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Hom-Tensor Categories and the Hom-Yang-Baxter Equation

We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation fits into this framework and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obtain a tensor category (respectively a braided tensor category) from a hom-tensor category (respectively a hom-braided category).


Introduction
Tensor categories were introduced by Bénabou in [3]. A basic example is the category of vector spaces over a field k. More interesting examples can be obtained from bialgebras. If A is an algebra and ∆ : A → A ⊗ A is a morphism of algebras, then the category of A-modules is a tensor category (with the tensor product induced by ∆ and trivial associativity constraint) if and only if A is a bialgebra.
The Yang-Baxter equation was introduced by Yang and Baxter (see [2], [24]). It has applications to knot invariants and it was intensively studied over the last thirty years.
Braided categories were introduced by Joyal and Street in [13]. The main example is the braid category; it satisfies a universal property for braided categories (see [15]). Other examples are obtained from quasitriangular Hopf algebras. Braided categories can be used to construct representations for the braid group and invariants for tangles, knots and 3-manifolds (see [23]). The braiding of a braided category satisfies a dodecagonal equation (see [15]) that may be regarded as a categorical analogue of the Yang-Baxter equation.
One natural question to ask is what type of categorical framework these hom-structures fit into. In the original concept of hom-bialgebra (see [20], [21]), two distinct linear maps twist the associative and co-associative structures of a bialgebra. When the two twisting maps are inverses to each other it was proved in [5] that the category of modules is a tensor category. The question that may be asked is what kind of categorical framework does a hom-bialgebra where two arbitrary linear maps twist the associative and co-associative structure fit into? Moreover, is there an analogue to the classical relationship between quasitriangular bialgebras and braided tensor categories for quasitriangular hom-bialgebras? It is these questions that motivated this paper and the concepts it contains.
There are two main objectives to this paper. The first one is to introduce a hom-analogue to a tensor category, called a hom-tensor category. In a hom-tensor category C the usual associator is replaced by a natural isomorphism a U,V,W : (U ⊗ V ) ⊗ F (W ) → F (U ) ⊗ (V ⊗ W ) that satisfies a generalized pentagonal equation (here F : C → C is a functor; when F is the identity functor we recover the definition of a tensor category without unit). We show that the category of modules over a hom-bialgebra (as it is posed in [20] and [21]) fits in the categorical framework of hom-tensor categories.
The second objective is to introduce a hom-analogue to a braided tensor category, called a hom-braided category. In a hom-braided category C we have a natural morphism c U,V : U ⊗ V → G(V ) ⊗ G(U ) that satisfies a generalization of the hexagonal axioms (where G : C → C is another functor). We show that this new categorical framework provides the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation (introduced by D. Yau in [26]) fits in the context of hombraided categories, and we prove that the category of Yetter-Drinfeld modules (introduced in [18]) over a hom-bialgebra with bijective structure map becomes a hom-braided category.
As applications to our theory, we give new proofs for Yau's result from [29] saying roughly that a quasitriangular hom-bialgebra H provides a solution for the hom-Yang-Baxter equation on any H-module and for the result in [18] saying that H H YD, the category of Yetter-Drinfeld modules (M, α M ) with α M bijective over a hom-bialgebra H with bijective structure map, is a quasi-braided category.
The structure of this paper is as follows. Section 2 begins with recalling some definitions and concepts of hom-structures necessary in presenting the upcoming results. We begin Section 3 by defining a homtensor category, and then we show how a hom-tensor category is the appropriate categorical framework for hom-bialgebras with arbitrary twisting maps. Section 4 introduces the notions of algebras in a hom-tensor category, left H-module hom-algebras over a hom-bialgebra H and a categorical analogue to a Yau twist. In Section 5 we define hom-braided categories and prove that they provide the right categorical framework for quasitriangular hom-bialgebras. In Section 6 we show how to regard the hom-Yang-Baxter equation in the categorical framework of hom-braided categories. In Section 7 the category of Yetter-Drinfeld modules over a hom-bialgebra as seen in [18] is organized under the framework of a hom-braided category. Finally, in Section 8 we show that under certain conditions one can obtain a tensor category (respectively a braided tensor category) from a hom-tensor category (respectively a hom-braided category).

