On the braided Connes-Moscovici construction

In $1998$, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In $2010$, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra $H$ in a braided category $\mathcal B$, they associate a paracocyclic object in $\mathcal B$. In this paper we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for $H$ and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with $H$ to that associated with an $H$-module coalgebra via a categorical version of the Connes-Moscovici trace.


Introduction
Cyclic (co)homology of algebras was introduced in the 1980s by Connes [3,4] and Tsygan [11] independently.To any algebra is associated a cocyclic vector space (that is, a cocyclic object in the category of vector spaces) whose cohomology is called the cyclic cohomology of the algebra.The notion of a (co)cyclic object in a category, introduced by Connes [3], is a generalization of the notion of a (co)simplicial object in that category.
Cyclic cohomology has been considered in various versions and generalizations.In particular, in [6], Connes and Moscovici defined the Hopf cyclic cohomology by associating a cocyclic vector space to a Hopf algebra H over C endowed with a modular pair in involution (that is, a pair (δ, σ) where δ : H → C is a character and σ ∈ H is a grouplike element verifying the modular pair condition δ(σ) = 1 and a certain involutivity condition).Also, in [5], they relate the Hopf cyclic cohomology of H to the cyclic cohomology of an H-module algebra by means of a trace map.
Braided monoidal categories were defined by Street and Joyal in the 1980s and appeared in many areas of mathematics such as low-dimensional topology and representation theory.Several generalizations of cyclic (co)homology were introduced in the braided setting.In this paper, we focus on the braided generalization of the Connes-Moscovici construction due to Khalkhali and Pourkia [8].Let H be a Hopf algebra in a braided monoidal category B = (B, ⊗, 1).A modular pair for H is a pair (δ, σ), where δ : H → 1 is an algebra morphism and σ : 1 → H is a coalgebra morphism such that δσ = id 1 .
Our first main result is a complete computation (by means of the Penrose graphical calculus) of the powers (up to n + 1) of the paracocyclic operator τ n (δ, σ) associated with a modular pair (δ, σ) for H, see Theorem 2. Next, assume that B has a twist θ.We introduce the notion of a θ-twisted modular pair in involution for H (see Section 4.1) and prove (see Corollary 4) that if (δ, σ) is such a pair, then the associated paracocyclic operator satisfies the following twisted cocyclicity condition: for all n ∈ N, (τ n (δ, σ)) n+1 = θ H ⊗n .
When B is further k-linear, we derive (co)cyclic k-modules from a θ-twisted modular pair (δ, σ) by composing CM • (H, δ, σ) with the functors Hom B (1, −) and Hom B (−, 1), see Section 4.3.Note that if B is symmetric, then a braided modular pair in involution in the sense of [8] is a id B -twisted modular pair in involution, where id B is the trivial twist of B, and so the associated paracocyclic operator satisfies the cocyclicity condition.
Let H be a Hopf algebra in braided category B with a twist θ.Our second main result is the construction of traces à la Connes-Moscovici.More precisely, let C be a H-module coalgebra, that is, a coalgebra in the category of right H-modules in B. Inspired by a construction of Akrami and Majid [1], we associate to C a paracocyclic object C • (C) in B. We introduce the notion of a δ-invariant σ-trace for C and derive from each such trace a natural transformation from CM • (H, δ, σ) to C • (C), see Theorem 7.This generalizes the standard Connes-Moscovici trace.We provide examples of traces in the case where B is a ribbon category and H is its coend (see Section 5.3).
The paper is organized as follows.In Section 2, we review braided monoidal categories, Hopf algebras, and graphical calculus.Section 3 is devoted to preliminaries on simplicial, paracyclic, and cyclic objects in a category.In Sections 4 and 5, we state our main results and their corollaries.Sections 6 and 7 are devoted to the proofs of Theorems 2 and 7.In Appendix, we provide an alternative proof (by using the Penrose graphical calculus) of the fact that the object CM • (H, δ, σ) defined in [8] is paracocyclic.
Throughout the paper, k denotes any commutative ring.The class of objects of a category B is denoted by Ob(B).
2.4.Graphical calculus.Throughout this paper, we will use the Penrose graphical calculus.For a systematic treatment, one may consult [12].The diagrams are to be read from bottom to top.In a monoidal category B, the diagrams are made of arcs colored by objects of B and of boxes, colored by morphisms of B. Arcs colored by 1 may be omitted in the pictures.The identity morphism of an object X, a morphism f : X → Y in B, and its composition with a morphism g : Y → Z in B are represented respectively as The tensor product of two morphisms f : X → Y and g : U → V is represented by placing a picture of f to the left of the picture of g: Any diagram represents a morphism.For example, the diagram PSfrag replacements The morphism associated to a diagram depends only on the isotopy class of the diagram representing it.For example, the following level-exchange property: When B is braided with braiding τ , we depict Axioms (1) and ( 2) for τ say that for all X, Y, Z ∈ Ob(B), PSfrag replacements Naturality of the braiding and the level-exchange property imply that for any two morphisms f : X → Y and g : When B is braided with a twist θ = {θ X : X → X} X∈Ob(B) , we denote the twist by The multiplication and unit are depicted by m = and u = , so that the associativity and unitality axioms rewrite graphically as = and = = .
Here, it is understood that the arcs are colored by the underlying object of the algebra.An algebra morphism between algebras (A, m, u) and  A coalgebra morphism between coalgebras (C, ∆, ε) and 2.7.Graphical calculus and iterated (co)multiplications.Let (A, m, u) and (C, ∆, ε) be an algebra and a coalgebra in a monoidal category B. For any n ∈ N, we define the n-th multiplication m n : A ⊗n → A and the n-th comultiplication ∆ n : C → C ⊗n inductively by: For n ≥ 1, we depict them as and the following compatibility relations hold: Graphically, these rewrite as = , = , = , and = ∅.
A bialgebra morphism between two bialgebras A et A ′ is a morphism A → A ′ in B which is both an algebra and a coalgebra morphism.
2.9.Categorical Hopf algebras.A Hopf algebra in B is a sextuple (H, m, u, ∆, ε, S), where (H, m, u, ∆, ε) is a bialgebra in B and S : H → H is an isomorphism in B, called the antipode, which satisfies The antipode and its inverse are depicted by S = and S −1 = .
Graphically, the antipode axiom is rewritten as = = .
A useful feature of antipodes is that it is anti-multiplicative: and anti-comultiplicative: .
A Hopf algebra morphism between two Hopf algebras is a bialgebra morphism between them.
With composition inherited from B, left A-modules and morphisms between them form a category A Mod.
When B is braided and A is a bialgebra in B, the category A Mod is monoidal: the unit object of A Mod is the pair (1, ε), the monoidal product of two left A-modules (M, r) and (M ′ , r ′ ) is given by the pair (M ⊗ M ′ , s), where , and the monoidal product of morphisms is inherited from B.
With composition inherited from B, left C-comodules and morphisms between them form a category C Comod.
When B is braided and C is a bialgebra in B, the category C Comod is monoidal: the unit object of C Comod is the pair (1, u), the monoidal product of two left C-comodules (N, γ) and (N ′ , γ ′ ) is given by the pair (N ⊗ N ′ , δ), where , and the monoidal product of morphisms is inherited from B.
satisfying some conditions.Briefly, these say that the associated left/right dual functors coincide as monoidal functors (see [12, Chapter 1] for more details).The latter implies that the dual morphism f * : We extend the graphical calculus for monoidal categories (see Section 2.4) to pivotal categories by orienting arcs.If an arc colored by X is oriented upwards, the represented object in source/target of corresponding morphism is X * .For example, id X , id X * , and a morphism f : The morphisms ev X , ev X , coev X , and coev X are respectively depicted by while the right twist of X is defined by The left and the right twist are natural isomorphisms with inverses .
The left twist θ l = {θ l X : X → X} X∈Ob(B) and the right twist θ r = {θ r X : X → X} X∈Ob(B) are twists for B in the sense of Section 2.3.
A ribbon category is a braided pivotal category B whose left and right twist coincide.Then θ = θ l = θ r is called the twist of B.
2.17.Coends.Let C and D be any categories and F : C op × C → D a functor.A dinatural transformation between F and an object D in D is a function d that assigns to any object X in C a morphism d X : F (X, X) → D such that for all morphisms f : X → Y in C the following diagram commutes: A coend of a functor F : C op × C → D is a pair (C, i) where C is an object of D and i is a dinatural transformation from F to C, which is universal among all dinatural transformations.More precisely, for any dinatural transformation d from F to D, there exists a unique morphism ϕ : C → D in D such that d X = ϕi X for all X ∈ Ob(C).A coend (C, i) of a functor F , if it exists, is unique up to a unique isomorphism commuting with the dinatural transformation.
2.18.Coend of a pivotal category.Let B be a pivotal category.The coend of B, if it exists, is the coend (H, i) of the functor F : B op × B → B defined by We depict the universal dinatural transformation i = {i X : Note that H is a coalgebra in B with comultiplication ∆ : H → H ⊗H and counit ε : H → 1, which are unique morphisms such that, for all X ∈ Ob(B), PSfrag replacements The coalgebra (H, ∆, ε) coacts on the objects in B via the universal coaction defined for any X ∈ Ob(B) by δ X = (id X ⊗ i X ) • (coev X ⊗ id X ).We will denote it graphically as Note that δ H is the right coadjoint coaction of H on H (see Section 2.14).If B is braided, then H is a Hopf algebra in B. Its unit is u = δ 1 : 1 → H and its multiplication m : H⊗H → H and antipode S : H → H are characterized as follows: for all X, Y ∈ Ob(B), PSfrag replacements We refer to [12,Chapter 6] for details.

