Research article Special Issues

Comparison of dominant hand to non-dominant hand in conduction of reaching task from 3D kinematic data: Trade-off between successful rate and movement efficiency

  • This study aimed to investigate the effects of handedness on motion accuracies and to compare 3D kinematic data in reaching performance of dominant and non-dominant hand with the influence of movement speed and target locations. Twelve healthy young adults used self-selected and fast speed to reach for three different target locations as follows: frontal, ipsilateral and contralateral to the performing hand, with equal distance. Both hands were tested and kinematic parameters were recorded by 3D motion analysis system. Successful rate, reach path ratio, mean and peak velocity, the timing of peak velocity and ROM of joints were analyzed. Reach path ratio was smaller when using the dominant hand (p < 0.01) and fast speed (p < 0.01) to perform the movement, but the successful rate of the dominant hand was lower than non-dominant hand during fast speed reaching (99.1% vs 100%). Contralateral movement had lower velocity than the other two target locations, while velocity did not vary between non-dominant and dominant hand. The timing of peak velocity occurred significantly later for fast speed movements (p < 0.01). Trunk rotation was significantly smaller when using the dominant hand, fast movement speed or reaching to the ipsilateral target. The ROM of elbow and wrist flexion-extension decreased in contralateral reaching. The performance of the dominant hand and/or fast speed movements was more efficient with straighter hand path and less trunk rotation, but the successful rate decreased in dominant hand during fast speed movements. The timing of peak velocity occurred later during fast movement in both hands indicating a decreased feedback phase. Target location can influence movement strategy as reaching to contralateral target required more proximal movements and ipsilateral reaching used more distal segment movements.

    Citation: Xiang Xiao, Huijing Hu, Lifang Li, Le Li. Comparison of dominant hand to non-dominant hand in conduction of reaching task from 3D kinematic data: Trade-off between successful rate and movement efficiency[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1611-1624. doi: 10.3934/mbe.2019077

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  • This study aimed to investigate the effects of handedness on motion accuracies and to compare 3D kinematic data in reaching performance of dominant and non-dominant hand with the influence of movement speed and target locations. Twelve healthy young adults used self-selected and fast speed to reach for three different target locations as follows: frontal, ipsilateral and contralateral to the performing hand, with equal distance. Both hands were tested and kinematic parameters were recorded by 3D motion analysis system. Successful rate, reach path ratio, mean and peak velocity, the timing of peak velocity and ROM of joints were analyzed. Reach path ratio was smaller when using the dominant hand (p < 0.01) and fast speed (p < 0.01) to perform the movement, but the successful rate of the dominant hand was lower than non-dominant hand during fast speed reaching (99.1% vs 100%). Contralateral movement had lower velocity than the other two target locations, while velocity did not vary between non-dominant and dominant hand. The timing of peak velocity occurred significantly later for fast speed movements (p < 0.01). Trunk rotation was significantly smaller when using the dominant hand, fast movement speed or reaching to the ipsilateral target. The ROM of elbow and wrist flexion-extension decreased in contralateral reaching. The performance of the dominant hand and/or fast speed movements was more efficient with straighter hand path and less trunk rotation, but the successful rate decreased in dominant hand during fast speed movements. The timing of peak velocity occurred later during fast movement in both hands indicating a decreased feedback phase. Target location can influence movement strategy as reaching to contralateral target required more proximal movements and ipsilateral reaching used more distal segment movements.


    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 ω(HHH), 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, 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.

    Throughout the paper, we let k be a fixed field and char(k)=0 and Veck be the category of finite dimensional k-spaces. All the algebras and coalgebras, modules and comodules are supposed to be in Veck. For the comultiplication Δ of a k-space C, we use the Sweedler-Heyneman's notation: Δ(c)=c1c2 for any cC.

    Let (C,,I,a,l,r) and (C,,I,a,l,r) be two monoidal categories. Recall that a quasi-monoidal functor from C to C is a triple (F,F2,F0), where F:CC is a functor, F2:FFF is a natural transformation, and F0:IFI is a morphism in C.

    Furthermore, if the following equations hold for any X,Y,ZC:

    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,F2,F0) is called a monoidal functor.

    Let C be a category, F: CC be a functor. Recall from [16] or [17] that if there exist natural transformations δ: FFF and ε: FidC, such that the following identities hold

    Fδδ=δFδ,andidF=Fεδ=εFδ,

    then we call the triple (F,δ,ε) a comonad on C.

    Let XC, and (F,δ,ε) a comonad on C. If there exists a morphism ρX: XFX, satisfying

    FρXρX=δXρX,andεXρX=idX,

    then we call the couple (X,ρX) an F-comodule.

