On quasi-monoidal comonads and their corepresentations

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 ^[2,3,4]. 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.

2.   Preliminaries

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 $c\in C$.

2.1.   Quasi-monoidal functor

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 $X, Y, Z \in \mathcal {C}$:

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

2.2.   Monoidal comonad

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

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$$\rightarrow$$FX$, satisfying

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 $X, Y \in \mathcal {C}$:

$\left\{\begin{array}{l} (C1)\; F(F_2(X, Y)) {\circ} F_2(FX, FY) {\circ} (\delta_X {\otimes} \delta_Y) = \delta_{X {\otimes} Y} {\circ} F_2(X, Y), \\ (C2)\; {\varepsilon}_{X {\otimes} Y} {\circ} F_2(X, Y) = {\varepsilon}_X {\otimes} {\varepsilon}_Y, \\ (C3)\; F(F_0) {\circ} F_0 = \delta_I {\circ} F_0, \\ (C4)\; {\varepsilon}_I\circ F_0 = id_I. \end{array}\right.$

2.3.   Convolution product

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:

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

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

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 $\mathcal {C}$,

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 ${{\alpha}^\sharp}^{-1} = ({\alpha}^{\ast-1})^\sharp$.

3.   Quasi-monoidal comonads

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:

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

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

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

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

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

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

Proof. $\Leftarrow)$: Since we have

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

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

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

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.

4.   Coquasitriangular structures

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

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

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

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

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

which implies Diagram (4.1) holds.

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

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

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:

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

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

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

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 $B_{{\alpha}_B}^{p, p}$.

5.   Gauge transformations

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

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

and define the followings natural transformations:

and

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

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

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

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

Thirdly, for Eq (4.2), we have

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

where

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

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

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

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

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 $\overline{\mathcal{H}}^{i, j}(Vec_k)$

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

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

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

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

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).

Conflict of interest

The authors declare there is no conflict of interest.