Structural properties of urban bus and subway networks of Madrid

  • The goal of this research is to estimate different parameters in the urban bus and the subway networks of Madrid. The obtained results will allow learning more about both types of networks: modularity, most important stops, sensitivity in the district networks (districts with highest and lowest sensitivity), bus line concentration by detected communities, communication capacity for these networks (districts with the greatest and less number of inner and external communications), and relation between network and dweller density by district. This study can help to improve the transport networks: reducing the district sensitivity, adding new stops or routes, etc.

    Citation: Mary Luz Mouronte, Rosa María Benito. Structural properties of urban bus and subway networks of Madrid[J]. Networks and Heterogeneous Media, 2012, 7(3): 415-428. doi: 10.3934/nhm.2012.7.415

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  • The goal of this research is to estimate different parameters in the urban bus and the subway networks of Madrid. The obtained results will allow learning more about both types of networks: modularity, most important stops, sensitivity in the district networks (districts with highest and lowest sensitivity), bus line concentration by detected communities, communication capacity for these networks (districts with the greatest and less number of inner and external communications), and relation between network and dweller density by district. This study can help to improve the transport networks: reducing the district sensitivity, adding new stops or routes, etc.


    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 [2,3,4]. 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 $ F $ is 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 [7,8,9,10,11] 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 multiplication are replaced by a weaker property, called coquasi-associativity. The multiplication is associative up to conjugation by a convolution invertible linear form $ \omega \in (H {\otimes} H {\otimes} H)^\ast $, called the coassociator. Note that the definition of a coquasi-bialgebra is not selfdual, and the category of (left or right) comodules 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 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 $ F $ is 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, $ \pi $-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.

    Throughout the paper, we let $ k $ be a fixed field and $ char(k) = 0 $ and $ Vec_k $ be the category of finite dimensional $ k $-spaces. All the algebras and coalgebras, modules and comodules are supposed to be in $ Vec_k $. For the comultiplication $ {\Delta} $ of a $ k $-space $ C $, we use the Sweedler-Heyneman's notation: $ \Delta(c) = \sum c_{1}{\otimes} c_{2} $ for any $ c\in C $.

    Let $ (\mathcal {C}, {\otimes}, I, a, l, r) $ and $ (\mathcal {C}', {\otimes}', I', a', l', r') $ be two monoidal categories. Recall that a quasi-monoidal functor from $ \mathcal {C} $ to $ \mathcal {C}' $ is a triple $ (F, F_2, F_0) $, where $ F:\mathcal {C}\rightarrow \mathcal {C}' $ is a functor, $ F_2: F\otimes' F \rightarrow F\otimes $ is a natural transformation, and $ F_0:I'\rightarrow FI $ is a morphism in $ \mathcal {C}' $.

    Furthermore, if the following equations hold for any $ X, Y, Z \in \mathcal {C} $:

    $ F2(X,YZ)(idFXF2(Y,Z))aFX,FY,FZ=F(aX,Y,Z)F2(XY,Z)(F2(X,Y)idFZ),
    $
    (2.1)
    $ F(lX)F2(I,X)(F0idFX)=lFX,
    $
    (2.2)
    $ F(rX)F2(X,I)(idFXF0)=rFX,
    $
    (2.3)

    then $ F = (F, F_2, F_0) $ is called a monoidal functor.

    Let $ \mathcal {C} $ be a category, $ F $: $ \mathcal {C}

    \mathcal {C} $ be a functor. Recall from [16] or [17] that if there exist natural transformations $ \delta $: $ F \rightarrow FF $ and $ \varepsilon $: $ F \rightarrow id_{\mathcal {C}} $, such that the following identities hold

    $ F\delta \circ \delta = \delta F\circ \delta,\; \; \mbox{and}\; \; id_F = F\varepsilon\circ \delta = \varepsilon F\circ \delta, $

    then we call the triple $ (F, \delta, \varepsilon) $ a comonad on $ \mathcal {C} $.

    Let $ X \in \mathcal {C} $, and $ (F, \delta, \varepsilon) $ a comonad on $ \mathcal {C} $. If there exists a morphism $ \rho^X $: $ X

    FX $, satisfying

    $ F{\rho^X}\circ \rho^X = \delta_X\circ \rho^X ,\; \; \mbox{and}\; \; \varepsilon_X\circ \rho^X = id_X, $

    then we call the couple $ (X, \rho^X) $ an F-comodule.

    A morphism between $ F $-comodules $ g $: $ X \rightarrow X' $ is called $ F $-colinear, if $ g $ satisfies: $ Fg \circ \rho^X = \rho^{X'} \circ g $. The category of $ F $-comodules is denoted by $ \mathcal {C}^F $.

    Let $ (\mathcal {C}, {\otimes}, I, a, l, r) $ be a monoidal category, $ (F, \delta, {\varepsilon}) $ be a comonad on $ \mathcal {C} $, and $ (F, F_2, F_0): \mathcal {C}\rightarrow \mathcal {C} $ be a monoidal functor. Then recall from [18] or [19] that $ F $ is called a monoidal comonad (or a bicomonad) on $ \mathcal {C} $ if $ \delta $ and $ {\varepsilon} $ are both monoidal natural transformations, i.e. the following compatibility conditions hold for any $ X, Y \in \mathcal {C} $:

    $ \left\{(C1)F(F2(X,Y))F2(FX,FY)(δXδY)=δXYF2(X,Y),(C2)εXYF2(X,Y)=εXεY,(C3)F(F0)F0=δIF0,(C4)εIF0=idI.

    \right. $

    Given a category $ \mathcal {C} $ and a positive integer $ n $, we denote $ \mathcal {C}^n = \mathcal {C} \times\mathcal {C} \times \cdots \times \mathcal {C} $ the $ n $-tuple cartesian product of $ \mathcal {C} $. If $ F $ is a comonad on $ \mathcal {C} $, then $ F^{\times n} $ (the $ n $-tuple cartesian product of $ F $) is a comonad on $ \mathcal {C}^n $, and we have $ {\mathcal {C}^n}^{F^{\times n}} = (\mathcal {C}^F)^n $.

    Assume that $ U: \mathcal {C}^F\rightarrow \mathcal {C} $ is the forgetful functor and $ P, Q: \mathcal {C}^n\rightarrow \mathcal {D} $ are functors. Then from [[9], Proposition 4.1], we have the following results.

    Lemma 2.1. There is a canonical bijection:

    $ Nat(PU^{\times n},QU^{\times n})\cong Nat(PF^{\times n},Q). $

    Proof. Define $?^\flat:Nat(PU^{\times n}, QU^{\times n})\rightarrow Nat(PF^{\times n}, Q) $, $ f\mapsto f^\flat $, by

    $ f(X1,,Xn):P(FX1××FXn)f(FX1,,FXn)Q(FX1××FXn)Q(εX1,,εXn)Q(X1××Xn),
    $

    and $?^\sharp:Nat(PF^{\times n}, Q)\rightarrow Nat(PU^{\times n}, QU^{\times n}) $, $ {\alpha}\mapsto {\alpha}^\sharp $, by

    $ α(M1,,Mn):P(M1××Mn)P(ρM1,,ρMn)P(FM1××FMn)α(M1,,Mn)Q(M1××Mn),
    $

    for any $ f \in Nat(PU^{\times n}, QU^{\times n}) $, $ \alpha \in Nat(PF^{\times n}, Q) $ and $ X_i \in \mathcal {C} $, $ (M_i, \rho^{M_i}) \in \mathcal {C}^F $. It is easy to check that $?^\flat $ and $?^\sharp $ are well defined and are inverse with each other.

