BiHom Hopf algebras viewed as Hopf monoids
aa r X i v : . [ m a t h . QA ] M a r BIHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS
GABRIELLA B ¨OHM AND JOOST VERCRUYSSE
Abstract.
We introduce monoidal categories whose monoidal products of any pos-itive number of factors are lax coherent and whose nullary products are oplax coher-ent. We call them
Lax + Oplax -monoidal. Dually, we consider Lax Oplax + -monoidalcategories which are oplax coherent for positive numbers of factors and lax coherentfor nullary monoidal products. We define Lax + Oplax + - duoidal categories with com-patible Lax + Oplax - and Lax Oplax + -monoidal structures. We introduce comonoidsin Lax + Oplax -monoidal categories, monoids in Lax Oplax + -monoidal categories andbimonoids in Lax + Oplax + - duoidal categories.Motivation for these notions comes from a generalization of a construction in [8].This assigns a Lax + Oplax + -duoidal category D to any symmetric monoidal category V . The unital BiHom -monoids, counital
BiHom -comonoids, and unital and couni-tal
BiHom -bimonoids of [11] in V are identified with the monoids, comonoids andbimonoids in D , respectively. Introduction
In recent years, lead by diverse motivations, several different generalizations ofHopf algebra have been proposed. Their similar features naturally raise the questionwhether they are instances of the same, more general notion. In [6] several examples,such as groupoids, Hopf monoids in braided monoidal categories, Hopf algebroids overcentral base algebras and weak Hopf algebras, were unified as Hopf monoids in theduoidal (called 2-monoidal in [1]) endohom category of a naturally Frobenius mapmonoidale in a suitable monoidal bicategory. In [4] also Hopf group algebras [13],Hopf categories [2] and Hopf polyads [7] were shown to fit this framework.The aim of the current paper is an interpretation of the (unital and counital)
BiHom -bimonoids in [11] as bimonoids in a category with some generalized duoidal structure.Recall that a
BiHom -monoid in a monoidal category ( V , ⊗ , I ) (whose coherence naturalisomorphisms are omitted) consists of an object a together with morphisms α, β ∶ a → a and µ ∶ a ⊗ a → a such that α and β commute, both of them preserve the multiplication µ , and, instead of the associativity of µ , the first diagram of a ⊗ a ⊗ a µ ⊗ (cid:15) (cid:15) a ⊗ a ⊗ a ⊗ ⊗ β o o α ⊗ ⊗ / / a ⊗ a ⊗ a ⊗ µ (cid:15) (cid:15) a ⊗ a µ / / a a ⊗ a µ o o a η ⊗ (cid:15) (cid:15) β / / a a α o o ⊗ η (cid:15) (cid:15) a ⊗ a µ / / a a ⊗ a µ o o ( ∗ )commutes. A BiHom -monoid ( a, α, β, µ ) is said to possess a unit η ∶ I → a if it ispreserved by α and β and also the second diagram above commutes. Diagrams with Date : March 2020. reversed arrows define (counital)
BiHom -comonoids . The (unital and counital)
BiHom -bimonoids in a symmetric monoidal category consist of a (unital)
BiHom -monoid struc-ture and a (counital)
BiHom -comonoid structure on the same object a (with possiblydifferent endomorphism parts α, β and κ, ν ) such that the multiplication µ (and theunit η ) are morphisms of (counital) BiHom -comonoids; equivalently, the comultiplica-tion (and the counit) are morphisms of (unital)
BiHom -monoids.One can see several attempts in the literature aiming at a description of (unital)
BiHom -monoids, (counital)
BiHom -comonoids and (unital and counital)
BiHom -bi-monoids, respectively, as monoids, comonoids and bimonoids in a suitable category.All of these ideas originate from the same construction in [8]. In [8], to any monoidalcategory ( V , ⊗ , I ) , a category is associated whose objects are pairs consisting of anobject a of V and an automorphism α of a . The morphisms ( a, α ) → ( a ′ , α ′ ) aremorphisms φ ∶ a → a ′ in V such that φ.α = α ′ .φ . It is equipped with a monoidalstructure with monoidal unit ( I, ) , monoidal product ( a, α )⊗( a ′ , α ′ ) ∶ = ( a ⊗ a ′ , α ⊗ α ′ ) and associativity and unitality coherence isomorphisms α ⊗ ⊗ α ′′− ∶ (( a, α ) ⊗ ( a ′ , α ′ )) ⊗ ( a ′′ , α ′′ ) → ( a, α ) ⊗ (( a ′ , α ′ ) ⊗ ( a ′′ , α ′′ )) ,α ∶ ( I, ) ⊗ ( a, α ) → ( a, α ) ,α ∶ ( a, α ) ⊗ ( I, ) → ( a, α ) . The monoids (respectively, comonoids) in this monoidal category are those unital
BiHom -monoids (respectively, those counital
BiHom -comonoids) in V whose two con-stituent endomorphisms are equal automorphisms. They are called in [8] monoidal Hom -monoids (respectively, monoidal
Hom -comonoids). Any possible symmetry on ( V , ⊗ , i ) is inherited by this monoidal category of [8]. Then the bimonoids therein arethose unital and counital BiHom -bimonoids in V whose comonoid part contains twocopies of an automorphism and the monoid part contains two copies of its inverse.They are called in [8] monoidal Hom -bimonoids.The above construction from [8] was widely generalized in [11, Section 2]. In the par-ticular case which is relevant here, the objects are triples consisting of an object a of V and two commuting automorphisms α and β of a . The resulting category also admitsa similar (symmetric) monoidal structure (where half of the occurring morphisms α isreplaced by β ). The monoids, comonoids and bimonoids in this (symmetric) monoidalcategory cover also those unital BiHom -monoids, counital
BiHom -comonoids and uni-tal and counital
BiHom -bimonoids, respectively, in which two possibly unrelated au-tomorphisms occur. Such
BiHom -objects are called in [11] monoidal
BiHom -monoids, monoidal
BiHom -comonoids and monoidal
BiHom -bimonoids respectively.So the essential observation, originated in [8], is that these monoidal
Hom - and
BiHom -structures are monoids, comonoids and bimonoids in a suitable (symmetric)monoidal category. Consequently, the standard machinery applies to them. Thisexplains why many aspects of this theory have been obtained in recent literature witha mild adaptation of the classical proofs.However, more general
BiHom -structures, with not necessarily invertible endomor-phism parts, are not covered by this construction, and it is the main aim of this paperto provide a categorical construction that covers these cases.Firstly, let us remark that the category whose objects consist of an object a ofa symmetric monoidal category V together with four commuting automorphisms of IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 3 a carries a duoidal structure (termed in [1]). One monoidal structure isdefined in terms of one half of the automorphisms and a second monoidal structureis defined in terms of the other half. The compatibility morphisms are provided bythe symmetry of V . The monoids, comonoids and bimonoids in this duoidal cate-gory finally cover also those unital BiHom -monoids, counital
BiHom -comonoids andunital and counital
BiHom -bimonoids, respectively, in which four possibly different,but still invertible endomorphisms occur. (A construction of this flavor occurred in[14]. However, that construction is based on four monoidal natural automorphisms ofthe identity functor on V . This seems to be quite restrictive since the identity func-tor on the category of vector spaces, for example, admits no other monoidal naturalautomorphism but the identity.)In order to deal with BiHom -structures with not necessarily invertible endomor-phism parts, one should give up the invertibility of the coherence natural transfor-mations and use the unbiased variants of monoidal category [12]. Although there is ameaningful theory of duoidal categories with one lax and one oplax monoidal struc-ture [10, Section 4], for the description of
BiHom -structures neither the lax nor theoplax variant looks suitable but rather a certain mixture of both. Indeed, for theformulation of the diagrams of ( ∗ ), one needs coherence morphisms of the kind ( a ⊗ b ) ⊗ c a ⊗ b ⊗ c o o / / a ⊗ ( b ⊗ c ) I ⊗ a / / a a ⊗ I. o o Those on the left are of the oplax type, while those on the right are of the lax type.In this paper we define unbiased monoidal categories of the above mixed type.As a first step, we consider
Lax + - (respectively, Oplax + - ) monoidal categories withmonoidal products of only positive number of factors with lax (respectively, oplax)coherence morphisms. We define cosemigroups (respectively, semigroups) in suchmonoidal categories. We also define Lax + Oplax + -duoidal categories which possess com-patible Lax + - and Oplax + -monoidal structures. Semigroups in a Lax + Oplax + -duoidalcategory are shown to constitute a Lax + -monoidal category; dually, cosemigroups in a Lax + Oplax + -duoidal category are shown to constitute an Oplax + -monoidal category. Sowe can define bisemigroups in a Lax + Oplax + -duoidal category as cosemigroups in thecategory of semigroups; equivalently, as semigroups in the category of cosemigroups.Next we introduce so called Lax Oplax + -monoidal categories, which have n -foldmonoidal products for any non-negative integer n such that the monoidal products ofpositive number of factors are oplax coherent while the 0-fold monoidal product is laxcoherent. Forgetting about 0-fold monoidal products, a Lax Oplax + -monoidal categorycan be seen Oplax + -monoidal. We define monoids in Lax Oplax + -monoidal categories.Dually, we introduce so called Lax + Oplax -monoidal categories, again with n -foldmonoidal products for any non-negative integer n such that the monoidal products ofpositive number of factors are lax coherent while the 0-fold monoidal product is oplaxcoherent. We define comonoids in Lax + Oplax -monoidal categories. Finally we define Lax + Oplax + -duoidal categories with compatible Lax + Oplax - and Lax Oplax + -monoidalstructures. Monoids in them are shown to constitute a Lax Oplax + -monoidal categoryand, dually, comonoids in them are shown to constitute a Lax + Oplax -monoidal cate-gory. So we can define bimonoids in them as comonoids in the category of monoids;equivalently, as monoids in the category of comonoids. GABRIELLA B ¨OHM AND JOOST VERCRUYSSE
Generalizing the construction in [8], we associate a
Lax + Oplax + -duoidal category D to any symmetric monoidal category V . We identify the semigroups, cosemigroupsand bisemigroups in D with the BiHom -monoids,
BiHom -comonoids and the
BiHom -bimonoids in V , respectively. We also identify the monoids, comonoids and bimonoidsin D with the unital BiHom -monoids, counital
BiHom -comonoids and the unital andcounital
BiHom -bimonoids in V , respectively, in the sense of [11].The consequence of our construction is twofold. Firstly, by extending the microcosm hosting bimonoids [3], it provides a suitable categorical framework where BiHom -objects with not necessarily invertible endomorphisms can be studied. On the otherhand, our proposed setting goes beyond the known framework of symmetric monoidalcategories, and even of duoidal categories. In this way results for such objects canno longer directly be obtained from the existing theory of monoids, comonoids abimonoids in duoidal categories, let alone in (symmetric) monoidal categories. Indeed,one would need first to extend this theory to the framework of
Lax + Oplax + -duoidalcategories, which is more involved than the classical case. This indicates that thetheory of BiHom -objects with not necessarily invertible endomorphisms is itself moreinvolved; no longer a straightforward generalization of the classical theory.The same axioms defining our various lax and oplax monoidal categories can beused to define monoids of the same lax and oplax type in any Gray monoid. Suchmonoids can be seen as the 0-cells of a 2-category. If the ambient Gray monoid is asymmetric strict monoidal 2-category, then this 2-category of monoids is again strictmonoidal. In this case we can also define duoids of various lax and oplax type, assuitably lax monoids in the strict monoidal 2-category of suitably lax monoids.However, not to distract attention from the main aim to describe
BiHom -structures,we do not work at the level of generality of the previous paragraph. We restrict ourstudy to the symmetric strict monoidal 2-category
Cat of categories, functors, andnatural transformations.We do not assume that the monoidal categories in the paper are strict monoidal but— relying on coherence — we omit explicitly denoting their Mac Lane type coherencenatural isomorphisms.
Acknowledgement.
GB is grateful for the financial support by the Hungarian Na-tional Research, Development and Innovation Office NKFIH (grant K124138). JVthanks the FNRS (National Research Fund of the French speaking community inBelgium) for support via the MIS project ‘Antipode’ (grant F.4502.18).1.
Some combinatorial background
As a preparation for the constructions of the subsequent sections, we begin withrecording some notation and a few technical results that will be heavily applied later.
In the 2-category
Cat of categories, functors and natural transformations wedenote by ⋅ the composition of functors and the corresponding Godement productof natural transformations. Cat is strict monoidal via the Cartesian product × andsymmetric via the flip maps. We often denote the Cartesian product (of any ofcategories, functors, and natural transformations) by juxtaposition. The Cartesianproduct of n copies of the same category A is also denoted by A n . By A we mean thesingleton category for any category A . For all non-negative integers n, p , consider IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 5 the particular components
Cat np Ô⇒ τ np ( Cat p ) n × n / / Cat p × p (cid:15) (cid:15) ( Cat n ) p × p (cid:15) (cid:15) Cat n × n / / Cat of the symmetry. They are 2-natural transformations whose component at any doublesequence of categories { A ij } i = ,...,pj = ,...,n is the flip functor ( A , . . . , A n ) , . . . , ( A p , . . . , A pn ) → ( A , . . . , A p ) , . . . , ( A n , . . . , A pn ) , ( a , . . . , a n ) , . . . , ( a p , . . . , a pn ) ↦ ( a , . . . , a p ) , . . . , ( a n , . . . , a pn ) . They satisfy the equalities τ np ⋅ τ pn = τ n = τ np ⋅ ( τ nk . . . τ nk p ) = τ n ∑ pi = k i (1.1)for all non-negative integers n, p and k , . . . , k p . We adopt the convention that empty sums are equal to zero. The set of non-negative integers is denoted by N and the set of (strictly) positive integers is denotedby N + .The following maps N → N will occur frequently. Z ( m ) ∶ = { m >
01 if m = m ∶ = m + Z ( m ) . For n ∈ N and a sequence of non-negative integers { m , . . . , m n } we denote the sumand the number of zeros, respectively by M ∶ = n ∑ i = m i and Z ∶ = n ∑ i = Z ( m i ) . Clearly, M + Z ≥ n and thus M + Z = n = { k ij } i = ,...,nj = ,...,mi (also in the particularcase when n = K i ∶ = m i ∑ j = k ij Z i ∶ = m i ∑ j = Z ( k ij ) K ∶ = n ∑ i = K i Z ∶ = n ∑ i = Z i . We introduce two associated double sequences with i = , . . . , n and j = , . . . , m i : ̃ k ij ∶ = { k ij if m i >
00 if m i = ̂ k ij ∶ = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ k ij if K i >
01 if K i = , m i > , j =
10 if K i = , m i > , j >
11 if m i = . We put ̃ Z ∶ = n ∑ i = m i ∑ j = Z (̃ k ij ) = Z + n ∑ i = Z ( m i ) = Z + n ∑ i = Z ( K i + Z i ) . GABRIELLA B ¨OHM AND JOOST VERCRUYSSE
Note that n ∑ i = m i ∑ j = (̂ k ij + Z (̂ k ij )) = n ∑ i = m i ∑ j = (̃ k ij + Z (̃ k ij )) = n ∑ i = K i + Z i + Z ( K i + Z i ) = K + ̃ Z. For any non-negative integer k , and for any functor F from the singleton category to an arbitrary category A , we introduce the functor A k [ k ] / / A k ∶ = ⎧⎪⎪⎨⎪⎪⎩ A k / / A k if k > F / / A if k = Lemma 1.3.
