Hopf Parametric Adjoint Objects through a 2-adjunction of the type Adj-Mnd
aa r X i v : . [ m a t h . C T ] J a n Hopf Parametric Adjoint Objects through a 2-adjunction of thetype Adj-Mnd
Adrian Vazquez-Marquez ∗ Universidad Incarnate Word, Campus Baj´ıo
Abstract
In this article Hopf parametric adjunctions are defined and analysed within the context of the 2-adjunctionof the type
Adj - Mnd . In order to do so, the definition of adjoint objects in the 2-category of adjunctions andin the 2-category of monads for
Cat are revised and characterized. This article finalises with the applicationof the obtained results on current categorical characterization of Hopf Monads.In memory of Lecter and Cosmo
Introduction
In 2002, I. Moerdijk [5] characterized the liftings of a monoidal structure to the category of Eilenberg-Moorealgebras, for a related initial monad. This characterization lead to the definition of a opmonoidal monad.In 2011, A. Brugui`eres, S. Lack and A. Virelizier [1] characterized the liftings of a closed monoidal structurethrough the concept of a Hopf monad. These two examples will be analysed in the context of higher categorytheory.This article belong to a series where 2-adjunctions of the type
Adj - Mnd are applied to classical monadtheory. In this installment, the author analyse adjoint objects and parametric adjunctions within this context.On the last subject, that of parametric adjunctions, this article is mainly based upon the ideas laid outin the seminal article of A. Brugui`eres et. al. [1]. In order to apply the 2-adjunction
Adj - Mnd , the ideasare developed into a 2-categorical framework, cf. [3]. It is in this framework that the Hopf monad concept,for a monoidal closed structure, is extended to the concept of Hopf 1-cells and adjoint parametric objects oncertain 2-categories.Without any further ado, the structure of the article is given.In chapter 1, the 2-categorical structure needed for the rest of the article is given, namely the constructionof the 2-adjunction Φ M E ⊣ Ψ M E .In chapter 2, adjoints objects in the 2-categories Adj R ( Cat ) and
Mnd ( Cat ) are revised and character-ized. The characterization of such objects is done based on [1] which is suitable for the 2-categorical contextof the article.In chapter 3, the concept of (left) Hopf 1-cells is defined within the 2-category
Adj R ( Cat ) and it is usedin order to construct the Hopf parametric adjoint objects in that 2-category. The concept of antipode is ∗ Contact: [email protected]
Mnd ( Cat ) and it is used inorder to construct the corresponding Hopf parametric adjoint objects for that 2-category.In chapter 5, the condition for being Hopf 1-cell and the related struture, that of a parametric adjointobject, is analysed through the 2-adjunction. The condition for a 1-cell to be Hopf 1-cell is preserved andthen Hopf 1-cells, in each 2-category, are compared using the 2-adjunction. At the end, using the isomor-phism of categories, induced by the 2-adjunction, a bijection of Hopf parametric adjoint objects is found.In chapter 6, remaining concepts and statements are done in order to get the main theorem of the ar-ticle which gives a bijection between Hopf parametric liftings, mimicking those liftings for the closure ofa monoidal structure [1], and certain parametric adjoint objects is given. This chapter finalises with thecorresponding application to Hopf monads in a monoidal category which was the main inspiration for thisextension to a 2-categorical context.A list of the notations and conventions taken in this article is given as follows. Consider an adjunction L ⊣ R , its unit and counit are denoted as η RL and ε LR , respectively. This notation might be complicated butrefrain one from running out of, and a posteriori very needed, the finite set of greek letters. Nevertheless,the notation will be simplyfied whenever possible. For example, if the adjunction comes from a free-forgetfulcase, i.e. F S ⊣ U S , the unit can be written as η UFS or when a parametric adjunction is involved, on P , F P ⊣ G P its unit can be written as η GFP . The direction of the adjunction L ⊣ R will be taken as the directionof its left adjoint functor L , therefore the domain category of the adjunction is the domain category of L .The triangular identity given by ε LR L ◦ L η RL = 1 L will be refered to as the triangular identity associated to L .For the 2-category Cat , of small categories and functors, the notation C will be used instead.The notation 1 ∗ will be used for cases like 1 P E whenever the context allows it. Also, in the cartesianmonoidal structure for Cat , whenever possible L × P will be understood as L × P , for example.In the proof of the communativity for a diagram, arguments based on the naturality property of a certaintransformation will ommited whenever possible, in order to spare for the numerous details. The pastingcomposition of 2-cells will be denoted as · p . The 2-category context needed for this article is the following 2-adjunctionΦ M E ⊣ Ψ M E : Adj R ( Cat ) −→ Mnd ( Cat )the subindex E refers to the Eilenberg-Moore objects, since Cat admits the construction of algebras, andthe superindex M refers to the monad case [3]. Whenever possible, one or both indexes will be dropped. R ( C ) The n-cells for the domain 2-category,
Adj R ( C ), are the following:i) The 0-cells are adjunctions L ⊣ R : C −→ X .ii) The 1-cells are of the form (