Preliminaries
We work over a base field k. An unlabeled tensor product means either a functor ⊗ : C × C → C on a category C or the tensor product over k. For a comultiplication ∆ : C → C ⊗ C on a k-vector space C we use a Sweedler-type notation ∆(c) = c (1) ⊗ c (2) , for c ∈ C. Unless otherwise specified, the (co)algebras ((co)associative or not) that will appear in what follows are not supposed to be (co)unital, and a multiplication µ : V ⊗ V → V on a k-vector space V is denoted by juxtaposition: µ(v ⊗ v ′ ) = vv ′ .
We will use the following terminology for categories. A pre-tensor category is a category satisfying all the axioms of a tensor category in [15] except for the fact that we do not require the existence of a unit object. If (C, ⊗, a) is a pre-tensor category, a quasi-braiding c in C is a family of natural morphisms c V,W : V ⊗ W → W ⊗ V in C satisfying all the axioms of a braiding in [15] except for the fact that we do not require c V,W to be isomorphisms; in this case, (C, ⊗, a, c) is called a quasi-braided pre-tensor category.
Definition 2.1. A hom-associative k-algebra is a triple (A, m A , α A ), where A is a k-vector space, m A : A ⊗ A → A is a k-linear map denoted by m A (a ⊗ b) = ab, for all a, b ∈ A, and α A : A → A is a k-linear map satisfying the following conditions, for all a, b, c ∈ A: Let (A, m A , α A ) and (B, m B , α B ) be two hom-associative k-algebras. A morphism of hom-associative algebras f : Definition 2.3. A hom-coassociative k-coalgebra is a triple (C, ∆ C , ψ C ), where C is a k-vector space, ∆ C : C → C ⊗ C and ψ C : C → C are k-linear maps satisfying the following conditions: Let (C, ∆ C , ψ C ), (D, ∆ D , ψ D ) be two hom-coassociative k-coalgebras. A morphism of hom-coassociative k-coalgebras g : is a hom-coassociative k-coalgebra, ∆ B is a morphism of hom-associative k-algebras, α B is a morphisms of hom-coassociative k-coalgebras and ψ B is a morphism of hom-associative k-algebras (in particular we have Remark 2.5. The following statement is equivalent to Definition 2.4. A hom-bialgebra is a homassociative k-algebra (B, m B , α B ) together with two k-linear maps ∆ B : B → B ⊗ B and ψ B : Remark 2.6. In the literature, most of the results about hom-bialgebras use the extra assumption that ψ B = α B or ψ B = α −1 B (see [5], [18], [29]). We treat the general situation, to cover both cases of interest. We recall now the so-called "twisting principle" or "Yau twisting".
Proposition 2.7. Let (A, µ) be an associative k-algebra and α : A → A an algebra endomorphism. Define a new multiplication µ α : is a hom-associative k-algebra, denoted by A α and called the Yau twist of A.
Definition 2.13. Let A = (A, m A , α A ) be a hom-associative k-algebra.
(i) Suppose that h · m = 0 for any A-module (M, α M ) and for all m ∈ M implies h = 0. Then we say that A is nondegenerate.
(ii) Suppose that h · m = 0 for any A-module (M, α M ) and for all m ∈ α M (M ) implies h = 0. Then we say that A is strongly nondegenerate.
Lemma 2.14. Let A be a nondegenerate hom-associative k-algebra and x ∈ A⊗ A such that x·(u ⊗ v) = 0 for all u ∈ U and all v ∈ V and for all left A-modules (U, α U ) and (V, α V ). Then x = 0. A similar result is true for y ∈ A ⊗3 . Similar results are true for strongly nondegenerate algebras.
Fix a left A-module (U, α U ) and fix a k-basis {e i } i∈I for U . Consider the set of k-linear applications e * i ∈ U * determined by e * i (e j ) = δ j i . For every left A-module (V, α V ) and for every u ∈ U , v ∈ V and i ∈ I we have: Since A is nondegenerate we get that for every i ∈ I and every u ∈ U we have that n p=1 e * i (a p ·u)b p = 0 ∈ A. But {b p } 1≤p≤n are linearly independent, so for every 1 ≤ p ≤ n we have e * i (a p · u) = 0 for all i ∈ I and for all u ∈ U . Now since {e i } i∈I is a k-basis for U we must have that a p · u = 0 for all 1 ≤ p ≤ n and for all u ∈ U . But U can be any A-module and A is nondegenerate which implies that a p = 0 for all 1 ≤ p ≤ n and so x = n p=1 a p ⊗ b p = 0.