Simplicial, paracyclic, and cyclic objects
In this section we recall the notions of (co)simplicial, para(co)cyclic, and (co)cyclic objects in a category.
For n ∈ N and 0 ≤ j ≤ n, the j-th codegeneracy σ n j : n + 1 → n is the unique increasing surjection from [n + 1] onto [n] which sends both j and j + 1 to j.
It is well known (see [10,Lemma 5.1]) that morphisms in ∆ are generated by cofaces {δ n i } 0≤i≤n,n∈N * and codegeneracies {σ n j } 0≤j≤n,n∈N subject to the simplicial relations (SR): (SR) 3.2.The paracyclic category.The paracyclic category ∆C ∞ is defined as follows.The objects of ∆C ∞ are the nonnegative integers n ∈ N. The morphisms are generated by morphisms {δ n i } n∈N * ,0≤i≤n , called cofaces, morphisms {σ n j } n∈N,0≤j≤n , called codegeneracies, and isomorphisms {τ n : n → n} n∈N , called paracocyclic operators, satisfying the simplicial relations (SR) and the following paracyclic compatibility relations (PCR): (PCR) 3.3.The cyclic category.The cyclic category ∆C is defined as follows.The objects of ∆C are the nonnegative integers n ∈ N. The morphisms are generated by morphisms {δ n i } n∈N * ,0≤i≤n , called cofaces, morphisms {σ n j } n∈N,0≤j≤n , called codegeneracies, and isomorphisms {τ n : n → n} n∈N , called cocyclic operators, which satisfy the relations (SR), (PCR), and the cyclicity condition (CC): Note that ∆C is a quotient of ∆C ∞ .
3.4.(Co)simplicial, para(co)cyclic, and (co)cyclic objects in a category.Let C be any category.A simplicial object in C is a functor and a cocyclic object in C is a functor ∆C → C. A (co)simplicial/para(co)cyclic/(co)cyclic object in the category of sets (respectively, of k-modules) are called (co)simplicial/para(co)cyclic/(co)cyclic sets (respectively, k-modules).
A morphism between two (co)simplicial/para(co)cyclic/(co)cyclic objects is a natural transformation between them.One often denotes the image of a morphism f under a (co)simplicial/para(co)cyclic/(co)cyclic by the same letter f .Since the categories ∆, ∆C ∞ , ∆C are defined by generators and relations, a (co)simplicial/para(co)cyclic/(co)cyclic object in a category is entirely determined by the images of the generators satisfying the corresponding relations.For example, a paracocyclic object Y in C may be seen as a family Y , called codegeneracies, and isomorphisms {τ n : Y n → Y n } n∈N , called paracocyclic operators, subject to the relations (SR) and (PCR).Similarly, a morphism Clearly, the composition of a (co)simplicial/para(co)cyclic/(co)cyclic object X in C with a functor F : C → D is a (co)simplicial/para(co)cyclic/(co)cyclic object F X in D. In particular, useful examples are provided by the covariant and the contravariant Hom-functors Hom C (I, −) and Hom C (−, I), where I is an object of C. In this case, we denote:

Proof. Let us prove (a). By composition with Hom
is the k-linear morphism given by f → θ Yn f .The naturality of θ and the fact that Then, using the functoriality of Y • and the hypothesis that Part (b) is proved similarly.
3.5.Cyclic (co)homology.To any cyclic k-module X : ∆C op → Mod k , one can associate a bicomplex CC(X) (see [13]).The n-th cyclic homology HC n (X) of X is defined as the n-th homology of the total chain complex associated to the chain bicomplex CC(X).
A morphism between cyclic k-modules induces a levelwise morphism in cyclic homology.Similarly, to any cocyclic k-module Y : ∆C → Mod k , one can associate a cochain bicomplex CC(Y ), obtained by a construction dual to the one of a chain bicomplex.The n-th cyclic cohomology HC n (Y ) of Y is defined as the n-th cohomology of the total cochain complex associated to the cochain bicomplex CC(Y ).A morphism between cocyclic k-modules induces a levelwise morphism in cyclic cohomology.

Modular pairs and braided Connes-Moscovici construction
In this section, B is a braided monoidal category and H is a Hopf algebra in B. We provide a braided generalization of the notion of a modular pair in involution for H and then compute the powers of the paracocyclic operator associated to such a pair (see Theorem 2 and its corollaries).

IVAN BARTULOVI Ć
4.1.Modular pairs.A modular pair for H is a pair (δ, σ) where δ : H → 1 is an algebra morphism and σ : 1 → H is a coalgebra morphism such that δσ = id 1 .For instance, (ε, u) is a modular pair for H, where ε : H → 1 and u : 1 → H are the counit and unit of H, respectively.Given a twist θ for B, a θ-twisted modular pair in involution for H is a modular pair (δ, σ) for H such that PSfrag replacements Here we use the graphical conventions from Section 2.4.If H is involutive Hopf algebra in B in the sense that S 2 = θ H , then (ε, u) is a θ-twisted modular pair in involution for H.
Note that if B is symmetric with a trivial twist id B (see Section 2.3), then an id B -twisted modular pair in involution corresponds to a braided modular pair in involution in the sense of [8].

