) that, the right hand side of (8.4) is equal to

$\xi^{D}_{D(M),K(M)}\psi^{Q}_{D(M),D(K(M))}((\lambda_{D})_{M}\mu_{D(M)}).$

Hence, equation (8.4) is equivalent to

(8.5) $\displaystyle\begin{split}\mu_{M}(&\mbox{\rm
id\,}_{Q}\triangleright(\mbox{ev}_{D})_{M})=\\\
&=\phi^{Q}_{D^{\rm{la}}(D(M)),K(M)}\big{(}\xi^{D}_{D(M),K(M)}\psi^{Q}_{D(M),D(K(M))}((\lambda_{D})_{M}\mu_{D(M)})\big{)}\end{split}$

It follows from (2.18) that, the right hand side of (8.5) is equal to

$\displaystyle=\phi^{Y}_{D^{\rm{la}}(D(M)),K(M)}\big{(}\xi^{D}_{D(M),K(M)}\big{)}\big{(}\psi^{Q}_{D(M),D(K(M))}((\lambda_{D})_{M}\mu_{D(M)})\triangleright\mbox{\rm
id\,}\big{)}$ $\displaystyle=\alpha^{-1}_{Y\triangleright
D(M),K(M)}(\phi^{Y}_{D(M),D(K(M))}(\mbox{\rm id\,}))\tilde{d}^{-1}_{Y,D(M)}$
$\displaystyle\big{(}\psi^{Q}_{D(M),D(K(M))}((\lambda_{D})_{M}\mu_{D(M)})\triangleright\mbox{\rm
id\,}\big{)}$ $\displaystyle=\alpha^{-1}_{Y\triangleright
D(M),K(M)}(\phi^{Y}_{D(M),D(K(M))}(\mbox{\rm
id\,}))D^{\rm{la}}(\psi^{Q}_{D(M),D(K(M))}((\lambda_{D})_{M}\mu_{D(M)})\triangleright\mbox{\rm
id\,})$ $\displaystyle\tilde{d}^{-1}_{Q,D(M)}$
$\displaystyle=\alpha^{-1}_{Q\triangleright
D(M),K(M)}(\phi^{Y}_{D(M),D(K(M))}(\mbox{\rm
id\,})(\psi^{Q}_{D(M),D(K(M))}((\lambda_{D})_{M}\mu_{D(M)})\triangleright\mbox{\rm
id\,}))$ $\displaystyle\tilde{d}^{-1}_{Q,D(M)}$
$\displaystyle=\alpha^{-1}_{Q\triangleright
D(M),K(M)}(\phi^{Q}_{D(M),D(K(M))}(\psi^{Q}_{D(M),D(K(M))}((\lambda_{D})_{M}\mu_{D(M)})))\tilde{d}^{-1}_{Q,D(M)}$
$\displaystyle=\alpha^{-1}_{Q\triangleright
D(M),K(M)}((\lambda_{D})_{M}\mu_{D(M)})\tilde{d}^{-1}_{Q,D(M)}$
$\displaystyle=\alpha^{-1}_{Q\triangleright
D(M),K(M)}(D(\mu_{M})d^{-1}_{Q,M})\tilde{d}^{-1}_{Q,D(M)}$
$\displaystyle=\mu_{M}\alpha^{-1}_{Q\triangleright D(M),Q\triangleright
M}(d^{-1}_{Q,M})\tilde{d}^{-1}_{Q,D(M)}=\mu_{M}(\mbox{\rm
id\,}_{Q}\triangleright(\mbox{ev}_{D})_{M}).$

Here $Y=\underline{\operatorname{Hom}}(D(M),D(K(M)))$. The second equality
follows from the definition of the isomorphisms $\xi^{D}$ given in (2.19).
Here $\tilde{d}$ is the module structure of the functor $D^{\rm{la}}$. The
third equality follows from the naturality of $\tilde{d}$, the fourth one
follows from the naturality of $\alpha$, the fifth equality follows from
(2.18), the sixth equality follows from the fact that $\mu$ is a module
natural transformation. The last equality follows from (2.11). ∎

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