Hom-Tensor Categories
We introduce a new type of categories called hom-tensor categories, which have a tensor functor ⊗ : C × C → C with the usual associativity condition replaced by a more relaxed condition (see Definition 3.1). We show that hom-bialgebras fit very nicely in this framework. Unlike the tensor category introduced in [5], a hom-tensor category can be associated even to hom-bialgebras for which α A is not necessary bijective. (1) C is a category.
for all morphisms f, g ∈ Hom (C). (4) a X,Y,Z : (X ⊗ Y )⊗F (Z) → F (X)⊗(Y ⊗ Z) is a natural isomorphism that satisfies the "Pentagon" axiom as seen in Figure 1 for all objects X, Y, Z, T ∈ C. We call a the hom-associativity constraint of the hom-tensor category. (5) G : C → C is a covariant functor such that G (U ⊗ V ) = G (U ) ⊗ G (V ) for all objects U, V ∈ C and G (f ⊗ g) = G (f ) ⊗ G (g) for all morphisms f, g ∈ Hom (C).
(6) There exists a natural transformation Φ : id C → G. Figure 1. The "Pentagon" axiom for the hom-associativity constraint a Remark 3.2. Note that we do not require the existence of a unit in the category, and so a more appropriate name for the structure we defined would be hom-pre-tensor category. However, in order to simplify the terminology, we prefer to call it hom-tensor category.
Remark 3.3. One may relax the above definition, by removing some of the axioms. For instance, one may remove the condition Φ M⊗N = Φ M ⊗ Φ N , which is used later in only one place, in the last section. The importance of the functor G will become apparent later, when we talk about hom-braided categories.
We present now a first class of examples of hom-tensor categories.

Proof. A straightforward verification.
The following proposition gives the relation between hom-tensor categories and hom-bialgebras. (iii) The hom-tensor product of (U, α U ) and (V, α V ) is given by , for all u ∈ U , v ∈ V . (iv) F : H-mod → H-mod is the covariant functor defined by F ((U, α U )) = U ψ , α U ψ , where U ψ = U as a k-vector space (we will denote an element of U ψ = U asū), and the H-module where U α = U as a k-vector space (we will denote an element of U α = U as u), and the H-module Remark 3.6. Notice that as k-vector spaces we have To begin, we want to show that the tensor product is well-defined in H . Let (U, α U ) , (V, α V ) ∈ Ob (H ). Our first claim is that (U ⊗ V, α U⊗V ) ∈ Ob (H ). We need to check that both conditions (10) and (11) hold for U ⊗ V .
When using a certain property to establish a particular equality within the following computations, we will indicate that property above the corresponding equal sign. So, suppose that h ∈ H, u ∈ U and v ∈ V . Then checking for property (10) Next, suppose that h, h ′ ∈ H, u ∈ U and v ∈ V . Checking for condition (11) So condition (11) holds for U ⊗ V . Thus (U ⊗ V, α U⊗V ) ∈ Ob (H ).
Thus the two expressions above are equal. Next .
. Now that the hom-tensor product is well-defined for objects and morphisms in H one can easily show that the remaining properties for being functorial are satisfied. That is, if we let f ∈ Hom H-mod (S, T ), Next we will show that F : H → H is a covariant functor. We want to show for any object (U, We check condition (10): Let h, h ′ ∈ H and u ∈ U where (U, α U ) is an object in H-mod. Then checking condition (11) results in Next we claim that F maps morphisms to morphisms in H . Let f ∈ Hom H-mod (V, W ) andv ∈ V ψ . We will prove our claim by checking the following two properties: Relation (22) is a consequence of the following computations: Relation (23) holds by the following sequence of equalities: The fact that F preserves compositions of morphisms and the identity morphism in H-mod is obvious. Therefore, F is indeed a covariant functor.
Next we need to check that As noticed in Remark 3.6, we already have that identification at the level of k-vector spaces. We need to show that the H-module structure is preserved, too. Let u ∈ U , v ∈ V , h ∈ H. Then: with left H-action given by We prove that the two left H-actions coincide: Next we need to check that and (X, α X ) are objects in H-mod. For the left hand side of (24) we have that For the right hand side of (24) we have that Thus, the hom-associativity constraint a satisfies the "Pentagon" axiom from Figure 1.
