aa r X i v : . [ m a t h . C T ] A ug SKEW MONOIDAL MONOIDS
K. SZLACH ´ANYI
Abstract.
Skew monoidal categories are monoidal categories with non-invertible ‘coherence’morphisms. As shown in a previous paper bialgebroids over a ring R can be characterized asthe closed skew monoidal structures on the category Mod - R in which the unit object is R R .This offers a new approach to bialgebroids and Hopf algebroids. Little is known about skewmonoidal structures on general categories. In the present paper we study the one-object case:skew monoidal monoids (SMM). We show that they possess a dual pair of bialgebroids describingthe symmetries of the (co)module categories of the SMM. These bialgebroids are submonoids oftheir own base and are rank 1 free over the base on the source side. We give various equivalentdefinitions of SMM, study the structure of their (co)module categories and discuss the possibleclosed and Hopf structures on a SMM. Three definitions of skew monoidal monoids
Skew monoidal monoids are skew monoidal categories, as defined in [4], in which the underlyingcategory has only one object, so it is a monoid.If A denotes the monoid and 1 denotes its unit element then a skew-monoidal structure on A amounts to have a monoid morphism A × A → A , h a, b i 7→ a ∗ b , the skew monoidal product,( ab ) ∗ ( cd ) = ( a ∗ c )( b ∗ d )(1) 1 ∗ γ , η , ε of A , the ”coherence” morphisms, satisfying the naturality conditions γ ( a ∗ ( b ∗ c )) = (( a ∗ b ) ∗ c ) γ (3) ηa = (1 ∗ a ) η (4) aε = ε ( a ∗ γ ∗ γ (1 ∗ γ ) = γ (6) γη = η ∗ εγ = 1 ∗ ε (8) ( ε ∗ γ (1 ∗ η ) = 1(9) εη = 1 . (10)If the skew monoidal product were associative, so γ were the identity, and it had a unit thenthe Eckman-Hilton argument would force a ∗ b = ab and A would be commutative. Since γ is notassumed even to be invertible, skew monoidal monoids have a good chance to be non-trivial. Lemma 1.1.
A skew monoidal monoid A = h A, ∗ , γ, η, ε i is the same as the data consisting of (i) a monoid A , Supported by the Hungarian Scientific Research Fund, OTKA 108384. (ii) two monoid endomorphisms T : A → A and Q : A → A such that (11) T ( a ) Q ( b ) = Q ( b ) T ( a ) a, b ∈ A (iii) an element γ ∈ A satisfying the intertwiner relations γT ( a ) = T ( a ) γ (12) γT Q ( a ) = QT ( a ) γ (13) γQ ( a ) = Q ( a ) γ (14) for all a ∈ A (iv) and elements η, ε ∈ A satisfying ηa = T ( a ) η (15) aε = εQ ( a )(16) for all a ∈ A .These data are required, furthermore, to obey the skew monoidality relations Q ( γ ) γT ( γ ) = γ (17) γη = Q ( η )(18) εγ = T ( ε )(19) Q ( ε ) γT ( η ) = 1(20) εη = 1 . (21) Proof.
Assume h A, ∗ , γ, η, ε i is a SMM. Define T := 1 ∗ and Q := ∗
1. Then T and Q aremonoid endomorphisms by (1), (2) and satisfies (11) by the interchange law, the latter being aconsequence of (1), too. Replacing two of the a, b, c by 1 in the expression a ∗ ( b ∗ c ) we obtain T ( c ), T Q ( b ) and Q ( a ), respectively. In a similar way ( a ∗ b ) ∗ c leads to T ( c ), QT ( b ) and Q ( a ).Therefore the naturality condition (3) alone implies (12), (13) and (14). Equations (15) and (16) areobvious transcriptions of (4) and (5). The same can be said about the remaining 5 skew-monoidalityrelations.Vice versa, assume that the data h A, T, Q, γ, η, ε i satis fies the 11 axioms (11-21). Then a skewmonoidal product on the one-object category A can be defined by a ∗ b := Q ( a ) T ( b ). The verificationof the skew monoidal monoid axioms is now easy. (cid:3) Let
Mon denote the 2-category of monoids as a monoidal sub-2-category of the cartesian monoidal2-category
Cat . So the objects of
Mon are the monoids, the 1-cells are the monoid homomorphisms f : A → B and the 2-cells b : f → g : A → B are the elements b ∈ B satisfying the intertwinerproperty bf ( a ) = g ( a ) b for all a ∈ A .As in every skew monoidal category (see [4, Lemma 2.6]) we can introduce a monad h T, µ, η i anda comonad h Q, δ, ε i on the object A of Mon where multiplication µ and the comultiplication δ aredefined by µ := Q ( ε ) γ (22) δ := γT ( η ) . (23) KEW MONOIDAL MONOIDS 3
It has also been shown in that Lemma that γ : T Q → QT is a mixed distributive law. In ourone-object category the defining relations of the distributive law take the form Q ( µ ) γT ( γ ) = γµ (24) Q ( γ ) γT ( δ ) = δγ (25) γη = Q ( η )(26) εγ = T ( ε ) . (27)The last two of these coincide with the triangle relations (18-19), the first two are simple conse-quences of the pentagon relation (17).The SMM axioms can be reformulated using µ and δ instead of γ . Although the number ofaxioms gets larger, it reveals SMM-s as something like a bimonoid. Proposition 1.2.