J, V, λ JV , ρ JV ) : L ⊣ R −→ L ⊣ R and depicted as2 DX Y J / / L (cid:15) (cid:15) R O O L (cid:15) (cid:15) R O O V / / where C DX Y J / / L (cid:15) (cid:15) L (cid:15) (cid:15) V / / λ JV (cid:7) (cid:7) C DX Y R O O J / / V / / R O O ρ JV - - are mates and such that ρ JV is an isomorphism. The inverse of ρ JV will be denoted as δ JV or ̺ JV .Because of the previous, the notation can be shorten to ( J, V, λ JV ) or even to ( J, V ) : L ⊣ R −→ L ⊣ R ,whenever the left mate is understood or unimportant. Since the right mate is an isomorphism, the2-category will be denoted as Adj R ( C ).Note: In general, the mate of a natural transformation ϑ : LF −→ GL might be denoted as R m R ( ϑ ) = ε LR · p ϑ · p η RL = RGε LR ◦ RϑR ◦ η RL F R : F R −→ RG. iii) The 2-cells are comprissed of a pair of natural transformations ( α, β ) where α : J −→ J ′ and β : V −→ V ′ such that one of the following equivalent requirements is fulfilled † ) βL ◦ λ JV = λ J ′ V ′ ◦ Lα ‡ ) Rβ ◦ ρ JV = ρ J ′ V ′ ◦ αR ( C ) The n -cells for the 2-category Mnd ( C ) are described as follows.i) The 0-cells are monads ( C , S, µ S , η S ), whose short notation is ( C , S ).ii) The 1-cells are pairs of the form ( B, ψ B ) : ( C , S ) −→ ( D , T ) C DC D B / / T (cid:15) (cid:15) S (cid:15) (cid:15) B / / ψ B (cid:8) (cid:8) where this natural transformation fulfills the following equations3 B ◦ µ T B = Bµ S ◦ ψ B S ◦ T ψ B ψ B ◦ η T B = Bη S these equations might be refered to as the compatibility , with the product and the unit of the monads, conditions .iii) The 2-cells θ : ( A, ψ A ) −→ ( B, ψ B ) are just natural transformations θ : A −→ B : C −→ D such thatthe following equation takes place ψ B ◦ T θ = θS ◦ ψ A Φ M E The 2-functor Φ M E : Adj R ( C ) −→ Mnd ( C ) is defined on n -cells as followsi) For the 0-cell, L ⊣ R : C −→ X , Φ M E ( L ⊣ R ) = ( C , RL, Rε LR L, η RL ). That is to say, the monad inducedon the domain of the adjunction.ii) For the 1-cell, ( J, V, λ JV ) : L ⊣ R −→ L ⊣ R ,Φ M E ( J, V, λ JV ) = ( J, ̺ JV L ◦ Rλ JV ) : ( C , RL ) −→ ( D , RL )it will be useful to provide the following notation, Φ( λ JV ) = ̺ JV L ◦ Rλ JV .iii) For the 2-cell, ( α, β ), Φ M E ( α, β ) = α . Ψ M E The 2-functor Ψ M E : Mnd ( C ) −→ Adj R ( C ) can be constructed if the initial 2-category admits the constructionof algebras [6], which is certainly the case for Cat . It is defined on n -cells as follows.i) For ( C , S ), Ψ M E ( C , S ) = F S ⊣ U S : C −→ C S . The category C S is the Eilenberg-Moore category for themonad S , on C , and the adjunction is the usual free-forgetful adjunction.ii) For ( B, ψ B ) : ( C , S ) −→ ( D , T ).Ψ M E ( B, ψ B ) = ( B, b B, λ B b B ) : F S ⊣ U S −→ F T ⊣ U T as in C DC S D T B / / F T (cid:15) (cid:15) U T O O F S (cid:15) (cid:15) U S O O b B / / where the functor b B : C S −→ D T is defined as b B ( M, k M ) = (cid:0) BU S ( M, k M ) , BU S ε UFS ( M, k M ) · ψ B U S ( M, k M ) (cid:1) (1)for ( M, k M ) in C S . On morphisms, b B ( m ) = Bm . The left mate λ B b B fulfills the following equation U T λ B b B = ψ B . It will be useful to make the following notation Ψ( ψ B ) = λ B b B L ψ ( B ) := b B , where the author is considering that L stands for lifting . Also, the notation B ψ is particularly useful. The author will use any of these nota-tions that suits better to the context at hand.The bar over the morphism m means that, although m is in C , it fulfills an additional requirement foralgebras. For example, in U S ( m ) = m , this requirement is forgotten.iii) For θ : ( A, ψ A ) −→ ( B, ψ B ), Φ M E ( θ ) = ( θ, b θ ), where b θ : b A −→ b B : C S −→ D T and whose component, at( M, k M ) in C S , is b θ ( M, k M ) = θM Φ M E ⊣ Ψ M E The 2-adjunction can be completed, along the previous pair of 2-functors, by giving the following unit andcounit.i) The component of the unit η ΨΦ : 1 Adj R ( C ) −→ Ψ M E Φ M E , on the 0-cell L ⊣ R , is given by η ΨΦ ( L ⊣ R ) := (1 C , K RL D ) = L ⊣ R −→ F RL ⊣ U RL where K RL D : D −→ C RL is the comparison functor. The notation for this functor tries to codify as muchits domain as its codomain, in order to minimize possible confusions.ii) The component of the counit ε ΦΨ : Φ M E Ψ M E −→ Mnd ( C ) , on the 0-cell ( C , S ) is ε ΦΨ ( C , S ) := (1 C , S ) : ( C , S ) −→ ( C , S )All of the previous data make the following Proposition. Proposition 1.5.1
There exists a 2-adjunction
Adj R ( C ) Mnd ( C ) Φ M E / / Ψ M E o o (cid:3) In this section, the definitions of adjoint objects are developed as much in
Adj R ( C ) as in Mnd ( C ). R ( C ) In this subsection, a characterization of adjoint objects in the 2-category
Adj R ( C ) is given. These adjointobjects corresponds to the usual definition of an adjoint object in a general 2-category A , nevertheless thedefinition is reviewed in order to characterize these structures. Definition 2.1.1 An adjoint object in Adj R ( C ) is comprised of the following.i) A pair of 1-cells ( J, V, λ JV ) : L ⊣ R −→ L ⊣ R , ( K, W, λ KW ) : L ⊣ R −→ L ⊣ R . i) A pair of 2-cells called unit and counit , respectively ( η KJ , η WV ) : (1 C , D ) −→ ( KJ, W V )( ε JK , ε V W ) : (
JK, V W ) −→ (1 X , Y ) such that they fulfill the following triangular identities ( ε JK J ◦ Jη KJ , ε V W V ◦ V η WV ) = (1 J , V )( Kε JK ◦ η KJ K, W ε
V W ◦ η WV W ) = (1 K , W )Similar to Theorem 3.13 in [1], this type of adjoint object can be characterized by the existence of anatural transformation inverse. Proposition 2.1.2
Consider the following diagram in
Cat , where L ⊣ R and L ⊣ R , C DX Y J / / K o o L (cid:15) (cid:15) R O O L (cid:15) (cid:15) R O O V / / W o o Consider J ⊣ K , V ⊣ W as classical adjunctions and ( J, V, λ JV ) a morphism in Adj R ( C ) . The followingassertions are equivalent:i) Exists an adjoint object in Adj R ( C ) , where ( K, W ) is extended to a 1-cell ( K, W, λ KW ) , ( J, V, λ JV ) ⊣ ( K, W, λ KW ) . ii) λ JV is invertible.In such a case, λ KW is the mate of the inverse of λ JV . The natural transformation λ KW might be called adjoint of λ JV , the corresponding notation is λ KW = ad ( λ JV ) Proof: i ⇒ ii .The proposed inverse, for λ JV , is the following γ JV = ε V W LJ ◦ V λ KW J ◦ V Lη KJ (2)For example, γ JV ◦ λ JV = ε V W LJ ◦ V λ KW J ◦ V Lη KJ ◦ λ JV = ε V W LJ ◦ V λ KW J ◦ λ JV KJ ◦ LJη KJ = Lε JK J ◦ LJη KJ = L ( ε JK J ◦ Jη KJ )= L J = 1 LJ
6n the third equality, it was used the fact that ( ε JK , ε V W ) is a 2-cell in
Adj R ( C ). In the fifth one, thetriangular identity associated to J , for the adjunction J ⊣ K , was applied.In a similar way, λ JV ◦ γ JV = 1 V L . ii ⇒ i .Supposing the existence of the inverse, the natural transformation λ KW is defined as follows λ KW = W Lε JK ◦ W γ JV K ◦ η WV LK That is to say, λ KW = K m W ( γ JV ).In order for ( K, W, λ KW ) to be a morphism in Adj R ( C ), the mate of λ KW must be an isomorphic naturaltransformation. Then, consider the mate of λ KW ρ KW := RW ε LR ◦ RW Lε JK R ◦ RW γ JV KR ◦ Rη WV LKR ◦ η RL KR and its proposed inverse is δ KW := KRε
V W ◦ KRV ε LR W ◦ KRλ JV RW ◦ Kη RL JRW ◦ η KJ RW The equation ρ KW ◦ δ KW = 1 RW is proved as follows. ρ KW ◦ δ KW = RW ε LR ◦ RW Lε JK R ◦ RW γ JV KR ◦ Rη WV LKR ◦ η RL KR ◦ KRε