Next we show that the constraint map a is a morphism in H . First, we will check that for all u ∈ U , v ∈ V and w ∈ W , where (U, α U ), (V, α V ) and (W, α W ) are objects in H . We compute: Thus, a is a morphism in H . The fact that a is an isomorphism follows directly from the definition. Just like for F one can show that G is a functor and G(( Next we check that Φ U is a morphism of H-modules. Indeed, In other words Φ is a natural transformation between the functors id H-mod and G. It is obvious that We begin with proving (3). .
The H-action for the left hand side of (26) is equal to the H-action for the right hand side of (26) h · ψ (ū ⊗v) That is, which means that Since H is assumed to be nondegenerate and by using Lemma 2.14, this equation implies that Next we will show (4) for ∆ H . Since H is a hom-tensor category it means that the hom-associativity constraint is a morphism of left H-modules. This means that for For the left and respectively right hand side of this equality we have So, we must have or equivalently Since H is nondegenerate and by using again Lemma 2.14, this equation implies that (4) and

Algebras in Hom-Tensor Categories
Definition 4.1. Let (C, ⊗, F, G, a, Φ) be a hom-tensor category. An algebra in C is a pair (A, µ A ), where A ∈ Ob(C) and µ A : A ⊗ A → F G(A) is a morphism in C such that the diagram in Figure 2 is commutative.
Proposition 4.2. We denote by k-mod the category of k-vector spaces with its usual structure as a tensor category and we consider the hom-tensor category h(k-mod) as in Proposition 3.4. Then an algebra in h(k-mod) is exactly a hom-associative k-algebra.
We recall the following concept from [10].
We have an analogue of Proposition 4.2 for hom-semigroups (the proof is similar and will be omitted). Let Set be the category of sets with its usual structure as a pre-tensor category: the tensor product is the cartesian product of sets and the associativity constraint is defined by Consider the hom-tensor category h(Set) as in Proposition 3.4. Then an algebra in h(Set) is exactly a multiplicative hom-semigroup.
such that the following condition is satisfied: Proof. Let ((A, α A ), µ A ) be an algebra in H . This means that: By taking into account the structure of H as a hom-tensor category presented in Proposition 3.5, this means that In conclusion, The next result may be regarded as a categorical analogue of the Yau twisting.  Then ((A, α A ), m A ) is an algebra in the hom-tensor category h(C).

Proof. First, by α
finishing the proof.

Hom-Braided Categories
We introduce hom-braided categories and present their connection with quasitriangular hom-bialgebras.
Here τ : C × C → C × C is the functor defined by τ (V, W ) = (W, V ) for any pair of objects in a category C.
Definition 5.1. Let C = (C, ⊗, F, G, a, Φ) be a hom-tensor category. A hom-commutativity constraint d is a natural morphism d : ⊗ → ⊗τ (G × G). That is, for any V, W ∈ Ob(C ) we have the morphism Figure 3 commutes for all morphisms f , g ∈ C .
Next we introduce the analog of the hexagon axiom in the context of hom-tensor categories.
Definition 5.2. We say that the hom-commutativity constraint d satisfies the (H1) property if the diagram, as seen in Figure 4, commutes for all objects U , V and W of the category C. Furthermore, we say that the hom-commutativity constraint d satisfies the (H2) property if the diagram, as seen in Figure 5, commutes for all objects U , V and W of the category C .  Remark 5.4. Note that d is not required to be invertible and so a more appropriate name for the above structure would be hom-quasi-braided category. However, in order to simplify the terminology, we prefer to call it hom-braided category.
We present a first class of examples of hom-braided categories. Proof. We only prove that the first hexagonal relation satisfied by c −,− , namely for all U, V, W ∈ Ob(C), implies property (H1) for d −,− and leave the rest of the proof to the reader.
The following proposition gives the connection between hom-braided categories and quasitriangular hom-bialgebras.
Let (U, α U ) and (V, α V ) be objects in H-mod and let u ∈ U , v ∈ V . We claim that c U,V is a morphism in H . First we check that Let h ∈ H, u ∈ U and v ∈ V . We next check that (c U,V ) (h · (u ⊗ v)) = h · (c U,V ) (u ⊗ v): Therefore, c U,V is a morphism in H . A similar computation shows the naturality of c.
Next we will confirm that c U,V as it is defined satisfies the (H1) property. That is, we need to show that for all objects (U, α U ), (V, α V ) and (W, α W ) in H-mod.