A skew monoidal monoid is the same as the data consisting of • a monoid A , • two monoid endomorphisms T : A → A and Q : A → A • and elements µ , η , δ , ε of A subject to the following axioms.Intertwiner relations: For all a ∈ A µ T ( a ) = T ( a ) µ (28) η a = T ( a ) η (29) δ Q ( a ) = Q ( a ) δ (30) a ε = ε Q ( a )(31) µ δ T Q ( a ) = QT ( a ) µ δ (32) Commutation relations: For all a, b ∈ AT ( a ) Q ( b ) = Q ( b ) T ( a )(33) µQ ( a ) = Q ( a ) µ (34) δT ( a ) = T ( a ) δ (35) Bimonoid(-like) relations: µT ( µ ) = µ (36) µ η = 1(37) µ T ( η ) = 1(38) Q ( δ ) δ = δ (39) εδ = 1(40) Q ( ε ) δ = 1(41) Q ( µ ) µ δT ( δ ) = δ µ (42) ε µ = ε T ( ε )(43) δ η = Q ( η ) η (44) εη = 1(45) Proof.
Setting γ := µδ the (28-45) axioms imply the axioms of Lemma 1.1. Assuming the latteraxioms and defining µ and δ by (22) and (23) the axioms (28-45) are easily obtained. (cid:3) K. SZLACH´ANYI
Every SMM A contains distinguished submonoids Q ( A ) ⊂ F ⊃ S ( A ) ⊂ G ⊃ T ( A )(46)where F = Q ( A ) δ (47) G = µT ( A )(48)and S is the anti endomorphism(49) S : A → A op , S ( a ) = µT ( Q ( a ) η ) ≡ Q ( εT ( a )) δ . Furthermore,(i) the endomorphisms Q , S , T are all injective,(ii) π : A → A , π ( a ) := εaη is a common left inverse in Set of the endofunctions Q , S and T ,(iii) their images, Q ( A ), S ( A ) and T ( A ), pairwise commute,(iv) S ( A ) = F ∩ G .For example, (iv) can be shown as follows. By the two equivalent formulas in (49) it is clear that S ( A ) ⊂ F ∩ G . Vice versa, suppose x ∈ F ∩ G . Then there exist a, b ∈ A such that x = Q ( a ) δ = µT ( b ). Multiplying with ε , resp. η we obtain a = εx and b = xη . Introducing y := aη we canwrite Q ( y ) η = Q ( a ) Q ( η ) η = Q ( a ) δη = xη = b and therefore S ( y ) = µT ( Q ( y ) η ) = µT ( b ) = x . Thisproves F ∩ G ⊂ S ( A ).The following useful relations will also be used in the sequel without explicit mention: µS ( a ) = µT Q ( a )(50) S ( a ) µ = µT S ( a )(51) S ( a ) δ = QT ( a ) δ (52) δS ( a ) = QS ( a ) δ (53) S ( a ) η = Q ( a ) η (54) εS ( a ) = εT ( a ) . (55) 2. The category of modules
The category
Mod - A of modules over a skew monoidal monoid A = h A, ∗ , γ, η, ε i is by definition[4] the Eilenberg-Moore category A T of the canonical monad T . Since A T is just a category whilethe category of modules over a bialgebroid is always monoidal, the challenge is to find a monoidalstructure on A T . This problem has been solved for skew monoidal categories [5], [2, Theorem8.1] provided the underlying category has reflexive coequalizers and the skew monoidal productpreserves such coequalizers in the 2nd argument. As we shall see, in case of skew monoidal monoidsthese assumptions are not necessary.Since A has only one object, the objects of A T are just elements x ∈ A satisfying two equations xµ = xT ( x )(56) xη = 1 A . (57)The arrows t ∈ A T ( x, y ) are the elements t ∈ A such that(58) tx = yT ( t ) . For a skew monoidal category the monoidal product, called the horizontal tensor product anddenoted by ¯ ⊗ , of two T -algebras α : T M → M and β : T N → N is defined as follows. First we KEW MONOIDAL MONOIDS 5 introduce the notation µ M,N := ( ε M ∗ N ) ◦ γ M,R,N . Then the underlying object of M ¯ ⊗ N is givenby the reflexive coequalizer M ∗ T N µ M,N ✲✲ M ∗ β M ∗ N ✲✲ M ¯ ⊗ N and the T -action on M ¯ ⊗ N is given by unique factorization in the diagram R ∗ ( N ∗ T M ) R ∗ µ N,M ✲✲ R ∗ ( N ∗ α ) R ∗ ( N ∗ M ) R ∗ π ✲✲ R ∗ ( N ¯ ⊗ M ) ❄ ( β ∗ T M ) ◦ γ R,N,TM ❄ ( β ∗ M ) ◦ γ R,N,M ❄ ψ (59) N ∗ T M µ N,M ✲✲ N ∗ α N ∗ M π ✲✲ N ¯ ⊗ M provided T preserves reflexive coequalizers. In case of skew monoidal monoids all these diagramssimplify radically: all objects are the same and all µ -s are the same element (22) of A . Thus thehorizontal tensor product of x, y ∈ A T is the element x • y of A making the diagram in A A µ ✲✲ T ( y ) A y ✲✲ AA T ( µ ) ✲✲ T ( y ) A T ( y ) ✲✲ A ❄ Q ( x ) γ ❄ Q ( x ) γ ❄ x • y commutative. Now it is easy to check that(60) x • y = yQ ( x ) δ . Note that the 2nd row is the canonical split coequalizer of the T -algebra y , hence it is preserved by T . This proves that the horizontal tensor product of any pair of objects exists in A T for any skewmonoidal monoid A .It is now an easy exercise to determine the horizontal tensor product of arrows s ∈ A T ( x , y )and t ∈ A T ( x , y ). It is given by the formula(61) s ¯ ⊗ t = y Q ( s ) ηt ∈ A T ( x • x , y • y ) . Notice that we use different notations for horizontal tensor product of objects and of arrows. Thisis necessary since the same element of A can be an object and an (non-identity) arrow in A T . Proposition 2.1.