    Let $ P, Q, R:\mathcal {C}^n\rightarrow \mathcal {D} $ be functors. For any $ \alpha \in Nat(PF^{\times n}, Q) $ and $ \beta \in Nat(QF^{\times n}, R) $, define their convolution product $ \beta\ast{\alpha} \in Nat(PF^{\times n}, R) $ by setting, for any objects $ X_1, \cdots, X_n $ in $ \mathcal {C} $,

    $ βαX1,,Xn=βX1,,XnαFX1,,FXnP(δX1,,δXn).
    $

    We say that $ \alpha \in Nat(PF^{\times n}, Q) $ is $ \ast $-invertible if there exists $ \beta \in Nat(QF^{\times n}, P) $ such that $ {\beta} \ast {\alpha} = P(\varepsilon^{\times n}) \in Nat(PF^{\times n}, P) $ and $ {\alpha} \ast {\beta} = Q(\varepsilon^{\times n}) \in Nat(QF^{\times n}, Q) $. We denote $ {\beta} $ by $ {\alpha}^{\ast-1} $.

    Proposition 2.2. The $ \ast $-invertible elements in $ Nat(PF^{\times n}, Q) $ are in corresponding with the natural isomorphisms in $ Nat(PU^{\times n}, QU^{\times n}) $.

    Proof. Suppose that $ f \in Nat(PU^{\times n}, QU^{\times n}) $ is a natural isomorphism. Then we immediately get that $ (f^\flat)^{\ast-1} = (f^{-1})^\flat $.

    Conversely, if $ {\alpha} \in Nat(PF^{\times n}, Q) $ is $ \ast $-invertible, then $ {{\alpha}^\sharp}^{-1} = ({\alpha}^{\ast-1})^\sharp $.

    Suppose that $ (\mathcal {C}, {\otimes}, I, a, l, r) $ is a monoidal category, $ F:\mathcal {C}\rightarrow \mathcal {C} $ is a functor, $ (F, \delta, {\varepsilon}) $ is a comonad and $ (F, F_2, F_0) $ is a quasi-monoidal functor.

    Lemma 3.1. If we define the $ F $-coaction on $ I $ by $ F_0 $, anddefine the $ F $-coaction on $ M {\otimes} N $ (as the tensor product in $ \mathcal {C} $) for any $ (M, \rho^M), (N, \rho^N) \in \mathcal {C}^F $ by:

    $ \rho^{M {\otimes} N}: M {\otimes} N \mathop \to \limits^{{\rho^M {\otimes} \rho^N}} FM {\otimes} FN \mathop \to \limits^{{F_2(M,N)}} F(M {\otimes} N), $

    then $ (I, F_0) $ and $ (M {\otimes} N, \rho^{M {\otimes} N}) $ are all objects in $ \mathcal {C}^F $ if and only ifthe compatibility conditions Eqs (C1)–(C4) hold.

    Proof. It is straightforward to check that Eqs (C1) and (C2) hold if and only if $ (M {\otimes} N, \rho^{M {\otimes} N}) \in \mathcal {C}^F $, Eqs (C3) and (C4) hold if and only if $ (I, F_0)\in \mathcal {C}^F $.

    From now on, we always assume that the compatibility conditions Eqs (C1)–(C4) hold.

    We suppose that there are natural transformations $ \vartheta: (\_ {\otimes} \_) {\otimes} \_ {\circ} F^{\times 3} \Rightarrow \_ {\otimes} (\_ {\otimes} \_): \mathcal {C}^{\times 3} \rightarrow \mathcal {C} $, and $ \iota: I {\otimes} F\_ \Rightarrow \_: \mathcal {C} \rightarrow \mathcal {C} $, $ \kappa:F\_ {\otimes} I \Rightarrow \_: \mathcal {C} \rightarrow \mathcal {C} $. From Lemma 2.1, for any objects $ (M, \rho^M), (N, \rho^N), (P, \rho^P) \in \mathcal {C}^F $, $ {\vartheta}, {\iota}, {\kappa} $ can induce the following natural transformations

    $ AM,N,P=ϑM,N,P,LM=ιM,RM=κM.
    $

    Conversely, if there are natural transformations $ A : (\_ {\otimes} \_) {\otimes} \_ \Rightarrow \_ {\otimes} (\_ {\otimes} \_): \mathcal {C}^{\times 3} \rightarrow \mathcal {C} $ and $ L: I {\otimes} \_ \Rightarrow id: \mathcal {C} \rightarrow \mathcal {C} $, $ R:\_ {\otimes} I \Rightarrow id: \mathcal {C} \rightarrow \mathcal {C} $, then from Lemma 2.1, for any $ X, Y, Z \in \mathcal {C} $, they can induce natural transformations

    $ ϑX,Y,Z=AX,Y,Z,ιX=LX,κX=RX.
    $

    Next, we will discuss when $ A $ is the associativity constraint and $ L, R $ are the unit constraints in $ \mathcal {C}^F $.

    Lemma 3.2. $ A $, $ L $ and $ R $ are isomorphisms if and only if $ {\vartheta} $, $ {\iota} $ and $ {\kappa} $ are $ \ast $-invertible.

    Proof. Straightforward from Proposition 2.2.

    Lemma 3.3. $ A $ is $ F $-colinear if and only if $ {\vartheta} $ satisfies

    (3.1)

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

    Proof. $ \Leftarrow) $: Since the following diagram

    is commutative for any $ M, N, P \in \mathcal {C}^F $, $ A_{M, N, P} $ is $ F $-colinear.

    $ \Rightarrow) $: Notice that $ A_{FX, FY, FZ} $ is $ F $-colinear for any $ X, Y, Z \in \mathcal {C} $, then it follows

    $ F(εXεYεZ)FAFX,FY,FZρ(FXFY)FZ=F(εXεYεZ)ρFX(FYFZ)AFX,FY,FZ.
    $

    After a direct computation, we obtain (3.1).

    Lemma 3.4. $ A $ satisfies the Pentagon Axiom in $ \mathcal {C}^F $ if and only if $ {\vartheta} $ satisfies

    $ (idϑX,Y,Z)ϑW,FXFY,FZ(idF2id)(ϑFW,FFX,FFYid)(δWδ2Xδ2YδZ)=ϑW,X,YZ(ididF2)ϑFWFX,FY,FZ(F2idid)(δWδXδYδZ)
    $
    (3.2)

    for any $ W, X, Y, Z \in \mathcal {C} $.