Let u ∈ U , v ∈ V and w ∈ W . Computing the left hand side of (31) we have that Computing the right hand side of (31) we have that 12, the left hand side and the right hand side of (31) agree. So c U,V has the (H1) property. Next we will confirm that c U,V satisfies the (H2) property. That is, we need to show that Let u ∈ U , v ∈ V and w ∈ W . For the left hand side of (32) we have that For the right hand side of (32) we have that and so c U,V is a hom-braiding in H . Therefore H is a hom-braided category.
(B) ⇒ (A) Suppose that H = (H-mod, ⊗, F, G, a, Φ, c) is a hom-braided category, (H, m H , α H ) is a strongly nondegenerate hom-associative algebra and let R ∈ H ⊗ H be given as . We will show that conditions (14), (15) and (16)  First we will show that H satisfies condition (14). Let (U, α U ) and (V, α V ) be objects in H . Since H is a hom-braided category it means that the hom-braiding c U,V is a morphism of left H-modules. That is, for h ∈ H, u ∈ U , and v ∈ V , we have c U,V (h · (u ⊗ v)) = h · c U,V ((u ⊗ v)). Just like above, for the left hand side of this equality we have c U,V (h · (u ⊗ v)) = i t i h (2) · α V (v) ⊗ s i h (1) · α U (u), and for the (2) s i · α U (u), and so we must have Since H is strongly nondegenerate and by using Lemma 2.14, this equation implies that i t i h (2) (2) s i , or equivalently R∆ (h) = ∆ cop (h) R, for every h ∈ H. Thus, R satisfies (14).
Next we will show that R satisfies conditions (15) and (16). One can show that Since c U,V satisfies the (H2) property and H is strongly nondegenerate, by using again Lemma 2.14 we obtain ( Similarly one can show that (α H ψ H ⊗ ∆ H ) (R) = i,j s i s j ⊗ t j ⊗ t i . Therefore, by Remark 2.12, (H, R) is a quasitriangular hom-bialgebra. [26] from which we recall the following definition.

D. Yau introduced the hom-Yang-Baxter equation in
. We say that d V is a solution to the hom-Yang-Baxter equation with respect to α V if it satisfies the following condition: The goal of this section is to describe two categorical versions of Definition 6.1. The following lemma will be useful in proving some of these results. Lemma 6.2. Assume that d is a hom-commutativity constraint as in Definition 5.1. Then the commutative diagram in Figure 6 is equivalent to the commutative diagram in Figure 5. We will call the equation from Figure 6 the H ′ 2 property of the hom-commutativity constraint d. Figure 6. The H ′ 2 property Proof. First observe that the commutative diagrams for the (H2) property and the H ′ 2 property are identical except on two of their edges and the object between those edges. These exceptions are indicated by bold edges in Figure 6. So it suffices to show that the diagram in Figure 7 commutes. The hom-associativity constraint a is a natural isomorphism, so a −1 is also natural and the diagram in Figure 7 is a particular case of the naturality of a −1 plus the fact that F (Φ G(W ) ) = Φ F G(W ) . Thus the diagram in Figure 7 commutes and we have the equivalence between the properties (H2) and H ′ 2 . Figure 7. Naturality of a −1 and F (Φ G(W ) ) = Φ F G(W ) Definition 6.3. We say that a hom-commutativity constraint d has the hom-Yang-Baxter property if d satisfies the equation in Figure 8. Proof. Let C = (C, ⊗, F, G, a, Φ, d) be a hom-braided category. The commutative diagram in Figure 6 together with the hom-associativity constraint a being an isomorphism implies that the diagram in Figure 9 commutes for all objects U, V, W in C . Observe that the commutative diagram in Figure 9 is precisely the bold portion of the diagram in Figure 11.
Next, consider substituting the object G(V ) for U and the object G(U ) for V in the diagram of Figure 9. Then the commutative diagram in Figure 9 implies that the diagram in Figure 10 commutes for all objects U, V, W in C . Observe that the commutative diagram in Figure 10 is precisely the dashed portion of the diagram seen in Figure 11. Thus, to prove that the diagram in Figure 8 commutes it suffices to show that the outermost perimeter of the diagram seen in Figure 12 commutes.
Indeed, the square diagram indicated by 1# in Figure 12 commutes as a consequence of the naturality of d and the fact that G(d U,V ) = d G(U),G(V ) . While the diagram indicated by 2# in Figure 12 commutes because ⊗ is a functor. Therefore, the diagram in Figure 8 commutes.