Let A = h A, T, Q, γ, η, ε i be a skew monoidal monoid. Then its module category A T equipped with the horizontal tensor product is a strict monoidal category. In this category µ isthe underlying object of a comonoid ε ε ←− µ δ −→ µ • µ . The associated representable functor is a faithful strong monoidal functor A T ( µ, ) : A T → A Set A , x xT ( A ) to the category of A - A -bimodules (= A -bisets or ( A op × A ) -sets). K. SZLACH´ANYI
Proof.
Strict associativity of • and ¯ ⊗ is a routine calculation. So is unitality; the unit object is ε . By (36-37) the µ is an object of A T . It has a special property that for any object x ∈ A T theelement x ∈ A is also an arrow x : µ → x in A T . Indeed, compare (56) with (58). It follows thatthe functor A T ( µ, ) is faithful: If sx = tx for a pair s, t of parallel arrows x → y then s = t by(57). Now ε : µ → ε is an arrow of A T and so is δ : µ → µ • µ by (42). The comonoid axioms for h µ, δ, ε i reduce to the comonad axioms (39-41) after noticing that for any arrow t ∈ A T we have1 µ ¯ ⊗ t = t and t ¯ ⊗ µ = Q ( t ), as elements of A .Before turning to the monoidal structure of the functor A T ( µ, ) we need an explicit computationof the intertwiner set A T ( µ, x ). If s : µ → x is an intertiner then s = sµT ( η ) = xT ( sη ) ∈ xT ( A ).Vice versa, if a ∈ A then xT ( a ) µ = xµT ( a ) = xT ( xT ( a )), so xT ( a ) is an intertwiner µ → x .This proves the equality A T ( µ, x ) = xT ( A ), as subsets of A . In particular, A T ( µ, µ ) = µT ( A ) is amonoid containing both T ( A ) and S ( A ) as submonoids. This allows us to define the A - A -bimodulestructure on A T ( µ, x ) by right multiplication a · t · a := tS ( a ) T ( a ) , t ∈ A T ( µ, x ) , a , a ∈ A .
The comonoid structure of µ induces the following monoidal structure for the functor A T ( µ, ). A T ( µ, x ) ⊗ A A T ( µ, y ) → A T ( µ, x • y ) s ⊗ A t ( s ¯ ⊗ t ) ◦ δ = tQ ( s ) δ (62)and the arrow A → A T ( µ, ε ) in A Set A is given by 1 A ε which is well-defined since a · ε = εS ( a ) = εT ( a ) = ε · a by (55). This monoidal structure is strong: The inverse of (62) is r ∈ A T ( µ, x • y ) x ⊗ A yT ( rη ) ∈ A T ( µ, x ) ⊗ A A T ( µ, y )and the inverse of the identity constraint is r ∈ A T ( µ, ε ) rη ∈ A since A T ( µ, ε ) = εT ( A ). (cid:3) Obviously, the next task is to factorize the forgetful functor of the above Proposition through theforgetful functor of G -modules, where G = A T ( µ, µ ), and show that G is a right bialgebroid over A . The conspicuous property of this bialgebroid is that it is a submonoid of its own base monoid. Theorem 2.2.
The forgetful functor of Proposition 2.1 factorizes through the forgetful functor
Set G → A Set A of a right A -bialgebroid G via a fully faithful A T → Set G where the bialgebroid G isdefined by underlying monoid := A T ( µ, µ ) = µT ( A ) source s G : A → G, s G ( a ) := T ( a ) target t G : A op → G, t G ( a ) := S ( a ) comultiplication ∆ G : G → G ⊗ A G, ∆ G ( g ) := µ ⊗ A µT ( δgη ) counit ǫ G : G → A, ǫ G ( g ) := π ( g ) = εgη . Proof.
The verification of the right bialgebroid axioms is a short exercise. The functor A T → Set G is faithful since the functor A T → A Set A was already faithful. In order to prove fullness let f : xT ( A ) → yT ( A ) be a G -module map, f ( sg ) = f ( s ) g . Then f ( x ) µ = f ( x ) T ( x ) and, writing f ( x ) = yT ( t ) with a unique t ∈ A , we obtain yT ( t ) µ = yT ( tx ) hence yT ( t ) = tx . This proves that t ∈ A T ( x, y ) and f ( s ) = f ( xT ( sη )) = f ( x ) T ( sη ) = yT ( tsη ) = ts therefore the functor is full. (cid:3) KEW MONOIDAL MONOIDS 7
As usual for right bialgebroids we consider G as an A -bimodule via a · g · a = gS ( a ) T ( a ).Notice that µ is a free generator for G as right A -module, the G A is free of rank 1. This explains thesimple formula for the coproduct since G ⊗ A G ∼ = G at least as right A -modules. The left A -modulestructure is by far not so trivial. Definition 2.3.