    Proof. $ \Leftarrow) $: Since we have

    $ (idϑN,P,Q)(idρNρPρQ)ϑM,NP,Q(idF2id)(ρMρNρPρQ)(ϑM,N,Pid)(ρMρNρPid)=(idϑN,P,Q)ϑM,FNFP,FQ(idF(ρNρP)ρQ)(idF2id)(ϑFM,FN,FPid)(FρMFρNFρPρQ)(ρMρNρPid)=(idϑN,P,Q)ϑM,FNFP,FQ(idF2id)(ϑFM,FFN,FFPid)(δMδ2Nδ2PδQ)(ρMρNρPρQ)=ϑM,N,PQ(ididF2)ϑFMFN,FP,FQ(F2idid)(δMδNδPδQ)(ρMρNρPρQ)=ϑM,N,PQ(ididF2)(ρMρNρPρQ)ϑMN,P,Q(F2idid)(ρMρNρPρQ)
    $

    for any $ M, N, P, Q \in \mathcal {C}^F $, $ A $ satisfies the Pentagon Axiom.

    $ \Rightarrow) $: For any $ W, X, Y, Z \in \mathcal {C} $, we have cofree $ F $-comodules $ FW, FX, FY, FZ $. Consider the following Pentagon Axiom:

    $ AFW,FX,FYFZAFWFX,FY,FZ=(idAFX,FY,FZ)AFW,FXFY,FZ(AFW,FX,FYid).
    $

    Applying $ {\varepsilon}_W {\otimes} {\varepsilon}_X {\otimes} {\varepsilon}_Y {\otimes} {\varepsilon}_Z $ to both sides of the above identity, we get Diagram (3.2).

    Lemma 3.5. For any $ X \in \mathcal {C} $,

    (1) $ L $ is $ F $-colinear if and only if $ {\iota} $ satisfies

    (3.3)

    (2) $ R $ is $ F $-colinear if and only if $ {\kappa} $ satisfies

    (3.4)

    Proof. We only prove (1).

    $ \Leftarrow) $: From the following commutative diagram

    for any $ M \in \mathcal {C}^F $, $ L_{M} $ is $ F $-colinear.

    $ \Rightarrow) $: Conversely, since $ FX $ is an $ F $-comodule and $ L_{FX} $ is $ F $-colinear for any $ X \in \mathcal {C} $, it is directly to get Diagram (3.3).

    Lemma 3.6. $ A $, $ L $ and $ R $ satisfy the Triangle Axiom in $ \mathcal {C}^F $ if and only if $ {\vartheta} $, $ {\iota} $ and $ {\kappa} $ satisfy

    (3.5)

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

    Proof. $ \Leftarrow) $: For any $ M, N\in \mathcal {C}^F $, we compute

    $ (idMιN)(idMidIρN)ϑM,I,N(ρMF0ρN)=(idMιN)ϑM,I,FN(idFMF0δN)(ρMidIρN)=(idMεN)(κMidFN)(ρMidIρN)=(κMidN)(ρMidIidN)
    $

    thus the Triangle Axiom in $ \mathcal {C}^F $ holds.

    $ \Rightarrow) $: Conversely, for any $ X, Y \in \mathcal {C} $, since we have

    it is a direct computation to get Diagram (3.5).

    Definition 3.7. Let $ (\mathcal {C}, \otimes, I, a, l, r) $ be a monoidal category on which $ (F, {\delta}, {\varepsilon}) $ is a monad and $ (F, F_2, F_0) $ is a quasi-monoidal functor such that the compatible conditions Eqs (C1)–(C4) are satisfied. If there are $ \ast $-invertible natural transformations $ {\vartheta} $, $ {\iota} $ and $ {\kappa} $ satisfying (3.1)–(3.5), then we call $ (F, {\delta}, {\varepsilon}, F_2, F_0, {\vartheta}, {\iota}, {\kappa}) $ a quasi-monoidal comonad on $ \mathcal {C} $,

    Then by Lemma 3.1–3.6, one gets the following result.

    Theorem 3.8. Let $ (\mathcal {C}, \otimes, I, a, l, r) $ be a monoidal category on which $ (F, {\delta}, {\varepsilon}) $ is a monad and $ (F, F_2, F_0) $ is a quasi-monoidal functor such that the compatible conditions Eqs (C1)–(C4) is satisfied. Then there exist natural transformations $ {\vartheta} $, $ {\iota} $ and $ {\kappa} $ such that $ (F, {\delta}, {\varepsilon}, F_2, F_0, {\vartheta}, {\iota}, {\kappa}) $ is a quasi-monoidal comonad if and only if there are natural transformations $ A $, $ L $ and $ R $ such that $ (\mathcal {C}^F, {\otimes}, I, A, L, R) $ is a monoidal category.

    Example 3.9. Let $ (\mathcal {C}, \otimes, I, a, l, r) $ be a monoidal category on which $ (F, {\delta}, {\varepsilon}) $ is a monad and $ (F, F_2, F_0) $ is a quasi-monoidal functor such that the compatible conditions Eqs (C1)–(C4) are satisfied. If we define

    $ {\vartheta}_{X,Y,Z} = a^\flat_{X,Y,Z},\; \; {\iota}_{X} = l^\flat_{X},\; \; {\kappa}_{X,Y,Z} = r^\flat_{X,Y,Z} $

    for any $ X, Y, Z \in \mathcal{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, F_2, F_0) $ is a monoidal functor. That means, the quasi-monoidal comonad $ (F, {\delta}, {\varepsilon}, F_2, F_0, {\vartheta}, {\iota}, {\kappa}) $ is exactly a monoidal comonad.

    Example 3.10. Recall from [9] or [10], we consider the following monoidal category $ \overline{\mathcal{H}}^{i, j}(Vec_k) $ for any $ i, j \in \mathbb{Z} $:

    $ \bullet $ the objects of $ \overline{\mathcal{H}}^{i, j}(Vec_k) $ are pairs $ (X, {\alpha}_X) $, where $ X\in Vec_k $ and $ {\alpha}_X\in Aut_{k}(X) $;

    $ \bullet $ the morphism $ f:(X, {\alpha}_X)\rightarrow (Y, {\alpha}_Y) $ in $ \overline{\mathcal{H}}^{i, j}(Vec_k) $ is a $ k $-linear map from $ X $ to $ Y $ such that $ {\alpha}_Y\circ f = f\circ {\alpha}_X $;

    $ \bullet $ the monoidal structure is given by

    $ (X,{\alpha}_X)\otimes(Y,{\alpha}_Y) = (X\otimes Y,{\alpha}_X\otimes{\alpha}_Y), $

    and the unit is $ (k, id_{k}) $;

    $ \bullet $ the associativity constraint $ a $, the unit constraints $ l $ and $ r $ are given by

    $ aX,Y,Z:(xy)zαi+1X(x)(yαj1Z(z));lX(1kx)=αj+1X(x),rX(x1k)=αi+1X(x),XVeck.
    $

    Now assume that $ (H, {\alpha}_H) $ is an object in $ \overline{\mathcal{H}}^{i, j}(Vec_k) $, $ m_H: H{\otimes} H \rightarrow H $ (with notation $ m_H(a {\otimes} b) = ab $), $ \eta_H: k\rightarrow H $ (with notation $ \eta_H(1_k) = 1_H $), and $ \Delta_H: H \rightarrow H{\otimes} H $ (with notation $ \Delta_H(h) = h_1 {\otimes} h_2 $), and $ \varepsilon_H: H\rightarrow k $ are all morphisms in $ \overline{\mathcal{H}}^{i, j}(Vec_k) $. Further, we write

    $ \ddot{H} = \_{\otimes} H:\overline{\mathcal{H}}^{i,j}(Vec_k)\rightarrow \overline{\mathcal{H}}^{i,j}(Vec_k),\; \; \; \; (X, {\alpha}_X)\mapsto ( X {\otimes} H, {\alpha}_X {\otimes} {\alpha}_H) $

    for the right tensor functor of $ H $.

