On quasi-monoidal comonads and their corepresentations

Abstract

In this paper, we define and study quasi-monoidal comonads on a monoidal category. It generalize the (Hom type) coquasi-bialgebras to a non-braided setting. We investigate their corepresentations and their coquasitriangular structures. We also discuss their gauge equivalence relations.

Full text

Electronic
Research Archive
http://www.aimspress.com/journal/era
ERA, 30(8): 3153–3171.
DOI: 10.3934/era.2022160
Received: 20 March 2022
Revised: 12 May 2022
Accepted: 17 May 2022
Published: 17 June 2022
Research article
On quasi-monoidal comonads and their corepresentations
Dingguo Wang and Xiaohui Zhang*
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China.
*Correspondence: Email: zhangxh2015@qfnu.edu.cn, zxhhhhh@hotmail.com.
Abstract: In this paper, we define and study quasi-monoidal comonads on a monoidal category. It
generalize the (Hom type) coquasi-bialgebras to a non-braided setting. We investigate their corepre-
sentations and their coquasitriangular structures. We also discuss their gauge equivalence relations.
Keywords: quasi-monoidal comonads; coquasitriangular structures; gauge transformations;
monoidal categories
1. Introduction
The theory of (co)monads can be used as a tool in various fields of mathematics such as algebra,
logic or operational semantics, and theoretical computer science. Note that in algebra theory, there are
two different “bimonads”. On the one hand, bimonads and Hopf monads without monoidal structures
were introduced in [1], and developed in [2–4]. On the other hand, bimonads on monoidal categories
were introduced in [5]. In 2002, Moerdijk used an opmonoidal monad to define a bimonad. This
bimonad Fis both a monad and an opmonoidal functor satisfying the multiplication and the unit of F
are all monoidal natural transformations (see [5] for details). Although Moerdijk called his bimonad
“Hopf monad”, the antipode was not involved in his definition. In 2007, A. Bruguières and A. Virelizier
introduced the notion of Hopf monad with antipode in the rigid categories in [6], and then put it in the
non-dual monoidal categories [7]. We refer to [7–11] for the recent research on A. Bruguières and A.
Virelizier’s bimonads.
Quasi-bialgebras were introduced by V. G. Drinfel’d in [12]. The dual definition, a k-coquasi-
bialgebra H(or a Majid algebra), was introduced by S. Majid in [13]. The associativity of the multipli-
cation are replaced by a weaker property, called coquasi-associativity. The multiplication is associative
up to conjugation by a convolution invertible linear form ω∈(H⊗H⊗H)∗, called the coassociator.
Note that the definition of a coquasi-bialgebra is not selfdual, and the category of (left or right) co-
modules over a coquasi-bialgebra is a monoidal category with nontrivial associativity constraint and
nontrivial unit constraints. Coquasi-bialgebras in a braided monoidal category also have been studied

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in [14].
Taking into account the results proved A. Bruguières and A. Virelizier in [6], it is now very natural
to ask how to extend coquasi-bialgebras to the non-braided setting. This is the main motivation of the
present paper.
In this paper, we present a dual version of the second author’s results about quasi-bimonads which
appeared in [15]. We mainly provide a generalization of coquasi-bialgebras by introducing the notion
of quasi-monoidal comonad. Actually, a quasi-monoidal comonad Fis both a comonad and a quasi-
monoidal functor such that its corepresentations is a non-strict monoidal category. The notion of quasi-
monoidal comonad is very general. For example, the tensor functor of a (Hom-type) coquasi-bialgebras
and bicomonads are all special cases of quasi-monoidal comonads.
The paper is organized as follows. In Section 2 we recall some notions of comonads, quasi-monoidal
functors, π-categories and so on. In Section 3, we introduce the definition of quasi-monoidal comonads
and discuss their corepresentations. In Section 4, we mainly investigate the coquasitriangular structures
of a quasi-monoidal comonad. At last, we introduce the gauge equivalent relation on quasi-monoidal
comonads.
2. Preliminaries
Throughout the paper, we let kbe a fixed field and char(k)=0 and Veckbe the category of finite
dimensional k-spaces. All the algebras and coalgebras, modules and comodules are supposed to be in
Veck. For the comultiplication ∆of a k-space C, we use the Sweedler-Heyneman’s notation: ∆(c)=
Pc1⊗c2for any c∈C.
2.1. Quasi-monoidal functor
Let (C,⊗,I,a,l,r) and (C′,⊗′,I′,a′,l′,r′) be two monoidal categories. Recall that a quasi-monoidal
functor from Cto C′is a triple (F,F2,F0), where F:C→C′is a functor, F2:F⊗′F→F⊗is a
natural transformation, and F0:I′→FI is a morphism in C′.
Furthermore, if the following equations hold for any X,Y,Z∈ C:
F2(X,Y⊗Z)◦(idFX ⊗′F2(Y,Z)) ◦a′
FX,F Y,FZ
=F(aX,Y,Z)◦F2(X⊗Y,Z)◦(F2(X,Y)⊗′idFZ ),(2.1)
F(lX)◦F2(I,X)◦(F0⊗′idFX)=l′
FX ,(2.2)
F(rX)◦F2(X,I)◦(idFX ⊗′F0)=r′
FX ,(2.3)
then F=(F,F2,F0) is called a monoidal functor.
2.2. Monoidal comonad
Let Cbe a category, F:C→C be a functor. Recall from [16] or [17] that if there exist natural
transformations δ:F→FF and ε:F→idC, such that the following identities hold
Fδ◦δ=δF◦δ, and idF=Fε◦δ=εF◦δ,
then we call the triple (F, δ, ε) a comonad on C.
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Let X∈ C, and (F, δ, ε) a comonad on C. If there exists a morphism ρX:X→F X, satisfying
FρX◦ρX=δX◦ρX,and εX◦ρX=idX,
then we call the couple (X, ρX) an F-comodule.
A morphism between F-comodules g:X→X′is called F-colinear, if gsatisfies: Fg ◦ρX=ρX′◦g.
The category of F-comodules is denoted by CF.
Let (C,⊗,I,a,l,r) be a monoidal category, (F, δ, ε) be a comonad on C, and (F,F2,F0) : C→Cbe a
monoidal functor. Then recall from [18] or [19] that Fis called a monoidal comonad (or a bicomonad)
on Cif δand εare both monoidal natural transformations, i.e. the following compatibility conditions
hold for any X,Y∈ C:















(C1) F(F2(X,Y)) ◦F2(FX,FY)◦(δX⊗δY)=δX⊗Y◦F2(X,Y),
(C2) εX⊗Y◦F2(X,Y)=εX⊗εY,
(C3) F(F0)◦F0=δI◦F0,
(C4) εI◦F0=idI.
2.3. Convolution product
Given a category Cand a positive integer n, we denote Cn=C×C×···×Cthe n-tuple cartesian
product of C. If Fis a comonad on C, then F×n(the n-tuple cartesian product of F) is a comonad on
Cn, and we have CnF×n=(CF)n.
Assume that U:CF→ C is the forgetful functor and P,Q:Cn→ D are functors. Then from [ [9],
Proposition 4.1], we have the following results.
Lemma 2.1. There is a canonical bijection:
Nat(PU×n,QU×n)N at(PF×n,Q).
Proof. Define ?♭:Nat(PU×n,QU×n)→Nat(PF×n,Q), f7→ f♭, by
f♭
(X1,··· ,Xn):P(FX1× · · · × F Xn)f(F X1,··· ,F Xn)
//Q(FX1× · · · × F Xn)
Q(εX1,··· ,εXn)
//Q(X1× · · · × Xn),
and ?♯:Nat(PF×n,Q)→Nat(PU×n,QU×n), α7→ α♯, by
α♯
(M1,··· ,Mn):P(M1× · · · × Mn)
P(ρM1,··· ,ρMn)
//P(FM1× · · · × F Mn)α(M1,··· ,Mn)//Q(M1× · · · × Mn),
for any f∈N at(PU×n,QU ×n), α∈Nat(PF×n,Q) and Xi∈ C, (Mi, ρMi)∈ CF. It is easy to check that ?♭
and ?♯are well defined and are inverse with each other. □
Let P,Q,R:Cn→ D be functors. For any α∈Nat(PF×n,Q) and β∈N at(QF×n,R), define their
convolution product β∗α∈N at(PF×n,R) by setting, for any objects X1,· · · ,Xnin C,
β∗αX1,··· ,Xn=βX1,··· ,Xn◦αFX1,··· ,F Xn◦P(δX1,· · · , δXn).
We say that α∈Nat(PF×n,Q) is ∗-invertible if there exists β∈Nat(QF×n,P) such that β∗α=P(ε×n)∈
Nat(PF×n,P) and α∗β=Q(ε×n)∈Nat(QF×n,Q). We denote βby α∗−1.
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Proposition 2.2. The ∗-invertible elements in Nat(PF×n,Q)are in corresponding with the natural
isomorphisms in Nat(PU×n,QU×n).
Proof. Suppose that f∈Nat(PU×n,QU×n) is a natural isomorphism. Then we immediately get that
(f♭)∗−1=(f−1)♭.
Conversely, if α∈Nat(PF×n,Q) is ∗-invertible, then α♯−1=(α∗−1)♯.□
3. Quasi-monoidal comonads
Suppose that (C,⊗,I,a,l,r) is a monoidal category, F:C → C is a functor, (F, δ, ε) is a comonad
and (F,F2,F0) is a quasi-monoidal functor.
Lemma 3.1. If we define the F-coaction on I by F0, and define the F-coaction on M ⊗N (as the tensor
product in C) for any (M, ρM),(N, ρN)∈ CFby:
ρM⊗N:M⊗NρM⊗ρN
//FM ⊗F N F2(M,N)//F(M⊗N),
then (I,F0)and (M⊗N, ρM⊗N)are all objects in CFif and only if the compatibility conditions Eqs
(C1)–(C4) hold.
Proof. It is straightforward to check that Eqs (C1) and (C2) hold if and only if (M⊗N, ρM⊗N)∈ CF,
Eqs (C3) and (C4) hold if and only if (I,F0)∈ CF.□
From now on, we always assume that the compatibility conditions Eqs (C1)–(C4) hold.
We suppose that there are natural transformations ϑ: (_ ⊗_) ⊗_◦F×3⇒_⊗(_ ⊗_) : C×3→ C,
and ι:I⊗F_⇒_ : C→C,κ:F_⊗I⇒_ : C→C. From Lemma 2.1, for any objects
(M, ρM),(N, ρN),(P, ρP)∈ CF,ϑ, ι, κ can induce the following natural transformations
AM,N,P=ϑ♯
M,N,P,LM=ι♯
M,RM=κ♯
M.
Conversely, if there are natural transformations A: (_ ⊗_) ⊗_⇒_⊗(_ ⊗_) : C×3→ C and
L:I⊗_⇒id :C→C,R: _ ⊗I⇒id :C→C, then from Lemma 2.1, for any X,Y,Z∈ C, they can
induce natural transformations
ϑX,Y,Z=A♭
X,Y,Z, ιX=L♭
X, κX=R♭
X.
Next, we will discuss when Ais the associativity constraint and L,Rare the unit constraints in CF.
Lemma 3.2. A, L and R are isomorphisms if and only if ϑ,ιand κare ∗-invertible.
Proof. Straightforward from Proposition 2.2. □
Lemma 3.3. A is F -colinear if and only if ϑsatisfies
(FX ⊗FY)⊗FZ
δX⊗δY⊗δX

δX⊗δY⊗δX//(FFX ⊗FFY)⊗FFZ F2⊗id //F(F X ⊗F Y )⊗F FZ
F2

(FFX ⊗FFY)⊗FFZ
ϑFX,F Y,FZ

F((FX ⊗FY)⊗FZ)
FϑX,Y,Z

FX ⊗(FY ⊗FZ)id⊗F2
//FX ⊗F(Y⊗Z)F2
//F(X⊗(Y⊗Z))
(3.1)
for any X,Y,Z∈ C.
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Proof. ⇐): Since the following diagram
(M⊗N)⊗P
ρM⊗ρN⊗ρP

ρM⊗ρN⊗ρP
//(FM ⊗F N)⊗FP
δM⊗δN⊗δP

ϑ//M⊗(N⊗P)
ρM⊗ρN⊗ρP

(FM ⊗F N)⊗FP
F2⊗id

(FF M ⊗FFN)⊗FFP
F2⊗id

ϑ//FM ⊗(F N ⊗F P)
id⊗F2

F(M⊗N)⊗FP
F2

FρM⊗FρN⊗FρP
//F(FM ⊗F N)⊗FFP
F2

FM ⊗F(N⊗P)
F2

F(M⊗(N⊗P)) F(ρM⊗ρN⊗ρP)//F(FM ⊗(F N ⊗F P)) Fϑ//F(M⊗(N⊗P))
is commutative for any M,N,P∈ CF,AM,N,Pis F-colinear.
⇒): Notice that AF X,FY,FZ is F-colinear for any X,Y,Z∈ C, then it follows
F(εX⊗εY⊗εZ)◦FAF X,FY,FZ ◦ρ(FX⊗FY )⊗F Z
=F(εX⊗εY⊗εZ)◦ρFX ⊗(FY ⊗FZ )◦AFX,FY,FZ .
After a direct computation, we obtain (3.1). □
Lemma 3.4. A satisfies the Pentagon Axiom in CFif and only if ϑsatisfies
(id ⊗ϑX,Y,Z)◦ϑW,FX ⊗FY,FZ ◦(id ⊗F2⊗id)◦(ϑF W,FFX,F FY ⊗id) (3.2)
◦(δW⊗δ2
X⊗δ2
Y⊗δZ)
=ϑW,X,Y⊗Z◦(id ⊗id ⊗F2)◦ϑF W⊗FX,F Y,FZ ◦(F2⊗id ⊗id)◦(δW⊗δX⊗δY⊗δZ)
for any W,X,Y,Z∈ C.
Proof. ⇐): Since we have
(id ⊗ϑN,P,Q)◦(id ⊗ρN⊗ρP⊗ρQ)◦ϑM,N⊗P,Q◦(id ⊗F2⊗id)◦(ρM⊗ρN⊗ρP⊗ρQ)
◦(ϑM,N,P⊗id)◦(ρM⊗ρN⊗ρP⊗id)
=(id ⊗ϑN,P,Q)◦ϑM,F N⊗FP,FQ ◦(id ⊗F(ρN⊗ρP)⊗ρQ)◦(id ⊗F2⊗id)◦(ϑF M,F N,F P ⊗id)
◦(FρM⊗FρN⊗FρP⊗ρQ)◦(ρM⊗ρN⊗ρP⊗id)
=(id ⊗ϑN,P,Q)◦ϑM,F N⊗FP,FQ ◦(id ⊗F2⊗id)◦(ϑF M,FFN,FFP ⊗id)◦(δM⊗δ2
N⊗δ2
P⊗δQ)
◦(ρM⊗ρN⊗ρP⊗ρQ)
=ϑM,N,P⊗Q◦(id ⊗id ⊗F2)◦ϑFM ⊗FN,FP,FQ ◦(F2⊗id ⊗id)◦(δM⊗δN⊗δP⊗δQ)
◦(ρM⊗ρN⊗ρP⊗ρQ)
=ϑM,N,P⊗Q◦(id ⊗id ⊗F2)◦(ρM⊗ρN⊗ρP⊗ρQ)◦ϑM⊗N,P,Q◦(F2⊗id ⊗id)
◦(ρM⊗ρN⊗ρP⊗ρQ)
for any M,N,P,Q∈ CF,Asatisfies the Pentagon Axiom.
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⇒): For any W,X,Y,Z∈ C, we have cofree F-comodules FW,F X,FY,F Z. Consider the following
Pentagon Axiom:
AFW,FX,FY⊗F Z ◦AFW ⊗FX,FY,FZ
=(id ⊗AFX,F Y,FZ )◦AFW,FX⊗F Y,FZ ◦(AFW,FX,FY ⊗id).
Applying εW⊗εX⊗εY⊗εZto both sides of the above identity, we get Diagram (3.2). □
Lemma 3.5. For any X ∈ C,
(1) L is F -colinear if and only if ιsatisfies
I⊗FX
F0⊗δX