    A morphism between F-comodules g: XX is called F-colinear, if g satisfies: FgρX=ρXg. 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):CC be a monoidal functor. Then recall from [18] or [19] that F is called a monoidal comonad (or a bicomonad) on C if δ and ε are both monoidal natural transformations, i.e. the following compatibility conditions hold for any X,YC:

    {(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.

    Given a category C and a positive integer n, we denote Cn=C×C××C the n-tuple cartesian product of C. If F is 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:CFC is the forgetful functor and P,Q:CnD 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)Nat(PF×n,Q).

    Proof. Define ?:Nat(PU×n,QU×n)Nat(PF×n,Q), ff, by

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

    and ?:Nat(PF×n,Q)Nat(PU×n,QU×n), αα, by

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

    for any fNat(PU×n,QU×n), αNat(PF×n,Q) and XiC, (Mi,ρMi)CF. It is easy to check that ? and ? are well defined and are inverse with each other.

    Let P,Q,R:CnD be functors. For any αNat(PF×n,Q) and βNat(QF×n,R), define their convolution product βαNat(PF×n,R) by setting, for any objects X1,,Xn in C,

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

    We say that αNat(PF×n,Q) is -invertible if there exists βNat(QF×n,P) such that βα=P(ε×n)Nat(PF×n,P) and αβ=Q(ε×n)Nat(QF×n,Q). We denote β by α1.

    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 fNat(PU×n,QU×n) is a natural isomorphism. Then we immediately get that (f)1=(f1).

    Conversely, if αNat(PF×n,Q) is -invertible, then α1=(α1).

    Suppose that (C,,I,a,l,r) is a monoidal category, F:CC 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, anddefine the F-coaction on MN (as the tensor product in C) for any (M,ρM),(N,ρN)CF by:

    ρMN:MNρMρNFMFNF2(M,N)F(MN),

    then (I,F0) and (MN,ρMN) are all objects in CF 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 (MN,ρMN)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×3C, and ι:IF__:CC, κ:F_I_:CC. 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=ϑM,N,P,LM=ιM,RM=κM.

    Conversely, if there are natural transformations A:(__)__(__):C×3C and L:I_id:CC, R:_Iid:CC, then from Lemma 2.1, for any X,Y,ZC, 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 CF.

    Lemma 3.2. A, L and R are isomorphisms if and only if ϑ, ι and κ are -invertible.

    Proof. Straightforward from Proposition 2.2.

    Lemma 3.3. A is F-colinear if and only if ϑ satisfies

    (3.1)

    for any X,Y,ZC.

    Proof. ): Since the following diagram

    is commutative for any M,N,PCF, AM,N,P is F-colinear.

    ): Notice that AFX,FY,FZ is F-colinear for any X,Y,ZC, 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 CF if and only if ϑ 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,ZC.

    Proof. ): 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,QCF, A satisfies the Pentagon Axiom.

    ): For any W,X,Y,ZC, 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 εWεXεYεZ to both sides of the above identity, we get Diagram (3.2).

    Lemma 3.5. For any XC,

    (1) L is F-colinear if and only if ι satisfies

    (3.3)

    (2) R is F-colinear if and only if κ satisfies

    (3.4)

    Proof. We only prove (1).

    ): From the following commutative diagram

    for any MCF, LM is F-colinear.

    ): Conversely, since FX is an F-comodule and LFX is F-colinear for any XC, it is directly to get Diagram (3.3).

    Lemma 3.6. A, L and R satisfy the Triangle Axiom in CF if and only if ϑ, ι and κ satisfy

    (3.5)

    for any X,Y,ZC.

    Proof. ): For any M,NCF, we compute

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

    thus the Triangle Axiom in CF holds.

    ): Conversely, for any X,YC, since we have

    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, ι and κ satisfying (3.1)–(3.5), then we call (F,δ,ε,F2,F0,ϑ,ι,κ) a quasi-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, ι and κ such 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=aX,Y,Z,ιX=lX,κX,Y,Z=rX,Y,Z

    for any X,Y,ZC, 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 functor. 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,jZ:

    the objects of ¯Hi,j(Veck) are pairs (X,αX), where XVeck and αXAutk(X);

    the morphism f:(X,αX)(Y,αY) in ¯Hi,j(Veck) is a k-linear map from X to Y such that αYf=fαX;

    the monoidal structure is given by

    (X,αX)(Y,αY)=(XY,αXαY),

    and the unit is (k,idk);

    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,αH) is an object in ¯Hi,j(Veck), mH:HHH (with notation mH(ab)=ab), ηH:kH (with notation ηH(1k)=1H), and ΔH:HHH (with notation ΔH(h)=h1h2), and εH:Hk are all morphisms in ¯Hi,j(Veck). Further, we write

    ¨H=_H:¯Hi,j(Veck)¯Hi,j(Veck),(X,αX)(XH,αXαH)

    for the right tensor functor of H.