A right module M over a right A -bialgebroid G (or a left module over a leftbialgebroid) is called source-regular if the A -module obtained from M by restriction along thesource map s G : A → G is isomorphic to the right regular A -module A A (resp. the left regular A A ). Corollary 2.4.
Let A be a SMM. Then there is a source regular right bialgebroid G such that Mod - A ≡ A T is monoidally equivalent to the category Set reg G of source regular right G -modules.Proof. Clearly, the image of the forgetful functor A T ( µ, ) consist only of the source-regular mod-ules xT ( A ) for x ∈ ob A T .Let M be a source-regular right G -module and let m ∈ M be a free generator of M A , so A → M , a m · T ( a ) is a bijection. Then m · µ = m · T ( x ) for a unique x ∈ A . From the equations m · T ( xµ ) = m · µT ( µ ) = m · µ = m · T ( x ) µ = m · µT ( x ) = m · T ( xT ( x ))it follows that xµ = xT ( x ). Similarly, m · T ( xη ) = m · µT ( η ) = m implies xη = 1. This provesthat x is a T -algebra such that M ∼ = xT ( A ) as G -modules. (cid:3) Comodules and the dual bialgebroid
We shall be very brief about comodules since all what we have to say can be obtained from thestructure of modules by dualizing, i.e.,
Comod - A = A Q is nothing but Mod - A op , rev .The objects of A Q are elements u ∈ A satisfying Q ( u ) u = δu and εu = 1. The arrows s ∈A Q ( u, v ) are elements s ∈ A satisfying Q ( s ) u = vs . Proposition 3.1.
The Eilenberg-Moore category A Q of the comonad Q is a strict monoidal categoryw.r.t. the vertical tensor product defined for objects by u ⊚ v := µT ( v ) u and for arrows s : u → v , t : u → v by s ⊗ t := tεT ( s ) u . The element δ ∈ A is the underlying object of the monoid h δ, µ, η i in A Q and δ is a cogenerator in A Q . The endomorphism monoid of δ is the underlying monoid ofa source-regular left A -bialgebroid F defined byunderlying monoid : A Q ( δ, δ ) = Q ( A ) δ source s F : A → F, s F ( a ) = Q ( a ) target t F : A op → F, t F ( a ) = S ( a ) comultiplication ∆ F : A F A → A ( F ⊗ A F ) A , ∆ F ( f ) = Q ( εf µ ) δ ⊗ A δ counit ǫ F : A F A → A A A , ǫ F ( f ) = π ( f ) ≡ εf η where in the last two lines the A -bimodule structure of F is defined by a · f · a := Q ( a ) S ( a ) f . Theorem 3.2.
The forgetful functor A Q ( , δ ) : A op Q → A Set A is strong monoidal and factorsthrough a monoidal equivalence A op Q ∼ = reg F Set of A op Q with the category of source-regular left F -modules. In order to characterize A Q in terms of G we need to perform a duality transformation also onthe level of bialgebroids. Lemma 3.3.
The left dual G ∗ = Hom( G A , A A ) of the right A -bialgebroid G can be identified withthe left A -bialgebroid F by the non-degenerate pairing h , i : F × G → A, h f, g i := εf gη K. SZLACH´ANYI satisfying the following properties h f, a · g · a i = h f Q ( a ) , g i a (63) h a · f · a , g i = a h f, T ( a ) g i (64) h f, gg ′ i = h f (1) · h f (2) , g i , g ′ i (65) h f f ′ , g i = h f, h f ′ , g (1) i · g (2) i (66) h f, G i = π ( f )(67) h F , g i = π ( g ) . (68) Corollary 3.4.
The category A Q of comodules of the SMM A is monoidally equivalent to thecategory Set G reg of source-regular right G -comodules.Proof. By dualizing Corollary 2.4 every source-regular left F -module X ∈ reg F Set is isomorphic to Q ( A ) u for some u ∈ ob A Q . The right dual ∗ X = Hom( A X, A A ) is therefore isomorphic to ∗ [ Q ( A ) u ] = Hom( Q ( A ) u, A A ) = { ϕ : Q ( A ) u → A | ∃ a ∈ A, ϕ ( x ) = εxa } and left F -module structures on X and right G -comodule structures on ∗ X are in bijection via h f · x, ϕ i = h f, h x, ϕ (0) i · ϕ (1) i , f ∈ F, x ∈ X, ϕ ∈ ∗ X .
For source-regular F -modules X and Y and for a left A -module map h : X → Y its transpose ∗ h : ∗ Y → ∗ X is a right A -module map and h f · h ( x ) , ϕ i = h f, h x, ∗ h ( ϕ (0) ) i · ϕ (1) ih h ( f · x ) , ϕ i = h f, h x, ∗ h ( ϕ ) (0) i · ∗ h ( ϕ ) (1) i Therefore h is left F -linear precisely when ∗ h is right G -colinear. (cid:3) Together with F and G the monoid A contains also an image of the smash product G F . Indeed,the axiom (42) implies that for all g ∈ G , f ∈ Ff g = Q ( εf ) δµT ( gη ) = Q ( εf µ ) µδT ( δgη ) = µT ( δgη ) Q ( εf µ ) δ == g (2) S ( h f (2) , g (1) i ) f (1) = ( f (2) ⇀ g ) f (1) where f ⇀ g is the action that makes G a left module algebra over the cooposite (left bialgebroid)of F . The invariant submonoid is G F = { g ∈ G | f ⇀ g = π ( f ) · g } = T ( A ) . The smash product G F = G ⊗ S ( A ) F is the tensor product over the common submonoid S ( A ) = F ∩ G with multiplication rule( g f )( g ′ f ′ ) = g ( f (2) ⇀ g ′ ) f (1) f ′ = gg ′ (2) S ( h f (2) , g ′ (1) i ) f (1) f ′ . There is a remarkable connection between grouplike elements of G and the objects of A Q . Recallthat in a bialgebroid G an element g ∈ G is called grouplike if ∆ G ( g ) = g ⊗ A g and ǫ G ( g ) = 1. Lemma 3.5.