id⊗δX//I⊗FFX
ιFX

FI ⊗FFX F2
//F(I⊗FX)FιX
//FX.
(3.3)
(2) R is F-colinear if and only if κsatisfies
FX ⊗I
δX⊗F0

δX⊗id //FFX ⊗I
κFX

FFX ⊗FI F2
//F(FX ⊗I)FκX
//FX.
(3.4)
Proof. We only prove (1).
⇐): From the following commutative diagram
I⊗M
id⊗ρM

id⊗ρM
//I⊗FM
id⊗FρM

F0⊗id //FI ⊗F M
id⊗FρM

F2//F(I⊗M)
id⊗ρM

I⊗FM
ιM&&
id⊗δM//I⊗FF M
ιFM ''
F0⊗id //FI ⊗FF M F2//F(I⊗F M)
FιM
vv
MρM//FM
for any M∈ CF,LMis F-colinear.
⇒): Conversely, since FX is an F-comodule and LF X is F-colinear for any X∈ C, it is directly to
get Diagram (3.3). □
Lemma 3.6. A, L and R satisfy the Triangle Axiom in CFif and only if ϑ,ιand κsatisfy
(FX ⊗I)⊗FY
id⊗F0⊗δY

κX⊗id //X⊗FY id ⊗εY//X⊗Y
(FX ⊗FI)⊗FFY ϑX,I,F Y
//X⊗(I⊗FY )
id⊗ιY
OO(3.5)
for any X,Y,Z∈ C.
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Proof. ⇐): For any M,N∈ CF, we compute
(idM⊗ιN)◦(idM⊗idI⊗ρN)◦ϑM,I,N◦(ρM⊗F0⊗ρN)
=(idM⊗ιN)◦ϑM,I,F N ◦(idF M ⊗F0⊗δN)◦(ρM⊗idI⊗ρN)
=(idM⊗εN)◦(κM⊗idF N )◦(ρM⊗idI⊗ρN)
=(κM⊗idN)◦(ρM⊗idI⊗idN)
thus the Triangle Axiom in CFholds.
⇒): Conversely, for any X,Y∈ C, since we have
FX ⊗FY εX⊗εY//X⊗Y
(FX ⊗I)⊗FY AFX,I,FY
//
RFX ⊗id
66
FX ⊗(I⊗FY)
id⊗LFY
hh
,
it is a direct computation to get Diagram (3.5). □
Definition 3.7. Let (C,⊗,I,a,l,r) be a monoidal category on which (F, δ, ε) is a monad and (F,F2,F0)
is a quasi-monoidal functor such that the compatible conditions Eqs (C1)–(C4) are satisfied. If there are
∗-invertible natural transformations ϑ,ιand κsatisfying (3.1)–(3.5), then we call (F, δ, ε, F2,F0, ϑ, ι, κ)
aquasi-monoidal comonad on C,
Then by Lemma 3.1–3.6, one gets the following result.
Theorem 3.8. Let (C,⊗,I,a,l,r)be a monoidal category on which (F, δ, ε)is a monad and (F,F2,F0)
is a quasi-monoidal functor such that the compatible conditions Eqs (C1)–(C4) is satisfied. Then there
exist natural transformations ϑ,ιand κsuch that (F, δ, ε, F2,F0, ϑ, ι, κ)is a quasi-monoidal comonad
if and only if there are natural transformations A, L and R such that (CF,⊗,I,A,L,R)is a monoidal
category.
Example 3.9. Let (C,⊗,I,a,l,r) be a monoidal category on which (F, δ, ε) is a monad and (F,F2,F0)
is a quasi-monoidal functor such that the compatible conditions Eqs (C1)–(C4) are satisfied. If we
define
ϑX,Y,Z=a♭
X,Y,Z, ιX=l♭
X, κX,Y,Z=r♭
X,Y,Z
for any X,Y,Z∈ C, then Eq (3.2) holds because of the Pentagon Axiom of a; Eq (3.5) holds because of
the Triangle Axiom of a,l,r; Eqs (3.1), (3.3) and (3.4) hold if and only if (F,F2,F0) is a monoidal func-
tor. That means, the quasi-monoidal comonad (F, δ, ε, F2,F0, ϑ, ι, κ) is exactly a monoidal comonad.
Example 3.10. Recall from [9] or [10], we consider the following monoidal category Hi,j(Veck) for
any i,j∈Z:
•the objects of Hi,j(Veck) are pairs (X, αX), where X∈V eckand αX∈Autk(X);
•the morphism f: (X, αX)→(Y, αY) in Hi,j(V eck) is a k-linear map from Xto Ysuch that αY◦f=
f◦αX;
•the monoidal structure is given by
(X, αX)⊗(Y, αY)=(X⊗Y, αX⊗αY),
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and the unit is (k,idk);
•the associativity constraint a, the unit constraints land rare given by
aX,Y,Z: (x⊗y)⊗z7→ αi+1
X(x)⊗(y⊗α−j−1
Z(z));
lX(1k⊗x)=αj+1
X(x),rX(x⊗1k)=αi+1
X(x),∀X∈Veck.
Now assume that (H, αH) is an object in Hi,j(Veck), mH:H⊗H→H(with notation mH(a⊗b)=ab),
ηH:k→H(with notation ηH(1k)=1H), and ∆H:H→H⊗H(with notation ∆H(h)=h1⊗h2), and
εH:H→kare all morphisms in Hi,j(Veck). Further, we write
¨
H=_⊗H:Hi,j(Veck)→ Hi,j(Veck),(X, αX)7→ (X⊗H, αX⊗αH)
for the right tensor functor of H.
If we define the following structures on ¨
H:
•δ:¨
H→¨
H¨
Hand ϵ:¨
H→idHi,j(Veck)are defined by
δX:x⊗h7→ (αX(x)⊗h1)⊗α−1
H(h2),
ϵX:x⊗h7→ εH(h)α−1
X(x);
•¨
H2:¨
H⊗¨
H→¨
H⊗and ¨
H0:k→¨
H(k) are given by
¨
H2(X,Y):(x⊗a)⊗(y⊗b)7→ (x⊗y)⊗αi
H(a)αj
H(b),
¨
H0(1k)=1k⊗1H,
for any X,Y∈ Hi,j(Veck). Then obviously ¨
H=(¨