Next we introduce a variation on the definition of hom-braided categories.
Definition 6.5. Let C = (C, ⊗, F, G, a, Φ) be a hom-tensor category. We say that a hom-commutativity constraint d (as in Definition 5.1) satisfies the (wH1) property if the diagram in Figure 13 commutes for all objects U , V and W of the category C . Furthermore, we say that a hom-commutativity constraint d satisfies the (wH2) property if the diagram in Figure 14 commutes for all objects U , V and W of the category C . Figure 9. The commutative bold portion of Figure 11 F Figure 10. The commutative dashed portion of Figure 11 Definition 6.6. Let C = (C, ⊗, F, G, a, Φ) be a hom-tensor category. A weak hom-braiding d in C is a hom-commutativity constraint with the following conditions: (i) d satisfies (wH1) and (wH2); (ii) For all objects U and V in the category C , G (d U,V ) = d G(U),G(V ) . A weakly hom-braided category (C, ⊗, F, G, a, Φ, d) is a hom-tensor category with weak hom-braiding d.
Using arguments similar to those found in the proof for Lemma 6.2, one can show the following.
Lemma 6.7. The commutative diagram in Figure 15 is equivalent to the (wH2) property of a homcommutativity constraint d. We call the relation in Figure 15 the wH ′ 2 property of d. Figure 11. Proof of the hom-Yang-Baxter property Figure 12. Naturality for d and G(d U,V ) = d G(U),G(V ) Definition 6.8. Let C = (C, ⊗, F, G, a, Φ) be a hom-tensor category and d a hom-commutativity constraint. We say that d has the weak hom-Yang-Baxter property if d satisfies the equation in Figure 16. Proof. Let C = (C, ⊗, F, G, a, Φ, d) be a weakly hom-braided category. The commutative diagram in Figure 15 together with the hom-associativity constraint a being an isomorphism implies that the diagram in Figure 17 commutes for all objects U, V, W in C . Observe that the commutative diagram in Figure 17 is precisely the bold portion of the diagram seen in Figure 19. Figure 13. The (wH1) property Figure 14. The (wH2) property Next, consider substituting the object G(V ) for U , the object G(U ) for V and the object G(W ) for W in the diagram of Figure 17. Then the commutative diagram in Figure 17 implies that the diagram in Figure  18 commutes for all objects U, V, W in C . Observe that the commutative diagram in Figure 18 is precisely Figure 15. The commutative diagram for the wH ′ 2 property Figure 16. The weak hom-Yang-Baxter property the dashed portion of the diagram in Figure 19. Thus, to prove that the diagram in Figure 16 commutes it suffices to show that the outermost perimeter of the diagram in Figure 20 commutes. Indeed, the square diagram labeled 1# in Figure 20 commutes as a consequence of the naturality of d and the fact that G(Φ F (W ) ) = Φ GF (W ) and G(d U,V ) = d G(U),G(V ) . The square diagram labeled 2# commutes because ⊗ is a functor, d is natural and G(Φ G(V ) ) = Φ G 2 (V ) . Thus, the diagram in Figure 16 commutes. Figure 17. The commutative bold portion of Figure 19 We can give now the connection between hom-braided categories and weakly hom-braided categories. Proof. After comparing Definitions 5.3 and 6.6 for hom-braiding and weak hom-braiding, it suffices to show that the (H1) property implies the (wH1) property and the (H2) property implies the (wH2) property. Figure 18. The commutative dashed portion of Figure 19 ( Figure 19. Proof of the weak hom-Yang-Baxter property We prove that the (H1) property implies the (wH1) property. Observe that the outermost perimeter of the diagram in Figure 21, indicated by a dashed line, is the commutative diagram of the (H1) property from Figure 4. In addition, observe that the innermost 7-gon of the diagram in Figure 21, indicated by the bold line, is the diagram of the (wH1) property as seen in Figure 13. The plan is to show that each of the square portions of this diagram, labeled id 1 , id 2 , id 3 , 1#, 2#, 3# and 4# commute. Once this is established then the assumed commutativity of the dashed portion of the diagram in Figure 21, and the fact that (id U ⊗ id V ) ⊗ id F (W ) is an isomorphism, will imply the commutativity of the bold portion. That is, the (H1) property implies the (wH1) property.