    If we define the following structures on $ \ddot{H} $:

    $ \bullet $ $ \delta:\ddot{H}\rightarrow \ddot{H}\ddot{H} $ and $ \epsilon:\ddot{H}\rightarrow id_{\overline{\mathcal{H}}^{i, j}(Vec_k)} $ are defined by

    $ δX:xh(αX(x)h1)α1H(h2),ϵX:xhεH(h)α1X(x);
    $

    $ \bullet $ $ \ddot{H}_2:\ddot{H} {\otimes} \ddot{H} \rightarrow \ddot{H} {\otimes} $ and $ \ddot{H}_0: k\rightarrow \ddot{H}(k) $ are given by

    $ ¨H2(X,Y):(xa)(yb)(xy)αiH(a)αjH(b),¨H0(1k)=1k1H,
    $

    for any $ X, Y \in \overline{\mathcal{H}}^{i, j}(Vec_k) $. Then obviously $ \ddot{H} = (\ddot{H}, \delta, \epsilon) $ forms a comonad on $ \overline{\mathcal{H}}^{i, j}(Vec_k) $ if and only if $ (H, {\alpha}_H, \Delta_H, \varepsilon_H) $ is a Hom-coalgebra over $ k $, Eqs (C1)–(C4) hold if and only if $ m_H $ and $ \eta_H $ are all morphisms of Hom-coalgebras.

    Suppose that there are $ {\alpha}_H $-invariant convolution invertible linear forms $ \omega\in (H {\otimes} H {\otimes} H)^\ast $ and $ p, q \in H^\ast $, then we can define the following $ \ast $-invertible natural transformations

    $ ϑX,Y,Z:((xa)(yb))(zc)ω(α2iH(a),αi+jH(b),αj1H(c))(αiX(x)(α1Y(y)αj2Z(z))),ιX:1k(xa)p(a)αjX(x),κX:(xa)1kq(a)αiX(x),
    $

    where $ a, b, c \in H $, $ x \in X $, $ y \in Y $, $ z \in Z $ and $ X, Y, Z \in Vec_k $. Thus we immediately get that $ {\vartheta} $ satisfies Eq (3.1) if and only if $ \omega $ satisfies

    $ αH(a1)(b1c1)ω(a2,b2,c2)=ω(a1,b1,c1)(a2b2)αH(c2);
    $
    (3.6)

    $ {\vartheta} $ satisfies Eq (3.2) if and only if $ \omega $ satisfies

    $ ω(αH(a1),αH(b1),c1d1)ω(a2b2,αH(c2),αH(d2))=ω(b1,c1,αH(d1))ω(αH(a1),α1H(b21)α1H(c21),αH(d2))ω(αH(a2),b22,c22);
    $
    (3.7)

    $ {\iota} $ satisfies Eq (3.3) and $ {\kappa} $ satisfies Eq (3.4) if and only if $ p, q $ satisfy

    $ p(a1)1Ha2=αH(a1)p(a2),q(a1)a21H=αH(a1)q(a2);
    $
    (3.8)

    $ {\vartheta}, {\iota} $ and $ {\kappa} $ satisfy Eq (3.5) if and only if $ \omega $, $ p $ and $ q $ satisfy

    $ ω(a,1H,b)=q(a)p1(b).
    $
    (3.9)

    This means, $ \ddot{H} = (\ddot{H}, {\delta}, \epsilon, \ddot{H}_2, \ddot{H}_0, {\vartheta}, {\iota}, {\kappa}) $ forms a quasi-monoidal comonad on $ \overline{\mathcal{H}}^{i, j}(Vec_k) $ if and only if $ H = (H, {\alpha}_H, m_H, \eta_H, {\Delta}_H, {\varepsilon}_H, \omega, 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) = (\overline{\mathcal{H}}^{i, j}(Vec_k))^{\ddot{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((mn)p)=ω(α2i(m1),αi+j(n1),αj1(p1))αiM(m0)(α1N(n0)αj2P(p0)),LM(1km)=p(m1)αjM(m0),RM(m1k)=q(m1)αiM(m0),
    $

    where $ m \in M $, $ n \in N $, $ p \in P $, $ M, N, P \in Corep(H) $.

    Example 3.11. Under the consideration of Example 3.10, if all the Hom-structure maps $ {\alpha} $ are identity maps, then the Hom-coquasi-bialgebra is exactly the Majid algebra (also called a Majid algebra, see [13] for details) over $ k $.

    Example 3.12. Let $ B = (B, \mu, 1_B, {\Delta}, {\varepsilon}) $ be a bialgebra over $ k $, $ {\alpha}_B:B\rightarrow B $ be an endo-isomrophism. Recall that a $ k $-linear form $ g\in B^\ast $ is called

    (1) dual central if $ g(x_1)x_2 = x_1g(x_2) $ for any $ x \in B $;

    (2) dual group-like if it is convolution invertible and satisfies $ g(xy) = g(x)g(y) $ for any $ x, y \in B $;

    (3) $ {\alpha}_B $-invariant if $ g({\alpha}_B(x)) = g(x) $.

    Now suppose that $ p, q \in B^\ast $ are all dual central dual group-like and $ {\alpha}_B $-invariant linear forms. Define a $ k $-linear form $ \omega:B {\otimes} B{\otimes} B\rightarrow k $ by

    $ \omega(x,y,z) = p(x){\varepsilon}(y)q^{\ast-1}(z),\; \; \; \; {\rm{for\; any\; }}x,y,z \in B, $

    define the new multiplication $ \mu^{{\alpha}_B} $ and comultiplication $ {\Delta}^{{\alpha}_B} $ by

    $ μαB=αBμ,ΔαB=ΔαB.
    $

    Then it is a direct calculation to check that $ {\alpha}_B, \omega, p, q $ satisfy Eqs.(3.6) - (3.9) (under $ \mu^{{\alpha}_B} $ and $ {\Delta}^{{\alpha}_B} $), hence $ B^{p, q}_{{\alpha}_B} = (B, {\alpha}_B, \mu^{{\alpha}_B}, 1_B, {\Delta}^{{\alpha}_B} $, $ {\varepsilon}, \omega, p, q) $ forms a nontrivial Hom-coquasi-bialgebra.