For any sequence of non-negative integers { k , . . . , k m } , for any functor F from the singleton category to an arbitrary category A , and for the associatedfunctors of (1.2) , the following diagrams commute. A K [ k ] ... [ k m ] / / [̃ k ] ... [̃ k m ] ' ' ◆◆◆◆◆◆◆◆◆◆◆◆ A K + Z [ K + Z ] (cid:15) (cid:15) A K + ̃ Z A [ K ] / / [̃ k ] ... [̃ k m ] ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ A K + Z ( K )[̂ k ] ... [̂ k m ] (cid:15) (cid:15) A K + ̃ Z Cosemigroups in
Lax + -monoidal categories Following the microcosm principle [3, Section 4.3], in order to define semigroups andcosemigroups internally to some monoidal category, one only needs to have products ofpositive number of factors, not necessarily nullary ones. In this section we introducecosemigroups in monoidal categories with monoidal products of arbitrary positivenumber of factors and lax coherence morphisms between them. The dual situation isdiscussed in the next section.Throughout, we use the notation introduced in Section 1.
Definition 2.1. A Lax + -monoidal category consists of ● a category L ● for all positive integers n , a functor ● n ∶ L n → L ● for all positive integers n, k , . . . , k n , natural transformations L K ● k ⋯ ● k n / / ● K Ô⇒ Φ k ,...,k n L n ● n (cid:15) (cid:15) L L ● > > Ô⇒ ι L such that the diagrams ● n ⋅ ( ● m ⋯ ● m n ) ⋅ ( ● k ⋯ ● k m ⋯ ● k n ⋯ ● k nm n ) Φ m ,...,mn ⋅ / / ⋅ ( Φ k ,...,k m ⋯ Φ kn ,...,knmn ) (cid:15) (cid:15) ● M ⋅ ( ● k ⋯ ● k m ⋯ ● k n ⋯ ● k nm n ) Φ k ,...,knmn (cid:15) (cid:15) ● n ⋅ ( ● K ⋯ ● K n ) Φ K ,...,Kn / / ● K ● n ι ⋅ / / ⋅ ( ι ⋯ ι ) (cid:15) (cid:15) ■■■■■■■■■■■■■■■ ■■■■■■■■■■■■■■■ ● ⋅ ● n Φ n (cid:15) (cid:15) ● n ⋅ ( ● ⋯ ● ) Φ ,..., / / ● n IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 7 commute, for any positive integers n , { m i } labelled by i = , . . . , n , and { k ij } labelledby i = , . . . , n and j = , . . . , m i .A Lax + -monoidal category is normal if ι is invertible and strict normal if ι is theidentity natural transformation.More generally, Lax + -monoids can be defined in any Gray monoid — so in particularin any strict monoidal 2-category — by the same diagrams in Definition 2.1. Lax + -monoidal categories are then re-obtained by applying it to Cat .Throughout, in a
Lax + -monoidal category we denote by G a H the image of any object a under the functor ● ∶ L → L ; and we denote by a ● ⋯ ● a n the image of an object ( a , . . . , a n ) under the functor ● n ∶ L n → L . Proposition 2.2.
The Cartesian product LL ′ of Lax + -monoidal categories ( L , ● , Φ , ι ) and ( L ′ , ● ′ , Φ ′ , ι ′ ) is again Lax + -monoidal via ● the functors ( LL ′ ) n τ n / / L n L ′ n ● n ● n ′ / / LL ′ ● the natural transformations ιι ′ ∶ → ● ● ′ and ( LL ′ ) K τ k ⋯ τ kn / / τ K L k L ′ k ⋯ L k n L ′ k n ● k ● k ′ ⋯ ● k n ● k n ′ / / τ n (cid:15) (cid:15) ( LL ′ ) nτ n (cid:15) (cid:15) L K L ′ K ● K ● K ′ Ô⇒ Φ k ,...,k n Φ ′ k ,...,k n ● k ⋯ ● k n ● k ′ ⋯ ● k n ′ / / L n L ′ n ● n ● n ′ (cid:15) (cid:15) LL ′ where the unlabelled regions denote identity natural transformations.Proof. It is left to the reader to check that the stated datum satisfies the axioms inDefinition 2.1 (analogously to Example (6) on page 80 of [9]). (cid:3)
Definition 2.3. A Lax + -monoidal functor consists of ● a functor G ∶ L → L ′ ● for all positive integers n , a natural transformation Γ n ∶ G ⋅ ● n → ● n ′ ⋅ ( G ⋯ G ) such that for all sequences of positive integers { k , . . . , k n } the following diagramscommute. G ι ′ ⋅ (cid:24) (cid:24) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ⋅ ι / / G ⋅ ● Γ (cid:15) (cid:15) ● ′ ⋅ G G ⋅ ● n ⋅ ( ● k ⋯ ● k n ) Γ n ⋅ (cid:15) (cid:15) ⋅ Φ k ,...,kn / / G ⋅ ● K Γ K (cid:15) (cid:15) ● n ′ ⋅ ( G ⋯ G ) ⋅ ( ● k ⋯ ● k n ) ⋅ ( Γ k ⋯ Γ kn ) (cid:15) (cid:15) ● n ′ ⋅ ( ● k ′ ⋯ ● k n ′ ) ⋅ ( G ⋯ G ) Φ ′ k ,...,kn ⋅ / / ● K ′ ⋅ ( G ⋯ G ) GABRIELLA B ¨OHM AND JOOST VERCRUYSSE
Proposition 2.4. (1)
The composite of (composable)
Lax + -monoidal functors ( G, Γ ) and ( H, Ξ ) is again Lax + -monoidal via the natural transformations H ⋅ G ⋅ ● n ⋅ Γ n / / H ⋅ ● n ⋅ ( G ⋯ G ) Ξ n ⋅ / / ● n ⋅ ( H ⋯ H ) ⋅ ( G ⋯ G ) = ● n ⋅ ( H ⋅ G ) n . (2) The Cartesian product of
Lax + -monoidal functors ( G, Γ ) and ( H, Ξ ) is again Lax + -monoidal via the natural transformations ( LN ) n τ n / / ( GH ) n (cid:15) (cid:15) L n N n ● n ● n / / G n H n (cid:15) (cid:15) Ô⇒ Γ n Ξ n LN GH (cid:15) (cid:15) ( L ′ N ′ ) n τ n / / L ′ n N ′ n ● n ′ ● n ′ / / L ′ N ′ where the unlabelled region denotes the identity natural transformation.Proof. It is left to the reader to check that the stated data satisfy the axioms inDefinition 2.3. (cid:3)
Definition 2.5. A Lax + -monoidal natural transformation ( G, Γ ) → ( G ′ , Γ ′ ) is a nat-ural transformation ω ∶ G → G ′ for which the following diagram commutes for allpositive integers n . G ⋅ ● n ω ⋅ / / Γ n (cid:15) (cid:15) G ′ ⋅ ● n Γ ′ n (cid:15) (cid:15) ● n ′ ⋅ ( G ⋯ G ) ⋅ ( ω ⋯ ω ) / / ● n ′ ⋅ ( G ′ ⋯ G ′ ) Proposition 2.6.
All of the composites, the Godement products, and the Carte-sian products of (composable)
Lax + -monoidal natural transformations are again Lax + -monoidal. Consequently, there is a strict monoidal 2-category Lax + of Lax + -monoidalcategories, Lax + -monoidal functors and Lax + -monoidal natural transformations.Proof. It is left to the reader to check that the diagram of Definition 2.5 commutes inall of the stated cases. (cid:3)
Definition 2.7. A cosemigroup in a Lax + -monoidal category ( L , ● , Φ , ι ) consists of ● an object a of L and ● a morphism δ ∶ a → a ● a rendering commutative the first (coassociativity) diagram below. A morphism ofcosemigroups ( a, δ ) → ( a ′ , δ ′ ) is a morphism φ ∶ a → a ′ rendering commutative thesecond diagram. a ● a δ ● ι (cid:15) (cid:15) a δ o o δ / / a ● a ι ● δ (cid:15) (cid:15) ( a ● a ) ● G a H Φ , / / a ● a ● a G a H ● ( a ● a ) Φ , o o a δ / / φ (cid:15) (cid:15) a ● a φ ● φ (cid:15) (cid:15) a ′ δ ′ / / a ′ ● a ′ Theorem 2.8.
For any
Lax + -monoidal category ( L , ● , Φ , ι ) , the following categoriesare isomorphic. IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 9 (i)
The category of cosemigroups and their morphisms in ( L , ● , Φ , ι ) . (ii) The category of
Lax + -monoidal functors ( , , , ) → ( L , ● , Φ , ι ) and their Lax + -monoidal natural transformations.Consequently, Lax + -monoidal functors preserve cosemigroups.Proof. A Lax + -monoidal functor ( , , , ) → ( L , ● , Φ , ι ) as in part (ii) consists of ● an object a of L ● for all positive integers n , a morphism δ n from a to the n -fold product a ● ⋯ ● a such that δ = ι and for all sequences of positive integers { k , . . . , k n } the followingdiagram commutes. a δ n / / δ K (cid:15) (cid:15) a ● ⋯ ● a δ k ●⋯● δ kn (cid:15) (cid:15) a ● ⋯ ● a ( a ● ⋯ ● a ) ● ⋯ ● ( a ● ⋯ ● a ) Φ k ,...,kn o o (2.1)Comparing (2.1) at n = k = k = n = k = k =
2, we see that ( a, δ ) isa cosemigroup.Conversely, starting with a cosemigroup ( a, δ ) , we put δ ∶ = ι , δ ∶ = δ and for i > δ i + ∶ = ( a δ / / a ● a ι ● δ i / / G a H ● ( a ● ⋯ ● a ) Φ ,i / / a ● ⋯ ● a ) . (2.2)Then the coassociativity of the cosemigroup tells exactly that δ equals a δ / / a ● a δ ● ι / / ( a ● a ) ● G a H Φ , / / a ● a ● a. By induction on i , one proves that the morphism of (2.2) is equal to a δ i / / a ● ⋯ ● a δ ● ι ●⋯● ι / / ( a ● a ) ● G a H ● ⋯ ● G a H Φ , ,..., / / a ● ⋯ ● a (2.3)for all i > K , that the so defined family of morphisms { δ i } i ∈ N + renders commutative (2.1) for all sequences of positive integers { k , . . . , k n } . For K = n and k must be equal to 1 and thus (2.1) commutes by one of the unitalityaxioms in Definition 2.1. There are two kinds of induction step: ● replacing { k , . . . , k n } with { , k , . . . , k n } and ● replacing { k , . . . , k n } with { + k , . . . , k n } .Assuming that (2.1) commutes for the original sequence { k , . . . , k n } , it is seen to com-mute for the modified sequence using the Lax + -monoidal category axioms in Definition2.1 together with (2.2) in the first case and (2.3) in the second case.From the constructions, it is obvious that the cosemigroup constructed from the Lax + -monoidal functor induced by a cosemigroup is exactly the initial cosemigroup.In order to see triviality of the composite of the above constructions in the oppositeorder, note that (2.1) for n = k =
1, and k = i yields (2.2). A Lax + -monoidal natural transformation ( a, { δ i } i ∈ N + ) → ( a ′ , { δ ′ i } i ∈ N + ) is a morphism f ∶ a → a ′ such that for all positive integers i , a δ i / / f (cid:15) (cid:15) a ● ⋯ ● a f ●⋯● f (cid:15) (cid:15) a ′ δ ′ i / / a ′ ● ⋯ ● a ′ (2.4)commutes. Such a morphism f is clearly a cosemigroup morphism ( a, δ ) → ( a ′ , δ ′ ) .Conversely, for a cosemigroup morphism f ∶ ( a, δ ) → ( a ′ , δ ′ ) , the diagram of (2.4)commutes for i = δ = ι = δ ′ . For the morphisms (2.2) for i > i . (cid:3) We close this section by introducing a natural notion of comodule over a cosemi-group in the sense of Definition 2.7, without entering its deeper analysis.
Definition 2.9. A comodule of a cosemigroup ( a, δ ) in a Lax + -monoidal category ( L , ● , Φ , ι ) consists of ● an object x of L ● a morphism ̺ ∶ x → x ● a (the coaction )such that the (coassociativity) diagram on the left of x ● a ι ● δ (cid:15) (cid:15) x ̺ o o ̺ / / x ● a ̺ ● ι (cid:15) (cid:15) G x H ● ( a ● a ) Φ , / / x ● a ● a ( x ● a ) ● G a H Φ , o o x θ (cid:15) (cid:15) ̺ / / x ● a θ ● (cid:15) (cid:15) x ′ ̺ ′ / / x ′ ● a commutes. A morphism of comodules ( x, ̺ ) → ( x ′ , ̺ ′ ) is a morphism θ ∶ x → x ′ in L such that the diagram on the right above commutes. Example 2.10.
Comparing the coassociativity conditions of Definition 5.7 and Defi-nition 2.9, it becomes obvious that ( a, δ ) is a comodule over an arbitrary cosemigroup ( a, δ ) in any Lax + -monoidal category.Notice, however, that it is not clear if the forgetful functor, from the category ofcomodules over a cosemigroup ( a, δ ) in some Lax + -monoidal category ( L , ● ) to L , iscomonadic. In particular, there is no obvious candidate for its right adjoint. Indeed,the usual construction of ‘free’ comodules does not seem to work: for an arbitraryobject x of L , the comultiplication δ does not seem to induce a coaction on x ● a .3. Semigroups in
Oplax + -monoidal categories Dually to the previous section, here we introduce semigroups in monoidal categoriesin which monoidal products of positive number of factors are available together withoplax coherence morphisms.