    If we define the following structures on ¨H:

    δ:¨H¨H¨H and ϵ:¨Hid¯Hi,j(Veck) are defined by

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

    ¨H2:¨H¨H¨H and ¨H0:k¨H(k) are given by

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

    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,ΔH,εH) is a Hom-coalgebra over k, Eqs (C1)–(C4) hold if and only if mH and ηH are all morphisms of Hom-coalgebras.

    Suppose that there are αH-invariant convolution invertible linear forms ω(HHH) and p,qH, then we can define the following -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,cH, xX, yY, zZ and X,Y,ZVeck. Thus we immediately get that ϑ satisfies Eq (3.1) if and only if ω satisfies

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

    ϑ satisfies Eq (3.2) if and only if ω 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)

    ι satisfies Eq (3.3) and κ 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)

    ϑ,ι and κ satisfy Eq (3.5) if and only if ω, p and q satisfy

    ω(a,1H,b)=q(a)p1(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,ΔH,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((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 mM, nN, pP, M,N,PCorep(H).

    Example 3.11. Under the consideration of Example 3.10, if all the Hom-structure maps α 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,μ,1B,Δ,ε) be a bialgebra over k, αB:BB be an endo-isomrophism. Recall that a k-linear form gB is called

    (1) dual central if g(x1)x2=x1g(x2) for any xB;

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

    (3) αB-invariant if g(αB(x))=g(x).

    Now suppose that p,qB are all dual central dual group-like and αB-invariant linear forms. Define a k-linear form ω:BBBk by

    ω(x,y,z)=p(x)ε(y)q1(z),foranyx,y,zB,

    define the new multiplication μαB and comultiplication ΔαB by

    μαB=αBμ,ΔαB=ΔαB.

    Then it is a direct calculation to check that αB,ω,p,q satisfy Eqs.(3.6) - (3.9) (under μαB and ΔαB), hence Bp,qαB=(B,αB,μαB,1B,ΔαB, ε,ω,p,q) forms a nontrivial Hom-coquasi-bialgebra.

    Recall that a braiding in a monoidal category (C,,I,a,l,r) is a natural isomorphism τ: op:C×CC 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,ZC.

    Now let F be a quasi-monoidal comonad on C. Suppose that there is a natural transformation σ: (F×F)op:C×2C. From Lemma 2.1, for any objects M,N in CF, σ can induce a natural transformation

    τM,N=σM,N:MNρMρNFMFNσM,NNM.

    Conversely, if there exists τ:op:C×CC, then from Lemma 2.1, for any X,YC, τ can induce the following

    σX,Y=τX,Y:FXFYτFX,FYFYFXεYεXYX.

    Next we will discuss when τ is a braiding in CF.

    Lemma 4.1. τ is an isomorphism if and only if σ is -invertible.

    Proof. Straightforward from Proposition 2.2.

    Lemma 4.2. τ is F-colinear if and only if σ satisfies

    (4.1)

    for any X,YC.

    Proof. ): We compute

    for any M,NCF. Hence τM,N is F-colinear.

    ): Conversely, notice that τFX,FY is F-colinear for any X,YC, we have

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

    which implies Diagram (4.1) holds.

    Lemma 4.3. Diagram (B1) holds in CF if and only if σ satisfies

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

    for any X,Y,ZC.

    Proof. ): Take X=M, Y=N, Z=P for any F-comodules M,N,P. Multiplied by ρMρNρP right on both sides of Eq (4.2), we immediately get Diagram (B1).

    ): Since Diagram (B1) is commutative for any FX,FY,FZC, multiplied by εεε left on both sides of the above equation, we get Eq (4.2).

    Lemma 4.4. For any X,Y,ZC, Diagram (B2) holds in CF if and only if σ satisfies

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

    where ϑ1 means 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 σNat(FF,op), satisfying Eqs (4.1)–(4.3) for any X,Y,ZC, then σ is 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 CF is a braided monoidal category if and only if there exists a natural transformationσ:FFop 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,ZC, σ 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 C, and σ:F×2op 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(HH) is αH-invariant and convolution invertible, then we have the following -invertible natural transformation

    σX,Y:¨HX¨HYYX,(xa)(yb)Q(αiH(a),αjH(b))αji1Y(y)αij1X(x),

    where xX, yY and X,Y¯Hi,j(Veck). Thus we immediately get that σ satisfies Eq (4.1) if and only if Q satisfies

    Q(a1,b1)a2b2=b1a1Q(a2,b2),

    σ 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,cH. 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))¨H is a braided monoidal category.