    Recall that a braiding in a monoidal category $ (\mathcal {C}, \otimes, I, a, l, r) $ is a natural isomorphism $ {\tau} $: $ \otimes \Rightarrow \otimes^{op}: \mathcal {C} \times \mathcal {C}\rightarrow \mathcal {C} $ such that the following identities hold

    $ aY,Z,XτX,YZaX,Y,Z=(idYτX,Z)aY,X,Z(τX,YidZ),
    $
    (B1)
    $ a1Z,X,YτXY,Za1X,Y,Z=(τX,ZidY)a1X,Z,Y(idXτY,Z)
    $
    (B2)

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

    Now let $ F $ be a quasi-monoidal comonad on $ \mathcal {C} $. Suppose that there is a natural transformation $ \sigma $: $ \otimes{\circ}(F\times F) \Rightarrow \otimes^{op}: \mathcal {C}^{\times 2}\rightarrow \mathcal {C} $. From Lemma 2.1, for any objects $ M, N $ in $ \mathcal {C}^F $, $ \sigma $ can induce a natural transformation

    $ {\tau}_{M,N} = \sigma^\sharp_{M,N}: M\otimes N \mathop \to \limits^{{ \rho^M {\otimes} \rho^N} } FM\otimes FN \mathop \to \limits^{{\sigma_{M,N}}} N\otimes M. $

    Conversely, if there exists $ {\tau} : \otimes \Rightarrow \otimes^{op}: \mathcal {C} \times \mathcal {C}\rightarrow \mathcal {C} $, then from Lemma 2.1, for any $ X, Y \in \mathcal {C} $, $ {\tau} $ can induce the following

    $ \sigma_{X,Y} = {\tau}^\flat_{X,Y}: FX\otimes FY\mathop \to \limits^{{{\tau}_{FX,FY}}} FY \otimes FX \mathop \to \limits^{{{\varepsilon}_{Y} \otimes {\varepsilon}_{X}}} Y\otimes X. $

    Next we will discuss when $ {\tau} $ is a braiding in $ \mathcal {C}^F $.

    Lemma 4.1. $ {\tau} $ is an isomorphism if and only if $ \sigma $ is $ \ast $-invertible.

    Proof. Straightforward from Proposition 2.2.

    Lemma 4.2. $ {\tau} $ is $ F $-colinear if and only if $ {\sigma} $ satisfies

    (4.1)

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

    Proof. $ \Leftarrow) $: We compute

    for any $ M, N \in \mathcal {C}^F $. Hence $ {\tau}_{M, N} $ is $ F $-colinear.

    $ \Rightarrow) $: Conversely, notice that $ {\tau}_{FX, FY} $ is $ F $-colinear for any $ X, Y \in \mathcal {C} $, we have

    $ F(εYεX)FτFX,FYρFXFY=F(εYεX)ρFYFXτFX,FY,
    $

    which implies Diagram (4.1) holds.

    Lemma 4.3. Diagram (B1) holds in $ \mathcal {C}^F $ if and only if $ \sigma $ satisfies

    $ ϑY,Z,XσFX,FYFZ(idF2)ϑFFX,FFY,FFZ(δ2Xδ2Yδ2Z)=(idσX,Z)ϑY,FX,FZ(σFFX,FYid)(δ2XδYδZ)
    $
    (4.2)

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

    Proof. $ \Leftarrow) $: Take $ X = M $, $ Y = N $, $ Z = P $ for any $ F $-comodules $ M, N, P $. Multiplied by $ \rho^M {\otimes} \rho^N {\otimes} \rho^P $ right on both sides of Eq (4.2), we immediately get Diagram (B1).

    $ \Rightarrow) $: Since Diagram (B1) is commutative for any $ FX, FY, FZ \in \mathcal {C} $, multiplied by $ {\varepsilon} {\otimes} {\varepsilon} {\otimes} {\varepsilon} $ left on both sides of the above equation, we get Eq (4.2).

    Lemma 4.4. For any $ X, Y, Z \in \mathcal {C} $, Diagram (B2) holds in $ \mathcal {C}^F $ if and only if $ \sigma $ satisfies

    $ ϑ1Z,X,YσFXFY,FZ(F2id))ϑ1FFX,FFY,FFZ(δ2Xδ2Yδ2Z)=(σX,Zid)ϑ1FX,FZ,Y(idσFY,FFZ)(δXδYδ2Z),
    $
    (4.3)

    where $ {\vartheta}^{\ast -1} $ means the $ \ast $-inverse of $ {\vartheta} $.

    Proof. The proof is similar to Lemma 4.3.

    Definition 4.5. Let $ (F, {\delta}, {\varepsilon}, F_2, F_0, {\vartheta}, {\iota}, {\kappa}) $ be a quasi-monoidal comonad on a monoidal category $ \mathcal {C} $. If there is a $ \ast $-invertible natural transformation $ \sigma \in Nat(F\otimes F, \otimes^{op}) $, satisfying Eqs (4.1)–(4.3) for any $ X, Y, Z \in \mathcal {C} $, then $ {\sigma} $ is called a coquasitriangular structure of $ F $, and $ (F, \sigma) $ is called a coquasitriangular quasi-monoidal comonad.

    Combining Lemma 4.1–Definition 4.5, we obtain the following result.

    Theorem 4.6. Let $ (F, {\delta}, {\varepsilon}, F_2, F_0, {\vartheta}, {\iota}, {\kappa}) $ be a quasi-monoidal comonad on a monoidal category $ \mathcal {C} $. Then $ \mathcal {C}^F $ is a braided monoidal category if and only if there exists a natural transformation$ \sigma: F{\otimes} F \rightarrow {\otimes}^{op} $ such that $ (F, \sigma) $ is a coquasitriangular quasi-monoidal comonad. Further, the braiding in $ \mathcal {C}^F $is given by $ {\tau} = \sigma^\sharp $.

    Corollary 4.7. Let $ (F, {\sigma}) $ be a coquasitriangular quasi-monoidal comonad on a monoidal category $ \mathcal {C} $. Then for any $ X, Y, Z \in \mathcal {C} $, $ {\sigma} $ satisfies the following generalized Yang-Baxter equation:

    $ (idσX,Y)ϑZ,FX,FY(σFFX,FZid)ϑ1F3X,FFZ,FFY(idσF3Y,F3Z)ϑF4X,F4Y,F4Z(δ4Xδ4Yδ4Z)=ϑZ,Y,X(σFY,FZid)ϑ1FFY,FFZ,FX(idσFFX,F3Z)ϑF3Y,F3X,F4Z(σF4X,F4Yid)(δ4Xδ4Yδ4Z).
    $

    Proof. Straightforward.

    Example 4.8. If $ F $ is a monoidal comonad on $ \mathcal{C} $, and $ {\sigma}:{\otimes} {\circ} F^{\times 2} \Rightarrow {\otimes}^{op} $ is a $ \ast $-invertible natural transformation satisfying Eqs (4.1)–(4.3), then $ (F, {\sigma}) $ is exactly a coquasitriangular monoidal comonad (see [9], Definition 4.12).