V W ◦ KRV ε LR W ◦◦ KRλ JV RW ◦ Kη RL JRW ◦ η KJ RW = RW ε
V W ◦ RW V ε LR W ◦ RW λ JV RW ◦ RW Lε JK RLJRW ◦ RW ε LR LJRWRW LJKη RL JRW ◦ RW LJη KJ RW ◦ RW γ JV RW ◦ Rη WV LRW ◦ η RL RW = η RL RW ◦ Rη WV LRW ◦ RW γ JV RW ◦ RW λ JV RW ◦ RW V ε LR W ◦ RW ε
V W = η RL RW ◦ Rη WV LRW ◦ RW V ε LR W ◦ RW ε
V W = 1 RW In the third equality, the triangular identity associated to LJ of the composed adjunction LJ ⊣ KR wasused. In the fifth, the triangular identity associated to RW was applied.In a similar fashion, it can be proved that δ KW ◦ ρ KW = 1 KR . Therefore, ( K, W, λ KW ) is a morphism, or1-cell, in Adj R ( C ).Remains to prove that the pair ( η KJ , η WV ) : (1 C , X ) −→ ( KJ, W V, W λ JV ◦ λ KW J ) : L ⊣ R −→ L ⊣ R isa 2-cell in Adj R ( C ). In particular, it is required that W λ JV ◦ λ KW J ◦ Lη KJ = η WV L Therefore,
W λ JV ◦ λ KW J ◦ Lη KJ = W λ JV ◦ W Lε JK J ◦ W γ JV KJ ◦ η WV LKJ ◦ Lη KJ = W λ JV ◦ W L ( ε JK J ◦ Jη KJ ) ◦ W γ JV ◦ η WV L = W λ JV ◦ W γ JV ◦ η WV L = η WV L In the third equality, it was used the triangular identity, of J ⊣ K , associated to J .7hat the pair ( ε JK , ε V W ) : (
JK, V W, V λ KW ◦ λ JV K ) : L ⊣ R −→ L ⊣ R is a 2-cell in Adj R ( C ) isproved similarly. Finally, the triangular identities are fulfilled since composition, and whiskering, of 2-cellsin Adj R ( C ) are composed, and whiskered, componently as in Cat . (cid:3) ( C ) As in the previous section, a detailed account of adjoint objects in the 2-category
Mnd ( C ) is given. Definition 2.2.1 An adjoint object in Mnd ( C ) is comprised of the following items:i) A pair of 1-cells, ( J, ψ J ) : ( C , S ) −→ ( D , T ) , ( K, ψ K ) : ( D , T ) −→ ( C , S ) . ii) A pair of 2-cells, the unit and the counit of the adjoint object η KJ : (1 C , S ) −→ ( KJ, Kψ J ◦ ψ K J ) : ( C , S ) −→ ( C , S ) ε JK : ( JK, Jψ K ◦ ψ J K ) −→ (1 D , T ) : ( D , T ) −→ ( D , T ) such that they fulfill the triangular identities ε JK J ◦ Jη KJ = 1 J Kε JK ◦ η KJ K = 1 K This type of object can be characterised using the Theorem 3.13 in [1]. However, it is restated and provedagain within the context of this article.
Proposition 2.2.2
Consider the following adjunction J ⊣ K : C −→ D , such that J is part of the 1-cell ( C , S ) ( D , T ) ( J, ψ J ) / / (3) in Mnd ( C ) . Then the following assertions are equivalent:i) Exists an adjoint object in Mnd ( C ) , where K is extended to a 1-cell ( K, ψ K ) , ( J, ψ J ) ⊣ ( K, ψ K ) ii) ψ J is invertible.in such case, ψ K = ad ( ψ J ) .Proof: i ⇒ ii .The definition of the inverse ζ J , of ψ J , goes as follows8 J := ε JK T J ◦ Jψ K J ◦ JSη KJ The equality ψ J ◦ ζ J = 1 JS is proved ψ J ◦ ζ J = ψ J ◦ ε JK T J ◦ Jψ K J ◦ JSη KJ = ε JK JS ◦ JKψ J ◦ Jψ K J ◦ JSη KJ = ε JK JS ◦ Jη KJ S = 1 JS In the third equality, it was used the fact that η KJ is a 2-cell in the 2-category Mnd ( C ). In the fourth one,the triangular identity associated to J . The case ζ J ◦ ψ J = 1 TJ is done similarly. ii ⇒ i .Suppose that ψ J has an inverse, ζ J , then ψ K is defined as follows ψ K := KT ε JK ◦ Kζ J K ◦ η KJ SK First, it is proved that (
K, ψ K ) : ( D , T ) −→ ( C , S ) is a morphism in Mnd ( C ). In order to do so, thecompatibility with the products, i.e. ψ K ◦ µ S K = Kµ T ◦ ψ K T ◦ Sψ K , is checked. ψ K ◦ µ S K = KT ε JK ◦ Kζ J K ◦ η KJ SK ◦ µ S K = KT ε JK ◦ Kµ T JK ◦ KT ζ J K ◦ Kζ J SK ◦ η KJ SSK = Kµ T ◦ KT T ε JK ◦ KT ζ J K ◦ KT ε KJ JSK ◦ KT Jη KJ SK ◦ Kζ J SK ◦ η KJ SSK = Kµ T ◦ KT ε KJ T ◦ Kζ J KT ◦ η KJ SKT ◦ SKT ε JK ◦ SKζ J K ◦ Sη KJ SK = Kµ T ◦ ψ K T ◦ Sψ K . If (
J, ψ J ) : ( C , S ) −→ ( D , T ) is a morphism in Mnd ( C ), then ( J, ζ J ) : ( C , S ) −→ ( D , T ) is a morphism inthe Kleisli dual of Mnd ( C ). In particular, ζ J ◦ Jµ S = µ T J ◦ T ζ J ◦ ζ J S . This compatibility was used for thesecond equality. In the fifth equality, it was used the J triangular identity and the definition of ψ K .The compatibility of the units is left to the reader.Next, it is needed that η KJ : (1 C , S ) −→ ( KJ, Kψ J ◦ ψ K J ) : ( C , S ) −→ ( C , S ), be a 2-cell in Mnd ( C ), i.e. the following equality takes place Kψ J ◦ ψ K J ◦ Sη KJ = η KJ S . Kψ J ◦ ψ K J ◦ Sη KJ = Kψ J ◦ KT ε JK J ◦ Kζ J KJ ◦ η KJ SKJ ◦ Sη KJ = Kψ J ◦ KT ε JK J ◦ KT Jη KJ ◦ Kζ J ◦ η KJ S = Kψ J ◦ Kζ J ◦ η KJ S = η KJ S In the third equality, it was used the triangular identity associated to J . In the fourth one, the fact that ζ J is the inverse of ψ J was applied.Likewise, ε JK : ( JK, Jψ K ◦ ψ J K ) −→ (1 D , T ) : ( D , T ) −→ ( D , T ) is a 2-cell in Mnd ( C ). Since thecomposition of 2-cells in Mnd ( C ), and the whiskering, is done as in the subjacent 2-category Cat , then thetriangular identities are fulfilled. 9
In Example 3.12 in [1], the lift of an adjunction corresponds to an adjoint object in
Mnd ( C ). For exam-ple, conditions 3a-3d correspond to G and V , along with ζ and ξ , being morphisms in Mnd ( C ) and 3e-3fare the requirements for the unit and counit ( h, e ) being 2-cells in the same 2-category.The results on adjoint objects, using the 2-adjunction Φ M E ⊣ Ψ M E , can be combined. Take the 0-cells F S ⊣ U S in Adj R ( C ) and ( D , T ) in Mnd ( C ). Therefore, there exists an isomorphism of categories, naturalin F S ⊣ U S and ( D , T ) Hom
Adj R ( C ) (cid:0) F S ⊣ U S , Ψ M E ( D , T ) (cid:1) ∼ = Hom
Mnd ( C ) (Φ M E ( F S ⊣ U S ) , ( D , T ))in particular, there is a bijection between adjoint objects inside each category. If we take into account theproofs of Proposition 2.1.2 and Proposition 2.2.2 and the previous isomorphism of categories, we are left,without any need of a proof, with the following Theorem. Theorem 2.2.3
The following statements are equivalenti) There exist adjoint objects in
Mnd ( C ) of the form ( C , S ) ( D , T ) ( J, ψ J ) / / ( K, ψ K ) o o ii) There exist adjoint objects in Adj R ( C ) of the form C DC S D T J / / K o o F T (cid:15) (cid:15) U T O O F S (cid:15) (cid:15) U S O O b J / / b K o o iii) The natural transformation ψ J : T J −→ JS is invertible.iv) The natural transformation λ J b J : F T J −→ b JF S is invertible. (cid:3) R ( C ) Recalling that the main objective in this article is to characterize parametric adjoint objects as much in
Adj R ( C ) as in Mnd ( C ) and relate them through the 2-adjunction Φ M E ⊣ Ψ M E , therefore the definition andcharacterization of these structures in Cat has to be given.