4.2.
Powers of the paracocyclic operators.Let (δ, σ) be a modular pair for H.For any n ≥ 0, define the paracocyclic operator τ n (δ, σ) : Here we use the diagonal actions defined in Section 2.12.Note that the operators τ n (δ, σ) are the paracocyclic operators of a paracocyclic object in B associated with H and (δ, σ) (see Section 4.3).In the following theorem we compute the powers (up to n + 1) of τ n (δ, σ).
Theorem 2. For n ≥ 2 and 2 ≤ k ≤ n, we have: In addition, PSfrag replacements In the statements of the theorem, we use the diagonal actions together with the (co)adjoint (co)actions defined in Sections 2.13 and 2.14.Also, an integer k below an arc denotes the kth tensorand of H ⊗n .We prove Theorem 2 in Section 6 by induction and by using properties of modular pairs and twisted antipodes.
In the next corollary, we compute the (n + 1)-th power of the paracocyclic operator τ n (δ, σ) in terms of the (n + 1)-th power of the paracocyclic operator τ n (ε, u), where ε and u are the counit and unit of H. Corollary 3.For any modular pair (δ, σ) and any n ∈ N, Proof.We show the result by induction.For n = 0, this follows since the adjoint action on 1 is given by counit, since the coadjoint coaction on 1 is given by unit and the fact that τ 0 (ε, u) = τ 0 (δ, σ) = id 1 .Let us check the case n = 1.Using (5) and the fact that τ 1 (ε, u) = S, we obtain Suppose that the result is true for an n ≥ 1 and let us show it for n + 1.Indeed, we have PSfrag replacements Here (i) follows by applying (5) for the modular pair (δ, σ), (ii) by applying the result for n = 1, and (iii) by applying (5) for the modular pair (ε, u).
The next corollary states that the paracocyclic operator associated with a twisted modular pair in involution satisfies the "twisted cocyclicity condition".
Proof.The equality (τ n (δ, σ)) n+1 = θ H ⊗n is shown by induction.For n = 0, this follows by definition and the fact that Here (i) follows by Formula (5) of Theorem 2, (ii) follows by the fact that (δ, σ) is a θ-twisted modular pair in involution for H, (iii) follows by the naturality of the twist and the definitions of left coadjoint coaction and right adjoint action, (iv) follows by (co)associativity and the fact that δ is an algebra morphism and σ is a coalgebra morphism, (v) follows by the antipode axiom and (co)unitality.
Suppose that the statement is true for an n ≥ 1 and let us show it for n + 1.We have Here (i) follows from Formula (5) of Theorem 2, (ii) follows from Corollary 3, (iii) follows from the statement for n = 1 and the induction hypothesis, (iv) follows by the naturality of the braiding and from the axiom of the twist.Remark 5.If B is a symmetric monoidal category endowed with the trivial twist id B and (δ, σ) is a id B -twisted modular pair for H, then Corollary 4 gives that (τ n (δ, σ)) n+1 = id H ⊗n for all n ∈ N.This was first proved by Khalkhali and Pourkia in [8].4.3.Paracocyclic objects associated with modular pairs.Let (δ, σ) be a modular pair for H. Let us recall the paracocyclic object CM • (H, δ, σ) in B from [8] associated to this data.For any n ≥ 0, define For any n ≥ 1, define the cofaces {δ n i (σ) : H ⊗n−1 → H ⊗n } 0≤i≤n by setting δ 1 0 = u, δ 1 1 = σ, and for any n ≥ 2, For any n ≥ 0, define the codegeneracies {σ n j : H ⊗n+1 → H ⊗n } 0≤j≤n by σ n j = ...
For any n ≥ 0, the paracocyclic operators τ n (δ, σ) : H ⊗n → H ⊗n of CM • (H, δ, σ) are those defined in Section 4.2.Theorem 2 is useful to prove that CM • (H, δ, σ) is a paracocyclic object in B. We prove this in Appendix.In particular, we prove that for all n ∈ N, The next corollary derives (co)cyclic k-modules from CM • (H, δ, σ).It follows directly from Lemma 1 and Corollary 4.
Corollary 6.If B has a twist θ and (δ, σ) a θ-twisted modular pair in involution for H, then

Categorical Connes-Moscovici trace
In this section, B is a braided category with a twist θ and H is a Hopf algebra in B. We introduce traces (à la Connes-Moscovici) between paracocyclic objects associated with H (as in Section 4.3) and paracocyclic objects associated with an H-module coalgebra.We provide an explicit example of such traces using coends.We associate with C a paracocyclic object C • (C) in B. It is inspired by the construction of Akrami and Majid from [1].When B = Mod k is the category of k-modules, one recovers the cocyclic k-module implicitly defined in the work of Farinati and Solotar [7].When B is a symmetric monoidal category endowed with the trivial twist, then the underlying cosimplicial object of C • (C) is equal to the one considered in [2, Definition 2.2].

PSfrag replacements
It follows directly from the definition of a twist (see Section 2. Theorem 7. Let (δ, σ) be a modular pair for H and α be a δ-invariant σ-trace for C. Then the family {α n : We prove Theorem 7 in Section 7. The next corollary relates the cyclic (co)homologies associated with CM • (H, δ, σ) and C • (C). .This means that for any n ∈ N, there is a morphism where [f ] is a representative class of an n-th cyclic cocycle.This finishes the proof of the part (a).The proof of (b) is similar.
Here, the coaction denoted with a black dot is the universal coaction of H (see Section 2.18).The dual of (M, r) ∈ Ob(Mod H ) is given by (M * , r † ), where together with the (co)evaluation morphisms inherited from B: Note that the last equality in the definition of r † follows from the involutivity of H.

=
PSfrag replacements Here (i) follows by definition of f and a, (ii) follows from the isotopy of the graphical calculus for pivotal categories and the multiplicativity of the counit, (iii) by the naturality of the braiding, the fact that εS = ε, and the counitality, (iv) since universal coaction of H on itself is the right coadjoint coaction, (v) by the naturality of the braiding, the (co)unitality and the antipode axiom, (vi) follows by definition of f .
Next, f : H → C is a coalgebra morphism.Indeed, we have: Here (i) and (v) follow by definition of f , (ii) from the isotopy of the graphical calculus for pivotal categories and the multiplicativity of the counit, (iii) by the naturality of the braiding and the fact that universal coaction of H on itself is the right coadjoint coaction, (iv) follows by the naturality of the braiding, the (co)unitality, and the antipode axiom.Also, PSfrag replacements Here (i) follow from definitions of f and ε C and (ii) from the fact that εu = id 1 .
Finally, let us show that α is an ε-invariant u-trace.By definition of α and the fact that f : H → C is an H-module morphism, we have that PSfrag replacements Here (i) and (vi) follow from definition, (ii) and (v) follow by the fact that f : H → C is a coalgebra morphism, (iii) follows by the naturality of twist and the braiding, and (iv) follows by hypothesis on κ.
Any coalgebra morphism 1 → H satisfies the condition of Lemma 9. Another family of examples satisfying the condition of Lemma 9 is given as follows: for any X ∈ Ob(B), set Here, (i) and (vi) follow by definition of κ X , (ii) and (v) follow by definition of comultiplication of H, (iii) by the naturality of twists, and (iv) by the naturality of the braiding and isotopy invariance of graphical calculus.Thus κ X satisfies the condition of Lemma 9 and so ( 6) Remark 10.Note that if B is k-linear, then any linear combination of δ-invariant σ-traces is a δ-invariant σ-trace.In particular, an interesting example of an ε-invariant u-trace comes from topological field theory: if B is a ribbon fusion k-linear category and I is a representative set of simple objects of B, then α = k∈I dim(k)α k is an ε-invariant u-trace.
Here α k is defined in (6) and dim(k) = ev k coev k = ev k coev k is the dimension of k.