    Example 4.9. With the notations in Example 3.10, if $ Q \in (H {\otimes} H)^\ast $ is $ {\alpha}_H $-invariant and convolution invertible, then we have the following $ \ast $-invertible natural transformation

    $ {\sigma}_{X,Y}:\ddot{H}X {\otimes} \ddot{H}Y\rightarrow Y {\otimes} X,\; \; \; \\ (x {\otimes} a){\otimes} (y {\otimes} b)\mapsto Q({\alpha}_H^i(a),{\alpha}_H^j(b)){\alpha}_Y^{j-i-1}(y) {\otimes} {\alpha}_X^{i-j-1}(x), $

    where $ x \in X $, $ y \in Y $ and $ X, Y \in \overline{\mathcal{H}}^{i, j}(Vec_k) $. Thus we immediately get that $ {\sigma} $ satisfies Eq (4.1) if and only if $ Q $ satisfies

    $ \sum Q(a_{1}, b_{1})a_{2}b_{2} = \sum b_{1}a_{1} Q(a_{2}, b_{2}), $

    $ {\sigma} $ satisfies Eqs (4.2) and (4.3) if and only if $ Q $ satisfies

    $ ω(b1,c1,a1)Q(αH(a21),b21c21)ω(a22,b22,c22)=Q(a1,c1)ω(b1,α1H(a21),c2)Q(α1H(a22),b2),ω1(c1,a1,b1)Q(a21b21,αH(c21))ω1(a22,b22,c22)=Q(a1,c1)ω1(a2,α1H(c21),b1)Q(b2,α1H(c22)),
    $

    where $ a, b, c \in H $. That is, $ (\ddot{H}, {\sigma}) $ 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) = (\overline{\mathcal{H}}^{i, j}(Vec_k))^{\ddot{H}} $ is a braided monoidal category.

    Example 4.10. With the notations in Example 3.12, if $ p\in B^\ast $ is a dual central dual group-like $ {\alpha}_B $-invariant $ k $-linear form on a bialgebra $ B $, then we get a coquasi-bialgebra $ B_{{\alpha}_B}^{p, p} $. Now suppose that $ Q \in (B {\otimes} B)^\ast $ is the coquasitriangular structure over $ B $. If $ Q{\circ} ({\alpha}_B {\otimes} {\alpha}_B) = Q $, then after a straightforward compute we get that $ Q $ is also a coquasitriangular structure over the Hom-coquasi-bialgebra $ B_{{\alpha}_B}^{p, p} $.

    Let $ F = (F, {\delta}, {\varepsilon}, F_2, F_0) $ be a quasi-monoidal comonad on a monoidal category $ (\mathcal {C}, \otimes, I, a, l, r) $.

    Definition 5.1. A gauge transformation on $ F $ is a $ \ast $-invertible natural transformation $ \xi:F\otimes F \Rightarrow \otimes $.

    Using a gauge transformation $ \xi $ on $ F $, we can build a new quasi-monoidal comonad $ F^\xi $ as follows.

    Firstly, as a functor, $ F^\xi = F: \mathcal{C}\rightarrow \mathcal{C} $.

    Secondly, the comonad structure of $ F^\xi $ is $ F^\xi = F = (F, {\delta}, {\varepsilon}) $.

    Thirdly, the quasi-monoidal functor structure of $ F^\xi $ is given by:

    $ \bullet $ for any $ X, Y \in \mathcal{C} $, $ F^\xi_2: F {\otimes} F \Rightarrow F {\otimes} $ is defined as follows

    $ Fξ2(X,Y):FXFYδ2Xδ2YF3XF3YξFFXFFYF2F(FXFY)F(ξ1X,Y)F(XY)
    $
    (5.1)

    where $ \xi^{\ast -1} $ means the $ \ast $-inverse of $ \xi $;

    $ \bullet $ $ F^\xi_0 = F_0:FI \rightarrow I $.

    Proposition 5.2. With the above notations, $ \delta $ and $ {\varepsilon} $ are both monoidal natural transformations

    Proof. We only need to show the compatible conditions Eqs (C1)–(C4) hold.

    To prove Eq (C1), we compute

    for any $ X, Y \in \mathcal{C} $. The rest are straightforward.

    For any $ X, Y \in \mathcal{C} $, define the natural transformation $ {\vartheta}^\xi:(F {\otimes} F) {\otimes} F\Rightarrow \_ {\otimes} (\_{\otimes} \_) $ by

    $ ϑξX,Y,Z=(idξ1Y,Z)ξ1X,FYFZ(idF2)ϑFX,FFY,FFZ(ξFFXF3Y,F3Z)(F2id)(ξF3X,F4Yid)(δ3Xδ4Yδ3Z),
    $
    (5.2)

    and define the followings natural transformations:

    $ ιξX:IFXF0δXFIFFXξIFXιXX,
    $
    (5.3)

    and

    $ κξX:FXIδXF0FFXFIξFXIκXX.
    $
    (5.4)

    It is easy to get that $ {\vartheta}^\xi $, $ {\iota}^\xi $ and $ {\kappa}^\xi $ are all $ \ast $-invertible. Further, we have the following properties.

    Lemma 5.3. With the above notations, $ {\vartheta}^\xi $ satisfies Eqs (3.1) and (3.2).

    Proof. We only prove Eq (3.1). For any $ X, Y, Z \in \mathcal {C} $, we compute

    $ F(ϑξX,Y,Z)Fξ2(Fξ2id)(δXδYδZ)=F(idξ1Y,Z)F(ξ1X,FYFZ)F(idF2)F(ϑFX,FFY,FFZ)F(ξFFXF3Y,F3Z)F(F2id)F(ξF4X,F3Yid)F(δ3Xδ4Yδ3Z)F(ξ1FXFY,FZ)F2(δFXFYδFZ)ξF(FXFY),FFZ(δFXFYδFZ)(F(ξ1FX,FY)id)(F2id)(δFXδFYid)(ξFFX,FFYid)(δFXδFYid)(δXδYδZ)=F(idξ1Y,Z)F(ξ1X,FYFZ)F(idF2)F(ϑFX,FFY,FFZ)F(ξFFXF3Y,F3Z)F(F2id)F(δFXδ2FYδ3Z)F(ξ1FFXFFY,FZ)F2(δFFXFFYδFZ)ξF(FFXFFY),FFZ(δFFXFFYδFZ)(F2id)(δFXδFYid)(ξFFX,FFYid)(δ2Xδ2YδZ)=F(idξ1Y,Z)F(ξ1X,FYFZ)F(idF2)F(ϑFX,FFY,FFZ)F2(F2id)(δ2FXδ2FYδ2FZ)ξF3XF3Y,FFZ(F2id)(δFXδ2FYδFZ)(ξFFX,FFYid)(δ2Xδ2YδZ)=F(idξ1Y,Z)F(ξ1X,FYFZ)F(idF2)F2(idF2)(δXδ2Yδ2Z)ϑFX,FY,FZξFFXFFY,FFZ(F2id)(ξF3X,F3Yid)(δ3Xδ3Yδ2Z)=F(ξ1X,YZ)F(idF(ξ1Y,Z))F2(δXδFYFZ)(idF2)(idδYδZ)(εFXεFYεFZ)ϑFFX,FFY,FFZξF3XF3Y,F3Z(F2id)(ξF4X,F4Yid)(δ4Xδ4Yδ3Z)=F(ξ1X,YZ)F2(δXδYZ)ξFX,F(YZ)ξ1FFX,FF(YZ)(FδXFδYZ)(idFF(ξ1Y,Z))(idF(F2))(idF(δYδZ))(idF2)ϑFFX,FFY,FFZξF3X,F3YF3Z(F(idFFεFY)FFFεFZ)(F2id)(ξF4X,F5Yid)(δ4Xδ5Yδ4Z)=F(ξ1X,YZ)F2(δXδYZ)ξFX,F(YZ)(δXδYZ)(F(idξ1Y,Z))(idF2(FY,FZ))(idδYδZ)(idξFX,FY)(idξ1FFY,FFZ)(idδFYδFZ)ξ1FX,FFYFFZ(idF2(FFYFFZ))ϑFFX,F3Y,F3ZξF3X,F4YF4Z(F2id)(ξF4X,F5Yid)(δ4Xδ5Yδ4Z)=Fξ2(FX,FYFZ)(idFξ2(FY,FZ))ϑξFX,FY,FZ(δXδYδZ).
    $