The definition of a parametric adjunction is recalled along with the corresponding theorem that characterizesit, [4]. 10 efinition 3.1.1
Consider the following categories P , C and D . A parametric adjunction , by P , is a pairof functors of the form F : C × P −→ D ,G : P op × D −→ C , such that for any P in P , there is an adjunction F P ⊣ G P and, for p : P −→ Q , a conjugate morphism ofadjunctions e p : F P ⊣ G P −→ F Q ⊣ G Q . This parametric adjunction can be denoted as F ⊣ P G : C −→ D . Now, the corresponding characterizing theorem.
Theorem 3.1.2
Consider a functor F : C × P −→ D such that for every P in P there exists a functor G P : D −→ C and an adjunction
C D F P / / G P o o Therefore, exists a unique G : P op × D −→ C such that for PG ( P, ∼ ) := G P and for p op : P ′ −→ P , in P op , a natural transformation G ( p op , ∼ ) : G P ′ −→ G P , further denoted as G p op , such that G p op = G P ε FGP ′ ◦ G P F p G P ′ ◦ η GFP G P ′ . (cid:3) The departure from the parametric adjoint objects in
Cat to the 2-category realm is given by the comonoidal adjunction , cf. [5], and the Hopf adjunction, cf. [1]
Definition 3.1.3 A comonoidal adjunction is defined as an adjunction L ⊣ R : C −→ D where C and D are monoidal categories and L and R are comonoidal functors and the unit and counit η RL : 1 C −→ RL and ε LR : LR −→ D are natural comonoidal transformations. In [3] there is a characterization of this comonoidal adjunctions.
Proposition 3.1.4
The following statements are equivalenti) The adjunction L ⊣ R : C −→ D is comonoidal.ii) The following diagram
C × C CD × D D L × L (cid:15) (cid:15) R × R O O ⊗ D / / ⊗ C / / L (cid:15) (cid:15) R O O is a 1-cell, ( ⊗ C , ⊗ D , λ ⊗CD ) , in Adj R ( C ) . Let us remember the definition of a
Hopf operator in order to start the extension of this concepts to thecontext of 2-categories. 11 efinition 3.1.5
Let L ⊣ R : C −→ D be a comonoidal adjunction. The left Hopf operator , H is thefollowing natural transformation H ( λ ⊗CD ) : ⊗ D ( L × ε LR ) ◦ λ ⊗CD ( C × R ) : C × D −→ D (4)
C × D C × C CD × D D L ×D (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ⊗ D / / C× R / / L × L (cid:15) (cid:15) ⊗ C / / L (cid:15) (cid:15) λ ⊗CD (cid:5) (cid:5) L × ε LR (cid:127) (cid:127) The objective of this section is to extend the definition of a parametric adjunction to the 2-category ofadjunctions
Adj R ( C ).Consider a 1-cell in Adj R ( C ) of the form ( J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R C × P DX × Q Y L × e L (cid:15) (cid:15) R × e R O O V / / J / / L (cid:15) (cid:15) R O O Suppose that the functors J : C × P −→ D and V : X × Q −→ Y are part of classical parametric adjunc-tions, namely J ⊣ P K and V ⊣ Q W . There is no immediate translation of a parametric adjoint object tothe 2-category Adj R ( C ) due to a little obstacle. The problem arises with the possible definition of the 1-cell( K, W, λ KW ) where the opposite adjunction, for e L ⊣ e R , e R op ⊣ e L op : Q op −→ P op change the domain and thecodomain, therefore a 1-cell of the form ( K, W, λ KW ) cannot be defined.Hence, the objective can be changed to the study of what extension a parametric adjunction can bereasoning within the 2-category Adj R ( C ). For that, the following modifications of definitions, in [1], can begiven. Definition 3.2.1
Let ( J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R be a 1-cell in Adj R ( C ) . A left Hopf operator H on ( J, V, λ JV ) is a 1-cell in Adj R ( C ) of the form H ( J, V, λ JV ) := ( J ( C × e R ) , V, H ( λ JV )) : L × Q ⊣ R × Q −→ L ⊣ R where H ( λ JV ) is the following natural transformation C × Q C × P DX × Q Y L ×Q (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ V / / C× e R / / L × e L (cid:15) (cid:15) J / / L (cid:15) (cid:15) λ JV (cid:5) (cid:5) L × ε e L e R (cid:5) (cid:5) that is to say ( λ JV ) = ( L × ε e L e R ) · p λ JV = V ( L × ε e L e R ) ◦ λ JV ( C × e R ) . (5) Definition 3.2.2 A left Hopf Adj R ( C ) , ( J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R, is such that H ( λ JV ) is invertible. In such a case, the inverse is denoted as N ( λ JV ) . Consider a left Hopf 1-cell in
Adj R ( C ), ( J, V, λ JV ), therefore its left Hopf operator is invertible and so isthe following natural transformation, for any Q in Q , C C × C × Q C × P DX X × X × Q X × Q Y L (cid:15) (cid:15) ρ − X / / X × E Q / / X × Q / / V / / ρ − C / / C× E Q / / C× e R / / J / / L (cid:15) (cid:15) L × (cid:15) (cid:15) L ×Q (cid:15) (cid:15) L × e L (cid:15) (cid:15) L (cid:15) (cid:15) L × ε e L e R (cid:5) (cid:5) λ JV (cid:5) (cid:5) The functor E Q stands for evaluation at Q in Q . The previous natural transformation can be written asfollows C DX Y L (cid:15) (cid:15) V Q / / J e RQ / / L (cid:15) (cid:15) λ JV e RQ (cid:5) (cid:5) Remark : If J ⊣ P K then J ( C × e R ) ⊣ Q K ( e R op × D ).Due to the previous remark, there are two parametric adjunctions on Q , J ( C × e R ) ⊣ Q K ( e R op × D ) and V ⊣ Q W with corresponding adjunctions J e RQ ⊣ K e RQ and V Q ⊣ W Q , for any Q in Q , such that( J e RQ , V Q , λ JV e RQ )is a 1-cell in Adj R ( C ) and λ JV e RQ is invertible. If Proposition 2.1.2 is recalled for this situation, there existsan adjoint object in Adj R ( C ) ( J e RQ , V Q , λ JV e RQ ) ⊣ ( K e RQ , W Q , λ KW e RQ )where λ KW e RQ = ad ( λ JV e RQ ).The natural transformation λ KW e RQ : LK e RQ −→ W Q L : D −→ X can be extended to a dinatural transfor-mation of the form λ KW e R : LK ( e R op × D ) −→ W ( Q op × L ) : Q op × D −→ X This claim is stated as the following proposition 13 roposition 3.2.3