Proof of Theorem 2
Our strategy to compute the (n + 1)-th power of the paracocyclic operator τ n (δ, σ) is similar to the proof of cocyclicity condition from Connes and Moscovici in [6], where Hopf algebras over C are considered.We indeed proceed by induction.The difficulty here is that the paracocyclic operators involve the braiding.In our approach, based on graphical calculus, we manage to keep track the powers of paracocyclic operators.In Section 6.1 we list algebraic properties used in our proof of the equalities from Theorem 2. In Section 6.2 we show Formula (4).In Section 6.3 we show Formula (5).
Recall that H denotes a Hopf algebra in the braided monoidal category B, δ : H → 1 is an algebra morphism and σ : 1 → H is a coalgebra morphism such that δσ = id 1 .Given such a pair, we define the twisted antipode S : For brevity, we denote the twisted antipode S graphically by PSfrag replacements H ∼ .With this notation, we will rewrite Similarly, equation (4), which is to be proven, rewrites as 6.1.Preliminary facts.In this section we state several lemmas, which are used in the proof of Theorem 2. We mention that equalities (a) and (b) from Lemma 11 Let us show the relation (b).Indeed, by definition of S, the fact that comultiplication is an algebra morphism, the fact that δ is an algebra morphism, and the naturality of the braiding we have PSfrag replacements Let us show the relation (c).Indeed, this relation follows by the definition of S, the coassociativity, the antipode axiom, and the counitality: PSfrag replacements Now we show the equality (d).It follows by the part (a), the naturality of the braiding, the definition of S, the fact that δ is an algebra morphism, and the definition of left coadjoint coaction: PSfrag replacements The equality (e) is a consequence of the equality (d).To see this, compose the left hand side of (d) with the antipode S of H and use the definition of S. Finally, let us show the equation (f ).Indeed, this equation follows by the part (b), the fact that comultiplication is an algebra morphism, the (co)associativity, the naturality of the braiding, the part (c), and the unitality: PSfrag replacements Remark 12. Another useful property of the twisted antipode S is that ε S = δ.It follows by the definition of S, the fact that εS = ε, and the counitality.
The following lemma gives the expression of the paracocyclic operator τ n (δ, σ) in terms of τ n−1 (ε, u).
Proof.Let us first show the equation (a).Indeed, by definition of τ n (δ, σ), Lemma 11(a), the naturality of the braiding, inductive definition of the left diagonal action, and the definition of τ n−1 (ε, u), we have Further, we show the part (b).For n = 2, the statement follows by definition.From now on, suppose that n ≥ 3.By the definition of τ n (δ, σ), the definition of the left diagonal action, the coassociativity, and the definition of τ n−1 (ε, u), we have: The equalities stated in the following lemma are used in computation of squares of the paracocyclic operator τ n (δ, σ) in the case n ≥ 3. Lemma 14.For any n ≥ 2, we have: PSfrag replacements PSfrag replacements Proof.We begin by showing the equality (a).Let us first inspect the case n = 2.To see that the equality is true in this case, we use the definition of τ 2 (ε, u), the fact that comultiplication is an algebra morphism, the coassociativity, the anti-comultiplicativity of the antipode, and the naturality of the braiding: PSfrag replacements From now on, suppose that n ≥ 3.By definition of τ n (ε, u), the fact that comultiplication is an algebra morphism, the naturality of the braiding, the coassociativity, and the anticomultiplicativity of the antipode we have: PSfrag replacements Let us show the equality (b).Indeed, it follows by definition of τ n (ε, u), the naturality of the braiding and the associativity: PSfrag replacements Let us show the equality (c).It follows from Lemma 13(b) applied on δ = ε and σ = u and by the fact that multiplication is an algebra morphism: PSfrag replacements Finally, let us show the equality (d).Indeed, we have: PSfrag replacements Here (i) follows by definition of τ n (ε, u) and τ n−1 (ε, u), (ii) follows by inductive definition of left diagonal action, (iii) follows by the anti-multiplicativity of the antipode, (iv) follows by the fact that multiplication is an algebra morphism, (v) follows by the axiom of a module and the anti-comultiplicativity of the antipode, (vi) follows by the naturality of the braiding, the (co)associativity, and by the axiom of a module, (vii) follows by applying the antipode axiom twice and by the axiom of a module, (viii) follows by the fact that εS = ε, the naturality of the braiding, and definition of τ n−1 (ε, u).
The equalities from the following lemma show how the endomorphism m(id H ⊗ σ) interacts with the paracocyclic operator τ n (ε, u).These equalities are intensively used while proving Formula (5) by using Formula (4) of Theorem 2.