    Thus the conclusion holds.

    Lemma 5.4. With the above notations, $ {\iota}^\xi $ satisfies Eq (3.3) and $ {\kappa}^\xi $ satisfies Eq (3.4).

    Proof. We only prove Eq (3.3). For any $ X \in \mathcal {C} $, we have

    which implies Eq (3.3).

    Lemma 5.5. With the above notations, $ {\vartheta}^\xi $ and $ {\iota}^\xi $, $ {\kappa}^\xi $ satisfy Eq (3.5).

    Proof. For any $ X, Y \in \mathcal {C} $, we obtain

    $ (idιξY)(ϑξX,I,FY)(idF0δY)=(idιY)(idξI,FFY)(idF0δY)(idξ1I,FY)ξ1X,FI,FFY(idF2)ϑFX,FFI,F3YξFFXF3I,F4Y(F2id)(ξF3X,F4Iid)(δ3Xδ4Iδ3FY)(δXF0δY)=ξ1X,Y(idFιY)(idF2)ϑFX,FI,FFYξFFXFFI,F3Y(F2id)(ξF3X,F3Iid)(δ3Xδ3Iδ2FY)(δXF0δY)=ξ1X,Y(idιFY)ϑFX,I,FFY(idF0δY)ξFFXI,FFY(F2id)(ξF3X,FIid)(δ3XδIδ2Y)(δXF0id)=(idεY)ξ1X,FY(κFXid)ξFFXI,FFY(F2id)(δFXF0id)(ξFFX,Iid)(δ2XF0δ2Y)=(idεY)ξ1X,FYξFX,FFY(κFFXid)(δFXidid)(ξFFX,Iid)(δ2XF0δ2Y)=(idεY)(κXid)(ξFX,Iid)(δXF0id)=(idεY)(κξXid)
    $

    hence Eq (3.5) holds.

    Theorem 5.6. $ F^\xi = (F, {\delta}, {\varepsilon}, F^\xi_2, F_0, {\vartheta}^\xi, {\iota}^\xi, {\kappa}^\xi) $ is a quasi-monoidal comonad.

    Remark 5.7. $ (\mathcal{C}^{F^\xi}, {\otimes}, I, A^\xi, L^\xi, R^\xi) $ is a monoidal category, where $ A^\xi = ({\vartheta}^\xi)^\sharp $, $ L^\xi = ({\iota}^\xi)^\sharp $, $ R^\xi = ({\kappa}^\xi)^\sharp $.

    Now consider a coquasitriangular quasi-monoidal comonad $ (F, {\sigma}) $. For any gauge transformation $ \xi $ on $ F $, for any $ X, Y \in \mathcal{C} $, define

    $ σξX,Y:FXFYδ2δ2F3XF3YξFFXFFYσFYFXξ1YX.
    $
    (5.5)

    Proposition 5.8. With the above notations, $ \sigma^\xi $ is a coquasitriangular structure of $ F^\xi $. Thus $ F^\xi $ is a coquasitriangular quasi-monoidal comonad. Hence $ \mathcal{C}^{F^\xi} $ is a braided monoidal category with the braiding $ {\tau}^\xi = (\sigma^\xi)^\sharp $.

    Proof. Firstly, it is straightforward to get that $ \sigma^\xi $ is $ \ast $-invertible.

    Secondly, to prove Eq (4.1), for any $ X, Y \in \mathcal{C} $, we compute

    $ Fξ2σξFX,FY(δXδY)=F(ξ1Y,X)F2(δYδX)ξFY,FXξ1FFY,FFX(δ2Yδ2X)σFX,FYξFFX,FFY(δ2Xδ2Y)=F(ξ1Y,X)F2σFFX,FFYξF3X,F3Y(δ3Xδ3Y)=F(ξ1Y,X)F(σFX,FY)F(ξFFX,FFY)F(ξ1F3X,F3Y)F(δ3Xδ3Y)F2(δXδY)=F(ξ1Y,X)F(σFX,FY)F(ξFFX,FFY)F(δ2Xδ2Y)F(ξ1FX,FY)F2(δFXδFY)ξFFX,FFY(δ2Xδ2Y)=F(σξY,X)Fξ2(FX,FY)(δXδY).
    $

    Thirdly, for Eq (4.2), we have

    $ ϑξY,Z,XσξFX,FYFZ(idFξ2)ϑξFFX,FFY,FFZ(δ2Xδ2Yδ2Z)=(idξ1Z,X)ξ1Y,FZFX(idF2)ϑFY,FFZ,FFXξFFYF3Z,F3X(F2id)(ξF3YF4Zid)(δ3Yδ4Zδ3X)ξ1FYFZ,FXσFFX,F(FYFZ)ξF3X,FF(FYFZ))(δ2FXδ2FYFZ)(idF(ξ1FY,FZ))(idF2)(idξF3Y,F3Z)(idδ2FYδ2FZ)(idξ1F2Y,F2Z)ξ1FFX,F3YF3Z(idF2)ϑF3X,F4Y,F4ZξF4XF5Y,F5Z(F2id)(ξF5X,F6Yid)(δ5Xδ6Yδ5Z)=(idξ1Z,X)ξ1Y,FZFX(idF2)ϑFY,FFZ,FFXσF3X,FFYF3Z(idF2)ϑF4X,F3Y,F4Z(δ2FFXδ2FYδ2FFZ)ξF3XFFY,F3Z(F2id)(ξF4X,F3Yid)(δ4Xδ3Yδ3Z)=(idξ1Z,X)(idσFX,FZ)ξ1Y,FFXFFZ(idF2)ϑFY,F3X,F3Z(σF4X,FFYid)(δ2FFXδFYδFFZ)ξF3XFFY,F3Z(F2id)(ξF4X,F3Yid)(δ4Xδ3Yδ3Z)=(idξ1Z,X)(idσFX,FZ)ξ1Y,FFXFFZ(idF2)ϑFY,F3X,F3ZξF2YF4X,F4Z(F(σF4X,FFY)id)(F2id)(δ4FXδ3FYδFY)(ξFFX,FFYid)(idδ2Zδ2Z)=(idσξX,Z)ϑξY,FX,FZ(σξFFX,FFid)(δ2XδYδZ).
    $

    At last, we can prove Eq (4.3) in a similar way. Thus the conclusion holds.