The bijection A ∼ → G , a µT ( a ) , restricts to a bijection A Q ∼ → Gr ( G ) between theset of objects u ∈ A Q and the set of grouplike elements of the bialgebroid G . This bijection lifts toan isomorphism of monoids: A Q ∼ → Gr ( G ) op . Dually, the map x Q ( x ) δ defines an isomorphismof monoids A T ∼ → Gr ( F ) op from the monoid of objects of A T to the opposite of the monoid ofgrouplikes in F . KEW MONOIDAL MONOIDS 9
We do not know other objects than the tensor powers of µ in A T and those of δ in A Q . Theknown part of A T , for example, contains the simplicial object ε ✛ ε µ ✛ Q ( ε ) ✲ δ ✛ ε µ • µ ✛ Q ( ε ) ✲ Q ( δ ) ✛ Q ( ε ) ✲ δ ✛ ε µ • µ • µ . . . associated to the comonad ¯ ⊗ µ but it contains also endoarrows g ∈ G = A T ( µ, µ ) at all occurenceof the object µ . E.g., there are arrows µ : µ → µ , µ ¯ ⊗ µ : µ • µ → µ • µ , . . . which do not belongto the above simplicial object. There is also another simplicial object ∆ op → A T associated to thecomonad 1 µ ¯ ⊗ . Dually, the comodule category A Q contains cosimplicial objects and endoarrows f ∈ F of the cogenerator object δ . The conjecture is that the category generated by these objectsand arrows exhausts all of A T and A Q , respectively, when A is the free SMM.4. Skew monoidal monoids are source regular bialgebroids
In the previous sections we have seen that every SMM A determines a dual pair of source regularbialgebroids F and G . In this section we show that every source regular bialgebroid determines askew monoidal structure on its base monoid and these two constructions are inverses of each other,up to isomorphisms.First we need a precise notion of isomorphism of SMM-s. Consider a skew monoidal functor h A, T, Q, γ, η, ε i Φ −→ h A ′ , T ′ , Q ′ , γ ′ , η ′ , ε ′ i . Such a functor consists of a monoid morphism φ : A → A ′ and elements φ , φ ∈ A ′ satisfying the intertwiner relations (expressing naturality of φ )(69) φ Q ′ φ ( a ) = φQ ( a ) φ and φ T ′ φ ( a ) = φT ( a ) φ and the identities (being the 3 skew monoidal functor axioms) φ ( γ ) φ T ′ ( φ ) = φ Q ′ ( φ ) γ ′ (70) φ Q ′ ( φ ) η ′ = φ ( η )(71) φ ( ε ) φ T ′ ( φ ) = ε ′ (72) Definition 4.1.
An isomorphism Φ : A ∼ → A ′ of skew monoidal monoids is a skew monoidal functor h φ, φ , φ i : A → A ′ such that φ : A → A ′ is an isomorphism of monoids and the elements φ and φ of A ′ are invertible.In particular, an isomorphism of SMM-s is both a skew monoidal and a skew opmonoidal functor.Therefore it determines not only a monad morphism σ := φ Q ′ ( φ ) : T ′ φ → φT (73) σµ ′ = φ ( µ ) σT ′ ( σ )(74) ση ′ = φ ( η )(75) but a comonad morphism τ := T ′ ( φ − ) φ − : φQ → Q ′ φ (76) δ ′ τ = Q ′ ( τ ) τ φ ( δ )(77) ε ′ τ = φ ( ε )(78)as well. Proposition 4.2.
Let A and A ′ be skew monoidal monoids and let G , resp. G ′ , denote the right A -bialgebroid associated to A by Theorem 2.2 and the right A ′ -bialgebroid associated to A ′ . If Φ :
A → A ′ is an isomorphism of skew monoidal monoids then the pair h ϕ, ϕ i , where ϕ : G → G ′ g τ φ ( g ) τ − ϕ : A → A ′ , a φ − φ ( a ) φ is an isomorphism of bialgebroids from G to G ′ , i.e., the following identities hold for all a ∈ A and g ∈ G : ϕ ( T ( a )) = T ′ ( ϕ ( a ))(79) ϕ ( S ( a )) = S ′ ( ϕ ( a ))(80) ǫ G ′ ( ϕ ( g )) = ϕ ( ǫ G ( g ))(81) ϕ ( g ) (1) ⊗ A ′ ϕ ( g ) (2) = ϕ ( g (1) ) ⊗ A ′ ϕ ( g (2) ) . (82) Proof.
Notice that for g ∈ G the expression σ − φ ( g ) σ is a well-defined composition of arrows µ ′ → φ ( µ ) σ → φ ( µ ) σ → µ ′ in A ′ T ′ , therefore defines an element in G ′ . But ϕ was defined using τ instead of σ − . Fortunately, the difference is only an inner automorphism of G ′ , induced by T ′ ( φ ),so ϕ : G → G ′ is well-defined.The verification of the relations (79 - 82) requires too much place to account for in full detail.But it is straightforward using the (co)monad morphism relations and, occasionaly, the fact thatsince φ is an isomorphism, expressions like φ Q ′ ( φ ) φ − commute with φ ( g ) for g ∈ G . (cid:3) Theorem 4.3.