Definition 3.1. An Oplax + -monoidal category consists of ● a category R ● for all positive integers n , a functor ○ n ∶ R n → R IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 11 ● for all positive integers n, k , . . . , k n , natural transformations R K ○ k ⋯ ○ k n / / ○ K Ô⇒ Ψ k ,...,k n R n ○ n (cid:15) (cid:15) R L ○ > > Ô⇒ υ L satisfying the so-called coassociativity and counitality axioms encoded in the diagramswhich are obtained from the diagrams of Definition 2.1 by reversing the arrows.Again, Definition 3.1 can be extended in a straightforward way to define Oplax + -monoids in any Gray monoid — so in particular in any strict monoidal 2-category. ( R , ○ , Ψ , υ ) is an Oplax + -monoidal category if and only if ( R op , ○ op , Ψ op , υ op ) is a Lax + -monoidal category.Throughout, in an Oplax + -monoidal category we denote by I a J the image of anyobject a under the functor ○ ∶ R → R ; and we denote by a ○ ⋯ ○ a n the image of anobject ( a , . . . , a n ) under the functor ○ n ∶ R n → R .Dually to Proposition 2.2, the Cartesian product of Oplax + -monoidal categoriescarries a canonical Oplax + -monoidal structure. Definition 3.2. An Oplax + -monoidal functor consists of ● a functor G ∶ R → R ′ ● for all positive integers n , a natural transformation ○ n ′ ⋅ ( G ⋯ G ) → G ⋅ ○ n such that the diagrams of Definition 2.3 with reversed arrows commute.Symmetrically to Proposition 2.4, the composites and the Cartesian products of(composable) Oplax + -monoidal functors are again Oplax + -monoidal via the evidentnatural transformations. Definition 3.3. An Oplax + -monoidal natural transformation is a natural transforma-tion for which the diagrams of Definition 2.5 with reversed arrows commute.Symmetrically to Proposition 2.6, all of the composites, the Godement products,and the Cartesian products of (composable) Oplax + -monoidal natural transformationsare again Oplax + -monoidal. Consequently, there is a strict monoidal 2-category Oplax + of Oplax + -monoidal categories, Oplax + -monoidal functors and Oplax + -monoidal naturaltransformations. Definition 3.4. A semigroup in an Oplax + -monoidal category ( R , ○ , Ψ , υ ) is a cosemi-group in the Lax + -monoidal category ( R op , ○ op , Ψ op , υ op ) . That is, it consists of ● an object a of R and ● a morphism µ ∶ a ○ a → a rendering commutative the first (associativity) diagram below. A morphism of semi-groups is a morphism φ rendering commutative the second diagram. ( a ○ a ) ○ I a J µ ○ υ (cid:15) (cid:15) a ○ a ○ a Ψ , o o Ψ , / / I a J ○ ( a ○ a ) υ ○ µ (cid:15) (cid:15) a ○ a µ / / a a ○ a µ o o a ○ a µ / / φ ○ φ (cid:15) (cid:15) a φ (cid:15) (cid:15) a ′ ○ a ′ µ ′ / / a ′ Theorem 3.5.
For any
Oplax + -monoidal category ( R , ○ , Ψ , υ ) , the following cate-gories are isomorphic. (i) The category of semigroups and their morphisms in ( R , ○ , Ψ , υ ) . (ii) The category of
Oplax + -monoidal functors ( , , , ) → ( R , ○ , Ψ , υ ) and their Oplax + -monoidal natural transformations.Consequently, Oplax + -monoidal functors preserve semigroups.Proof. It follows by the application of Theorem 2.8 to the
Lax + -monoidal category ( R op , ○ op , Ψ op , υ op ) . (cid:3) Dually to comodules in Definition 2.9, one may consider modules over semigroupsin the following sense.
Definition 3.6. A module of a semigroup ( a, µ ) in an Oplax + -monoidal category ( R , ○ , Ψ , υ ) is a comodule of the cosemigroup ( a, µ ) in the Lax + -monoidal category ( R op , ○ op , Ψ op , υ op ) . Thus it consists of ● an object x of R ● a morphism ̺ ∶ x ○ a → x (the action )such that the (associativity) diagram on the left of I x J ○ ( a ○ a ) υ ○ µ (cid:15) (cid:15) x ○ a ○ a Ψ , o o Ψ , / / ( x ○ a ) ○ I a J ̺ ○ υ (cid:15) (cid:15) x ○ a ̺ / / x x ○ a ̺ o o x ○ a ̺ / / θ ○ (cid:15) (cid:15) x θ (cid:15) (cid:15) x ′ ○ a ̺ ′ / / x ′ commutes. A morphism of modules ( x, ̺ ) → ( x ′ , ̺ ′ ) is a morphism θ ∶ x → x ′ in R suchthat the diagram on the right above commutes.4. Bisemigroups in
Lax + Oplax + -duoidal categories In order to define bisemigroups in some category, one needs both a
Lax + -monoidalstructure — for the cosemigroup part to live in — and an Oplax + -monoidal structure —to host the semigroup part. Also a suitable compatibility of these monoidal structuresis needed so that the category of cosemigroups inherits the Oplax + -monoidal structureand, dually, the category of semigroups inherits the Lax + -monoidal structure. Thenwe can define bisemigroups by the usual principle: as cosemigroups in the categoryof semigroups; equivalently, as semigroups in the category of cosemigroups. In thissection we formulate a suitable compatibility between Lax + - and Oplax + -monoidalstructures which allows us to carry out this program. Definition 4.1. A Lax + Oplax + -duoidal category consists of IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 13 ● a category D ● a Lax + -monoidal structure ( D , ● , Φ , ι ) ● an Oplax + -monoidal structure ( D , ○ , Ψ , υ ) ● for all positive integers n and p , natural transformations D np τ n p / / ○ n ⋯ ○ n (cid:15) (cid:15) Ô⇒ ξ pn D pn ● p ⋯ ● p / / D n ○ n (cid:15) (cid:15) D p ● p / / D satisfying the following equivalent compatibility conditions, for all positive integers n, p, k , . . . , k p .(i) ● ( ○ n , ξ n ) is a Lax + -monoidal functor ( D , ● , Φ , ι ) n → ( D , ● , Φ , ι ) ; ● Ψ k ,...,k p and υ are Lax + -monoidal natural transformations.Succinctly, (( D , ● , Φ , ι ) , ( ○ , ξ ) , Ψ , υ ) is an Oplax + -monoid in the strict monoidal2-category Lax + of Proposition 2.6.(ii) ● ( ● n , ξ n ) is an Oplax + -monoidal functor ( D , ○ , Ψ , υ ) n → ( D , ○ , Ψ , υ ) ; ● Φ k ,...,k p and ι are Oplax + -monoidal natural transformations.Succinctly, (( D , ○ , Ψ , υ ) , ( ● , ξ ) , Φ , ι ) is a Lax + -monoid in the strict monoidal2-category Oplax + of Section 3.(iii) The following diagrams commute. ○ n ⋅ ( ● p ⋯ ● p ) ⋅ ( ● k ⋯ ● k p ⋯ ● k ⋯ ● k p ) ⋅ τ nK ⋅ ( Φ k ,...,kp ⋯ Φ k ,...,kp ) ⋅ / / ○ n ⋅ ( ● K ⋯ ● K ) ⋅ τ nKξ Kn (cid:15) (cid:15) ○ n ⋅ ( ● p ⋯ ● p ) ⋅ τ np ⋅ ( ● k ⋯ ● k ⋯ ● k p ⋯ ● k p ) ⋅ ( τ nk ⋯ τ nk p ) ξ pn ⋅ ⋅ (cid:15) (cid:15) ● p ⋅ ( ○ n ⋯ ○ n ) ⋅ ( ● k ⋯ ● k ⋯ ● k p ⋯ ● k p ) ⋅ ( τ nk ⋯ τ nk p ) ⋅ ( ξ k n ⋯ ξ kpn ) (cid:15) (cid:15) ● p ⋅ ( ● k ⋯ ● k p ) ⋅ ( ○ n ⋯ ○ n ) Φ k ,...,kp ⋅ / / ● K ⋅ ( ○ n ⋯ ○ n ) ○ n ⋅ ( ι ⋯ ι ) / / ι ⋅ (cid:22) (cid:22) ✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲✲ ○ n ⋅ ( ● ⋯ ● ) ξ n (cid:15) (cid:15) ● ⋅ ○ n ● n ⋅ ( ○ p ⋯ ○ p ) ⋅ ( ○ k ⋯ ○ k p ⋯ ○ k ⋯ ○ k p ) ● n ⋅ ( ○ K ⋯ ○ K ) ⋅ ( Ψ k ,...,kp ⋯ Ψ k ,...,kp ) o o ○ p ⋅ ( ● n ⋯ ● n ) ⋅ τ pn ⋅ ( ○ k ⋯ ○ k p ⋯ ○ k ⋯ ○ k p ) ξ np ⋅ O O ○ p ⋅ ( ● n ⋯ ● n ) ⋅ ( ○ k ⋯ ○ k ⋯ ○ k p ⋯ ○ k p ) ⋅ τ pn ○ p ⋅ ( ○ k ⋯ ○ k p ) ⋅ ( ● n ⋯ ● n ) ⋅ ( τ k n . . . τ k p n ) ⋅ τ pn ⋅ ( ξ nk ⋯ ξ nkp ) ⋅ O O ○ K ⋅ ( ● n ⋯ ● n ) ⋅ τ Knξ nK ⋅ O O Ψ k ,...,kp ⋅ ⋅ o o ○ ⋅ ● n ξ n (cid:15) (cid:15) υ ⋅ (cid:23) (cid:23) ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ ● n ⋅ ( ○ ⋯ ○ ) ⋅ ( υ ⋯ υ ) / / ● n Definition 4.2. A Lax + Oplax + -duoidal functor ( D , ● , ○ ) → ( D ′ , ● ′ , ○ ′ ) is defined as(i) an Oplax + -monoidal 1-cell (( G, Γ ● ) , Γ ○ ) ∶ (( D , ● ) , ○ ) → (( D ′ , ● ′ ) , ○ ′ ) in thestrict monoidal 2-category Lax + of Proposition 2.6; equivalently,(ii) a Lax + -monoidal 1-cell (( G, Γ ○ ) , Γ ● ) ∶ (( D , ○ ) , ● ) → (( D ′ , ○ ′ ) , ● ′ ) in the strictmonoidal 2-category Oplax + of Section 3; equivalently,(iii) ● a functor G ∶ D → D ′ ● a Lax + -monoidal structure { Γ ● n ∶ G ⋅ ● n → ● n ′ ⋅ ( G ⋯ G )} n ∈ N ● a Oplax + -monoidal structure { Γ ○ n ∶ ○ p ′ ⋅ ( G ⋯ G ) → G ⋅ ○ p } p ∈ N rendering commutative the following diagram for all positive integers n, p . ○ p ′ ⋅ ( G ⋯ G ) ⋅ ( ● n ⋯ ● n ) ⋅ τ pn ⋅ ( Γ ● n ⋯ Γ ● n ) ⋅ / / Γ ○ p ⋅ ⋅ (cid:15) (cid:15) ○ p ′ ⋅ ( ● n ′ ⋯ ● n ′ ) ⋅ ( G ⋯ G ) ⋅ τ pn ○ p ′ ⋅ ( ● n ′ ⋯ ● n ′ ) ⋅ τ pn ⋅ ( G ⋯ G ) ξ ′ np ⋅ / / ● n ′ ⋅ ( ○ p ′ ⋯ ○ p ′ ) ⋅ ( G ⋯ G ) ⋅ ( Γ ○ p ⋯ Γ ○ p ) (cid:15) (cid:15) G ⋅ ○ p ⋅ ( ● n ⋯ ● n ) ⋅ τ pn ⋅ ξ np / / G ⋅ ● n ⋅ ( ○ p ⋯ ○ p ) Γ ● n ⋅ / / ● n ′ ⋅ ( G ⋯ G ) ⋅ ( ○ p ⋯ ○ p ) (4.1)It follows immediately from Definition 4.2 and Proposition 2.4 (1) that the com-posite of Lax + Oplax + -duoidal functors is again Lax + Oplax + -duoidal via the composite Lax + -monoidal structure in Proposition 2.4 (1) and the symmetrically constructedcomposite Oplax + -monoidal structure. Definition 4.3. A Lax + Oplax + -duoidal natural transformation is defined as(i) an Oplax + -monoidal 2-cell in the strict monoidal 2-category Lax + of Proposition2.6; equivalently,(ii) a Lax + -monoidal 2-cell in the strict monoidal 2-category Oplax + of Section 3;equivalently,(iii) a natural transformation which is both Oplax + -monoidal and Lax + -monoidal.That is, it renders commutative both the diagram of Definition 2.5 and itssymmetric counterpart with reversed arrows and inverted colors. IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 15
Theorem 4.4.
For any
Lax + Oplax + -duoidal category ( D , ( ● , Φ , ι ) , ( ○ , Ψ , υ ) , ξ ) , thefollowing assertions hold. (1) The category of cosemigroups in the
Lax + -monoidal category ( D , ● , Φ , ι ) is Oplax + -monoidal via the structure ( ○ , Ψ , υ ) . (2) The category of semigroups in the
Oplax + -monoidal category ( D , ○ , Ψ , υ ) is Lax + -monoidal via the structure ( ● , Φ , ι ) .Proof. We only prove (1), part (2) follows symmetrically.By Theorem 2.8 and Definition 4.1 (i), any sequence { a , ⋯ , a n } of cosemigroups in ( D , ● , Φ , ι ) determines a Lax + -monoidal functor; and any sequence { f ∶ a → a ′ , ⋯ , f n ∶ a n → a ′ n } of cosemigroup morphisms determines a Lax + -monoidal natural transforma-tion a ⋯ a n ' ' Ô⇒ f ⋯ f na ′ ⋯ a ′ n D n ○ n / / D . That is, ○ n determines a functor to the category of cosemigroups in ( D , ● , Φ , ι ) fromthe n th Cartesian power of this category.Again by Theorem 2.8 and Definition 4.1 (i), the natural transformations Ψ and υ — if evaluated at cosemigroups — yield morphisms of cosemigroups. Thus they canbe seen as natural transformations between functors connecting Cartesian powers ofcategories of cosemigroups. They satisfy the axioms in Definition 3.1 by construction. (cid:3) Definition 4.5. A bisemigroup in a Lax + Oplax + -duoidal category ( D , ● , ○ , ξ ) is(i) a semigroup in the Oplax + -monoidal category of cosemigroups in ( D , ● ) — seeTheorem 4.4 (1); equivalently,(ii) a cosemigroup in the Lax + -monoidal category of semigroups in ( D , ○ ) — seeTheorem 4.4 (2); equivalently,(iii) ● an object a ● a semigroup ( a, µ ) in ( D , ○ ) ● a cosemigroup ( a, δ ) in ( D , ● ) subject to the compatibility condition encoded in the commutative diagram a ○ a µ / / δ ○ δ (cid:15) (cid:15) a δ / / a ● a. ( a ● a ) ○ ( a ● a ) ξ / / ( a ○ a ) ● ( a ○ a ) . µ ● µ O O A morphism of bisemigroups is a semigroup morphism in the category of cosemigroupsin ( D , ● ) ; equivalently, a cosemigroup morphism in the category of semigroups in ( D , ○ ) ; equivalently, a morphism in D which is both a cosemigroup morphism in ( D , ● ) and a semigroup morphism in ( D , ○ ) . Theorem 4.6.
For any
Lax + Oplax + -duoidal category ( D , ● , ○ , ξ ) the following cate-gories are isomorphic. (i) The category of bisemigroups and their morphisms in ( D , ● , ○ , ξ ) . (ii) The category of
Lax + Oplax + -duoidal functors ( , , , ) → ( D , ● , ○ , ξ ) and their Lax + Oplax + -duoidal natural transformations.Consequently, Lax + Oplax + -duoidal functors preserve bisemigroups.Proof. By Definition 4.5, the category of part (i) is isomorphic to the category ofsemigroups in the category of cosemigroups in ( D , ● ) . So by Theorem 3.5, it isisomorphic to the category of Oplax + -monoidal functors and Oplax + -monoidal naturaltransformations from to the category of cosemigroups in ( D , ● ) . Then by Theorem2.8, it is further equivalent to the category of Oplax + -monoidal 1-cells and Oplax + -monoidal 2-cells from ( , ) to ( D , ● ) in Lax + . By Definition 4.2 and Definition 4.3this is isomorphic to the category of part (ii). (cid:3) Theorem 4.7.