In the previous context, there exists a dinatural transformation of the form λ KW e R : LK ( e R op × D ) −→ W ( Q op × L ) : Q op × D −→ X defined, on ( Q, D ) in Q op ×D , as λ KW e R ( Q, D ) := λ KW e RQ D and such that for any ( q op , d ) : ( Q ′ , D ) −→ ( Q, D ′ ) in Q op × D , the following diagram commutes LK ( e RQ ′ , D ′ ) W ( Q ′ , LD ′ ) LK ( e RQ ′ , D ) W ( Q, LD ′ ) LK ( e RQ, D ) W ( Q, LD ) LK (( e Rq ) op ,D ) % % ▲▲▲▲▲▲▲▲▲▲ λ KW e R ( Q,D ) / / W ( Q,Ld ) rrrrrrrrrr LK ( e RQ ′ ,d ) rrrrrrrrrr λ KW e R ( Q ′ ,D ′ ) / / W ( q op ,LD ′ ) % % ▲▲▲▲▲▲▲▲▲▲ Proof :First, recall that K (cid:0) ( e Rq ) op , ∼ (cid:1) := K e RQ ε JK e RQ ′ ◦ K e RQ J e R q K e RQ ′ ◦ η KJ e RQ K e RQ ′ since J ( C × e R ) ⊣ Q K ( e R op × D ). There exists a similar expression for W ( q op , ∼ ). Second, the followingequation takes place due to the naturality of all of the involved components W Q Ld · W Q ε V WQ ′ LD · W Q V q W Q ′ LD · η WV Q W Q ′ LD · λ KW e RQ D = W Q ε V WQ ′ LD ′ · W Q V q W Q ′ LD ′ · η WV Q W Q ′ LD ′ · λ KW e RQ ′ D ′ · LK e RQ ′ d Therefore, it is left to prove that the following equation takes place λ KW e RQ D · LK e RQ ε JK e RQ ′ · LK e RQ J e R q K e RQ ′ D · Lη KJ e RQ K e RQ ′ D = W Q ε V WQ ′ LD · W Q V q W Q ′ LD · η WV Q W Q ′ LD · λ KW e RQ ′ D This is done by the following process: λ KW e RQ D · LK e RQ ε JK e RQ ′ · LK e RQ J e R q K e RQ ′ D · Lη KJ e RQ K e RQ ′ D = λ KW e RQ D · LK e RQ ε JK e RQ ′ · LK ( e R q ) op J e RQ ′ K e RQ ′ D · Lη KJ e RQ ′ K e RQ ′ D = λ KW e RQ D · LK ( e R q ) op D · K e RQ ′ ε JK e RQ ′ D · η KJ e RQ ′ K e RQ ′ D = λ KW e RQ D · LK ( e R q ) op D = W q op LD · λ KW e RQ ′ D = W Q ε V WQ LD · η WV Q W Q LD · W q op LD · λ KW e RQ ′ D = W Q ε V WQ LD · W Q V Q W q op LD · η WV Q W Q ′ LD · λ KW e RQ ′ D = W Q ε V WQ ′ LD · W Q V q W Q ′ LD · η WV Q W Q ′ LD · λ KW e RQ ′ D The first equality takes place due to the Proposition 3.1.2 where J e R q and K ( e R q ) op are conjugate morphisms.The third one, uses the triangular identity associated to K e RQ . The fourth one, is due to the fact that( V q , W q op ) is a 2-cell in Adj R ( C ). The fifth one uses the triangular identity associated to W Q . The seventhis related to the fact that V q and W q op are conjugate. The rest of the equalities have to do with an involvednaturality and therefore the details are spare for those.14 The mate of λ KW e RQ , ρ KW e RQ , and the inverse of this last one ̺ KW e RQ can also be extended to a dinaturaltransformation. Corollary 3.2.4
The transformation defined, for ( Q, Y ) in Q op × Y , as ̺ KW e R ( Q, Y ) := ̺ KW e RQ Y is dinatural, i.e. for any ( q op , y ) in Q op × Y , the following diagram commutes RW ( Q ′ , Y ′ ) K ( e RQ ′ , RY ′ ) RW ( Q ′ , Y ) K ( e RQ, RY ′ ) RW ( Q, Y ) K ( e RQ, RY ) RW ( q op ,Y ) % % ▲▲▲▲▲▲▲▲▲▲▲ ̺ KW e R ( Q,Y ) / / K ( e RQ,Ry ) rrrrrrrrrr RW ( Q ′ , y ) rrrrrrrrrrr ̺ KW e R ( Q ′ ,Y ′ ) / / K (( e Rq ) op ,RY ′ ) % % ▲▲▲▲▲▲▲▲▲▲ (cid:3) The inverse of the mate of the dinatural transformation λ KW e R is the dinatural transformation ̺ KW e R thenthe following is a 1-cell in Adj R ( C )( K ( e R op × D ) , W, λ KW e R ) : Q op × L ⊣ Q op × R −→ L ⊣ R Therefore, using a left Hopf 1-cell, an object similar to a parametric adjunction could be obtained. Thisresult is summarized, and the corresponding process, into the following statement and definition.
Theorem 3.2.5
Consider a left Hopf 1-cell of the form ( J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R and a pair of classical parametric adjunctions J ⊣ P K and V ⊣ Q W . Then we havei) ( J ( C × e R ) , V, λ JV e R ) : L × Q ⊣ R × Q −→ L ⊣ R .ii) ( K ( e R op × D ) , W, λ KW e R ) : Q op × L ⊣ Q op × R −→ L ⊣ R .as 1-cells in Adj R ( C ) and for each Q in Q , an adjoint object ( J e RQ , V Q , λ JV e RQ ) ⊣ ( K e RQ , W Q , λ KW e RQ ) Therefore, this structure might be defined as a
Hopf parametric adjoint object , in
Adj R ( C ) , and denotedas (cid:0) J ( C × e R ) , V, λ JV e R (cid:1) ⊣ Q (cid:0) K ( e R op × D ) , W, λ KW e R (cid:1) (cid:3) .3 The Antipode Similar to the definition of an antipode in [1], the following corollary is stated in order to define this conceptfor a left Hopf 1-cell.