Let A be a monoid. Then there is a bijection between isomorphism classes of skewmonoidal monoids h A, T, Q, µ, η, δ, ε i with underlying monoid A and source regular right bialgebroids h G, A, s G , t G , ∆ G , ǫ G i over A .Proof. The construction of G from a SMM A given in Theorem 2.2 serves as the object map of afunctor from the category of SMM-s, with arrows being the isomorphisms of Definition 4.1, intothe category of source regular bialgebroids. The arrow map is provided by Proposition 4.2.Next we construct a map from source regular bialgebroids to SMM-s. Source regularity is pre-cisely the statement that the source map s G : A → G , as a 1-cell in Mon , has a right adjoint. Wechoose a right adjoint s G : G → A with unit η : id A → s G s G and counit µ : s G s G → id G . Thenboth s G and s G are injective and a µs G ( a ) is a bijection A → G with inverse g s G ( g ) η . Fromthe adjunction relations it follows that T := s G s G is a monad on A with multiplication s G ( µ ) andunit η . This proves 5 of the 18 SMM axioms of Proposition 1.2, namely (28), (29), (36), (37) and(38).Let F := Hom( G A , A A ) with h f, g i denoting evaluation of f ∈ F on g ∈ G . Then F is a left A -module via h a · f, g i = a h f, g i and every f ∈ F is uniquely determined by its value on µ since h f, µs G ( a ) i = h f, µ i a . This defines a bijection J : F → A , J ( f ) := h f, µ i . Using the comonoidstructure of G we make F into a monoid by (66) and (68). Then defining(83) ε := J (1 F ) , δ := J − (1 A ) , s F ( a ) := J − ( aε ) , s F ( f ) := J ( δf ) KEW MONOIDAL MONOIDS 11 we obtain that J ( s F ( a )) = aε = J ( a · F ), hence s F ( a ) = a · F and h f s F ( a ) , g i = h f, h s F ( a ) , g (1) i · g (2) i = h f, a · g i == h f, gt G ( a ) i . (84)We deduce that s F : A → F and s F : F → A are monoid morphisms and s F is left adjoint tothe source map s F with unit δ and counit ε . It follows that Q := s F s F is a comonad on A withcomultiplication s F ( δ ) and counit ε . This proves 5 more SMM axioms.The canonical pairing can now be written as(85) h f, g i = εs F ( f ) s G ( g ) η . The target map of the dual (would-be-)bialgebroid F can be introduced by(86) h t F ( a ) , g i := h F , s G ( a ) g i and proving h f t F ( a ) , g i = h f, h t F ( a ) , g (1) i · g (2) i = h f, ε G ( s G ( a ) g (1) ) · g (2) i = h f, ε G ( g (1) ) · t G ( a ) g (2) i == h f, t G ( a ) g i (87)it follows that it is a monoid morphism t F : A op → F and t F ( a ) commutes with s F ( b ) for all a, b ∈ A . What used to be the antiendomorphism S in previous Sections can now be written as(88) s F t F ( a ) = h δt F ( a ) , µ i = h δ, t G ( a ) µ i = s G t G ( a ) . There is one more property of the pairing that we need: h t F ( a ) f, g i = h t F ( a ) , h f, g (1) i · g (2) i ( ) = h F , h f, g (1) i · s G ( a ) g (2) i == h F , h f, ( s G ( a ) g ) (1) i · ( s G ( a ) g ) (2) i == h f, s G ( a ) , g i . (89)The basic commutation relations of a SMM structure can now be proven as follows. Since s F s F ( a ) = h δs F ( a ) , µ i ( ) = h δ, µt G ( a ) i ( ) = s G ( µ ) s G t G ( a ) η , we can conclude that s F s F ( a ) s G ( g ) = s G ( µ ) s G t G ( a ) s G s G s G ( g ) η = s G ( µ ) s G s G s G ( g ) s G t G ( a ) η == s G ( g ) s F s F ( a ) . (90)Similarly, from the equality s G s G ( a ) = h δ, s G ( a ) µ i ( ) = h t F ( a ) δ, µ i = εs F t F ( a ) s F ( δ )we infer s F ( f ) s G s G ( a ) = εs F s F s F ( f ) s F t F ( a ) s F ( δ ) = εs F t F ( a ) s F s F s F ( f ) s F ( δ ) == s G s G ( a ) s F ( f ) . (91)Equations (90) and (91) prove, redundantly, the 3 SMM axioms (33), (34) and (35). It remains to show the intertwiner relation (32) for γ = s G ( µ ) s F ( δ ) and the last 4 bimonoidaxioms. For that first one proves the relations analogous to (50), (52), (54) and (55), namely s F t F ( a ) η = s F s F ( a ) η (92) µt G ( a ) = µs G s F s F ( a )(93) εs G t G ( a ) = εs G s G ( a )(94) t F ( a ) δ = s F s G s G ( a ) δ . (95)Then s G ( µ ) s F ( δ ) s G s G s F s F ( a ) ( ) = s G ( µ ) s G s G s F s F ( a ) s F ( δ ) ( ) == s G ( µ ) s G t G ( a ) s F ( δ ) ( ) = s G ( µ ) s F t F ( a ) s