H, δ, ϵ) forms a comonad on Hi,j(Veck) if and only
if (H, αH,∆H, εH) is a Hom-coalgebra over k, Eqs (C1)–(C4) hold if and only if mHand ηHare all
morphisms of Hom-coalgebras.
Suppose that there are αH-invariant convolution invertible linear forms ω∈(H⊗H⊗H)∗and
p,q∈H∗, then we can define the following ∗-invertible natural transformations
ϑX,Y,Z: ((x⊗a)⊗(y⊗b)) ⊗(z⊗c)7→ ω(α2i
H(a), αi+j
H(b), αj−1
H(c))(αi
X(x)⊗(α−1
Y(y)⊗α−j−2
Z(z))),
ιX: 1k⊗(x⊗a)7→ p(a)αj
X(x), κX: (x⊗a)⊗1k7→ q(a)αi
X(x),
where a,b,c∈H,x∈X,y∈Y,z∈Zand X,Y,Z∈Veck. Thus we immediately get that ϑsatisfies Eq
(3.1) if and only if ωsatisfies
XαH(a1)(b1c1)ω(a2,b2,c2)=Xω(a1,b1,c1)(a2b2)αH(c2); (3.6)
ϑsatisfies Eq (3.2) if and only if ωsatisfies
Xω(αH(a1), αH(b1),c1d1)ω(a2b2, αH(c2), αH(d2))
=Xω(b1,c1, αH(d1))ω(αH(a1), α−1
H(b21)α−1
H(c21), αH(d2))ω(αH(a2),b22 ,c22); (3.7)
ιsatisfies Eq (3.3) and κsatisfies Eq (3.4) if and only if p,qsatisfy
Xp(a1)1Ha2=αH(a1)p(a2),Xq(a1)a21H=αH(a1)q(a2); (3.8)
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ϑ, ι and κsatisfy Eq (3.5) if and only if ω,pand qsatisfy
ω(a,1H,b)=q(a)p∗−1(b).(3.9)
This means, ¨
H=(¨
H, δ, ϵ, ¨
H2,¨
H0, ϑ, ι, κ) forms a quasi-monoidal comonad on Hi,j(Veck) if and only
if H=(H, αH,mH, ηH,∆H, εH, ω, p,q) forms a Hom-coquasi-bialgebra over k(see [20] for the dual
definition). Further, from Theorem 3.10, one get that Corep(H)=(Hi,j(Veck)) ¨
H, the category of right
H-Hom-comodules, is a monoidal category and its associativity constraint, unit constraints are given
as follows:
AM,N,P((m⊗n)⊗p)=Pω(α2i(m1), αi+j(n1), αj−1(p1))αi
M(m0)⊗(α−1
N(n0)⊗α−j−2
P(p0)),
LM(1k⊗m)=Pp(m1)αj
M(m0),RM(m⊗1k)=Pq(m1)αi
M(m0),
where m∈M,n∈N,p∈P,M,N,P∈Core p(H).
Example 3.11. Under the consideration of Example 3.10, if all the Hom-structure maps αare identity
maps, then the Hom-coquasi-bialgebra is exactly the coquasi-bialgebra (also called a Majid algebra,
see [13] for details) over k.
Example 3.12. Let B=(B, µ, 1B,∆, ε) be a bialgebra over k,αB:B→Bbe an endo-isomrophism.
Recall that a k-linear form g∈B∗is called
(1) dual central if g(x1)x2=x1g(x2) for any x∈B;
(2) dual group-like if it is convolution invertible and satisfies g(xy)=g(x)g(y) for any x,y∈B;
(3) αB-invariant if g(αB(x)) =g(x).
Now suppose that p,q∈B∗are all dual central dual group-like and αB-invariant linear forms. Define
ak-linear form ω:B⊗B⊗B→kby
ω(x,y,z)=p(x)ε(y)q∗−1(z),for any x,y,z∈B,
define the new multiplication µαBand comultiplication ∆αBby
µαB=αB◦µ, ∆αB= ∆ ◦αB.
Then it is a direct calculation to check that αB, ω, p,qsatisfy Eqs.(3.6) - (3.9) (under µαBand ∆αB),
hence Bp,q
αB=(B, αB, µαB,1B,∆αB,ε, ω, p,q) forms a nontrivial Hom-coquasi-bialgebra.
4. Coquasitriangular structures
Recall that a braiding in a monoidal category (C,⊗,I,a,l,r) is a natural isomorphism τ:⊗⇒⊗op :
C × C → C such that the following identities hold
aY,Z,X◦τX,Y⊗Z◦aX,Y,Z=(idY⊗τX,Z)◦aY,X,Z◦(τX,Y⊗idZ),(B1)
a−1
Z,X,Y◦τX⊗Y,Z◦a−1
X,Y,Z=(τX,Z⊗idY)◦a−1
X,Z,Y◦(idX⊗τY,Z)(B2)
for any X,Y,Z∈ C.
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Now let Fbe a quasi-monoidal comonad on C. Suppose that there is a natural transformation σ:
⊗ ◦ (F×F)⇒ ⊗op :C×2→ C. From Lemma 2.1, for any objects M,Nin CF,σcan induce a natural
transformation
τM,N=σ♯
M,N:M⊗NρM⊗ρN
//FM ⊗F N σM,N//N⊗M.
Conversely, if there exists τ:⊗⇒⊗op :C × C → C, then from Lemma 2.1, for any X,Y∈ C,τcan
induce the following
σX,Y=τ♭
X,Y:FX ⊗FY τF X,FY //FY ⊗F X εY⊗εX//Y⊗X.
Next we will discuss when τis a braiding in CF.
Lemma 4.1. τis an isomorphism if and only if σis ∗-invertible.
Proof. Straightforward from Proposition 2.2. □
Lemma 4.2. τis F-colinear if and only if σsatisfies
FX ⊗FY
δX⊗δY

δX⊗δY//FFX ⊗FFY σFX,FY //F Y ⊗F X
F2

FFX ⊗FFY F2
//F(FX ⊗FY)FσX,Y
//F(Y⊗X)
(4.1)
for any X,Y∈ C.
Proof. ⇐): We compute
M⊗N
ρM⊗ρN