For any bisemigroup ( a, δ, µ ) in an arbitrary Lax + Oplax + -duoidal cat-egory ( D , ( ● , Φ , ι ) , ( ○ , Ψ , υ ) , ξ ) the following assertions hold. (1) The category of modules — in the sense of Definition 3.6 — of the semigroup ( a, µ ) in the Oplax + -monoidal category ( D , ○ , Ψ , υ ) admits the Lax + -monoidalstructure ( ● , Φ , ι ) . (2) The category of comodules — in the sense of Definition 2.9 — of the cosemi-group ( a, δ ) in the Lax + -monoidal category ( D , ● , Φ , ι ) admits the Oplax + -monoidal structure ( ○ , Ψ , υ ) .Proof. We only prove part (1), part (2) is verified symmetrically.For any sequence of ( a, µ ) -modules {( x i , ̺ i )} i = ,...,n , there is an ( a, µ ) -module ( x ● ⋯ ● x n ) ○ a ○ δ n / / ( x ● ⋯ ● x n ) ○ a ● n ξ n / / ( x ○ a ) ● ⋯ ● ( x n ○ a ) ̺ ●⋯● ̺ n / / x ● ⋯ ● x n . Indeed, one checks by induction on l that the l -fold comultiplication of (2.2) for a ○ a comes out as a ○ a δ l ○ δ l / / a ● l ○ a ● l ξ l / / ( a ○ a ) ● l . The induction step uses the top rightdiagram in part (iii) of Definition 4.1 for n =
2, and the top left diagram of Definition4.1 (iii) for n = p = k = k = l . Since by Definition 4.5 the multiplication µ is a morphism of cosemigroups, we obtain the commutative diagram a ○ a µ (cid:15) (cid:15) δ n ○ δ n / / a ● n ○ a ● n ξ n / / ( a ○ a ) ● nµ ● n (cid:15) (cid:15) a δ n / / a ● n for all non-negative integers n . Together with the bottom right diagram of Definition4.1 (iii), this proves the commutativity of the leftmost region of Figure 1. The regionlabelled by (BR) in Figure 2 also commutes by the bottom right diagram of Definition4.1 (iii). The regions marked by (BL) in Figure 1 and Figure 2 both commute by thebottom left diagram of Definition 4.1 (iii) at the respective values ( n, p = , k = , k = ) and ( n, p = , k = , k = ) . The remaining regions commute by evident naturality.Whenever each action ̺ i is associative, the right verticals of the diagrams in Figure1 and Figure 2 are equal. Hence also the paths on their left hand sides are equal,proving the associativity of the stated action on x ● ⋯ ● x n .The ● -monoidal products of module morphisms are easily seen to be morphisms ofmodules. The natural transformation ι — if evaluated at a module — is a morphism x ● ⋯ ● x n J ○ ( a ○ a ) ○ ( δ n ○ δ n ) * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ υ ○ µ (cid:15) (cid:15) ( x ● ⋯ ● x n ) ○ a ○ a ○ δ n ○ δ n (cid:15) (cid:15) Ψ , o o I x ● ⋯ ● x n J ○ ( a ● n ○ a ● n ) ξ n ○ ξ n (cid:15) (cid:15) (BL) ( x ● ⋯ ● x n ) ○ a ● n ○ a ● nξ n (cid:15) (cid:15) Ψ , o o ( x ○ a ○ a ) ● ⋯ ● ( x n ○ a ○ a ) Ψ , ● ⋯ ● Ψ , (cid:15) (cid:15) ( I x J ● ⋯ ● I x n J ) ○ ( a ○ a ) ● n ξ n / / ( υ ● ⋯ ● υ ) ○ µ ● n (cid:15) (cid:15) ( I x J ○ ( a ○ a )) ● ⋯ ● ( I x n J ○ ( a ○ a )) ( υ ○ µ ) ● ⋯ ● ( υ ○ µ ) (cid:15) (cid:15) ( x ● ⋯ ● x n ) ○ a ○ δ n / / ( x ● ⋯ ● x n ) ○ a ● n ξ n / / ( x ○ a ) ● ⋯ ● ( x n ○ a ) ̺ ● ⋯ ● ̺ n (cid:15) (cid:15) x ● ⋯ ● x n Figure 1.
Associativity of the action on ● -products of modules — first part ( x ● ⋯ ● x n ) ○ a ) ○ I a J ( ○ δ n ) ○ (cid:15) (cid:15) ( x ● ⋯ ● x n ) ○ a ○ a ○ δ n ○ δ n (cid:15) (cid:15) Ψ , o o (( x ● ⋯ ● x n ) ○ a ● n ) ○ I a J ○ I δ n J / / ξ n ○ (cid:15) (cid:15) (( x ● ⋯ ● x n ) ○ a ● n ) ○ I a ● n J ξ n ○ (cid:15) (cid:15) ( x ● ⋯ ● x n ) ○ a ● n ○ a ● nξ n (cid:15) (cid:15) Ψ , o o (BL) ( x ○ a ○ a ) ● ⋯ ● ( x n ○ a ○ a ) Ψ , ● ⋯ ● Ψ , (cid:15) (cid:15) (( x ○ a ) ● ⋯ ● ( x n ○ a )) ○ I a J ○ I δ n J / / ( ̺ ● ⋯ ● ̺ n ) ○ υ (cid:15) (cid:15) (( x ○ a ) ● ⋯ ● ( x n ○ a )) ○ I a ● n J ( ̺ ● ⋯ ● ̺ n ) ○ υ (cid:15) (cid:15) ○ ξ n / / (( x ○ a ) ● ⋯ ● ( x n ○ a )) ○ I a J ● n ξ n / / ( ̺ ● ⋯ ● ̺ n ) ○ υ ● n (cid:15) (cid:15) (BR) (( x ○ a ) ○ I a J ) ● ⋯ ● (( x n ○ a ) ○ I a J ) ( ̺ ○ υ ) ● ⋯ ● ( ̺ n ○ υ ) (cid:15) (cid:15) ( x ● ⋯ ● x n ) ○ a ○ δ n / / ( x ● ⋯ ● x n ) ○ a ● n ( x ● ⋯ ● x n ) ○ a ● n ξ n / / ( x ○ a ) ● ⋯ ● ( x n ○ a ) ̺ ● ⋯ ● ̺ n (cid:15) (cid:15) x ● ⋯ ● x n Figure 2.
Associativity of the action on ● -products of modules — second part IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 19 of modules by the top right diagram of Definition 4.1 (iii). In order to see that forall non-negative integers p, k , . . . k p , the component of Φ k ,...,k p at any modules is amorphism of modules, use (2.1) and the top left diagram of Definition 4.1 (iii) for n = (cid:3) Proposition 4.8.
For any bisemigroup ( a, δ, µ ) in an arbitrary Lax + Oplax + -duoidalcategory the following assertions hold. (1) (( a, δ ) , µ ) is a semigroup in the Oplax + -monoidal category of comodules overthe cosemigroup ( a, δ ) (cf. Theorem 4.7 (2)). (2) (( a, µ ) , δ ) is a cosemigroup in the Lax + -monoidal category of modules over thesemigroup ( a, µ ) (cf. Theorem 4.7 (1)).Proof. We only prove part (1), part (2) follows symmetrically.By Example 2.10, ( a, δ ) is an ( a, δ ) -comodule. The compatibility diagram of Def-inition 4.5 (iii) can be interpreted as µ being a morphism of ( a, δ ) -comodules. Asso-ciativity of µ in the category of ( a, δ ) -comodules follows from the associativity of themonoid ( a, µ ) . (cid:3) Definition 4.9. A Hopf module over a bisemigroup ( a, δ, µ ) in a Lax + Oplax + -duoidalcategory ( D , ● , ○ , ξ ) is defined by the following equivalent data.(i) A module over the semigroup (( a, δ ) , µ ) in the category of comodules over thecosemigroup ( a, δ ) in ( D , ● ) .(ii) A comodule over the cosemigroup (( a, µ ) , δ ) in the category of modules overthe semigroup ( a, µ ) in ( D , ○ ) .(iii) ● An object x of D , ● a module ( x, ν ) over the semigroup ( a, µ ) in ( D , ○ ) , ● a comodule ( x, ̺ ) over the cosemigroup ( a, δ ) in ( D , ● ) ,rendering commutative the following diagram. x ○ a ν / / ̺ ○ δ (cid:15) (cid:15) x ̺ / / x ● a ( x ● a ) ○ ( a ● a ) ξ / / ( x ○ a ) ● ( a ○ a ) ν ● µ O O Example 4.10.
By Example 2.10 and its dual counterpart, and by the compatibilitydiagram of Definition 4.5 (iii), ( a, δ, µ ) is a Hopf module over an arbitrary bisemigroup ( a, δ, µ ) in a Lax + Oplax + -duoidal category.5. Comonoids in
Lax + Oplax -monoidal categories For the definition of monoids and comonoids — that is, to define units for multipli-cations and counits for comultiplications — we also need nullary monoidal products.They may be related to the higher monoidal products in possibly different ways.There are well-known and well-studied structures called lax and oplax monoidal cat-egories, see e.g. [9]. However, the gadgets motivating this paper, namely, unital
BiHom -monoids and counital
BiHom -comonoids, do not seem to fit these well-studiedsituations. In this section we show that counital
BiHom -comonoids can be describedas comonoids in monoidal categories in which the monoidal products of positive num-bers of factors come with lax compatibiliy morphisms, but the nullary part is oplax coherent. The dual situation, suitable to describe unital
BiHom -monoids as monoids,will be discussed in the next section.
Definition 5.1. A Lax + Oplax -monoidal category consists of ● a category L ● for any non-negative integer n , a functor ● n ∶ L n → L ● for all non-negative integers n, k , . . . , k n , and for the functors L k i [ k i ] / / L k i ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ L k i / / L k i if k i > ● / / L if k i = , and using the notation of Section 1, natural transformations L K ● k ⋯ ● k n / / [ k ] ⋯ [ k n ] ( ( PPPPPPPPPPPPPP ● K Ô⇒ Φ k ,...,k n L n ● n (cid:15) (cid:15) L K + Z Ô⇒ φ k ,...,k n ● K + Z ( ( PPPPPPPPPPPPPPP
L L ● > > Ô⇒ ι L such thatΦ k ,...,k n = K = , φ k ,...,k n = Z = , φ = ι ⋅ , and the diagram ● n ι ⋅ / / ⋅ ( ι ⋯ ι ) (cid:15) (cid:15) ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ● ⋅ ● n Φ n (cid:15) (cid:15) ● n ⋅ ( ● ⋯ ● ) Φ ,..., / / ● n (5.1)as well as the regions of the diagram of Figure 3 commute, for all non-negative integers n , { m i } labelled by i = , . . . , n , and { k ij } labelled by i = , . . . , n and j = , . . . , m i .Remark that the diagram of Figure 3 reduces to that of Definition 2.1 if Z = φ k ,...,k n , and those instances ofΦ k ,...,k n for which Z >
0, a
Lax + Oplax -monoidal category can be regarded as a Lax + -monoidal category.A Lax + Oplax -monoidal category is (strict) normal if it is (strict) normal as a Lax + -monoidal category.Again, Definition 5.1 can be extended in a straightforward way to define Lax + Oplax -monoids in any Gray monoid — so in particular in any strict monoidal 2-category.Throughout, in a Lax + Oplax -monoidal category we denote by j the image of thesingle object of under the functor ● ∶ → L ; and keep the earlier notation G a H forthe image of any object a under the functor ● ∶ L → L ; and a ● ⋯ ● a n for the imageof an object ( a , . . . , a n ) under the functor ● n ∶ L n → L . n ⋅ ( ● m ⋯ ● m n ) ⋅ ( ● k ⋯ ● k nm n ) Φ m ,...,mn ⋅ / / ⋅ ( Φ k ,...,k m ⋯ Φ kn ,...,knmn ) (cid:15) (cid:15) ● M + ∑ ni = Z ( m i ) ⋅ ([ m ] ⋯ [ m n ]) ⋅ ( ● k ⋯ ● k nm n ) ● M ⋅ ( ● k ⋯ ● k nm n ) φ m ,...,mn ⋅ o o Φ k ,...,knmn (cid:15) (cid:15) ● M + ∑ ni = Z ( m i ) ⋅ ( ● ̃ k ⋯ ● ̃ k nm n ) Φ ̃ k ,..., ̃ knmn (cid:15) (cid:15) ● K + ̃ Z ⋅ ([̃ k ] ⋯ [̃ k nm n ]) Lemma . ● n ⋅ ( ● K + Z ⋯ ● K n + Z n ) ⋅ ([ k ] ⋯ [ k nm n ]) Φ K + Z ,...,Kn + Zn ⋅ / / ● K + ̃ Z ⋅ ([ K + Z ] ⋯ [ K n + Z n ]) ⋅ ([ k ] ⋯ [ k nm n ]) Lemma . ● K + Z ⋅ ([ k ] ⋯ [ k nm n ]) φ K + Z ,...,Kn + Zn ⋅ o o ● K + ̃ Z ⋅ ([̂ k ] ⋯ [̂ k nm n ]) ⋅ ([ K ] ⋯ [ K n ]) ● n ⋅ ( ● K ⋯ ● K n ) ⋅ ( φ k ,...,k m ⋯ φ kn ,...,knmn ) O O Φ K ,...,Kn / / ● K + ∑ ni = Z ( K i ) ⋅ ([ K ] ⋯ [ K n ]) φ ̂ k ,..., ̂ knmn ⋅ O O ● K φ K ,...,Kn o o φ k ,...,knmn O O Figure 3.
Axioms of
Lax + Oplax -monoidal category Analogously to Proposition 2.2, the following holds.
Proposition 5.2.