    Example 4.10. With the notations in Example 3.12, if pB is a dual central dual group-like αB-invariant k-linear form on a bialgebra B, then we get a coquasi-bialgebra Bp,pαB. Now suppose that Q(BB) is the coquasitriangular structure over B. If Q(αBαB)=Q, then after a straightforward compute we get that Q is also a coquasitriangular structure over the Hom-coquasi-bialgebra Bp,pαB.

    Let F=(F,δ,ε,F2,F0) be a quasi-monoidal comonad on a monoidal category (C,,I,a,l,r).

    Definition 5.1. A gauge transformation on F is a -invertible natural transformation ξ:FF.

    Using a gauge transformation ξ on F, we can build a new quasi-monoidal comonad Fξ as follows.

    Firstly, as a functor, Fξ=F:CC.

    Secondly, the comonad structure of Fξ is Fξ=F=(F,δ,ε).

    Thirdly, the quasi-monoidal functor structure of Fξ is given by:

    for any X,YC, Fξ2:FFF is defined as follows

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

    where ξ1 means the -inverse of ξ;

    Fξ0=F0:FII.

    Proposition 5.2. With the above notations, δ and ε 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,YC. The rest are straightforward.

    For any X,YC, define the natural transformation ϑξ:(FF)F_(__) 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 ϑξ, ιξ and κξ are all -invertible. Further, we have the following properties.

    Lemma 5.3. With the above notations, ϑξ satisfies Eqs (3.1) and (3.2).

    Proof. We only prove Eq (3.1). For any X,Y,ZC, 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, ιξ satisfies Eq (3.3) and κξ satisfies Eq (3.4).

    Proof. We only prove Eq (3.3). For any XC, we have

    which implies Eq (3.3).

    Lemma 5.5. With the above notations, ϑξ and ιξ, κξ satisfy Eq (3.5).

    Proof. For any X,YC, 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. Fξ=(F,δ,ε,Fξ2,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,YC, define

    σξX,Y:FXFYδ2δ2F3XF3YξFFXFFYσFYFXξ1YX. (5.5)

    Proposition 5.8. With the above notations, σξ is a coquasitriangular structure of Fξ. Thus Fξ is a coquasitriangular quasi-monoidal comonad. Hence CFξ is a braided monoidal category with the braiding τξ=(σξ).

    Proof. Firstly, it is straightforward to get that σξ is -invertible.

    Secondly, to prove Eq (4.1), for any X,YC, 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 Fξ.

    Theorem 5.9. CF and CFξ are isomorphic as monoidal categories.Further, if F is a coquasitriangular quasi-monoidal comonad, then CF and CFξare braided isomorphic.

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

    E=(E,Eξ2,E0):(CF,,I,A,L,R)(CFξ,,I,Aξ,Lξ,Rξ),

    where

    E(M):=MasanFcomodule,E(f):=f,E0=idI,

    and Eξ2(M,N):E(M)E(N)E(MN) is given by

    Eξ2(M,N)=ξ:MNρMρNFMFNξM,NMN.

    Obviously 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 σ is a coquasitriangular structure of F, then from Theorem 5.6, (Fξ,σξ) 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 (E,Eξ2,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(HH) satisfying χ(αHαH)=χ, then we have the following -invertible natural transformation in ¯Hi,j(Veck)

    ξX,Y:¨HX¨HYXY,(xa)(yb)χ(αiH(a),αjH(b))α1X(x)α1Y(y),

    where a,bH, xX, yY and X,Y¯Hi,j(Veck). It is not hard to check that ¨Hξ2, ϑξ, ιξ and κξ in Eqs (5.1)–(5.4) are deduced from the following

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

    where χ1 means the convolution inverse of χ, 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, Hχ=(H,αH,mχ,1H,Δ,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,αBμ,1B,ΔαB,ε) is a Hom-bialgebra, and it can be seen as a Hom-coquasi-bialgebra BαB=(B,αB,αBμ,1H,ΔαB,ε,εεε,ε,ε). If there are αB-invariant and dual central dual group-like k-linear forms p,qB, then we have the following gauge transformation χ(BB) by

    χ(a,b)=q1(a)p(b),where a,bB.

    Obviously BχαB=Bp,qαB.

    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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