ρM⊗ρN
//FM ⊗F N
FρM⊗FρN

σM,N//N⊗M
ρN⊗ρM

FM ⊗F N
F2

δM⊗δN//
=FρM⊗FρN//FF M ⊗FFN
F2

σFM,F N //FN ⊗F M
F2

F(M⊗N)F(ρM⊗ρN)//F(FM ⊗F N)FσM,N
//F(N⊗M)
for any M,N∈ CF. Hence τM,Nis F-colinear.
⇒): Conversely, notice that τFX,FY is F-colinear for any X,Y∈ C, we have
F(εY⊗εX)◦FτFX,F Y ◦ρFX⊗FY =F(εY⊗εX)◦ρF Y⊗FX ◦τF X,FY ,
which implies Diagram (4.1) holds. □
Lemma 4.3. Diagram (B1) holds in CFif and only if σsatisfies
ϑY,Z,X◦σFX,F Y⊗FZ ◦(id ⊗F2)◦ϑFFX,FFY,FF Z ◦(δ2
X⊗δ2
Y⊗δ2
Z) (4.2)
=(id ⊗σX,Z)◦ϑY,F X,FZ ◦(σFFX,F Y ⊗id)◦(δ2
X⊗δY⊗δZ)
for any X,Y,Z∈ C.
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Proof. ⇐): Take X=M,Y=N,Z=Pfor any F-comodules M,N,P. Multiplied by ρM⊗ρN⊗ρP
right on both sides of Eq (4.2), we immediately get Diagram (B1).
⇒): Since Diagram (B1) is commutative for any FX,FY,FZ ∈ C, multiplied by ε⊗ε⊗εleft on
both sides of the above equation, we get Eq (4.2). □
Lemma 4.4. For any X,Y,Z∈ C, Diagram (B2) holds in CFif and only if σsatisfies
ϑ∗−1
Z,X,Y◦σFX ⊗FY,FZ ◦(F2⊗id)) ◦ϑ∗−1
FFX,FF Y,FFZ ◦(δ2
X⊗δ2
Y⊗δ2
Z) (4.3)
=(σX,Z⊗id)◦ϑ∗−1
FX,F Z,Y◦(id ⊗σFY,FF Z )◦(δX⊗δY⊗δ2
Z),
where ϑ∗−1means the ∗-inverse of ϑ.
Proof. The proof is similar to Lemma 4.3. □
Definition 4.5. Let (F, δ, ε, F2,F0, ϑ, ι, κ) be a quasi-monoidal comonad on a monoidal category C. If
there is a ∗-invertible natural transformation σ∈Nat(F⊗F,⊗op), satisfying Eqs (4.1)–(4.3) for any
X,Y,Z∈ C, then σis called a coquasitriangular structure of F, and (F, σ) is called a coquasitriangular
quasi-monoidal comonad.
Combining Lemma 4.1–Definition 4.5, we obtain the following result.
Theorem 4.6. Let (F, δ, ε, F2,F0, ϑ, ι, κ)be a quasi-monoidal comonad on a monoidal category C.
Then CFis a braided monoidal category if and only if there exists a natural transformation σ:F⊗F→
⊗op such that (F, σ)is a coquasitriangular quasi-monoidal comonad. Further, the braiding in CFis
given by τ=σ♯.
Corollary 4.7. Let (F, σ)be a coquasitriangular quasi-monoidal comonad on a monoidal category C.
Then for any X,Y,Z∈ C,σsatisfies the following generalized Yang-Baxter equation:
(id ⊗σX,Y)◦ϑZ,FX,F Y ◦(σFFX,FZ ⊗id)◦ϑ∗−1
F3X,FF Z,FFY ◦(id ⊗σF3Y,F3Z)
◦ϑF4X,F4Y,F4Z◦(δ4
X⊗δ4
Y⊗δ4
Z)
=ϑZ,Y,X◦(σFY,F Z ⊗id)◦ϑ∗−1
FF Y,FFZ,F X ◦(id ⊗σFFX,F3Z)◦ϑF3Y,F3X,F4Z
◦(σF4X,F4Y⊗id)◦(δ4
X⊗δ4
Y⊗δ4
Z).
Proof. Straightforward. □
Example 4.8. If Fis a monoidal comonad on C, and σ:⊗ ◦ F×2⇒ ⊗o p is a ∗-invertible natural
transformation satisfying Eqs (4.1)–(4.3), then (F, σ) is exactly a coquasitriangular monoidal comonad
(see [9], Definition 4.12).
Example 4.9. With the notations in Example 3.10, if Q∈(H⊗H)∗is αH-invariant and convolution
invertible, then we have the following ∗-invertible natural transformation
σX,Y:¨
HX ⊗¨
HY →Y⊗X,(x⊗a)⊗(y⊗b)7→ Q(αi
H(a), αj
H(b))αj−i−1
Y(y)⊗αi−j−1
X(x),
where x∈X,y∈Yand X,Y∈ Hi,j(Veck). Thus we immediately get that σsatisfies Eq (4.1) if and
only if Qsatisfies XQ(a1,b1)a2b2=Xb1a1Q(a2,b2),
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σsatisfies Eqs (4.2) and (4.3) if and only if Qsatisfies
Xω(b1,c1,a1)Q(αH(a21),b21 c21)ω(a22 ,b22,c22 )
=XQ(a1,c1)ω(b1, α−1
H(a21),c2)Q(α−1
H(a22),b2),
Xω∗−1(c1,a1,b1)Q(a21b21 , αH(c21))ω∗−1(a22,b22,c22)
=XQ(a1,c1)ω∗−1(a2, α−1
H(c21),b1)Q(b2, α−1
H(c22)),
where a,b,c∈H. That is, ( ¨
H, σ) forms a coquasitriangular quasi-monoidal comonad if and only
if (H,Q) is a coquasitriangular Hom-coquasi-bialgebra. Further, from Theorem 4.6, one get that
Corep(H)=(Hi,j(Veck)) ¨
His a braided monoidal category.
Example 4.10. With the notations in Example 3.12, if p∈B∗is a dual central dual group-like αB-
invariant k-linear form on a bialgebra B, then we get a coquasi-bialgebra Bp,p
αB. Now suppose that
Q∈(B⊗B)∗is the coquasitriangular structure over B. If Q◦(αB⊗αB)=Q, then after a straightforward
compute we get that Qis also a coquasitriangular structure over the Hom-coquasi-bialgebra Bp,p
αB.
5. Gauge transformations
Let F=(F, δ, ε, F2,F0) be a quasi-monoidal comonad on a monoidal category (C,⊗,I,a,l,r).
Definition 5.1. Agauge transformation on Fis a ∗-invertible natural transformation ξ:F⊗F⇒ ⊗.
Using a gauge transformation ξon F, we can build a new quasi-monoidal comonad Fξas follows.
Firstly, as a functor, Fξ=F:C→C.
Secondly, the comonad structure of Fξis Fξ=F=(F, δ, ε).
Thirdly, the quasi-monoidal functor structure of Fξis given by:
•for any X,Y∈ C,Fξ
2:F⊗F⇒F⊗is defined as follows
Fξ
2(X,Y) : FX ⊗FYδ2
X⊗δ2
Y
//F3X⊗F3Yξ//FFX ⊗FFY F2//F(FX ⊗FY )
F(ξ∗−1
X,Y)
//F(X⊗Y) (5.1)
where ξ∗−1means the ∗-inverse of ξ;
•Fξ
0=F0:FI →I.
Proposition 5.2. With the above notations, δand εare both monoidal natural transformations
Proof. We only need to show the compatible conditions Eqs (C1)–(C4) hold.
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To prove Eq (C1), we compute
FX ⊗FY
δX⊗δY

δ4
X⊗δ4
Y
((
δ2
X⊗δ2
Y//F3X⊗F3Y
FδδX⊗FδδY

ξ//FFX ⊗FFY
δδX⊗δδY

δFX ⊗δF Y //F3X⊗F3YF2//F(FFX ⊗FFY )
F(FδX⊗FδY)
uu
Fξ∗−1

FFX ⊗FFY
ξ

F5X⊗F5Yξ//F4X⊗F4YF2//F(F3X⊗F3Y)
Fξ∗−1
))
F(FX ⊗FY)
F(δX⊗δY)