IVAN BARTULOVI Ć
Proof.Let us prove the part (a).Indeed, we have: Sfrag replacements Here (i) follows by definition of τ n (ε, u), (ii) follows by the anti-multiplicativity of antipode, (iii) follows by the naturality of the braiding and the fact that multiplication is an algebra morphism, (iv) follows by the anti-comultiplicativity of antipode, (v) follows by the fact that σ is a coalgebra morphism and by the axiom of a module and (vi) follows by definition of τ n (ε, u) and the naturality of the braiding.
Let us now show the part (b).Indeed, by definition of τ n (ε, u) and the left diagonal action, by the naturality of the braiding, and the associativity, we have: Before passing to the proof of Theorem 2, let us state another auxilary lemma.
Lemma 16.We have the following assertions: PSfrag replacements The equality (b) from Lemma 16 is intensively used while proving both of the equalities from Theorem 2. The equality (c) from Lemma 16 is particularly used in final steps of the computation of τ n (δ, σ) n+1 .
Proof.Let us first show the part (a).We first show that the morphism from part (a) is an algebra morphism.By using definition of the left coadjoint coaction, the bialgebra compatibility axiom, the fact that δ is an algebra morphism, the anti-multiplicativity of the antipode of H, and the naturality of the braiding, we have: Similarly, by using definition of the left coadjoint coaction, the fact that unit is a coalgebra morphism, the fact that δ is an algebra morphism, and by the fact that Su = u, we have: Let us now show that the morphism from (a) is a coalgebra morphism.Indeed, by definition of the left coadjoint coaction, the fact that δ is an algebra morphism, the naturality of the braiding, the coassociativity, the antipode axiom, and the (co)unitality, we have: Furthermore, by definition of the left coadjoint coaction, the fact that δ is an algebra morphism, the naturality of the braiding, the (co)unitality, and the antipode axiom we have:

This completes the proof of the part (a).
Let us show the part (b) by induction.For n = 1, we prove the statement as follows.By using the part (a), the definition of the coadjoint coaction, the fact that δ is an algebra morphism, the naturality of the braiding, the coassociativity, the antipode axiom, and the (co)unitality we have: Suppose that the statement is true for an n ≥ 1 and let us show it for n + 1.We have: PSfrag replacements which shows the desired statement.Here (i) and (viii) both follow by inductive definition of left diagonal action and left coadjoint coaction, (ii) follows by the fact that δ is an algebra morphism, (iii) follows by the coassociativity, (iv) follows by the naturality of the braiding and the case n = 1, (v) and (vii) both follow by the naturality of the braiding and by the coassociativity and (vi) follows by the induction hypothesis.Finally, we show the part (c) by induction.For n = 1, the statement follows by definition of right adjoint action and the fact that σ is a coalgebra morphism.Suppose that the statement is true for n ≥ 1 and let us show it for n + 1.Indeed, by using inductive definition of the left and the right diagonal actions, the anti-multiplicativity of the antipode, the fact that σ is a coalgebra morphism, the naturality of the braiding, and the induction hypothesis, we have: PSfrag replacements PSfrag replacements PSfrag replacements . Indeed, this follows by the definition of S, the fact that σ is a coalgebra morphism, and since the (δ, σ) is a modular pair: PSfrag replacements 6.2.Proof of Formula (4).The proof of Formula (4) of Theorem 2 is divided into several steps.For n = k = 2, it suffices to calculate the square of τ 2 (δ, σ).For n ≥ 3, we first calculate the square and then derive formulas for the remaining powers.
6.2.1.Squares of τ n (δ, σ) for n ≥ 2. Let us first show that Formula (4) is true in the case n = k = 2. Indeed, we have: Here (i) follows by definition of τ 2 (δ, σ), (ii) follows from Lemma 13(a) for n = 2 and since τ 1 (ε, u) = S, (iii) follows by the fact that multiplication is an algebra morphism and associativity, (iv) by the anti-multiplicativity of the antipode, (v) follows by the associativity and the naturality of the braiding and (vi) follows by the antipode axiom, the naturality of the braiding, the (co)unitality, and the fact that τ 1 (ε, u) = S.

Proof of Formula
If n = 1, then we have Here (i) follows by definition τ 1 (δ, σ), (ii) follows from Lemma 11(b), (iii) follows by the naturality of the braiding, the associativity, and Remark 17, (iv) follows by definition of right adjoint action, the fact that σ is a coalgebra morphism, and Lemma 11(e).

Proof of Theorem 7
In order to show that the family {α n : H ⊗n → C ⊗n+1 } n∈N is a morphism between the paracocyclic objects CM • (H, δ, σ) and C • (C) in B, we will directly check that Note that we abusively use the same notation for cofaces, codegeneracies, and paracocyclic operators of two different constructions.These should be understood from context.Roughly described, the equalities ( 7) and ( 8) follow by the fact that C is a coalgebra in the category of right H-modules.In order to show the equality (9)