The Cartesian product LL ′ of Lax + Oplax -monoidal categories ( L , ● , Φ , φ, ι ) and ( L ′ , ● ′ , Φ ′ , φ ′ , ι ′ ) is again Lax + Oplax -monoidal via ● the functors ( LL ′ ) n τ n / / L n L ′ n ● n ● n ′ / / LL ′ ● the natural transformations ιι ′ ∶ → ● ● ′ , ( LL ′ ) K τ k ⋯ τ kn / / [ k ][ k ] ′ ⋯ [ k n ][ k n ] ′ (cid:15) (cid:15) τ K * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ L k L ′ k ⋯ L k n L ′ k n ● k ● k ′ ⋯ ● k n ● k n ′ / / τ n (cid:15) (cid:15) ( LL ′ ) nτ n (cid:15) (cid:15) L K L ′ K Ô⇒ Φ k ,...,k n Φ ′ k ,...,k n ● k ⋯ ● k n ● k ′ ⋯ ● k n ′ / / [ k ] ⋯ [ k n ][ k ] ′ ⋯ [ k n ] ′ (cid:15) (cid:15) L n L ′ n ● n ● n ′ (cid:15) (cid:15) ( LL ′ ) K + Z τ ( K + Z ) / / L K + Z L ′ K + Z ● K + Z ● K + Z ′ / / LL ′ ( LL ′ ) K τ K / / [ k ][ k ] ′ ⋯ [ k n ][ k n ] ′ (cid:15) (cid:15) L K L ′ K Ô⇒ φ k ,...,k n φ ′ k ,...,k n [ k ] ⋯ [ k n ][ k ] ′ ⋯ [ k n ] ′ (cid:15) (cid:15) ● K ● K ′ (cid:27) (cid:27) L n L ′ n ( LL ′ ) K + Z τ ( K + Z ) / / L K + Z L ′ K + Z ● K + Z ● K + Z ′ / / LL ′ where the unlabelled regions denote identity natural transformations.Proof. We leave it to the reader to check that the stated datum satisfies the axiomsin Definition 5.1. (cid:3)
Definition 5.3. A Lax + Oplax -monoidal functor consists of ● a functor G ∶ L → L ′ ● for all non-negative integers n , a natural transformation Γ n ∶ G ⋅ ● n → ● n ′ ⋅ ( G ⋯ G ) such that for all sequences of non-negative integers { k , . . . , k n } , for the sequence ofnatural transformations G k i ⋅ [ k i ] ⟦ k i ⟧ / / [ k i ] ′ ⋅ G k i ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ G k i / / G k i if k i > G ⋅ ● Γ / / ● ′ if k i = i = , . . . , n , and using the notation of Section 1, the following diagrams commute. G ⋅ ι / / ι ′ ⋅ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄ G ⋅ ● Γ (cid:15) (cid:15) ● ′ ⋅ G IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 23 G ⋅ ● n ⋅ ( ● k ⋯ ● k n ) Γ n ⋅ (cid:15) (cid:15) ⋅ Φ k ,...,kn / / G ⋅ ● K + Z ⋅ ([ k ] ⋯ [ k n ]) Γ K + Z ⋅ (cid:15) (cid:15) G ⋅ ● K ⋅ φ k ,...,kn o o Γ K (cid:15) (cid:15) ● n ′ ⋅ ( G ⋯ G ) ⋅ ( ● k ⋯ ● k n ) ⋅ ( Γ k ⋯ Γ kn ) (cid:15) (cid:15) ● K + Z ′ ⋅ ( G ⋯ G ) ⋅ ([ k ] ⋯ [ k n ]) ⋅ (⟦ k ⟧ ⋯ ⟦ k n ⟧) (cid:15) (cid:15) ● n ′ ⋅ ( ● k ′ ⋯ ● k n ′ ) ⋅ ( G ⋯ G ) Φ ′ k ,...,kn ⋅ / / ● K + Z ′ ⋅ ([ k ] ′ ⋯ [ k n ] ′ ) ⋅ ( G ⋯ G ) ● K ′ ⋅ ( G ⋯ G ) φ ′ k ,...,kn ⋅ o o Forgetting about the nullary part of the structure,
Lax + Oplax -monoidal functorscan be seen Lax + -monoidal.Analogously to Proposition 2.4, the following holds. Proposition 5.4. (1)
The composite of (composable)
Lax + Oplax -monoidal func-tors ( G, Γ ) and ( H, Ξ ) is again Lax + Oplax -monoidal via the natural transfor-mations H ⋅ G ⋅ ● n ⋅ Γ n / / H ⋅ ● n ⋅ ( G ⋯ G ) Ξ n ⋅ / / ● n ⋅ ( H ⋯ H ) ⋅ ( G ⋯ G ) = ● n ⋅ ( H ⋅ G ) n . (2) The Cartesian product of
Lax + Oplax -monoidal functors ( G, Γ ) and ( H, Ξ ) isagain Lax + Oplax -monoidal via the natural transformations ( LN ) n τ n / / ( GH ) n (cid:15) (cid:15) L n N n ● n ● n / / G n H n (cid:15) (cid:15) Ô⇒ Γ n Ξ n LN GH (cid:15) (cid:15) ( L ′ N ′ ) n τ n / / L ′ n N ′ n ● n ′ ● n ′ / / L ′ N ′ where the unlabelled region denotes the identity natural transformation.Proof. It is left to the reader to check that the stated datum in both parts satisfiesthe axioms in Definition 5.3. (cid:3)
Definition 5.5. A Lax + Oplax -monoidal natural transformation ( G, Γ ) → ( G ′ , Γ ′ ) isa natural transformation ω ∶ G → G ′ for which the following diagram commutes forall non-negative integers n . G ⋅ ● n ω ⋅ / / Γ n (cid:15) (cid:15) G ′ ⋅ ● n Γ ′ n (cid:15) (cid:15) ● n ′ ⋅ ( G ⋯ G ) ⋅ ( ω ⋯ ω ) / / ● n ′ ⋅ ( G ′ ⋯ G ′ ) Lax + Oplax -monoidal natural transformations are in particular Lax + -monoidal.Analogously to Proposition 2.6, the following holds. Proposition 5.6.
All of the composites, the Godement products, and the Carte-sian products of (composable)
Lax + Oplax -monoidal natural transformations are again Lax + Oplax -monoidal. Consequently, there is a strict monoidal 2-category Lax + Oplax of Lax + Oplax -monoidal categories, Lax + Oplax -monoidal functors and Lax + Oplax -monoidal natural transformations, admitting a strict monoidal forgetful 2-functor tothe 2-category Lax + of Proposition 2.6.Proof. The straightforward check of the axiom in Definition 5.5 in each of the statedcases is left to the reader. (cid:3)
Definition 5.7. A comonoid in a Lax + Oplax -monoidal category ( L , ● , Φ , φ, ι ) consistsof ● a cosemigroup ( a, δ ) in the Lax + -monoidal category ( L , ● , Φ , ι ) and ● a morphism ε ∶ a → j in L rendering commutative the first (counitality) diagram below. A comonoid morphism ( a, δ, ε ) → ( a ′ , δ ′ , ε ′ ) is a morphism of cosemigroups φ ∶ ( a, δ ) → ( a ′ , δ ′ ) renderingcommutative the second diagram as well. a ● a ε ● (cid:15) (cid:15) a ι (cid:15) (cid:15) δ o o δ / / a ● a ● ε (cid:15) (cid:15) j ● a G a H φ , o o φ , / / a ● j a ε / / φ (cid:15) (cid:15) ja ′ ε ′ / / j Theorem 5.8.
For any
Lax + Oplax -monoidal category ( L , ● , Φ , φ, ι ) , the followingcategories are isomorphic. (i) The category of comonoids and their morphisms in ( L , ● , Φ , φ, ι ) . (ii) The category of
Lax + Oplax -monoidal functors ( , , , , ) → ( L , ● , Φ , φ, ι ) and their Lax + Oplax -monoidal natural transformations.Consequently, Lax + Oplax -monoidal functors preserve comonoids.Proof. A Lax + Oplax -monoidal functor ( , , , , ) → ( L , ● , Φ , φ, ι ) as in part (ii) con-sists of ● an object a of L ● for all non-negative integers n , a morphism δ n from a to the n -fold product a ● n ∶ = a ● ⋯ ● a for n >
1, to a ● ∶ = G a H for n =
1, and to a ● ∶ = j for n = δ = ι (5.2)and for all sequences of non-negative integers { k , . . . , k n } , and for the double se-quences {̂ a pq } p = ,...,nq = ,...,kp of elements in L , and {⟨ pq ⟩} p = ,...,nq = ,...,kp of morphisms in L , given by ̂ a pq = { a if k p > j if k p = , a ⟨ pq ⟩ / / ̂ a pq = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ a / / a if k p > a δ / / j if k p = . IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 25 the following diagram commutes. a ● nδ k ●⋯● δ kn (cid:15) (cid:15) a δ n o o δ K + Z (cid:15) (cid:15) a δ K (cid:15) (cid:15) a ● K + Z ⟨ ⟩ ●⋯● ⟨ k ⟩ ●⋯● ⟨ n ⟩ ●⋯● ⟨ nk n ⟩ (cid:15) (cid:15) a ● k ● ⋯ ● a ● k n Φ k ,...,kn / / ̂ a ● ⋯ ● ̂ a k ● ⋯ ● ̂ a n ● ⋯ ● ̂ a nk n a ● Kφ k ,...,kn o o (5.3)If Z = ( a, δ ) is a cosemigroup. Evaluating the region on the right of (5.3)at n = k = k = n = k = k = δ is a counit for it.Conversely, if ( a, δ, ε ) is a comonoid in ( L , ● , Φ , φ, ι ) then we put δ ∶ = ε , δ ∶ = ι , δ ∶ = δ and for i > δ i + iteratively as in (2.2) (so that it satisfies (2.3)). Bythe following similar procedure to that in the proof of Theorem 2.8, one checks byinduction on K + Z that (5.3) commutes for the so defined family of morphisms, forall sequences of non-negative integers { k , . . . , k n } .If K + Z = n = K + Z = n = k is either 0 or 1. If k = = φ = ι ⋅ Lax + Oplax -monoidal category. If k = φ = Lax + Oplax -monoidal category and the upper triangle of (5.1) for n = ● replacing { k , . . . , k n } with { , k , . . . , k n } , ● replacing { k , . . . , k n } with { , k , . . . , k n } , and ● replacing { k , . . . , k n } with { + k , . . . , k n } if k > { k , . . . , k n } then it also commutes for the modified sequence in each of the listedcases.These constructions are mutually inverse bijections by (5.2) and by (5.3) at n = k =
1, and k = i (see the proof of Theorem 2.8).A Lax + Oplax -monoidal natural transformation ( a, { δ i } i ∈ N ) → ( a ′ , { δ ′ i } i ∈ N ) is obvi-ously a comonoid morphism ( a, δ , δ ) → ( a ′ , δ ′ , δ ′ ) . Conversely, a comonoid morphism ( a, δ, ε ) → ( a ′ , δ ′ , ε ′ ) is compatible with the unary parts δ = ι = δ ′ of the corresponding Lax + Oplax -monoidal functors by the naturality of ι and it is compatible with theirhigher components by Theorem 2.8. (cid:3) Definition 5.9. A comodule of a comonoid ( a, δ, ε ) in a Lax + Oplax -monoidal cate-gory ( L , ● , Φ , φ, ι ) is a comodule ( x, ̺ ∶ x → x ● a ) in the sense of Definition 2.9 overthe underlying cosemigroup ( a, δ ) such that also the following (counitality) diagramcommutes. x ̺ / / ι (cid:15) (cid:15) x ● a ● ε (cid:15) (cid:15) G x H φ , / / x ● j A morphism of comodules over a comonoid is a morphism of comodules over theunderlying cosemigroup in the sense of Definition 2.9. Example 5.10.
Comparing the counitality conditions of Definition 5.7 and Defini-tion 5.9, it follows immediately from Example 2.10 that ( a, δ ) is a comodule over anarbitrary comonoid ( a, δ, ε ) in any Lax + Oplax -monoidal category.Generalizing a construction in [8], we obtain the following example of Lax + Oplax -monoidal category (in whose presentation below the notation of Section 1 is used). Theorem 5.11.
Associated to any monoidal category ( V , ⊗ , I ) , there is a strictlynormal Lax + Oplax -monoidal category ( L , ⊗ , Φ , φ, ) as follows. ● In the category L , an object consists of an object a of V together with twomorphisms α, β ∶ a → a such that α ⋅ β = β ⋅ α .A morphism ( a, α, β ) → ( a ′ , α ′ , β ′ ) is a morphism ϑ ∶ a → a ′ such that ϑ ⋅ α = α ′ ⋅ ϑ and ϑ ⋅ β = β ′ ⋅ ϑ . ● For n = we put ( I, , ) ∶ → L , for n = we take the identity functor L → L and for any n > we take the n -fold monoidal product functor L n → L , {( a , α , β ) , . . . , ( a n , α , β n )} { ϑ ,...,ϑ n } / / {( a ′ , α ′ , β ′ ) , . . . , ( a ′ n , α ′ , β ′ n )} ↦ ( a ⊗ ⋯ ⊗ a n , α ⊗ ⋯⊗ α n , β ⊗ ⋯ ⊗ β n ) ϑ ⊗⋯⊗ ϑ n / / ( a ′ ⊗ ⋯ ⊗ a ′ n , α ′ ⊗ ⋯ ⊗ α ′ n , β ′ ⊗ ⋯ ⊗ β ′ n ) . ● For any sequence of non-negative integers { k , . . . , k n } , and for any collectionof objects {( a ij , α ij , β ij )} i = ,...,nj = ,...,ki , the natural transformation Φ k ,...,k n is taken tobe ⊗ ni = ⊗ k i j = α ∑ np = i + ( k p − ) ij β ∑ i − p = ( k p − ) ij ∶ ⊗ ni = ⊗ k i j = a ij → ⊗ ni = ⊗ k i j = a ij , and the natural transformation φ k ,...,k n is taken to be ⊗ ni = ⊗ k i j = α ∑ np = i + Z ( k p ) ij β ∑ i − p = Z ( k p ) ij ∶ ⊗ ni = ⊗ k i j = a ij → ⊗ ni = ⊗ k i j = a ij . Proof.
Obviously, Φ k ,...,k n = K = φ k ,...,k n = Z =
0. Since also Φ n andΦ ,..., are trivial for all positive integers n , we only need to check the commutativityof the regions of the diagram of Figure 3 for the stated natural transformations.For each region of Figure 3 this means a comparison of endomorphisms of ⊗ ni = ⊗ m i j = ⊗ k ij l = a ijl for non-negative integers n , { m i } i = ,...,n , and { k ij } i = ,...,nj = ,...,mi ; and for all collections {( a ijl , α ijl , β ijl )} i = ,...,nj = ,...,m i l = ,...,k ij of objects in L . Each of these endomorphisms is a monoidalproduct of endomorphisms of the individual factors a ijl , and each of the occurringendomorphisms of a ijl is a composite of copies of the commuting morphisms α ijl and β ijl . So we only need to compare the resulting exponents of α ijl and of β ijl for allvalues of i, j, l . We only do it for β ijl in each case; the computations for α ijl aresymmetric. IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 27
Then the four regions of the diagram of Figure 3 amount to the respective conditions i − ∑ p = ( m p − ) + i − ∑ p = m p ∑ q = (̃ k pq − ) + j − ∑ q = (̃ k iq − ) = j − ∑ q = ( k iq − ) + i − ∑ p = ( K p + Z p − ) j − ∑ q = Z ( k iq ) + i − ∑ p = ( K p + Z p − ) = i − ∑ p = ( K p − ) + i − ∑ p = m p ∑ q = Z (̂ k pq ) + j − ∑ q = Z (̂ k iq ) i − ∑ p = Z ( m p ) + i − ∑ p = m p ∑ q = (̃ k pq − ) + j − ∑ q = (̃ k iq − ) = i − ∑ p = m p ∑ q = ( k pq − ) + j − ∑ q = ( k iq − ) + i − ∑ p = Z ( K p + Z p ) i − ∑ p = Z p + j − ∑ q = Z ( k iq ) + i − ∑ p = Z ( K p + Z p ) = i − ∑ p = Z ( K p ) + i − ∑ p = m p ∑ q = Z (̂ k pq ) + j − ∑ q = Z (̂ k iq ) for all i = , . . . , n and j = , . . . , m i , which are easily checked to hold. (cid:3) A straightforward application of Definition 5.7 yields the following.
Theorem 5.12.