F ( δ ) ( ) = s G ( µ ) s F s F s G s G ( a ) s F ( δ ) ( ) == s F s F s G s G ( a ) s G ( µ ) s F ( δ )which is precisely the intertwiner relation (32). Finally, the remaining bimonoid relations can beshown by the calculations εη ( ) = h F , G i = ε G (1 G ) = 1 A εs G ( µ ) ( ) = h F , µ i = ǫ G ( µ ) = ǫ G ( s G ( ǫ G ( µ )) µ ) ( ) = h F , s G ( ε ) µ i = ( ) = h t F ( ε ) , µ i ( ) = εs F t F ( ε ) ( ) = εs G t G ( ε ) ( ) = εs G s G ( ε ) s F ( δ ) η ( ) = h δ , G i = h δ, h δ, G i · G i ( ) = h δ, t G ( η ) i = ( ) = s G t G ( η ) η ( ) = s F t F ( η ) η ( ) = s F s F ( η ) ηs F ( δ ) s G ( µ ) ( ) = h δ , µ i = h δ, h δ, µ (1) µ (1 ′ ) i · µ (2) µ (2 ′ ) i == h δ, h δ, µ i · ( µs G s F ( δ )) i = h δ, s G ( µ ) · µ s G s G s G s F ( δ ) s G s F ( δ ) i == h δ, s G ( µ ) · µ i s G s G s F ( δ ) s F ( δ ) = h δ, µ t G s G ( µ ) i s G s G s F ( δ ) s F ( δ ) = ( ) = h δ, µ S G s F s F s G ( µ ) i s G s G s F ( δ ) s F ( δ ) == s G ( µ ) s F s F s G ( µ ) s G s G s F ( δ ) s F ( δ )where for the last relation, in passing from the first line to the second, we used the coproductformula ∆ G ( µ ) = µ ⊗ A µT ( δ ) which, in the present construction, can be easily proven by pairing itwith arbitrary f , f ′ ∈ F . This finishes the construction of a skew monoidal monoid A = h A, s G s G , s F s F , s G ( µ ) , η, s F ( δ ) , ε i from the data of a source regular right A -bialgebroid G and from a choice of adjunction data s G , µ , η for the left adjoint s G .If G is obtained from a SMM structure on A , let us say A ′ , then s G has another right adjoint,the inclusion s ′ G : G ֒ → A with counit µ ′ and unit η ′ . Then there is an invertible 2-cell z : s G → s ′ G in Mon such that µ ′ s G ( z ) = µ and zη = η ′ . Then (83) yields ε ′ = εz − , δ ′ = s F ( z ) δ , s ′ F ( a ) = s F ( a )and s ′ F ( f ) = zs F ( f ) z − . It follows that the monoid automorphism φ ( a ) = zaz − together with φ = Q ′ ( z − ) T ′ ( z − ) and φ = z is an isomorphism of skew monoidal monoids from A to A ′ , that is to say the triple h φ, φ , φ i satisfies equations (69), (70), (71) and (72). KEW MONOIDAL MONOIDS 13
Going in the opposite direction suppose that we start from a source regular bialgebroid G ,perform the above construction of a SMM A and then apply Theorem 2.2 to construct a bialgebroid G ′ . This bialgebroid G ′ is nothing but the submonoid of A generated by s G ( µ ) and s G s G ( A ). Butthis is nothing but the image of the map s G . Therefore restricting s G to its image (together withthe identity A → A ) is a bialgebroid isomorphism G ∼ → G ′ . (cid:3) Closed and Hopf skew monoidal monoids
Let M be the category of all (small) right A -modules. Then the A -bialgebroid G of a SMM A determines a closed skew monoidal structure on M with skew monoidal product(96) M ⊛ N := M ⊗ S ( N ⊗ T G )where ⊗ S refers to tensoring over A w.r.t the left A -action a · g = gS ( a ) on G and ⊗ T w.r.t. theother left A -action a · g = T ( a ) g . That is to say, M ⊛ N is the right A -set the elements of whichare equivalence classes [ m, n, g ] of elements h m, n, g i ∈ M × N × G w.r.t. the equivalence relationgenerated by h m · b, n · a, g i ∼ h m, n, T ( a ) gS ( b ) i m ∈ M, n ∈ N, g ∈ G, a, b ∈ A .
The A -action is given by [ m, n, g ] · a := [ m, n, gT ( a )]. The skew unit object R is the right regular A -module and the coherence morphisms are γ L,M,N : L ⊛ ( M ⊛ N ) → ( L ⊛ M ) ⊛ N [ l, [ m, n, g ] , h ] [[ l, m, h (1) ] , n, gh (2) ] η M : M → R ⊛ Mm [1 A , m, G ] ε M : M ⊛ R → M [ m, a, g ] m · π ( T ( a ) g ) . The skew monoidal product M ⊛ N being a colimit, both endofunctors M ⊛ and ⊛ N on M = Set A have right adjoints therefore ⊛ is a closed skew monoidal structure.Let M ⊂ M be the full subcategory of rank 1 free A -modules. For each object M ∈ M choosea free generator ξ M ∈ M , so that ν M : R → M, a ξ M · a is an isomorphism in M . For two objects M, N ∈ M define ν M,N : M ⊛ N → G, [ m, n, g ] T ( ν − N ( n )) gS ( ν − M ( m ))which is an isomorphism in M with inverse ν − M,N ( g ) = [ ξ M , ξ N , g ]. Since G ∈ M with ξ G = µ , thecategory M is a skew monoidal subcategory of M . Lemma 5.1.