    Now consider the corepresentations of $ F $ and $ F^\xi $.

    Theorem 5.9. $ \mathcal{C}^F $ and $ \mathcal{C}^{F^\xi} $ are isomorphic as monoidal categories.Further, if $ F $ is a coquasitriangular quasi-monoidal comonad, then $ \mathcal{C}^F $ and $ \mathcal{C}^{F^\xi} $are braided isomorphic.

    Proof. For any morphism $ f $ and objects $ M, N $ in $ \mathcal{C} $, the monoidal functor is defined as follows

    $ \mathbb{E} = (\mathbb{E},\mathbb{E}^\xi_2,\mathbb{E}_0):(\mathcal{C}^F,{\otimes}, I,A,L,R)\rightarrow ( \mathcal{C}^{F^\xi},{\otimes},I,A^\xi,L^\xi,R^\xi ), $

    where

    $ \mathbb{E}(M): = M\; \; {\rm{as\; an}}\; F -{\rm{comodule}},\; \; \mathbb{E}(f): = f,\; \; \mathbb{E}_0 = id_I, $

    and $ \mathbb{E}^\xi_2(M, N): \mathbb{E}(M) {\otimes} \mathbb{E}(N)\rightarrow \mathbb{E}(M {\otimes} N) $ is given by

    $ Eξ2(M,N)=ξ:MNρMρNFMFNξM,NMN.
    $

    Obviously $ \mathbb{E} $ is well-defined.

    Now we will check relation (2.1). Indeed, we have

    $ Eξ2(M,NP)(idEξ2(N,P))AξM,N,P=ξM,NP(idF2)(ρMρNρP)(idξN,P)(idρNρP)(idξ1N,P)ξ1M,FNFP(idF2)ϑFM,FFN,FFPξFFMF3N,F3P(F2id)(ξF3M,F4Nid)(δ3Mδ4Nδ3P)(ρMρNρP)=ξM,NP(ρMF2)ξ1M,FNFP(idF2)ϑFM,FFN,FFPξFFMF3N,F3P(F2id)(ξF3M,F4Nid)(δ3Mδ4Nδ3P)(ρMρNρP)=ξM,NPξ1FM,F(NP)(δMδNP)(idF2)ϑFM,FN,FPξFFMFFN,FFP(F2id)(ξF3M,F3Nid)(δ3Mδ3Nδ2P)(ρMρNρP)=ϑM,N,PξFMFN,FP(F2id)(ξF2M,F2Nid)(δ2Mδ2NδP)(ρMρNρP)=E(AM,N,P)Eξ2(MN,P)(Eξ2(M,N)id),
    $

    which implies Eq (2.1).

    Further, we can obtain (2.2) and (2.3) by straightforward computation. Hence the conclusion holds.

    Moreover, if $ {\sigma} $ is a coquasitriangular structure of $ F $, then from Theorem 5.6, $ (F^\xi, {\sigma}^\xi) $ is also a coquasitriangular quasi-monoidal comonad. Then we have

    $ Eξ2(N,M)τξM,N=ξN,M(ρNρM)ξ1N,MσFM,FNξFFM,FFN(δ2Mδ2N)(ρMρN)=(εNεM)σFM,FNξFFM,FFN(δ2Mδ2N)(ρMρN)=σFM,FN(ρNρM)ξM,N(ρMρN)=E(τM,N)Eξ2(M,N),
    $

    which implies $ (\mathbb{E}, \mathbb{E}^\xi_2, \mathbb{E}_0) $ is a braided monoidal functor.

    Example 5.10. With the notations in Example 3.10, if there is a convolution invertible linear form $ \chi \in (H {\otimes} H)^\ast $ satisfying $ \chi {\circ} ({\alpha}_H {\otimes} {\alpha}_H) = \chi $, then we have the following $ \ast $-invertible natural transformation in $ \overline{\mathcal{H}}^{i, j}(Vec_k) $

    $ \xi_{X,Y}: \ddot{H}X {\otimes} \ddot{H}Y\rightarrow X {\otimes} Y,\; \; \; \\ (x {\otimes} a) {\otimes} (y {\otimes} b) \mapsto \chi({\alpha}_H^i(a),{\alpha}_H^j(b)) {\alpha}_X^{-1}(x) {\otimes} {\alpha}_Y^{-1}(y), $

    where $ a, b \in H $, $ x \in X $, $ y \in Y $ and $ X, Y \in \overline{\mathcal{H}}^{i, j}(Vec_k) $. It is not hard to check that $ \ddot{H}_2^\xi $, $ {\vartheta}^\xi $, $ {\iota}^\xi $ and $ {\kappa}^\xi $ in Eqs (5.1)–(5.4) are deduced from the following

    $ mχ(ab)=χ1(a1,b1)α2H(a21)α2H(b21)χ(a22,b22),
    $

    where $ \chi^{\ast-1} $ means the convolution inverse of $ \chi $, and

    $ ωχ(a,b,c)=χ1(b11,c11)χ1(αH(a11),α1H(b121)c12)ω(a12,α1H(b122),c21)χ(a21b21,αH(c22))χ(a22,b22),pχ(a)=p(a1)χ(1H,a2),qχ(a)=q(a1)χ(a2,1H),
    $

    respectively. Thus from Example 3.10 and Theorem 5.6, $ H^\chi = (H, {\alpha}_H, m^\chi, 1_H, {\Delta}, {\varepsilon}, \omega^\chi, p^\chi, q^\chi) $ is also a Hom-coquasi-bialgebra.

    Example 5.11. With the notations in Example 3.12, note that the $ B_{{\alpha}_B} = (B, {\alpha}_B, {\alpha}_B {\circ}{\mu}, 1_B, {\Delta} {\circ} {\alpha}_B, {\varepsilon}) $ is a Hom-bialgebra, and it can be seen as a Hom-coquasi-bialgebra $ B_{{\alpha}_B} = (B, {\alpha}_B, {\alpha}_B {\circ}\mu, 1_H, {\Delta} {\circ} {\alpha}_B, {\varepsilon}, {\varepsilon}{\otimes}{\varepsilon}{\otimes}{\varepsilon}, {\varepsilon}, {\varepsilon}) $. If there are $ {\alpha}_B $-invariant and dual central dual group-like $ k $-linear forms $ p, q \in B^\ast $, then we have the following gauge transformation $ \chi\in (B {\otimes} B)^\ast $ by

    $ χ(a,b)=q1(a)p(b),where a,bB.
    $

    Obviously $ B_{{\alpha}_B}^\chi = B_{{\alpha}_B}^{p, q} $.

    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).

    The authors declare there is no conflict of interest.

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