FX ⊗FY
δ3
X⊗δ3
Y
22
δX⊗δY
//FFX ⊗FFY
F2
vv
δFX ⊗δF Y
((
F2//F(FX ⊗FY)
F(δ2
X⊗δ2
Y)
OO
F(FFX ⊗FFY)
F(ξ)

F(FX ⊗FY)
δFX ⊗FY
,,
Fξ∗−1

F3X⊗F3Y
F2
,,
F(FX ⊗FY)
F(δX⊗δY)

F(X⊗Y)δX⊗Y
//FF(X⊗Y)FF(F X ⊗F Y )
FF ξ∗−1
ooF(FFX ⊗FFY),
F2
oo
for any X,Y∈ C. The rest are straightforward. □
For any X,Y∈ C, define the natural transformation ϑξ: (F⊗F)⊗F⇒_⊗(_ ⊗_) by
ϑξ
X,Y,Z=(id ⊗ξ∗−1
Y,Z)◦ξ∗−1
X,FY ⊗FZ ◦(id ⊗F2)◦ϑF X,FFY,FFZ ◦(ξFFX⊗F3Y,F3Z) (5.2)
◦(F2⊗id)◦(ξF3X,F4Y⊗id)◦(δ3
X⊗δ4
Y⊗δ3
Z),
and define the followings natural transformations:
ιξ
X:I⊗FX F0⊗δX//FI ⊗FFX ξ//I⊗FX ιX//X,(5.3)
and
κξ
X:FX ⊗IδX⊗F0//FFX ⊗FI ξ//FX ⊗IκX//X.(5.4)
It is easy to get that ϑξ,ιξand κξare all ∗-invertible. Further, we have the following properties.
Lemma 5.3. With the above notations, ϑξsatisfies Eqs (3.1) and (3.2).
Proof. We only prove Eq (3.1). For any X,Y,Z∈ C, we compute
F(ϑξ
X,Y,Z)◦Fξ
2◦(Fξ
2⊗id)◦(δX⊗δY⊗δZ)
=F(id ⊗ξ∗−1
Y,Z)◦F(ξ∗−1
X,FY ⊗FZ )◦F(id ⊗F2)◦F(ϑF X,FFY,FF Z )◦F(ξFFX⊗F3Y,F3Z)
◦F(F2⊗id)◦F(ξF4X,F3Y⊗id)◦F(δ3
X⊗δ4
Y⊗δ3
Z)◦F(ξ∗−1
FX ⊗FY,FZ )◦F2
◦(δFX ⊗FY ⊗δF Z )◦ξF(FX⊗FY ),F FZ ◦(δF X⊗F Y ⊗δFZ )◦(F(ξ∗−1
FX,F Y )⊗id)
◦(F2⊗id)◦(δF X ⊗δFY ⊗id)◦(ξFFX,F F Y ⊗id)◦(δF X ⊗δFY ⊗id)◦(δX⊗δY⊗δZ)
=F(id ⊗ξ∗−1
Y,Z)◦F(ξ∗−1
X,FY ⊗FZ )◦F(id ⊗F2)◦F(ϑF X,FFY,FF Z )◦F(ξFFX⊗F3Y,F3Z)
◦F(F2⊗id)◦F(δF X ⊗δ2
FY ⊗δ3
Z)◦F(ξ∗−1
FFX⊗FFY,F Z )◦F2◦(δFFX⊗FF Y ⊗δFZ )
◦ξF(FFX⊗FFY),F FZ ◦(δFFX⊗F FY ⊗δF Z )◦(F2⊗id)◦(δF X ⊗δFY ⊗id)◦(ξFFX,FFY ⊗id)
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◦(δ2
X⊗δ2
Y⊗δZ)
=F(id ⊗ξ∗−1
Y,Z)◦F(ξ∗−1
X,FY ⊗FZ )◦F(id ⊗F2)◦F(ϑF X,FFY,FF Z )◦F2◦(F2⊗id)◦(δ2
FX ⊗δ2
FY
⊗δ2
FZ )◦ξF3X⊗F3Y,FF Z ◦(F2⊗id)◦(δF X ⊗δ2
FY ⊗δF Z )◦(ξFFX,FFY ⊗id)◦(δ2
X⊗δ2
Y⊗δZ)
=F(id ⊗ξ∗−1
Y,Z)◦F(ξ∗−1
X,FY ⊗FZ )◦F(id ⊗F2)◦F2◦(id ⊗F2)◦(δX⊗δ2
Y⊗δ2
Z)◦ϑFX,F Y,FZ
◦ξFFX⊗FFY,F F Z ◦(F2⊗id)◦(ξF3X,F3Y⊗id)◦(δ3
X⊗δ3
Y⊗δ2
Z)
=F(ξ∗−1
X,Y⊗Z)◦F(id ⊗F(ξ∗−1
Y,Z)) ◦F2◦(δX⊗δFY⊗F Z )◦(id ⊗F2)◦(id ⊗δY⊗δZ)
◦(εFX ⊗εF Y ⊗εFZ )◦ϑFFX,FF Y,FFZ ◦ξF3X⊗F3Y,F3Z◦(F2⊗id)◦(ξF4X,F4Y⊗id)
◦(δ4
X⊗δ4
Y⊗δ3
Z)
=F(ξ∗−1
X,Y⊗Z)◦F2◦(δX⊗δY⊗Z)◦ξFX,F(Y⊗Z)◦ξ∗−1
FFX,FF (Y⊗Z)◦(FδX⊗FδY⊗Z)◦(id
⊗FF(ξ∗−1
Y,Z)) ◦(id ⊗F(F2)) ◦(id ⊗F(δY⊗δZ)) ◦(id ⊗F2)◦ϑFFX,FF Y,FFZ ◦ξF3X,F3Y⊗F3Z
◦(F(id ⊗FFεFY )⊗FFFεFZ )◦(F2⊗id)◦(ξF4X,F5Y⊗id)◦(δ4
X⊗δ5
Y⊗δ4
Z)
=F(ξ∗−1
X,Y⊗Z)◦F2◦(δX⊗δY⊗Z)◦ξFX,F(Y⊗Z)◦(δX⊗δY⊗Z)◦(F(id ⊗ξ∗−1
Y,Z)) ◦(id ⊗F2(FY,FZ))
◦(id ⊗δY⊗δZ)◦(id ⊗ξFX,F Y )◦(id ⊗ξ∗−1
FF Y,FFZ )◦(id ⊗δF Y ⊗δFZ )◦ξ∗−1
FX,F FY⊗F FZ
◦(id ⊗F2(FFY ⊗FFZ)) ◦ϑFFX,F3Y,F3Z◦ξF3X,F4Y⊗F4Z◦(F2⊗id)◦(ξF4X,F5Y⊗id)
◦(δ4
X⊗δ5
Y⊗δ4
Z)
=Fξ
2(FX,FY ⊗F Z)◦(id ⊗Fξ
2(FY,FZ)) ◦ϑξ
FX,F Y,FZ ◦(δX⊗δY⊗δZ).
Thus the conclusion holds. □
Lemma 5.4. With the above notations, ιξsatisfies Eq (3.3) and κξsatisfies Eq (3.4).
Proof. We only prove Eq (3.3). For any X∈ C, we have
I⊗FX
id⊗δX