PSfrag replacements
, which indeed proves the claim for n + 1.Here (i) follows by the inductive definition of the left diagonal action, (ii) follows by the induction hypothesis, (iii) follows by the right module axiom and (iv) follows by the naturality of the braiding and the fact that the comultiplication ∆ C : C → C ⊗ C is an H-linear morphism.
7.1.Proof of the equality (7).Let us show the equality (7).If n = 1 and i = 0, then the equality (7) writes as α 1 δ 1 0 = δ 1 0 α 0 , which follows by definitions and the right module axiom.Let n = i = 1.In this case, the equality (7) writes as α 1 δ 1 1 = δ 1 1 α 0 , which is exactly the condition that α : 1 → C is a σ-trace.This shows the equality (7) for n = 1 and 0 ≤ i ≤ 1.Now let n ≥ 2 and i = 0.By definitions, the naturality of the braiding, the axiom of a right module, and the coassociativity, we have: Let n ≥ 2 and 1 ≤ i ≤ n − 1.In this case, equation (7) follows by definitions, the coassociativity, and the fact that ∆ C : C → C ⊗ C is an H-linear morphism: Finally, let n ≥ 2 and i = n.Equation (7) in this case follows by the case n ≥ 2 and i = 0, which is written above, the equation ( 9), which is proven in Section 7.3, and by the paracyclic compatibility relation τ n δ n 0 = δ n n (see Section 3.2).Indeed, we have 7.2.Proof of the equality (8).Let us show the equality (8).We consider the three following cases: j = 0, 1 ≤ j ≤ n − 1, and j = n.In each case, the desired equality follows by definition, the counitality, the fact that the counit ε C : C → 1 is an H-linear morphism, and the naturality of the braiding.Indeed, if j = 0, then we have Finally, if j = n, then we have 7.3.Proof the equality (9).Let us verify that equation ( 9) holds.When n = 0, this holds since twist morphisms are natural, θ 1 = id 1 , and since τ 0 (δ, σ) = id 1 .Indeed, Let us check it for the case n = 1.Indeed, we have Here (i) and (viii) follow by definition, (ii) follows by the fact that α is δ-invariant, (iii) follows from the fact that the comultiplication ∆ C : C → C ⊗ C is H-linear.The equality (iv) follows by the (co)associativity, (v) follows by the antipode axiom and the (co)unitality, (vi) follows from the fact that α is a σ-trace and (vii) follows by the naturality of the braiding.
Finally, let us check the equality (9) when n ≥ 2. Indeed, we have Here (i) and (x) follow by definition, (ii) follows by applying Lemma 18, (iii) follows from the fact that α is δ-invariant, (iv) follows from the fact that the comultiplication ∆ C : C → C ⊗ C is H-linear.The equality (v) follows from the coassociativity and the right module axiom, (vi) follows by the antipode axiom, the (co)unitality, and the right module axiom, (vii) follows by the coassociativity and the naturality of the braiding, (viii) follows from the fact that α is a σ-trace and finally, (ix) follows by the naturality of the braiding.

Appendix
In this appendix, we verify paracyclic compatibility relations (PCR) of the paracocyclic object CM • (H, δ, σ) defined in Section 4.3.Note that the verification of simplicial relations (SR) for this object is an easy task.One can show it graphically, by using the level-exchange property (see Section 2.4), the coassociativity, and the counitality.Also, the relation τ n σ n i = σ n i−1 τ n+1 for 1 ≤ i ≤ n follows from the bialgebra axiom and by the naturality of the braiding.In this appendix, we show that  If n = 1 and i = 1, the relation rewrites as τ 1 δ 1 1 = δ 1 0 τ 0 and it follows by definitions, the fact that σ is a coalgebra morphism, the fact that (δ, σ) is a modular pair and by the antipode axiom: If n = 2 and i = 1, the relation rewrites as τ 2 δ 2 1 = δ 2 0 τ 1 and it follows by definitions, Lemma 11a), the coassociativity, the antipode axiom, and the naturality of the braiding: PSfrag replacements If n = 2 and i = 2, the relation rewrites as τ 2 δ 2 2 = δ 2 1 τ 1 and it follows by definitions, the fact that σ is a coalgebra morphism, and the fact that multiplication is an algebra morphism: PSfrag replacements Now let n ≥ 3.For i = 1, the relation rewrites as τ n δ n 1 = δ n 0 τ n−1 and it is true since Here (i) and (ix) follow by definition, (ii) follows by inductive definition of the left diagonal action, (iii) and (viii) follow from Lemma 11(a), (iv) and (vi) follow by the naturality of the braiding and the coassociativity, (v) follows by the anti-comultiplicativity of the antipode, (vii) follow by the antipode axiom, the counitality, and the naturality of the braiding.For 2 ≤ i ≤ n − 1, the relation τ n δ n i = δ n i−1 τ n−1 is a consequence of the fact that multiplication is an algebra morphism and the naturality of the braiding.Let us check τ n δ n i = δ n i−1 τ n−1 for i = n.The relation follows by definition, Lemma 13b), by the fact that σ is a coalgebra morphism, and by the fact that the comultiplication is an algebra morphism:  = τ 0 (δ, σ)σ 0 0 .
Here (i) follows by using Formula (5) of Theorem 2 and the definition of σ 0 0 , (ii) follows from definition of right adjoint action and left coadjoint action of H on itself, the naturality of the braiding, and the fact that δ is an algebra morphism and σ is a coalgebra morphism.The equality (iii) follows by applying twice the fact that the counit is an algebra morphism, (iv) follows by the fact that εS = ε and since σ is a coalgebra morphism, (v) follows by the fact that δ is an algebra morphism and by the antipode axiom, (vi) follows by definition of τ 0 (δ, σ) and σ 0 0 .