Corollary 3.3.1
The following transformation ψ K e R := ̺ KW e R · p λ KW e R = ̺ KW e R ( Q op × L ) ◦ Rλ KW e R : RLK ( e R op × D ) −→ K ( e R op × RL ) is dinatural. According to [1], there is a certain bijection of dinatural transformations, which is now rewritten in thiscontext for a left Hopf 1-cell in
Adj R ( C ). Proposition 3.3.2
There is a bijection between the following dinatural transformationsi) ψ K e R : RLK ( e R op × D ) −→ K ( e R op × RL ) : Q op × D −→ C .ii) σ K e R : RLK ( e R op e L op × D ) −→ K ( P op × RL ) : P op × D −→ C .This last dinatural transformation is called antipode . (cid:3) ( C ) In this chapter, the definitions made in the previous section are recalled but this time monads are used. Theobjective, in this section as in the whole article, is to give an extension of a classical parametric adjunction J ⊣ P K within the 2-categorical context of Mnd ( C ). Consider for this case the following 0-cells ( C , S ), ( D , T ) and ( P , E ). For any functor J : C × P −→ D onecan think of a 1-cell, in
Mnd ( C ), of the form( J, ψ J ) : ( C × P , S × E ) −→ ( D , T )If one wishes to construct a parametric adjoint object then there must exist a functor K : P op × D −→ C that can be extended to a 1-cell in Mnd ( C ), but such an extension presents a problem. Since E is a monad on P , E op is a comonad on P op , therefore it cannot be proposed a 1-cell ( K, ψ K ) : ( P op × D , E op × T ) −→ ( C , S ),neither in Mnd ( C ) or in the comonad dual of Mnd ( C ) in order to complete a possible parametric adjunction.In the same way as before, a modification of the functors J and K has to be made in order to achive theproposed objective. Definition 4.1.1
Consider a 1-cell, in
Mnd ( C ) , of the form ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) The left Hopf operator on ( J, ψ J ) is the following 1-cell in Mnd ( C ) H ( J, ψ J ) := ( J ( C × U E ) , H ( ψ J )) = ( J ( C × U E ) , S × P E ) −→ ( D , T ) where H ( ψ J ) is the following natural tranformation × P E C × P DC × P D S × U E " " ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉ J / / C× U E / / S × E (cid:15) (cid:15) J / / T (cid:15) (cid:15) ψ J (cid:5) (cid:5) S × U E ε FUE (cid:127) (cid:127) that is to say, H ( ψ J ) = ( S × U E ε FUE ) · p ψ J = J ( S × U E ε FUE ) ◦ ψ J ( C × U E ) Definition 4.1.2
Consider a 1-cell ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) in Mnd ( C ) . The left fusion operator is the following 1-cell in Mnd ( C ) F ( J, ψ J ) := ( J ( C × E ) , F ( ψ J )) = ( J ( C × E ) , S × P ) −→ ( D , T ) where F ( ψ J ) is the following natural transformation C × P C × P DC × P D S × E (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ J / / C× E / / S × E (cid:15) (cid:15) J / / T (cid:15) (cid:15) ψ J (cid:5) (cid:5) S × µ E (cid:5) (cid:5) that is to say F ( ψ J ) = ( S × µ E ) · p ψ J = J ( S × µ E ) ◦ ψ J ( C × E ) Definition 4.1.3 A left Hopf Mnd ( C ) , is of the form ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) suchthat H ( ψ J ) is invertible. In such case, the inverse is denoted as N ( ψ J ) . Definition 4.1.4 A left fusion Mnd ( C ) , ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) is such that F ( ψ J ) isinvertible. In such case, the inverse is denoted as G ( ψ J ) . Remark:
Later on, it will be checked that the Hopf and fusion 1-cell will be equivalent, which in turnwill ease the difference with the Hopf monad definition on [1].Consider a classical parametric adjunction J ⊣ P K and a left Hopf 1-cell ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ). Since H ( ψ J ) is invertible so is the following natural transformation for any ( M, k M ). C C × C × P E C × P DC C × C × P E C × P E Y S (cid:15) (cid:15) ρ − C / / C× E Mk / / C× U E / / J / / ρ − C / / C× E Mk / / C× U E / / J / / S × (cid:15) (cid:15) S ×P E (cid:15) (cid:15) S × E (cid:15) (cid:15) T (cid:15) (cid:15) S × U E ε FUE (cid:5) (cid:5) ψ J (cid:5) (cid:5) The previous invertible natural transformation is denoted as ψ JUE ( M, k M ) or ψ JM : T J M −→ J M S : C −→ D , emark : If J ⊣ P K then J ( C × U E ) ⊣ P E K ( U Eop × D ).For any Eilenberg-Moore algebra (
M, k M ), there is an adjunction J M ⊣ K M such that the following is a1-cell in Mnd ( C ) ( J M , ψ JM ) : ( C , S ) −→ ( D , T )and ψ JM is invertible. If the Proposition 2.2.2 is applied, an adjoint object in Mnd ( C ) is obtained( J M , ψ JM ) ⊣ ( K M , ψ KM )where ψ KM = ad ( ψ JM ). The last natural transformation can be further extended. Proposition 4.1.5
The transformation ψ KM , on ( M, k M ) , can be extended to the following dinatural trans-formation ψ KUE : SK ( U Eop × D ) −→ K ( U Eop × D )(( P E ) op × T ) : ( P E ) op × D −→ C . Proof:
Define ψ KUE for (cid:0) ( M, k M ) , D (cid:1) , in ( P E ) op × D , as ψ KUE (cid:0) ( M, k M ) , D (cid:1) := ψ KM D. The proof of the commutativity for the corresponding morphism p op : ( M ′ , k M ′ ) −→ ( M, k M ) is left to thereader. (cid:3) In the context of the previous Proposition, the dinatural transformation can be denoted as ψ KUE := ad ( H ( ψ J )) or ψ KUE := H ♯ ( ψ J ).The previous proccess can be summarized into the following Theorem. Theorem 4.1.6
Consider a classical parametric adjunction J ⊣ P K and a left Hopf 1-cell in Mnd ( C ) ofthe form ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) Then the followingi) ( J ( C × U E ) , ψ JUE ) : (
C × P E , S × P E ) −→ ( D , T ) ,ii) ( K ( U Eop × D ) , ψ KUE ) : (cid:0) ( P E ) op × D , ( P E ) op × T (cid:1) −→ ( C , S ) are 1-cells in Mnd ( C ) and for each ( M, k M ) in P E there is an adjoint object ( J M , ψ JM ) ⊣ ( K M , ψ KM ) Therefore, this structure might be defined as a
Hopf parametric adjoint object , in