F0⊗δX//FI ⊗FX
id⊗FδX

ξ
''
δI⊗δFX //FFI ⊗F3Xξ//F I ⊗FFX
F2

δI⊗δFX //FFI ⊗F3X
F2

I⊗FFX
F0⊗id
88
FI ⊗F3X
ξ''
I⊗FX
id⊗δX

F0⊗δX
66
F(I⊗FX)
FιX
vv
F(F0⊗δX)//F(FI ⊗FFX)
F(δI⊗δFX )
vv
F(ξ∗−1)

I⊗FFX
ιFX
ww
F(FFI ⊗F3X)
F(ξ∗−1)
((
F(I⊗FX)
F(F0⊗δX)

FX F(I⊗FX)
FιX
ooF(FI ⊗FFX)
Fξ
oo
which implies Eq (3.3). □
Lemma 5.5. With the above notations, ϑξand ιξ,κξsatisfy Eq (3.5).
Proof. For any X,Y∈ C, we obtain
(id ⊗ιξ
Y)◦(ϑξ
X,I,FY )◦(id ⊗F0⊗δY)
=(id ⊗ιY)◦(id ⊗ξI,FF Y )◦(id ⊗F0⊗δY)⊗(id ⊗ξ∗−1
I,FY )◦ξ∗−1
X,FI ,FF Y ◦(id ⊗F2)◦ϑF X,FFI,F3Y
◦ξFFX⊗F3I,F4Y◦(F2⊗id)◦(ξF3X,F4I⊗id)◦(δ3
X⊗δ4
I⊗δ3
FY )⊗(δX⊗F0⊗δY)
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=ξ∗−1
X,Y◦(id ⊗FιY)◦(id ⊗F2)◦ϑFX,F I,F FY ◦ξFFX⊗FFI,F3Y◦(F2⊗id)◦(ξF3X,F3I⊗id)
◦(δ3
X⊗δ3
I⊗δ2
FY )◦(δX⊗F0⊗δY)
=ξ∗−1
X,Y◦(id ⊗ιFY )◦ϑF X,I,FF Y ◦(id ⊗F0⊗δY)◦ξFFX⊗I,FF Y ◦(F2⊗id)◦(ξF3X,F I ⊗id)
◦(δ3
X⊗δI⊗δ2
Y)◦(δX⊗F0⊗id)
=(id ⊗εY)◦ξ∗−1
X,FY ◦(κF X ⊗id)◦ξFFX⊗I,FFY ◦(F2⊗id)◦(δF X ⊗F0⊗id)◦(ξFFX,I⊗id)
◦(δ2
X⊗F0⊗δ2
Y)
=(id ⊗εY)◦ξ∗−1
X,FY ◦ξF X,FF Y ◦(κFFX ⊗id)◦(δF X ⊗id ⊗id)◦(ξFFX,I⊗id)◦(δ2
X⊗F0⊗δ2
Y)
=(id ⊗εY)◦(κX⊗id)◦(ξF X,I⊗id)◦(δX⊗F0⊗id)=(id ⊗εY)◦(κξ
X⊗id)
hence Eq (3.5) holds. □
Theorem 5.6. Fξ=(F, δ, ε, Fξ
2,F0, ϑξ, ιξ, κξ)is a quasi-monoidal comonad.
Remark 5.7.(CFξ,⊗,I,Aξ,Lξ,Rξ) is a monoidal category, where Aξ=(ϑξ)♯,Lξ=(ιξ)♯,Rξ=(κξ)♯.
Now consider a coquasitriangular quasi-monoidal comonad (F, σ). For any gauge transformation ξ
on F, for any X,Y∈ C, define
σξ
X,Y:FX ⊗FY δ2⊗δ2
//F3X⊗F3Yξ//FFX ⊗FFY σ//FY ⊗F X ξ∗−1
//Y⊗X.(5.5)
Proposition 5.8. With the above notations, σξis a coquasitriangular structure of Fξ. Thus F ξis
a coquasitriangular quasi-monoidal comonad. Hence CFξis a braided monoidal category with the
braiding τξ=(σξ)♯.
Proof. Firstly, it is straightforward to get that σξis ∗-invertible.
Secondly, to prove Eq (4.1), for any X,Y∈ C, we compute
Fξ
2◦σξ
FX,F Y ◦(δX⊗δY)
=F(ξ∗−1
Y,X)◦F2◦(δY⊗δX)◦ξFY,F X ◦ξ∗−1
FF Y,FFX ◦(δ2
Y⊗δ2
X)◦σFX,F Y ◦ξFFX,FFY ◦(δ2
X⊗δ2
Y)
=F(ξ∗−1
Y,X)◦F2◦σFFX,FF Y ◦ξF3X,F3Y◦(δ3
X⊗δ3
Y)
=F(ξ∗−1
Y,X)◦F(σFX,FY )◦F(ξFFX,FFY )◦F(ξ∗−1
F3X,F3Y)◦F(δ3
X⊗δ3
Y)◦F2◦(δX⊗δY)
=F(ξ∗−1
Y,X)◦F(σFX,FY )◦F(ξFFX,FFY )◦F(δ2
X⊗δ2
Y)◦F(ξ∗−1
FX,F Y )◦F2◦(δFX ⊗δF Y )
◦ξFFX,FF Y ◦(δ2
X⊗δ2
Y)
=F(σξ
Y,X)◦Fξ
2(FX,FY )◦(δX⊗δY).
Thirdly, for Eq (4.2), we have
ϑξ
Y,Z,X◦σξ
FX,F Y⊗F Z ◦(id ⊗Fξ
2)◦ϑξ
FFX,FF Y,FFZ ◦(δ2
X⊗δ2
Y⊗δ2
Z)
=(id ⊗ξ∗−1
Z,X)◦ξ∗−1
Y,FZ⊗FX ◦(id ⊗F2)◦ϑFY,FFZ,FFX ◦ξFF Y⊗F3Z,F3X◦(F2⊗id)
◦(ξF3Y⊗F4Z⊗id)◦(δ3
Y⊗δ4
Z⊗δ3
X)◦ξ∗−1
FY ⊗FZ,F X ◦σFFX,F(FY⊗F Z)◦ξF3X,F F(FY ⊗FZ))
◦(δ2
FX ⊗δ2
FY ⊗FZ )◦(id ⊗F(ξ∗−1
FY,FZ )) ◦(id ⊗F2)◦(id ⊗ξF3Y,F3Z)◦(id ⊗δ2
FY ⊗δ2
FZ )
◦(id ⊗ξ∗−1
F2Y,F2Z)◦ξ∗−1
FFX,F3Y⊗F3Z◦(id ⊗F2)◦ϑF3X,F4Y,F4Z◦ξF4X⊗F5Y,F5Z
◦(F2⊗id)◦(ξF5X,F6Y⊗id)◦(δ5
X⊗δ6
Y⊗δ5
Z)
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=(id ⊗ξ∗−1
Z,X)◦ξ∗−1
Y,FZ⊗FX ◦(id ⊗F2)◦ϑFY,FFZ,FFX ◦σF3X,FF Y⊗F3Z◦(id ⊗F2)
◦ϑF4X,F3Y,F4Z◦(δ2
FFX ⊗δ2
FY ⊗δ2
FF Z )◦ξF3X⊗F FY,F3Z◦(F2⊗id)◦(ξF4X,F3Y⊗id)
◦(δ4
X⊗δ3
Y⊗δ3
Z)
=(id ⊗ξ∗−1
Z,X)◦(id ⊗σFX,F Z )◦ξ∗−1
Y,FFX⊗F FZ ◦(id ⊗F2)◦ϑF Y,F3X,F3Z◦(σF4X,FFY ⊗id)
◦(δ2
FFX ⊗δFY ⊗δF FZ )◦ξF3X⊗F F Y,F3Z◦(F2⊗id)◦(ξF4X,F3Y⊗id)◦(δ4
X⊗δ3
Y⊗δ3
Z)
=(id ⊗ξ∗−1
Z,X)◦(id ⊗σFX,F Z )◦ξ∗−1
Y,FFX⊗F FZ ◦(id ⊗F2)◦ϑF Y,F3X,F3Z◦ξF2Y⊗F4X,F4Z
◦(F(σF4X,FF Y )⊗id)◦(F2⊗id)◦(δ4
FX ⊗δ3
FY ⊗δF Y )◦(ξFFX,F FY ⊗id)◦(id ⊗δ2
Z⊗δ2
Z)
=(id ⊗σξ
X,Z)◦ϑξ
Y,FX,FZ ◦(σξ
FFX,FF ⊗id)◦(δ2
X⊗δY⊗δZ).
At last, we can prove Eq (4.3) in a similar way. Thus the conclusion holds. □