Clearly, the square portions labeled id 1 , id 2 and id 3 of the diagram in Figure 21 commute. Additionally, both the square portions labeled 1# and 4# commute since ⊗ is a functor. More precisely, we have Furthermore, the square portion labeled 2# commutes since the hom-associativity constraint a is natural and F (Φ W ) = Φ F (W ) . Finally, the square portion labeled 3# commutes since the hom-commutativity constraint d is natural and which gives 3#, by using again the fact that ⊗ is a functor. A similar argument shows that the (H2) property implies the (wH2) property.
Remark 6.11. The above proof (i.e. the diagram in Figure 21) also shows that the converse of Proposition 6.10 is true if the natural transformation Φ is assumed to be an isomorphism.
is a hom-braiding. By Proposition 6.10, it is also a weak hom-braiding, so, by Proposition 6.9, the diagram in Figure 16 is commutative. We write the diagram in Figure Figure 16 reads which is exactly the hom-Yang-Baxter equation for B with respect to α M .

Yetter-Drinfeld modules
Throughout this section, H = (H, m H , ∆ H , α H , ψ H ) will be a hom-bialgebra for which α H = ψ H and α H is bijective. We recall the following concept and results from [18].   Figure 21. The (H1) property implies the (wH1) property Remark 7.6. Similarly to what we did in Proposition 6.13, we can reobtain the relation (36) in Proposition 7.4 as a consequence of our theory. Indeed, since B −,− is a hom-braiding, it is also a weak hom-braiding, so the diagram in Figure 16 Figure 16 reduces to which is exactly the hom-Yang-Baxter equation (36).

Hom-Tensor Categories versus Tensor Categories
We show that under certain conditions one can associate a pre-tensor category to a hom-tensor category.
Proposition 8.1. Let C = (C, ⊗, F, G, a, Φ) be a hom-tensor category. Suppose that Θ : id C → F is a natural isomorphism such that Θ M⊗N = Θ M ⊗ Θ N for all objects M, N ∈ C. Consider b U,V,W : Figure 22. Then b U,V,W is an associativity constraint for the pre-tensor category (C, ⊗, b).
Proof. Notice that b U,V,W is a natural isomorphism being a composition of natural isomorphisms. So we only need to check that b U,V,W satisfies the Pentagon axiom of an associativity constraint.
Consider the diagram in Figure 23. Observe that the inner bold portion of this diagram is the "Pentagon" axiom of the hom-associativity constraint a U,V,W and that the outer dashed portion of this diagram is the Pentagon axiom of the mapping b U,V,W . We will show that commutativity of the bold portion of the diagram implies the commutativity of the dashed portion of the diagram. This will be done by proving that each square portion of the diagram labeled 1#, 2#, 3#, 4# and 5# commutes for all objects U, V, W, X ∈ C.
We begin with the square portion of Figure 23 labeled 1#. We have: The first and third equality follow from the functoriality of the tensor product. The second equality follows from the definition of b U,V,W (see Figure 22).
Next we consider the square portion of Figure 23 labeled 2#. Notice that the diagram in Figure 24 commutes for all objects U, V, W, X ∈ C. Indeed, the portion of the diagram labeled ( * ) commutes by the definition of b U,V ⊗W,X , and the portion of the diagram labeled ( * * ) commutes by the naturality of a U,V ⊗W,X . Now we consider the square portion of Figure 23 labeled 3#. Computing we get that bU,V,W ⊗X Figure 23. Hom-associativity for a implies associativity for b Figure 24. The diagram for 2# The first and third equality are consequence of the functoriality of the tensor product, while the second equality follows from the definition of b V,W,X . Next we consider the square portion of Figure 23 labeled 4#. Notice that the diagram in Figure 25 commutes for all objects U, V, W, X ∈ C. Indeed, the portion of the diagram labeled (•) commutes by the definition of b U⊗V,W,X and the portion of the diagram labeled (••) commutes by the naturality of a U⊗V,W,X .
Notice that in order to glue together 4# and 5# we use that Θ U⊗V = Θ U ⊗ Θ V , Θ W ⊗X = Θ W ⊗ Θ X and the functoriality of ⊗.
Finally we consider the square portion of Figure 23 labeled 5#. Notice that the diagram in Figure 26 commutes for all objects U, V, W, X ∈ C. The portion of the diagram labeled (⋄) commutes by the definition of b U,V,W ⊗X and the portion of the diagram labeled (⋄⋄) commutes by the naturality of a U,V,W ⊗X .