Mnd ( C ) , and denotedas ( J ( C × U E ) , ψ JUE ) ⊣ P E ( K ( U Eop × D ) , ψ KUE ) (cid:3) .2 Antipode Analogous to the bijection in Proposition 3.3.2, consider the dinatural transformation ψ KUE : SK ( U Eop × D ) −→ K ( U Eop × D )( P op × T )whisker it with the functor F Eop × D and compose it with the natural transformation K ( ε UFEop × T ) to get σ := K ( ε UFEop × T ) ◦ ψ KUE ( F Eop × D ) : SK ( E op × D ) −→ K ( P op × T ) : P op × D −→ C whose component at ( P, D ) in P op × D , noting that ε UFEop P = ( η FUE P ) op = ( η E P ) op , is σ K ( P, D ) = K (cid:0) ( η E P ) op , T D (cid:1) · ψ KUE (( EP, ( µ E P ) op ) , D )therefore, σ K ( P, D ) : SK ( EP, D ) −→ K ( P, T D ).In [1], A. Brugieres et. al. called this natural transformation (left) antipode . As pointed out by them,there is a bijection between the dinatural transformations ψ KUE and σ K , where the inverse of the bijectionacts on σ K as follows ι := σ K ( U Eop × D ) ◦ SK ( η FUEop × D ) : SK ( U Eop × D ) −→ K ( U E op × D )whose component at the object (cid:0) ( M, k M ) , D (cid:1) , noting that η FUEop ( M, k M ) = ( ε FUE ( M, k M )) op = ( k M ) op , is ι (cid:0) ( M, k M ) , D (cid:1) = σ K ( M, D ) · SK ( k op M , D ) : SK ( M, D ) −→ K ( M, T D )reminiscent of the properties for ψ KUE , as a 1-cell in
Mnd ( C ), the equations that fulfills this antipode arethe following σ K ◦ µ S K ( E op × D ) = K (1 ∗ × µ T ) ◦ σ K (1 ∗ × T ) ◦ Sσ K ( E op × D ) ◦ SSK (cid:0) ( µ E ) op D (cid:1) σ K ◦ η S K ( E op × D ) = K ( ε UFEop × η T ) = K (cid:0) ( η E ) op × η T (cid:1) Compare these equations with those equivalent as in Proposition 3.8.b, [1]. Φ M E ⊣ Ψ M E Consider the 1-cell (
J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R in Adj R ( C ). This induces a 1-cell in Mnd ( C )of the form Φ M E ( J, V, λ JV ) = ( J, Φ( λ JV )) = ( J, ̺ JV ( L × e L ) ◦ Rλ JV ), where ̺ JV is the inverse of the mate ρ JV = R × e R m R ( λ JV ). Therefore H (Φ( λ JV )) = H ( ̺ JV ( L × e L ) ◦ Rλ JV )= J ( RL × e Rε e L e R ) ◦ ̺ JV ( L × e L )( C × e R ) ◦ Rλ JV ( C × e R )= ̺ JV ( L × Q ) ◦ R (cid:0) V ( L × ε e L e R ) ◦ λ JV ( C × e R ) (cid:1) = ̺ JV ( L × Q ) ◦ RH ( λ JV ) = Φ( H ( λ JV ))where the last equality takes place since R × e R m R ( λ JV ) = R ×Q m R ( H ( λ JV )). Then, the following propositioncan be stated. Proposition 5.1.1
Consider the 1-cell ( J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R in Adj R ( C ) , such that R reflects isomorphisms, then the following statements are equivalent: ) H ( λ JV ) is invertible, i.e. the 1-cell is left Hopf in Adj R ( C ) .ii) H (Φ( λ JV )) is invertible, i.e. the induced 1-cell Φ( λ JV ) is left Hopf in Mnd ( C ) .Proof : i ) ⇒ ii )If H ( λ JV ) is invertible, so is ̺ JV ( L × Q ) ◦ RH ( λ JV ) = Φ( H ( λ JV )) and the conclusion follows from theprevious equality. ii ) ⇒ i )If H (Φ( λ JV )) is invertible so is ρ JV ( L × Q ) ◦ H (Φ( λ JV )) = RH ( λ JV ), since R reflects isomorphisms H ( λ JV ) is invertible. (cid:3) The inverses are related as follows N (Φ( λ JV )) = RN ( λ JV ) ◦ ρ JV ( L × Q ) Φ M E ⊣ Ψ M E Using the unit of the 2-adjunction Φ M E ⊣ Ψ M E for the 1-cell ( J ( C × e R ) , V, H ( λ JV )) the following propositioncan be stated. Proposition 5.2.1
Consider the following list:i) L ⊣ R : C −→ X and the induced monad ( C , S ) .ii) L ⊣ R : D −→ Y and the induced monad ( D , T ) .iii) e L ⊣ e R : P −→ Q and the induced monad ( P , E ) .and suppose that ( J, V, λ JV ) , is a Hopf 1-cell. Therefore, there exists the following pair of commuting dia-gramms in Adj R ( C ) C × Q DX × Q YC × Q DC S × Q Q D T J ( C× e R ) / / L (cid:15) (cid:15) R O O L ×Q (cid:15) (cid:15) R ×Q O O V / / J ( C× e R ) / / F T (cid:15) (cid:15) U T O O F S × F Q (cid:15) (cid:15) U S × U Q O O [ J ( C× e R )] Hψ / / " " ❊❊❊❊❊❊❊❊❊❊❊❊❊ " " ❊❊❊❊❊❊❊❊❊❊❊❊❊❊ K S X × K QQ " " ❊❊❊❊❊❊❊❊❊❊❊❊❊ K T Y " " ❊❊❊❊❊❊❊❊❊❊❊❊❊❊ Q op × D CQ op × Y XQ op × D C ( Q op ) Q op × D T C S K ( e R op ×D ) / / L (cid:15) (cid:15) R O O Q op × L (cid:15) (cid:15) Q op × R O O W / / K ( e R op ×D ) / / F S (cid:15) (cid:15) U S O O F ∗ × F T (cid:15) (cid:15) U ∗ × U T O O [ K ( e R op ×D )] H♯ψ / / " " ❊❊❊❊❊❊❊❊❊❊❊❊❊ " " ❊❊❊❊❊❊❊❊❊❊❊❊❊❊ K T Y × K ∗Q op " " ❊❊❊❊❊❊❊❊❊❊❊❊❊ K S X " " ❊❊❊❊❊❊❊❊❊❊❊❊❊ (cid:3) Consider the 2-adjunction Φ M E ⊣ Ψ M E , this structure gives particular classes of isomorphisms of categories.Certain 0-cells are chosen in order to get an adequate isomorphism, for example, consider the followingmonads, 0-cells in Mnd ( C ), ( C , S ), ( P , E ) and ( D , T ) and construct the 0-cells F S × F E ⊣ U S × U E and F T ⊣ U T , in Adj R ( C ), therefore exists the following isomorphism of categories20 om Adj R ( C ) (cid:0) F S × P E ⊣ U S × P E , F T ⊣ U T (cid:1) ∼ = Hom
Mnd ( C ) (cid:0) ( C × P E , S × P E ) , ( D , T ) (cid:1) Similar isomorphisms exists for combinations of the 0-cells ( P E ) op × F T ⊣ ( P E ) op × U T , F T ⊣ U T and F S ⊣ U S .The following theorem can come forth which combines, through the 2-adjunction Φ M E ⊣ Ψ M E and thecorresponding isomorphisms, the parametric adjoint objects in Theorem 3.2.5 and Theorem 4.1.6. Theorem 5.2.2
Consider a left Hopf 1-cell ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) in Mnd ( C ) whose functoris part of a classical parametric adjunction J ⊣ P K . Therefore, there exists a bijection between the followingstructuresi) Hopf parametric adjunctions, in Adj R ( C ) , of the form (cid:16) J ( C × U E ) , (cid:2) J ( C × U E ) (cid:3) Hψ , λ JHψ (cid:17) ⊣ P E (cid:16) K ( U Eop × D ) , (cid:2) K ( U Eop × D ) (cid:3) H♯ψ , λ
KH♯ψ (cid:17) ii) Hopf parametric adjunctions, in
Mnd ( C ) , of the form (cid:0) J ( C × U E ) , H ( ψ J ) (cid:1) ⊣ P E (cid:0) K ( U Eop × D ) , H ♯ ( ψ J ) (cid:1) In order to lift a classical parametric adjunction, to some Eilenberg-Moore categories of algebras, there aresome further discussion and calculations to be done.