For any monoidal category ( V , ⊗ , I ) the following categories are pair-wise isomorphic. (1.i) The category of cosemigroups in the
Lax + -monoidal category L of Theorem5.11. (1.ii) The category of
BiHom -comonoids in V .and (2.i) The category of comonoids in the
Lax + Oplax -monoidal category L of Theorem5.11. (2.ii) The category of counital
BiHom -comonoids in V .Proof. In both categories of part (1), an object consists of ● an object ( a, α, β ) of L ● a morphism δ ∶ ( a, α, β ) → ( a ⊗ a, α ⊗ α, β ⊗ β ) of L such that the first diagram of a ⊗ a δ ⊗ (cid:15) (cid:15) a δ o o δ / / a ⊗ a ⊗ δ (cid:15) (cid:15) a ⊗ a ⊗ a Φ , = ⊗ ⊗ β / / a ⊗ a ⊗ a a ⊗ a ⊗ a Φ , = α ⊗ ⊗ o o a δ / / ϑ (cid:15) (cid:15) a ⊗ a ϑ ⊗ ϑ (cid:15) (cid:15) a ′ δ ′ / / a ′ ⊗ a ′ commutes. In both categories of part (1), a morphism (( a, α, β ) , δ ) → (( a ′ , α ′ , β ′ ) , δ ′ ) is a morphism ϑ ∶ ( a, α, β ) → ( a ′ , α ′ , β ′ ) in L such that the second diagram abovecommutes.In both categories of part (2), an object consists of ● an object (( a, α, β ) , δ ) in the isomorphic categories of part (1) ● a morphism ε ∶ ( a, α, β ) → ( I, , ) in L such that the first diagram of a ⊗ a ε ⊗ (cid:15) (cid:15) a δ o o δ / / a ⊗ a ⊗ ε (cid:15) (cid:15) a a φ , = β o o φ , = α / / a a ε / / ϑ (cid:15) (cid:15) Ia ′ ε ′ / / I commutes. In both categories of part (2), a morphism (( a, α, β ) , δ, ε ) → (( a ′ , α ′ , β ′ ) , δ ′ ,ε ′ ) is a morphism ϑ ∶ (( a, α, β ) , δ ) → (( a ′ , α ′ , β ′ ) , δ ′ ) in the isomorphic categories ofpart (1) such that the second diagram above commutes. (cid:3) A comodule in the sense of Definition 2.9 over a cosemigroup in the
Lax + -monoidalcategory L of Theorem 5.11 is precisely the same as a comodule in the sense of [11,Definition 5.3] over the corresponding BiHom -comonoid in Theorem 5.12 (1.ii). Acomodule in the sense of Definition 5.9 over a comonoid in the
Lax + Oplax -monoidalcategory L of Theorem 5.11 is precisely the same as a counital comodule in the senseof [11, Definition 5.3] over the corresponding counital BiHom -comonoid in Theorem5.12 (2.ii).The same category L of Theorem 5.11 also has a ‘biased’ monoidal structure inher-ited from ( V , ⊗ , I ) . In this monoidal category — let us denote it by M — the monoidalproduct and the monoidal unit are ⊗ = ⊗ , respectively, ⊗ = ( I, , ) of L . Mac Lane’scoherence natural isomorphisms are given by the same (omitted) morphisms as in V .A cosemigroup (respectively, comonoid) in ( M , ⊗ , ⊗ ) is a triple consisting of a cosemi-group (respectively, comonoid) in ( V , ⊗ , I ) together with two commuting cosemigroup(respectively, comonoid) endomorphisms. Cosemigroup (respectively, comonoid) mor-phisms in M are those cosemigroup (respectively, comonoid) morphisms in ( V , ⊗ , I ) which are compatible with the endomorphism parts too. Proposition 5.13.
Consider an arbitrary monoidal category ( V , ⊗ , I ) , the associatedstrictly normal Lax + Oplax -monoidal category L of Theorem 5.11, and the monoidalcategory M of the previous paragraph regarded as a Lax + Oplax -monoidal category.The identity functor M → L carries a Lax + Oplax -monoidal structure given by themorphisms Γ n ∶ = ⊗ ni = α n − ii ⋅ β i − i ∶ ⊗ ni = a i → ⊗ ni = a i , for all non-negative integers n and any sequence of objects {( a i , α i , β i )} i = ,...,n .Consequently, it preserves cosemigroups and comonoids. The action of the
Lax + Oplax -monoidal functor of Proposition 5.13 on cosemigroupsand comonoids was termed in [11, Proposition 5.9] the Yau twist . Proof.
A similar comparison of exponents as in the proof of Theorem 5.11 shows thecommutaivity of the diagrams in Definition 5.3. (cid:3)
Remark . In the category L of Theorem 5.11 take the full subcategory for whoseobjects ( a, α, β ) the morphisms α and β are invertible. It is monoidal with the asso-ciativity natural isomorphism α − ⊗ ⊗ β ′′ ∶ (( a ⊗ a ′ ) ⊗ a ′′ , ( α ⊗ α ′ ) ⊗ α ′′ , ( β ⊗ β ′ ) ⊗ β ′′ ) → ( a ⊗ ( a ′ ⊗ a ′′ ) ,α ⊗ ( α ′ ⊗ α ′′ ) ,β ⊗ ( β ′ ⊗ β ′′ )) , left unit natural isomorphism β − ∶ ( I ⊗ a, ⊗ α, ⊗ β ) → ( a, α, β ) and right unit natural isomorphism α − ∶ ( a ⊗ I, α ⊗ , β ⊗ ) → ( a, α, β ) . IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 29
This is the same monoidal category that was constructed in Section 2 of [11] un-der the name H ( − , ) , ( , − ) , ( Z × Z , V ) . By Theorem 5.12 the cosemigroups (respec-tively, comonoids) therein are those BiHom -comonoids (respectively, counital
BiHom -comonoids) whose endomorphism parts are automorphisms; thus we re-cover [11, Re-mark 3.5].Let us take also in the monoidal category M of Proposition 5.13 the full (monoidal)subcategory of those objects ( a, α, β ) whose morphisms α and β are invertible; this isthe monoidal category H ( Z × Z , V ) of [11, Section 2]. As shown in [11, Theorem 2.5],the Lax + Oplax -monoidal functor of Proposition 5.13 restricts to a strong monoidalisomorphism between these monoidal subcategories; hence it induces isomorphismsbetween the cosemigroups, as well as the comonoids in these monoidal categories.That is, analogously to [11, Claim 3.7], we obtain an isomorphism between(i) the full subcategory of the isomorphic categories in Theorem 5.12 (1) for whoseobjects (( a, α, β ) , δ ) the morphisms α and β are invertible;(ii) the category whose objects are triples consisting of a cosemigroup in V togetherwith two commuting cosemigroup automorphisms; and the morphisms are thecosemigroup morphisms which commute with both of these automorphisms.It induces an analogous isomorphism also between(i) the full subcategory of the isomorphic categories in Theorem 5.12 (2) for whoseobjects (( a, α, β ) , δ, ε ) the morphisms α and β are invertible;(ii) the category whose objects are triples consisting of a comonoid in V togetherwith two commuting comonoid automorphisms; and the morphisms are thecomonoid morphisms which commute with both of these automorphisms.Restricting further to those objects for which β = α − , the latter isomorphism reducesto [8, Proposition 1.13].6. Monoids in
Lax Oplax + -monoidal categories Dually to the previous section, here we introduce monoids in monoidal categoriesin which all monoidal products with non-negative number of factors are available;and the monoidal products of positive number of factors are oplax coherent and thenullary product is lax coherent. Unital
BiHom -monoids are described as monoids insuch monoidal categories.
Definition 6.1. A Lax Oplax + -monoidal category consists of ● a category R ● for all non-negative integers n , a functor ○ n ∶ R n → R ● for all non-negative integers n, k , . . . , k n , natural transformations R K ○ k ⋯ ○ k n / / [ k ] ⋯ [ k n ] ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ ○ K Ô⇒ Ψ k ,...,k n R n ○ n (cid:15) (cid:15) R K + Z Ô⇒ ψ k ,...,k n ○ K + Z ( ( PPPPPPPPPPPPPPP
R R ○ > > Ô⇒ υ R satisfying the so-called coassociativity and counitality axioms encoded in the diagramswhich are obtained from the diagrams of Definition 5.1 by reversing the arrows.Forgetting about the irrelevant structure, any Lax Oplax + -monoidal category canbe regarded as an Oplax + -monoidal category.A Lax Oplax + -monoidal category is (strictly) normal if it is (strictly) normal as an Oplax + -monoidal category. ( R , ○ , Ψ , ψ, υ ) is a Lax Oplax + -monoidal category if and only if ( R op , ○ op , Ψ op , ψ op ,υ op ) is a Lax + Oplax -monoidal category.Throughout, in a Lax Oplax + -monoidal category we denote by i the image of thesingle object of under the functor ○ ∶ → R and we keep our earlier notation I a J forthe image of any object a under the functor ○ ∶ R → R ; and a ○ ⋯ ○ a n for the imageof an object ( a , . . . , a n ) under the functor ○ n ∶ R n → R .Symmetrically to Proposition 5.2, the Cartesian product of Lax Oplax + -monoidalcarries a canonical Lax Oplax + -monoidal structure. Definition 6.2. A Lax Oplax + -monoidal functor ( R , ○ , Ψ , ψ, υ ) → ( R ′ , ○ ′ , Ψ ′ , ψ ′ , υ ′ ) consists of ● a functor G ∶ R → R ′ ● for all non-negative integers n , a natural transformation ○ n ′ ⋅ ( G ⋯ G ) → G ⋅ ○ n such that the diagrams of Definition 5.3 with reversed arrows commute.Analogously to Proposition 5.4, the composite of (composable) Lax Oplax + -monoidalfunctors is again Lax + Oplax -monoidal via the evident natural transformations; andthe Cartesian product of Lax Oplax + -monoidal functors is Lax + Oplax -monoidal viathe evident natural transformations too. Definition 6.3. A Lax Oplax + -monoidal natural transformation is a natural transfor-mation for which the diagrams of Definition 5.5 with reversed arrows commute.Symmetrically to Proposition 5.6, all of the composites, the Godement products,and the Cartesian products of (composable) Lax Oplax + -monoidal natural transfor-mations are again Lax Oplax + -monoidal. Consequently, there is a strict monoidal2-category Lax Oplax + of Lax Oplax + -monoidal categories, Lax Oplax + -monoidal func-tors and Lax Oplax + -monoidal natural transformations, admitting a strict monoidalforgetful 2-functor to the 2-category Oplax + of Section 3. Definition 6.4. A monoid in ( R , ○ , Ψ , ψ, υ ) is a comonoid in ( R op , ○ op , Ψ op , ψ op , υ op ) .That is, it consists of ● a semigroup ( a, µ ) and ● a morphism η ∶ i → a rendering commutative the first (unitality) diagram below. A morphism of monoids ( a, µ, η ) → ( a ′ , µ ′ , η ′ ) is a morphism of semigroups φ ∶ ( a, µ ) → ( a ′ , µ ′ ) rendering IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 31 commutative the second diagram too. i ○ a η ○ (cid:15) (cid:15) ψ , / / I a J υ (cid:15) (cid:15) a ○ i ψ , o o ○ η (cid:15) (cid:15) a ○ a µ / / a a ○ a µ o o i η / / a φ (cid:15) (cid:15) i η ′ / / a ′ Theorem 6.5.
For any
Lax Oplax + -monoidal category ( R , ○ , Ψ , ψ, υ ) , the followingcategories are isomorphic. (i) The category of monoids and their morphisms in ( R , ○ , Ψ , ψ, υ ) . (ii) The category of
Lax Oplax + -monoidal functors ( , , , , ) → ( R , ○ , Ψ , ψ, υ ) and their Lax Oplax + -monoidal natural transformations.Consequently, Lax Oplax + -monoidal functors preserve monoids.Proof. Apply Theorem 5.8 to the
Lax + Oplax -monoidal category ( R op , ○ op , Ψ op , ψ op ,υ op ) . (cid:3) Definition 6.6. A module of a monoid ( a, µ, η ) in a Lax Oplax + -monoidal category ( R , ○ , Ψ , ψ, υ ) is a comodule of the comonoid ( a, µ, η ) in the Lax + Oplax -monoidalcategory ( R op , ○ op , Ψ op , υ op ) . Equivalently, it is a module ( x, ̺ ∶ x ○ a → x ) in the senseof Definition 3.6 of the semigroup ( a, µ ) such that also the diagram x ○ i ψ , / / ○ η (cid:15) (cid:15) I x J υ (cid:15) (cid:15) x ○ a ̺ / / x commutes. A morphism of modules over a monoid is a morphism of modules over theunderlying semigroup in the sense of Definition 3.6. Theorem 6.7.
Associated to a monoidal category ( V , ⊗ , I ) , there is a strictly normal Lax Oplax + -monoidal category R as follows. ● As a category, R is identical to the category L of Theorem 5.11. ● For all non-negative integers n , we take the same functors ⊗ n ∶ R n → R as inTheorem 5.11. ● For any sequence of non-negative integers { k , . . . , k n } , and for any collectionof objects {( a ij , κ ij , ν ij )} i = ,...,nj = ,...,ki , the natural transformation Ψ k ,...,k n is taken tobe ⊗ ni = ( ⊗ k i j = κ ∑ np = i + ( k p − ) ij ν ∑ i − p = ( k p − ) ij ) ∶ ⊗ ni = ( ⊗ k i j = a ij ) → ⊗ ni = ( ⊗ k i j = a ij ) , and the natural transformation ψ k ,...,k n is taken to be ⊗ ni = ( ⊗ k i j = κ ∑ np = i + Z ( k p ) ij ν ∑ i − p = Z ( k p ) ij ) ∶ ⊗ ni = ( ⊗ k i j = a ij ) → ⊗ ni = ( ⊗ k i j = a ij ) . Proof.
Apply Theorem 5.11 to the monoidal category ( V op , ⊗ op , I op ) ; and take theopposite of the resulting Lax + Oplax -monoidal category. (cid:3) In complete analogy with Theorem 5.12 we get the following.
Theorem 6.8.
For any monoidal category ( V , ⊗ , I ) the following categories are pair-wise isomorphic. (1.i) The category of semigroups in the
Oplax + -monoidal category R of Theorem 6.7. (1.ii) The category of
BiHom -monoids in V .and (2.i) The category of monoids in the
Lax Oplax + -monoidal category R of Theorem5.11. (2.ii) The category of unital
BiHom -monoids in V . A module in the sense of Definition 3.6 over a semigroup in the
Oplax + -monoidalcategory R of Theorem 6.7 is precisely the same as a module in the sense of [11,Definition 4.1] over the corresponding BiHom -monoid in Theorem 6.8 (1.ii). A modulein the sense of Definition 6.6 over a monoid in the
Lax Oplax + -monoidal category R ofTheorem 6.7 is precisely the same as an unital module in the sense of [11, Definition4.1] over the corresponding unital BiHom -monoid in Theorem 6.8 (2.ii).Applying Proposition 5.13 to the opposite categories, we obtain the following ex-planation of the Yau twist of semigroups and monoids in [11, Proposition 5.9].
Proposition 6.9.