Next we turn our attention to the relation between hom-braided and quasi-braided categories. Figure 25. The diagram for 4# for all objects U, V ∈ C. Then c U,V is a quasi-braiding for the quasi-braided category (C, ⊗, b, c). Figure 27. Definition of c U,V : U ⊗ V → V ⊗ U Proof. Being a composition of natural morphisms, c U,V is a natural morphism. So we only need to check that c U,V satisfies the Hexagon axiom of a braiding. We check the first Hexagon axiom for c U,V .
Consider the diagram in Figure 28. Observe that the inner bold portion of this diagram is the (H1) property of the hom-braiding d U,V and the outer dashed portion of this diagram is the first Hexagon axiom property for c U,V . We will show that commutativity of the bold portion of the diagram implies the commutativity of the dashed portion of the diagram. This will be done by proving that each square portion of the diagram labeled 1#, 2#, 3#, 4#, 5#, 6# and 7# commutes for all objects U, V, W ∈ C.
We begin with the square portion of Figure 28 labeled 1#. Computing we get that Figure 28. The (H1) property implies the first Hexagon axiom The first and third equality follow from the functoriality of the tensor product. The second equality is the definition of c U,V . Next we consider the square portion of Figure 28 labeled 2#, which coincides with the diagram in Figure  29 by the functoriality of the tensor product. The portion of the diagram in Figure 29 labeled ( * ) commutes by the definition of b, and the portion labeled ( * * ) commutes by the naturality of a V,U,W . Now we consider the square portion of Figure 28 labeled 3#, which coincides with the diagram in Figure  30 by the functoriality of the tensor product. The portion of the diagram in Figure 30 labeled (•) commutes by the definition of c V,W and the functoriality of the tensor product, and the portion of the diagram labeled (••) commutes by the naturality of d U,W and the functoriality of the tensor product. The square portion labeled 4# of Figure 28 commutes by the definition of b U,V,W . Next we consider the square portion of Figure 28 labeled 5#. In Figure 31, the portion of the diagram labeled (⋄) commutes by the definition of c U,V ⊗W and the portion of the diagram labeled (⋄⋄) commutes by the naturality of d U,V ⊗W . Now we notice that the diagram in Figure 31 coincides with the square portion of Figure 28 labeled 5# by using the functoriality of the tensor product and the axiom Φ V ⊗W = Φ V ⊗ Φ W from the definition of a hom-tensor category. Next we consider the square portion of Figure 28 labeled 6#. In Figure 32 the portion of the diagram labeled (♮) commutes by naturality of b V,W,U and the portion of the diagram labeled (♮♮) commutes by the definition of b G(V ),G(W ),G(U) . Now we notice that the diagram in Figure 32 coincides with the square portion of Figure 28 labeled 6# by using the functoriality of the tensor product and the fact that G(Θ V ) = Θ G(V ) , which also allows us to glue 6# and 7# together. Finally we consider the square portion of Figure 28 labeled 7#. This is commutative by using the functoriality of ⊗, the fact that Φ G(U) = G(Φ U ) and because Θ G(V ) • Φ V = Φ F (V ) • Θ V . The last statement is a a consequence of the fact that Θ is a natural transformation and F (Φ V ) = Φ F (V ) (see Figure 33).
Thus the (H1) property for d U,V implies the first Hexagon axiom property for c U,V . Using similar arguments one can show that the (H2) property for d U,V implies the second Hexagon axiom property for c U,V . Therefore, c U,V is a quasi-braiding for the quasi-braided category (C, ⊗, b, c). Remark 8.3. Let again H = (H, m H , ∆ H , α H , ψ H ) be a hom-bialgebra for which α H = ψ H and α H is bijective. We can give a new proof of Proposition 7.3 based on the results obtained in this section. We proceed as follows. We know from Proposition 7.5 that H H YD is a hom-braided category, and it is obvious that its subcategory H H YD inherits this hom-braided structure. We apply Propositions 8.1 and 8.2 to this hom-braided category H H YD, in this case the natural isomorphism Θ : id C → F being the same as the natural isomorphism Φ : id C → G (we recall that we have F = G in this situation). Thus, we obtain that H H YD becomes a quasi-braided category, and it is very easy to see that its quasi-braided structure is exactly the one that appears in Proposition 7.3.