Consider the 1-cell (
J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R . Similar to the relation of the Hopf operators,there is the following relation between the fusion and Hopf operators F (Φ( λ JV )) = (cid:0) ̺ JV · p H ( λ JV ) (cid:1) ( C × e L ) (6)The following lemma is required. Lemma 6.1.1
Given an adjunction of the form e L ⊣ e R : P −→ Q and a natural transformation α : A e R −→ B e R , where A and B are arbitrary parallel functors, with domain Q . Therefore, α is invertible if α e L is so.In this case the inverse of the component αQ is the following α − Q = A e Rε e L e R Q · ( α e L ) − e RQ · Bη e R e L e RQ (7) Proof :The proof is similar to the Lemma 2.19 given in [1] but this time the following split fork is used e R e L e R e L e RQ e R e L e RQ e RQ e R e L e Rε e L e R Q / / e Rε e L e R e L e RQ / / e Rε e L e R Q / / (cid:3) The following proposition can be written.
Proposition 6.1.2
Consider the 1-cell ( J, V, λ JV ) : L × e L ⊣ R × e R −→ L ⊣ R , such that R reflectsisomorphisms, then the following statements are equivalenti) H ( λ JV ) is invertible, i.e. the 1-cell is left Hopf in Adj R ( C ) .ii) F (Φ( λ JV )) is invertible, i.e. the 1-cell is left fusion in Mnd ( C ) . roof : i ) ⇒ ii ) Clear by taking into consideration (6). ii ) ⇒ i ) In order to use the previous lemma, take C in C and the natural transformation α is given by (cid:0) ̺ JV · p H ( λ JV ) (cid:1) ( C, ∼ ) : RLJ ( C, e R ) −→ J ( RLC, e R ) , therefore ̺ JV · p H ( λ JV ) is invertible and so is RH ( λ JV ), since R reflects isomorphisms H ( λ JV ) is alsoinvertible. (cid:3) The inverses are related as follows RN ( λ JV ) ◦ ρ JV ( L × Q ) = RLJ ( C × e Rε e L e R ) ◦ G (Φ( λ JV ))( C × e R ) ◦ J ( RL × η e R e L e R )The reader is compelled to check the same expression in Lemma 2.18 [1]. Corollary 6.1.3
Consider a 1-cell ( J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) . Therefore F ( ψ J ) is invertible iff H (Ψ( ψ J )) is invertible, i.e. the 1-cell is Hopf iff is fusionable. This corollary allows the author to keep using the adjective Hopf without losing the generality of theresults.
Without any further ado, the main Theorem of the article is stated and proved.
Theorem 6.2.1
Consider a parametric adjunction J ⊣ P K , and 0-cells in Mnd ( C ) of the form ( C , S ) , ( D , T ) and ( P , E ) . There is a bijection between the following structuresi) Parametric Adjoint Liftings b J ⊣ P E b K , where ( J, b J, λ J b J ) is a Hopf 1-cell in Adj R ( C ) . This liftedparametric adjunction makes the following diagrams commutative C × P DC S × P E D T U S × U E O O J / / b J / / U T O O P op × D C ( P E ) op × D T C S U Eop × U T O O K / / b K / / U S O O ii) Hopf parametric adjunctions of the form ( J ( C × U E ) , H ( ψ J )) ⊣ P E (cid:0) K ( U Eop × D ) , ψ KUE (cid:1) where ψ KUE = ad ( H ( ψ J )) Proof :Induce the following left Hopf 1-cell (
J, ψ J ) : ( C × P , S × E ) −→ ( D , T ) in Mnd ( C ), where ψ J := Φ( λ J b J ).According to Theorem 5.2.2 there is a bijection between · ) Hopf parametric adjunctions in Mnd ( C ) of the form (cid:0) J ( C × U E ) , H ( ψ J ) (cid:1) ⊣ P E (cid:0) K ( U Eop × D ) , H ♯ ( ψ J ) (cid:1) ) Hopf parametric adjunctions in Adj R ( C ) of the form (cid:0) J ( C × U E ) , [ J ( C × U E )] Hψ , λ JHψ (cid:1) ⊣ P E (cid:0) K ( U Eop × D ) , [ K ( U Eop × D )] H♯ψ , λ
KH♯ψ (cid:1)
Taking into account this result, the bijection can be given as follows.The first lifting diagram can be seen as a 1-cell ( J, b J, λ J b J ) in Adj R ( C ) and the following 1-cell can beconstructed ( J ( C × U E ) , b J, H ( λ J b J )) : F S × P E ⊣ U S × P E −→ F T ⊣ U T . Using the Proposition 5.2.1 withthis last 1-cell the commutative diagram can be obtained C S × P E D T C S × ( P E ) ∗ D T C S × F ∗ (cid:15) (cid:15) [ J ( C× U E )] Hψ / / b J / / D T (cid:15) (cid:15) A similar argument, for the 1-cell ( K ( U Eop × D ) , b K ), gives the following commutative diagram( P E ) op × D T C S ( P Eop ) ∗ × C S C S F ∗ × D T (cid:15) (cid:15) [ K ( U Eop ×D )] H♯ψ / / b K / / C S (cid:15) (cid:15) These two last diagrams give the bijection of the forgetful diagrams with the components of the Hopfparametric adjunction in
Adj R ( C ). (cid:3) Consider the case a closed monoidal category ( C , ⊗ , (cid:3) , I ). In [3], there is a bijection between monoidalliftings, 1-cells in ( ⊗ , b ⊗ , λ ⊗ b ⊗ ) : ( F S × F S , U S × U S ) −→ ( F S ⊣ U S ) in Adj R ( C ), and opmonoidal monads,1-cells ( ⊗ , ψ ⊗ ) : ( C × C , S × S ) −→ S in Mnd ( C ).If the closure functor is to be lifted when ( ⊗ , b ⊗ , λ ⊗ b ⊗ ) is a Hopf 1-cell, the previous calculations show that( J ( C × U E ) , [ J ( C × U E )] Hψ , λ JHψ ) = ( ⊗ ( C × U S ) , [ ⊗ ( C × U S )] Hψ , λ ⊗ Hψ ) ∼ = ( ⊗ , b ⊗ , λ ⊗ b ⊗ )( K ( U Eop × D ) , [ K ( U Eop × D )] H♯ψ , λ
KH♯ψ ) = ( (cid:3) ( U Sop × C ) , [ (cid:3) ( U Sop × C )] H♯ψ , λ (cid:3)
H♯ψ )are the corresponding liftings, where ψ = Φ( λ ⊗ b ⊗ ). This article was intended to explore more examples in classic monad theory in order to prove, what theauthor considers, the relevance of the 2-adjunctions of the type
Adj - Mnd . This relevance will be significantif the recollection of a numerous quantity of useful examples is done.23he development of the article only used the left definition, nevertheless, the author hopes that the right and the left-right case can be completed without any complication whatsoever.As far as further work is concerned, there is a pair of possible connections. The first one, is to take theframework of multivariable adjunctions in [2] for further analysis using the 2-adjunction and the parametricobjects already defined.Second, there might be a further development on categorical duality provided by this parametric objects.
Acknowledgments
The author would like to thank to the Consejo Nacional de Ciencia y Tecnolog´ıa (CONACYT) for fi-nancial support through the grant SNI-59154. Special thanks, for useful comments on this article to T.Brzezinski, G. Bohm and B. Mesablishvili.The author would like to thank to J. Antonio L. Verver and Fernando Vega for their endless support andpatience.
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