Axiom ( 3 ) 2 . 5 .
for θ gives that for any X, Y ∈ Ob(B), Categorical algebras.An algebra in a monoidal category B is a triple (A, m, u), where A is an object of B, m : A ⊗ A → A and u : 1 → A are morphisms in B, called multiplication and unit respectively, which satisfy the associativity and unitality axioms:

The comultiplication and counit are depicted
)associativity and (co)unitality of m and ∆ imply that

2. 10 .
Categorical modules.Let (A, m, u) an algebra in a monoidal category B. A left Amodule in B is a pair (M, r), where r : A ⊗ M → M is a morphism in B, called the action of A on M, which satisfies r(m ⊗ id M ) = r(id A ⊗ r) and r(u ⊗ id M ) = id M .Graphically, the action r : A ⊗ M → M is denoted by axioms of a left A-module rewrite as PSfrag replacements

5. 1 .
Paracocyclic objects associated with coalgebras.Let C be a coalgebra in B.

..for n ≥ 1 .
3) that the paracocyclic operator for C • (C) satisfies the relation τ n+1 n = θ C ⊗n+1 for all n ∈ N. 5.2.Traces.Let C be an H-module coalgebra in B, that is, a coalgebra in the category of right H-modules in B. In other words, C is a coalgebra in B endowed with a right action r : C ⊗ H → C of H on C such that the comultiplication ∆ C and the counit ε C of C are both H-linear, that is, morphisms of right H-modules.By depicting the right action by linearity of ∆ C and ε C depicts as In this pictures, the red strands are colored by C and the black ones by H. Let δ : H → 1 be an algebra morphism and let σ : 1 → H be a coalgebra morphism.A δ-invariant σ-trace for C is a morphism α : 1 → C in B satisfying PSfrag replacements Given such a morphism, define for any n ∈ N the morphism α n : H ⊗n → C ⊗n+1 in B by setting α 0 = α and α n = Consider the paracocyclic object CM • (H, δ, σ) in B (see Section 4.3) and the paracocyclic object C • (C) in B associated to the coalgebra C in B (see Section 5.1).

.Lemma 9 .: 1
Mod H has a coend ((C, a), j), where C = H * ⊗H, the action a : C⊗H → C of H on C is computed by dinatural transformation j = {j (M,r) : (M, r) * ⊗(M, r) → (C, a)} (M,r)∈Mod H is given by j (M,r) 18, the coend C is a Hopf algebra in Mod H .In particular, it is a coalgebra in Mod H .The comultiplication ∆ C : C → C ⊗ C and the counit ε C : C → 1 of C are computedThe following lemma gives a way to produce an ε-invariant u-trace 1 → C, where ε and u denote the counit and unit of H.If a morphism κ :1 → H in B satisfies → C is an ε-invariant u-trace.Proof.Denote f = ε * ⊗ id H : H → C.Let us first check that f is a morphism between right H-modules (H, ε) and (C, a).Indeed,

:
H → H is a bialgebra morphism.(b) For all n ≥ 1, PSfrag replacements (i) follows by parts (a) and (b) of Lemma 13, (ii) follows by the associativity and from Lemma 14(a) for n−1, (iii) follows by the associativity and from Lemma 14(b) for n−1, (iv) follows from Lemma 14(c) for n − 1, (v) follows from Lemma 14(d) for n − 1 and (vi) follows by the counitality, definition of the twisted antipode S, and from Lemma 13(b) applied on δ = ε and σ = u for n − 1.

τ n− 1 . 8 . 2 .= σ = δ 1 1 . 8 . 3 .
The relation τ n δ n 0 = δ n n .For n = 1, the relation rewrites as τ 1 δ 1 0 = δ 1 1 and it follows by definition, the fact that δ is an algebra morphism, by εS = ε, and the fact that the unit is a coalgebra morphism: Now let n ≥ 2. By definition, the fact that δ is an algebra morphism, by εS = ε, the fact that the unit is a coalgebra morphism, the left module axiom, and the naturality of the braiding, we haveτ n δ n 0 = The relation τ n (δ, σ)σ n 0 = σ n n (τ n+1 (δ, σ)) 2 .In order to show the relation, one can use Theorem 2. We first prove the case n = 0.It is true since σ 0 0 (τ 1 (δ, σ)) 2.12.Diagonal actions.Let H be a bialgebra in a braided category B. The left diagonal action of H on H ⊗n is defined inductively by It follows from the definitions, that if σ : 1 → H is a coalgebra morphism, then ... ...Similarly, the right diagonal action of H on H ⊗n is defined inductively by Pivotal categories.A pivotal category is a monoidal category B such that each object X of B has a dual object X * and four morphisms ev [12,Traces from coends.Let B be a ribbon category with a coend (H, i), see Section 2.18.By[12, Chapter 6], the object H is a Hopf algebra in B which is involutive (that is S 2 = θ H , where θ is the twist of B, see Section 2.16).By Section 4.1, since H is involutive, the pair (ε, u) is a θ-twisted modular pair in involution for H. Since Mod H is braided isomorphic to the center of B (see[12, Section 6.5.3]for details), we obtain that Mod H is ribbon.The braiding of Mod H is given by [8, the equality from Remark 17 are already stated in[8, Proposition 4.3].In the lemma that follows, some properties of the twisted antipode are established.
in the case n ≥ 2, we will need the following computation: We prove the claim by induction.Let us first show it for n = 2. Indeed, by the right module axiom, the naturality of braiding and the fact that the comultiplication ∆ C : C → C ⊗ C is an H-linear morphism, we have Suppose that the claim is true for an n ≥ 2 and let us show it for n + 1.We have .Proof.