Consider an arbitrary monoidal category ( V , ⊗ , I ) , the associatedstrictly normal Lax Oplax + -monoidal category R of Theorem 6.7, and the monoidalcategory M of Proposition 5.13 regarded now as a Lax Oplax + -monoidal category. Theidentity functor M → R carries a Lax Oplax + -monoidal structure given by the samemorphisms in Proposition 5.13. Consequently, it preserves semigroups and monoids.Remark . In the coinciding categories R of Theorem 6.7 and L of Theorem 5.11take the full monoidal subcategory described in Remark 5.14. The category of semi-groups (resp. monoids) in it is isomorphic to the category whose objects are triplesconsisting of a semigroup (resp. monoid) in V together with two commuting semigroup(resp. monoid) automorphisms.7. Bimonoids in
Lax + Oplax + -duoidal categories In this section we introduce compatibility conditions between
Lax + Oplax -monoidal,and Lax Oplax + -monoidal structures on the same category. Similarly to Section 4,they imply that the category of comonoids w.r.t. the Lax + Oplax -monoidal structureinherits the Lax Oplax + -monoidal structure; symmetrically, the category of monoidsw.r.t. the Lax Oplax + -monoidal structure inherits the Lax + Oplax -monoidal structure.We define then bimonoids as monoids in the category of comonoids; equivalently, ascomonoids in the category of monoids. Unital and counital BiHom -bimonoids aredescribed as bimonoids in this sense.
Definition 7.1. A Lax + Oplax + -duoidal category consists of ● a category D ● a Lax + Oplax -monoidal structure ( D , ● , Φ , φ, ι ) ● a Lax Oplax + -monoidal structure ( D , ○ , Ψ , ψ, υ ) IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 33 ● for all non-negative integers n and p , natural transformations D np τ np / / ○ n ⋯ ○ n (cid:15) (cid:15) Ô⇒ ξ pn D pn ● p ⋯ ● p / / D n ○ n (cid:15) (cid:15) D p ● p / / D satisfying the following equivalent compatibility conditions, for all non-negative inte-gers n, p, k , . . . , k p .(i) ● ( ○ n , ξ n ) is a Lax + Oplax -monoidal functor ( D , ● , Φ , φ, ι ) n → ( D , ● , Φ , φ, ι ) ; ● Ψ k ,...,k p , ψ k ,...,k p and υ are Lax + Oplax -monoidal natural transformations.Succinctly, (( D , ● , Φ , φ, ι ) , ( ○ , ξ ) , Ψ , ψ, υ ) is a Lax Oplax + -monoid in the strictmonoidal 2-category Lax + Oplax of Proposition 5.6.(ii) ● ( ● n , ξ n ) is a Lax Oplax + -monoidal functor ( D , ○ , Ψ , ψ, υ ) n → ( D , ○ , Ψ , ψ, υ ) ; ● Φ k ,...,k p , φ k ,...,k p and ι are Lax Oplax + -monoidal natural transformations.Succinctly, (( D , ○ , Ψ , ψ, υ ) , ( ● , ξ ) , Φ , φ, ι ) is a Lax + Oplax -monoid in the strictmonoidal 2-category Lax Oplax + of Section 6.(iii) The diagrams ○ ⋅ ● n υ ⋅ % % ❑❑❑❑❑❑❑❑❑❑❑❑❑ ξ n (cid:15) (cid:15) ● n ⋅ ( ○ ⋯ ○ ) ⋅ ( υ ⋯ υ ) / / ● n ○ n ⋅ ( ι ⋯ ι ) / / ι ⋅ % % ❑❑❑❑❑❑❑❑❑❑❑❑❑ ○ n ⋅ ( ● ⋯ ● ) ξ n (cid:15) (cid:15) ● ⋅ ○ n and those of Figure 4 commute, for all non-negative integers n, p, k , . . . , k p ,for the families of functors D k i [ k i ] ● / / D k i ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ D k i / / D k i if k i > ● / / D if k i = D k i [ k i ] ○ / / D k i ∶ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ D k i / / D k i if k i > ○ / / D if k i = ○ n k i ⋅ ([ k i ] ● ) n ⟦ k i ⟧ ● n / / [ k i ] ● ⋅ ○ n k i ∶ = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ○ n k i / / ○ n k i if k i > ○ n ⋅ ● n ξ n / / ● if k i = [ k i ] ○ ⋅ ● n k i ⟦ k i ⟧ ○ n / / ● n k i ⋅ ([ k i ] ○ ) n ∶ = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ● n k i / / ● n k i if k i > ○ ξ n / / ● n ⋅ ○ n if k i = . n ⋅ ( ● p ⋯ ● p ) ⋅ ( ● k ⋯ ● k p ⋯ ● k ⋯ ● k p ) ⋅ τ nK ⋅ ( Φ k ,...,kp ⋯ Φ k ,...,kp ) ⋅ / / ○ n ⋅ ( ● K + Z ⋯ ● K + Z ) ⋅ ([ k ] ● ⋯ [ k p ] ● ⋯ [ k ] ● ⋯ [ k p ] ● ) ⋅ τ nK ○ n ⋅ ( ● K ⋯ ● K ) ⋅ τ nK ⋅ ( φ k ,...,kp ⋯ φ k ,...,kp ) ⋅ o o ξ Kn (cid:15) (cid:15) ○ n ⋅ ( ● p ⋯ ● p ) ⋅ τ np ⋅ ( ● k ⋯ ● k ⋯ ● k p ⋯ ● k p ) ⋅ ( τ nk ⋯ τ nk p ) ξ pn ⋅ ⋅ (cid:15) (cid:15) ○ n ⋅ ( ● K + Z ⋯ ● K + Z ) ⋅ τ n ( K + Z ) ⋅ ([ k ] ● ⋯ [ k ] ● ⋯ [ k p ] ● ⋯ [ k p ] ● ) ξ K + Zn ⋅ (cid:15) (cid:15) ● p ⋅ ( ○ n ⋯ ○ n ) ⋅ ( ● k ⋯ ● k ⋯ ● k p ⋯ ● k p ) ⋅ ( τ nk ⋯ τ nk p ) ⋅ ( ξ k n ⋯ ξ kpn ) (cid:15) (cid:15) ● K + Z ⋅ ( ○ n ⋯ ○ n ) ⋅ ([ k ] ● ⋯ [ k ] ● ⋯ [ k p ] ● ⋯ [ k p ] ● ) ⋅ (⟦ k ⟧ ● n ⋯ ⟦ k p ⟧ ● n ) (cid:15) (cid:15) ● p ⋅ ( ● k ⋯ ● k p ) ⋅ ( ○ n ⋯ ○ n ) Φ k ,...,kp ⋅ / / ● K + Z ⋅ ([ k ] ● ⋯ [ k p ] ● ) ⋅ ( ○ n ⋯ ○ n ) ● K ⋅ ( ○ n ⋯ ○ n ) φ k ,...,kp ⋅ o o ● n ⋅ ( ○ p ⋯ ○ p ) ⋅ ( ○ k ⋯ ○ k p ⋯ ○ k ⋯ ○ k p ) ● n ⋅ ( ○ K + Z ⋯ ○ K + Z ) ⋅ ([ k ] ○ ⋯ [ k p ] ○ ⋯ [ k ] ○ ⋯ [ k p ] ○ ) ⋅ ( Ψ k ,...,kp ⋯ Ψ k ,...,kp ) o o ⋅ ( ψ k ,...,kp ⋯ ψ k ,...,kp ) / / ● n ⋅ ( ○ K ⋯ ○ K ) ○ p ⋅ ( ● n ⋯ ● n ) ⋅ τ pn ⋅ ( ○ k ⋯ ○ k p ⋯ ○ k ⋯ ○ k p ) ξ np ⋅ O O ○ K + Z ⋅ ( ● n ⋯ ● n ) ⋅ τ ( K + Z ) n ⋅ ([ k ] ○ ⋯ [ k p ] ○ ⋯ [ k ] ○ ⋯ [ k p ] ○ ) ξ nK + Z ⋅ O O ○ p ⋅ ( ● n ⋯ ● n ) ⋅ ( ○ k ⋯ ○ k ⋯ ○ k p ⋯ ○ k p ) ⋅ τ pn ○ K + Z ⋅ ( ● n ⋯ ● n ) ⋅ ([ k ] ○ ⋯ [ k ] ○ ⋯ [ k p ] ○ ⋯ [ k p ] ○ ) ⋅ τ Kn ○ p ⋅ ( ○ k ⋯ ○ k p ) ⋅ ( ● n ⋯ ● n ) ⋅ ( τ k n . . . τ k p n ) ⋅ τ pn ⋅ ( ξ nk ⋯ ξ nkp ) ⋅ O O ○ K + Z ⋅ ([ k ] ○ ⋯ [ k p ] ○ ) ⋅ ( ● n ⋯ ● n ) ⋅ τ Kn ⋅ (⟦ k ⟧ ○ n ⋯ ⟦ k p ⟧ ○ n ) ⋅ O O Ψ k ,...,kp ⋅ ⋅ o o ψ k ,...,kp ⋅ ⋅ / / ○ K ⋅ ( ● n ⋯ ● n ) ⋅ τ Knξ nK O O Figure 4.
Axioms of
Lax + Oplax + -duoidal category IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 35
Forgetting about the irrelevant structure, any
Lax + Oplax + -duoidal category can beregarded as a Lax + Oplax + -duoidal category.It is immediate from Definition 7.1 (i) and Theorem 5.8 that for any Lax + Oplax + -duoidal category ( D , ● , ○ ) , ( i, ξ , ξ ) is a comonoid in ( D , ● ) . Dually, it follows fromDefinition 7.1 (ii) and Theorem 6.5 that ( j, ξ , ξ ) is a monoid in ( D , ○ ) . Definition 7.2. A Lax + Oplax + -duoidal functor ( D , ● , ○ ) → ( D ′ , ● ′ , ○ ′ ) is defined as(i) a Lax Oplax + -monoidal 1-cell (( G, Γ ● ) , Γ ○ ) ∶ (( D , ● ) , ○ ) → (( D ′ , ● ′ ) , ○ ′ ) in thestrict monoidal 2-category Lax + Oplax of Proposition 5.6; equivalently,(ii) A Lax + Oplax -monoidal 1-cell (( G, Γ ○ ) , Γ ● ) ∶ (( D , ○ ) , ● ) → (( D ′ , ○ ′ ) , ● ′ ) in thestrict monoidal 2-category Lax Oplax + of Section 6; equivalently,(iii) ● a functor G ∶ D → D ′ ● a Lax + Oplax -monoidal structure { Γ ● n ∶ G ⋅ ● n → ● n ′ ⋅ ( G ⋯ G )} n ∈ N ● a Lax Oplax + -monoidal structure { Γ ○ p ∶ ○ p ′ ⋅ ( G ⋯ G ) → G ⋅ ○ p } p ∈ N rendering commutative the diagram of (4.1) for all non-negative integers n, p .Forgetting about the nullary parts of the structures, Lax + Oplax + -duoidal functorscan be seen as Lax + Oplax + -duoidal functors.It follows immediately from Definition 7.2 (ii) and Proposition 5.4 (1) that thecomposite of Lax + Oplax + -duoidal functors is again Lax + Oplax + -duoidal via the com-posite Lax + Oplax -monoidal structure in Proposition 5.4 (1) and the symmetricallyconstructed composite Lax Oplax + -monoidal structure. Definition 7.3. A Lax + Oplax + -duoidal natural transformation is defined as(i) a Lax Oplax + -monoidal 2-cell in the strict monoidal 2-category Lax + Oplax ofProposition 5.6; equivalently,(ii) a Lax + Oplax -monoidal 2-cell in the strict monoidal 2-category Lax Oplax + ofSection 6; equivalently,(iii) a natural transformation which is both Lax Oplax + -monoidal and Lax + Oplax -monoidal. That is, it renders commutative both the diagram of Definition 5.5and its symmetric counterpart with reversed arrows and inverted colors. Lax + Oplax + -duoidal natural transformations are in particular Lax + Oplax + -duoidal. Theorem 7.4.
For any
Lax + Oplax + -duoidal category ( D , ( ● , Φ , φ, ι ) , ( ○ , Ψ , ψ, υ ) , ξ ) ,the following assertions hold. (1) The comonoids in the
Lax + Oplax -monoidal category ( D , ● , Φ , φ, ι ) constitutea Lax Oplax + -monoidal category with the structure ( ○ , Ψ , ψ, υ ) . (2) The monoids in the
Lax Oplax + -monoidal category ( D , ○ , Ψ , ψ, υ ) constitute a Lax + Oplax -monoidal category with the structure ( ● , Φ , φ, ι ) .Proof. In exactly the same way as we derived Theorem 4.4 from Theorem 2.8 andDefinition 4.1 (ii), part (1) follows from Theorem 5.8 and Definition 7.1 (ii). Part (2)follows symmetrically. (cid:3)
Definition 7.5. A bimonoid in a Lax + Oplax + -duoidal category ( D , ● , ○ , ξ ) is definedby the following equivalent data.(i) A monoid in the Lax Oplax + -monoidal category of comonoids in ( D , ● ) — seeTheorem 7.4 (1). (ii) A comonoid in the Lax + Oplax -monoidal category of monoids in ( D , ○ ) — seeTheorem 7.4 (2).(iii) ● An object a of D , ● a monoid ( a, µ, η ) in ( D , ○ ) ● a comonoid ( a, δ, ε ) in ( D , ● ) such that ( a, µ, δ ) is a bisemigroup in the Lax + Oplax + -duoidal category ( D , ● , ○ ,ξ ) and the following diagrams commute as well. a ○ a µ / / ε ○ ε (cid:15) (cid:15) a ε (cid:15) (cid:15) j ○ j ξ / / j i ξ / / η (cid:15) (cid:15) i ● i η ● η (cid:15) (cid:15) a δ / / a ● a i η / / ξ (cid:29) (cid:29) ❁❁❁❁❁❁❁❁ a ε (cid:15) (cid:15) j A morphism of bimonoids is a monoid morphism in the category of comonoids in ( D , ● ) ; equivalently, a comonoid morphism in the category of monoids in ( D , ○ ) ;equivalently, a morphism in D which is both a comonoid morphism in ( D , ● ) and amonoid morphism in ( D , ○ ) . Theorem 7.6.
For any
Lax + Oplax + -duoidal category ( D , ● , ○ , ξ ) the following cate-gories are isomorphic. (i) The category of bimonoids and their morphisms in ( D , ● , ○ , ξ ) . (ii) The category of
Lax + Oplax + -duoidal functors ( , , , ) → ( D , ● , ○ , ξ ) and their Lax + Oplax + -duoidal natural transformations.Consequently, Lax + Oplax + -duoidal functors preserve bimonoids.Proof. By Definition 7.5, the category of part (i) is isomorphic to the category ofmonoids in the category of comonoids in ( D , ● ) . So by Theorem 6.5, it is isomorphicto the category of Lax Oplax + -monoidal functors and Lax Oplax + -monoidal naturaltransformations from to the category of comonoids in ( D , ● ) . Then by Theorem 5.8,it is further equivalent to the category of Lax Oplax + -monoidal 1-cells and Lax Oplax + -monoidal 2-cells from ( , ) to ( D , ● ) in Lax + Oplax . By Definition 7.2 and Definition7.3 this is isomorphic to the category of part (ii). (cid:3) Analogously to Theorem 4.7, we have the following.
Theorem 7.7.