Now consider the corepresentations of Fand Fξ.
Theorem 5.9. CFand CFξare isomorphic as monoidal categories. Further, if F is a coquasitriangular
quasi-monoidal comonad, then CFand CFξare braided isomorphic.
Proof. For any morphism fand objects M,Nin C, the monoidal functor is defined as follows
E=(E,Eξ
2,E0):(CF,⊗,I,A,L,R)→(CFξ,⊗,I,Aξ,Lξ,Rξ),
where
E(M) :=Mas an F-comodule,E(f) :=f,E0=idI,
and Eξ
2(M,N) : E(M)⊗E(N)→E(M⊗N) is given by
Eξ
2(M,N)=ξ♯:M⊗NρM⊗ρN
//FM ⊗F N ξM,N//M⊗N.
Obviously Eis well-defined.
Now we will check relation (2.1). Indeed, we have
Eξ
2(M,N⊗P)◦(id ⊗Eξ
2(N,P)) ◦Aξ
M,N,P
=ξM,N⊗P◦(id ⊗F2)◦(ρM⊗ρN⊗ρP)◦(id ⊗ξN,P)◦(id ⊗ρN⊗ρP)◦(id ⊗ξ∗−1
N,P)
◦ξ∗−1
M,FN ⊗FP ◦(id ⊗F2)◦ϑFM,FFN,FFP ◦ξFF M ⊗F3N,F3P◦(F2⊗id)◦(ξF3M,F4N⊗id)
◦(δ3
M⊗δ4
N⊗δ3
P)◦(ρM⊗ρN⊗ρP)
=ξM,N⊗P◦(ρM⊗F2)◦ξ∗−1
M,FN ⊗FP ◦(id ⊗F2)◦ϑFM,FFN,FFP ◦ξFF M ⊗F3N,F3P◦(F2⊗id)
◦(ξF3M,F4N⊗id)◦(δ3
M⊗δ4
N⊗δ3
P)◦(ρM⊗ρN⊗ρP)
=ξM,N⊗P◦ξ∗−1
FM,F(N⊗P)◦(δM⊗δN⊗P)◦(id ⊗F2)◦ϑFM,F N,F P ◦ξFFM ⊗FFN,FFP ◦(F2⊗id)
◦(ξF3M,F3N⊗id)◦(δ3
M⊗δ3
N⊗δ2
P)◦(ρM⊗ρN⊗ρP)
=ϑM,N,P◦ξFM ⊗FN,FP ◦(F2⊗id)◦(ξF2M,F2N⊗id)◦(δ2
M⊗δ2
N⊗δP)◦(ρM⊗ρN⊗ρP)
=E(AM,N,P)◦Eξ
2(M⊗N,P)◦(Eξ
2(M,N)⊗id),
which implies Eq (2.1).
Further, we can obtain (2.2) and (2.3) by straightforward computation. Hence the conclusion holds.
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Moreover, if σis a coquasitriangular structure of F, then from Theorem 5.6, (Fξ, σξ) is also a
coquasitriangular quasi-monoidal comonad. Then we have
Eξ
2(N,M)◦τξ
M,N
=ξN,M◦(ρN⊗ρM)◦ξ∗−1
N,M◦σFM,FN ◦ξF F M,FFN ◦(δ2
M⊗δ2
N)◦(ρM⊗ρN)
=(εN⊗εM)◦σFM,FN ◦ξF F M,FFN ◦(δ2
M⊗δ2
N)◦(ρM⊗ρN)
=σFM,FN ◦(ρN⊗ρM)◦ξM,N◦(ρM⊗ρN)=E(τM,N)◦Eξ
2(M,N),
which implies (E,Eξ
2,E0) is a braided monoidal functor. □
Example 5.10. With the notations in Example 3.10, if there is a convolution invertible linear form
χ∈(H⊗H)∗satisfying χ◦(αH⊗αH)=χ, then we have the following ∗-invertible natural transformation
in Hi,j(Veck)
ξX,Y:¨
HX ⊗¨
HY →X⊗Y,(x⊗a)⊗(y⊗b)7→ χ(αi
H(a), αj
H(b))α−1
X(x)⊗α−1
Y(y),
where a,b∈H,x∈X,y∈Yand X,Y∈ Hi,j(Veck). It is not hard to check that ¨
Hξ
2,ϑξ,ιξand κξin Eqs
(5.1)–(5.4) are deduced from the following
mχ(a⊗b)=Xχ∗−1(a1,b1)α−2
H(a21)α−2
H(b21)χ(a22 ,b22),
where χ∗−1means the convolution inverse of χ, and
ωχ(a,b,c)=Xχ∗−1(b11,c11 )χ∗−1(αH(a11), α−1
H(b121)c12 )
ω(a12, α−1
H(b122),c21 )χ(a21b21 , αH(c22))χ(a22,b22 ),
pχ(a)=Xp(a1)χ(1H,a2),qχ(a)=Xq(a1)χ(a2,1H),
respectively. Thus from Example 3.10 and Theorem 5.6, Hχ=(H, αH,mχ,1H,∆, ε, ωχ,pχ,qχ) is also
a Hom-coquasi-bialgebra.
Example 5.11. With the notations in Example 3.12, note that the BαB=(B, αB, αB◦µ, 1B,∆◦αB, ε) is
a Hom-bialgebra, and it can be seen as a Hom-coquasi-bialgebra BαB=(B, αB, αB◦µ, 1H,∆◦αB, ε, ε ⊗
ε⊗ε, ε, ε). If there are αB-invariant and dual central dual group-like k-linear forms p,q∈B∗, then we
have the following gauge transformation χ∈(B⊗B)∗by
χ(a,b)=q∗−1(a)p(b),where a,b∈B.
Obviously Bχ
αB=Bp,q
αB.
Acknowledgments
The work was partially supported by the National Natural Science Foundation of China (No.
11801304, 11871301), and the Taishan Scholar Project of Shandong Province (No. tsqn202103060).
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3170
Conflict of interest
The authors declare there is no conflict of interest.
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