For any bimonoid ( a, ( δ, ε ) , ( µ, η )) in a Lax + Oplax + -duoidal category ( D , ( ● , Φ , φ, ι ) , ( ○ , Ψ , ψ, υ ) , ξ ) the following assertions hold. (1) The category of modules — in the sense of Definition 6.6 — over the underlyingmonoid ( a, µ, η ) in the Lax Oplax + -monoidal category ( D , ○ , Ψ , ψ, υ ) admits the Lax + Oplax -monoidal structure ( ● , Φ , φ, ι ) . (2) The category of comodules — in the sense of Definition 5.9 — over the un-derlying comonoid ( a, δ, ε ) in the Lax + Oplax -monoidal category ( D , ● , Φ , φ, ι ) admits the Lax Oplax + -monoidal structure ( ○ , Ψ , ψ, υ ) .Proof. We only prove part (1), part (2) is verified symmetrically.For any sequence of ( a, µ, η ) -modules {( x i , ̺ i )} i = ,...,n , the object x ● ⋯ ● x n withthe action ( x ● ⋯ ● x n ) ○ a ○ δ n / / ( x ● ⋯ ● x n ) ○ a ● n ξ n / / ( x ○ a ) ● ⋯ ● ( x n ○ a ) ̺ ●⋯● ̺ n / / x ● ⋯ ● x n IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 37 is an ( a, µ ) -module — in the sense of Definition 3.6 — by Theorem 4.7.Let us check its unitality whenever all of the actions ̺ i are unital. The diagram onthe right of part (iii) of Definition 7.1 for n = ι ∶ i → G i H and ξ .By Definition 7.1 (i), ( ○ , ξ ) ∶ → D is a Lax + Oplax -monoidal functor. Equivalently,by Theorem 5.8, ( i, ξ , ξ ) is a comonoid in ( D , ● ) . With these facts at hand, fromthe commutativity of the left half of the upper diagram of Figure 4 at n = p = k = k = l , the l -fold comultiplication i → i ● l as in (2.2) comes out as ξ l . ByDefinition 7.5, η is a comonoid morphism. Therefore the leftmost region of ( x ● ⋯ ● x n ) ○ i ψ , / / ○ ξ n ( ( PPPPPPPPPPPPP ○ η (cid:15) (cid:15) I x ● ⋯ ● x n J υ (cid:15) (cid:15) ξ n x x qqqqqqqqqqq ( x ● ⋯ ● x n ) ○ i ● n ξ n / / ○ η ● n (cid:15) (cid:15) ( x ○ i ) ● ⋯ ● ( x n ○ i ) ψ , ●⋯● ψ , / / ( ○ η ) ● ⋯ ● ( ○ η ) (cid:15) (cid:15) I x J ● ⋯ ● I x n J υ ● ⋯ ● υ & & ◆◆◆◆◆◆◆◆◆◆◆◆ ( x ● ⋯ ● x n ) ○ a ○ δ n / / ( x ● ⋯ ● x n ) ○ a ● n ξ n / / ( x ○ a ) ● ⋯ ● ( x n ○ a ) ̺ ● ⋯ ● ̺ n / / x ● ⋯ ● x n . commutes. The triangular region on the right commutes by the diagram on the leftin part (iii) of Definition 7.1; and the large pentagonal region at the top commutesby the right half of the lower diagram of Figure 4 for the values n , p = k = k =
0. The quadratic region at the bottom right commutes by the unitality of eachaction ̺ i , and the region on its left commutes by the naturality of ξ n . This proves theunitality of the action in the bottom row.The ● -monoidal products of module morphisms, as well as the natural transforma-tions ι and Φ k ,...,k p for all k i > ( a, µ, η ) -modules — are morphismsof ( a, µ ) -modules by Theorem 4.7. Then they are morphisms of ( a, µ, η ) -modules bydefinition. We leave it to the reader to check — similarly to the proof of Theorem4.7 — that the remaining natural transformations φ and Φ k ,...,k p with possibly zerovalues of k i — if evaluated at ( a, µ, η ) -modules — are morphisms of ( a, µ, η ) -modulestoo. (cid:3) Proposition 7.8.
For any bimonoid ( a, ( δ, ε ) , ( µ, η )) in an arbitrary Lax + Oplax + -duoidal category the following assertions hold. (1) (( a, δ ) , µ, η ) is a monoid in the Lax Oplax + -monoidal category of comodulesover the comonoid ( a, δ, ε ) (cf. Theorem 7.7 (2)). (2) (( a, µ ) , δ, ε ) is a comonoid in the Lax + Oplax -monoidal category of modulesover the monoid ( a, µ, η ) (cf. Theorem 7.7 (1)).Proof. By Example 5.10 ( a, δ ) is a comodule and by Proposition 4.8 µ is a morphismof comodules. Also η is a morphism of comodules by the second diagram of Definition7.5 (iii) which completes the proof of part (1). Part (2) follows symmetrically, usingnow the first diagram of Definition 7.5 (iii). (cid:3) Definition 7.9. A Hopf module over a bimonoid ( a, ( δ, ε ) , ( µ, η )) in an arbitrary Lax + Oplax + -duoidal category ( D , ● , ○ , ξ ) is defined by the following equivalent data.(i) A module over the monoid (( a, δ ) , µ, η ) in the category of comodules over thecomonoid ( a, δ, ε ) in ( D , ● ) (cf. Proposition 7.8 (1)). (ii) A comodule over the comonoid (( a, µ ) , δ, ε ) in the category of modules overthe monoid ( a, µ, η ) in ( D , ○ ) (cf. Proposition 7.8 (2)).(iii) ● An object x of D , ● a module ( x, ν ) over the monoid ( a, µ, η ) in ( D , ○ ) , ● a comodule ( x, ̺ ) over the comonoid ( a, δ, ε ) in ( D , ● ) ,rendering commutative the diagram of Definition 4.9 (iii). Example 7.10.
By Example 5.10 and its dual counterpart, and by the compatibilitydiagram of Definition 4.5 (iii), ( a, δ, µ ) is a Hopf module over an arbitrary bimonoid ( a, ( δ, ε ) , ( µ, η )) in any Lax + Oplax + -duoidal category.Our final task is a description of unital and counital BiHom -bimonoids as bimonoidsin a suitable
Lax + Oplax + -duoidal category. Theorem 7.11.
Associated to any symmetric monoidal category ( V , ⊗ , I, σ ) , there isa Lax + Oplax + -duoidal category D as follows. ● The objects of the category D consist of an object a of V and four pairwisecommuting endomorphisms α, β, κ, ν of a .The morphisms in D are those morphisms in V which commute with all of thefour endomorphisms of the source and target objects. ● The
Lax + Oplax -monoidal structure is given by the same data as in L of The-orem 5.11; in terms of the first two endomorphisms α, β at each object. ● The
Lax Oplax + -monoidal structure is given by the same data as in R of The-orem 6.7; in terms of the last two endomorphisms κ, ν at each object. ● For any non-negative integers n, p , the natural transformation ξ pn is given bythe unique component of the symmetry σ pn ∶ ⊗ n ⋅ ( ⊗ p ⋯⊗ p ) ⋅ τ np → ⊗ p ⋅ ( ⊗ n ⋯⊗ n ) .Proof. An easy substitution, using naturality of the symmetry σ , shows that theaxioms of Definition 7.1 hold; we leave it to the reader. (cid:3) As an immediate consequence of Theorem 5.12 and Theorem 6.8 we obtain thefollowing.
Theorem 7.12.
For any symmetric monoidal category ( V , ⊗ , I, σ ) the following cat-egories are pairwise isomorphic. (1.i) The category of bisemigroups in the
Lax + Oplax + -duoidal category D of Theorem7.11. (1.ii) The category of
BiHom -bimonoids in V .and (2.i) The category of bimonoids in the
Lax + Oplax + -duoidal category D of Theorem7.11. (2.ii) The category of unital and counital
BiHom -bimonoids in V . Next we explain the Yau twist of (unital and counital)
BiHom -bimonoids in [11,Proposition 5.9]. Analogously to the monoidal category M of Proposition 5.13, weassociate the following symmetric monoidal category S to any symmetric monoidalcategory ( V , ⊗ , I, σ ) . As a category, S is the same as D of Theorem 7.11. Its monoidalproduct is ⊗ = ⊗ of D and the monoidal unit is ⊗ = ( I, , , , ) . Mac Lane’s coher-ence natural isomorphisms are given by the same (omitted) morphisms as in V and IHOM HOPF ALGEBRAS VIEWED AS HOPF MONOIDS 39 the symmetry is given by σ . A bisemigroup (respectively, bimonoid) in S is a quintu-ple consisting of a bisemigroup (respectively, bimonoid) in ( V , ⊗ , I, σ ) together withfour commuting bisemigroup (respectively, bimonoid) endomorphisms. Bisemigroup(respectively, bimonoid) morphisms are those bisemigroup (respectively, bimonoid)morphisms in ( V , ⊗ , I, σ ) which are compatible with the endomorphism parts too. Proposition 7.13.
Consider an arbitrary symmetric monoidal category ( V , ⊗ , I, σ ) ,the corresponding Lax + Oplax + -duoidal category D of Theorem 7.11, and the symmetricmonoidal category S in the previous paragraph regarded as a Lax + Oplax + -duoidal cate-gory. The identity functor S → D is Lax + Oplax + -duoidal via the Lax + Oplax -monoidalstructure in Proposition 5.13 and the Lax Oplax + -monoidal structure in Proposition6.9. Consequently it preserves bisemigroups and bimonoids.Proof. The easy check of the commutativity of (4.1) for the stated data, using thenaturality of σ , is left to the reader. (cid:3) Remark . In the category D of Theorem 7.11, the full subcategory D × for whoseobjects ( a, α, β, κ, ν ) all morphisms α, β, κ and ν are invertible, carries a duoidal struc-ture. Its monoidal structures are as in Remark 5.14, and the compatibility morphismsare either trivial or given by the symmetry σ of V .The Lax + Oplax + -duoidal functor of Proposition 7.13 restricts to a strong duoidalisomorphism between D × and the full symmetric monoidal subcategory S × of S inProposition 7.13 with the same objects. This strong duoidal isomorphism D × ≅ S × gives rise to an isomorphism of the categories of bisemigroups (respectively, bimonoids)in the isomorphic duoidal categories D × and S × . This yields an isomorphism betweenthose BiHom -bialgebras (resp. unital and counital
BiHom -bialgebras) whose four en-domorphisms are invertible; and the bisemigroups (resp. bimonoids) in V togetherwith four commuting bisemigroup (resp. bimonoid) automorphisms.This amounts to saying that any bimonoid in D × is a Yau twist of a bimonoid in V .Moreover, for any bimonoid (( a, α, β, κ, ν ) , ( δ, ε ) , ( µ, η )) in D × ; and the correspondingbimonoid ( a, (̃ µ = µ ⋅ ( κ − ⊗ ν − ) , η ) , (̃ δ = ( α − ⊗ β − ) ⋅ δ, ε )) , the following assertionsare equivalent.(i) ( a, (̃ δ, ε ) , (̃ µ, η )) is a Hopf monoid in V . That is, there is a (unique) morphism χ ∶ a → a — the ‘antipode’ — rendering commutative the following diagram. a ̃ δ / / ̃ δ (cid:15) (cid:15) ε ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ a ⊗ a ⊗ χ / / a ⊗ a ̃ µ (cid:15) (cid:15) I η ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ a ⊗ a χ ⊗ / / a ⊗ a ̃ µ / / a (ii) The following ‘canonical morphism’ (see [5, (1.10)]) is invertible, for all objects y and a -modules ( x, ̺ ) in D × . x ⊗ y ⊗ a ⊗ ⊗ δ / / x ⊗ y ⊗ a ⊗ a ⊗ σ ⊗ / / x ⊗ a ⊗ y ⊗ a ̺ ⊗ ⊗ / / x ⊗ y ⊗ a (iii) (( a, α, β, κ, ν ) , ( µ, η ) , ( δ, ε )) satisfies the axioms of [11, Definition 6.9]. That is,there is a (unique) morphism χ ∶ a → a — the ‘ BiHom -antipode’ — rendering commutative the following diagram. a δ / / δ (cid:15) (cid:15) ε + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ a ⊗ a ⊗ χ / / a ⊗ a βν ⊗ ακ / / a ⊗ a µ (cid:15) (cid:15) I η + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ a ⊗ a χ ⊗ / / a ⊗ a βν ⊗ ακ / / a ⊗ a µ / / a The morphism χ occurring in parts (i) and (iii) is the same. Although the assertions inparts (ii) and (iii) (but not in (i)) are meaningful for any bimonoid in the Lax + Oplax + -duoidal category D of Theorem 7.11, there seems to be no reason to expect theirequivalence any longer. References [1] Marcelo Aguiar and Swapneel Mahajan,
Monoidal functors, species and Hopf algebras, volume29 of CRM Monograph Series. American Mathematical Society, Providence, RI, 2010. 1, 3[2] Eliezer Batista, Stefaan Caenepeel and Joost Vercruysse,
Hopf Categories,
Algebr. Represent.Theory 19 no. 5 (2016) 1173–1216. 1[3] John C. Baez and James Dolan,
Higher-Dimensional Algebra III. n-Categories and the Algebraof Opetopes,
Adv. in Math. 135 no. 2 (1998) 145–206. 4, 6[4] Gabriella B¨ohm,
Hopf polyads, Hopf categories and Hopf group monoids viewed as Hopf monads,
Theor. and Appl. of Categories 32 no. 37 (2017) 1229–1257. 1[5] Gabriella B¨ohm, Yuanyuan Chen and Liangyun Zhang,
On Hopf monoids in duoidal categories,
J. Algebra 394 (2013) 139–172. 39[6] Gabriella B¨ohm and Stephen Lack,
Hopf comonads on naturally Frobenius map-monoidales,
J.Pure Appl. Algebra 220 no. 6 (2016) 2177–2213. 1[7] Alain Brugui`eres,
Hopf Polyads,
Alg. Represent. Theory 20 no. 5 (2017) 1151–1188. 1[8] Stefaan Caenepeel and Isar Goyvaerts,
Monoidal Hom-Hopf algebras,
Comm. Algebra 39 (2011)2216–2240. 1, 2, 4, 26, 29[9] Brian Day and Ross Street,
Lax monoids, pseudo-operads, and convolution, in: “DiagrammaticMorphisms and Applications” David E. Radford, Fernando J. O. Souza and David N. Yetter(eds.) pp. 75–96 Contemp. Math. 318, 2003 7, 19[10] Brian Day and Ross Street,
Quantum categories, star autonomy, and quantum groupoids,
FieldsInstitute Comm. 43 (2004), 193-231. 3[11] Giacomo Grazianu, Abdenacer Makhlouf, Claudia Menini and Florin Panaite,
BiHom -Associative Algebras,
BiHom -Lie Algebras and
BiHom -Bialgebras,
SIGMA 11 (2015), 086, 34pages. 1, 2, 4, 28, 29, 32, 38, 39[12] Tom Leinster,
Higher Operads, Higher Categories,
Cambridge University Press 2004. 3[13] Vladimir Turaev,
Homotopy field theory in dimension 3 and crossed group-categories, preprintavailable at https://arxiv.org/abs/math/0005291. 1[14] Xiaohui Zhang and Dingguo Wang,
Cotwists of Bicomonads andBiHom-bialgebras,
Alg. Represent. Theory (2019) available online athttps://link.springer.com/article/10.1007/s10468-019-09888-2.
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