The functor ∇ : M → A which maps M f −→ N to ν − N ( f ( ξ M )) is a skew monoidalequivalence.Proof. Since all objects of M are isomorphic, ∇ is clearly an equivalence. The only question isstrong skew monoidality of ∇ . Omitting the details of a straightforward calculation we remark thatthe isomorphism ∇ ( M ) ⊛ ∇ ( N ) ∼ → ∇ ( M ⊛ N ) is given by the unique element α M,N ∈ A such that ξ M ⊛ N · α M,N = [ ξ M , ξ N , µ ]and choosing ξ R = 1 A , hence ν R = id A , the ∇ can be made strictly normal. (cid:3) Corollary 5.2.
Every SMM is equivalent, as skew monoidal categories, to a full skew monoidalsubcategory of a closed skew monoidal category.Remark . Although all objects of M are isomorphic to the skew monoidal unit, the componentsof γ , η and ε may have non-invertible components within M . In fact if any one of the componentsof η or ε is invertible then all of them are and A is a trivial SMM; see Lemma 5.9.Before discussing the closed structure of SMM-s we prove an embedding theorem relating SMM-sto SMC-s. In the following Theorem skew monoidal embedding means a fully faithful strong skewmonoidal functor. Theorem 5.4.
Let hM , ⊛ , R, γ, η, ε i be a skew monoidal category (SMC). Then for the existenceof a SMM A and a skew monoidal embedding E : A → M it is sufficient and necessary that R ⊛ R ∼ = R .Proof. Let E : A → M be a SM embedding of a SMM A . Let R ′ be the image of the single objectof A . Then R ∼ = R ′ and R ′ ⊛ R ′ ∼ = R ′ by strong skew monoidality of E . Hence R ⊛ R ∼ = R .Assuming that κ : R ⊛ R ∼ → R is an isomorphism we can define a skew monoidal structure onthe endomorphism monoid A = M ( R, R ) by setting a ∗ b := κ ◦ ( a ⊛ b ) ◦ κ − , a, b ∈ Aγ := κ ◦ ( κ ⊛ R ) ◦ γ R,R,R ◦ ( R ⊛ κ − ) ◦ κ − η := κ ◦ η R ε := ε R ◦ κ − Therefore A = h A, ∗ , γ, η, ε i is a skew monoidal monoid and E : A → M , E ( a ) = ( R a −→ R )is a skew monoidal embedding the strong skew monoidal structure of which is given by κ and bythe identity arrow R = −→ R . (cid:3) As a skew monoidal category A being (left or/and right) closed is equivalent, by Lemma 5.1, to M having this property. But M is a full skew monoidal subcategory of the closed M thereforeany type of closedness of M is the question of whether the corresponding internal hom functorsmap the unit object R into M . Since M ∗ ∼ = ⊗ T ( M ⊗ S G ) ∼ = ⊗ T G ∼ = ⊗ A ( T ( A ) A A ) ∗ M ∼ = ⊗ S ( M ⊗ T G ) ∼ = ⊗ S G ∼ = ⊗ A ( Q ( A ) A A )for all object M ∈ M , the right and left internal homs are[ M, N ] r ∼ = N T ( A ) (97) [ M, N ] ℓ ∼ = N Q ( A ) . (98)respectively, where, e.g. N T ( A ) denotes the set N equipped with right A -action h n, a i 7→ n · T ( a ).This leads to the following result. Proposition 5.5.
For a SMM A the following conditions are equivalent: (i) A is right closed, i.e., the T : A → A has a right adjoint in the 2-category Mon . (ii) The right internal hom [ R, R ] r ∼ = A T ( A ) belongs to M , i.e., A is rank 1 free as right A -module via T . KEW MONOIDAL MONOIDS 15 (iii) F A ≡ S ( A ) F is rank 1 free.Also, the following conditions are equivalent: (i) A is left closed, i.e., the Q : A → A has a right adjoint in the 2-category Mon . (ii) The left internal hom [ R, R ] ℓ ∼ = A Q ( A ) belongs to M , i.e., A is rank 1 free as right A -module via Q . (iii) G T Q ( A ) is a rank 1 free right A -module. Comparing left and right closedness, especially property (iii), we see that closedness is utterlyasymmetic. In a closed SMM the bialgebroid F is both source-regular and target-regular but G isnot target-regular. In order to get a symmetric notion let us say that A is ‘biclosed’ if it is rightclosed and ‘left coclosed’, i.e., require that T has a right adjoint and Q has a left adjoint. Then alittle inspection shows that A is ‘biclosed’ precisely when both F and G are target-regular.So far, in this Section, we have been working within the skew monoidal category M = Set A associated to the bialgebroid G . There is another construction associated to a bialgebroid, thebimonad or opmonoidal monad acting on the bimodule category A Set A . It is this latter languageon which Hopf algebroids [3] and Hopf monads [1] can be defined.Identifying A Set A with Set E , where E := A op × A , the opmonoidal monad is given by tensoringwith the E -monoid G , ˆ T X := X ⊗ E G ˆ µ X : ( x ⊗ E g ) ⊗ E h x ⊗ E gh ˆ η X : x x ⊗ E G and the opmonoidal structure isˆ T X,Y : ( X ⊗ A Y ) ⊗ E G → ( X ⊗ E G ) ⊗ A ( Y ⊗ E G )( x ⊗ y ) ⊗ g ( x ⊗ g (1) ) ⊗ ( y ⊗ g (2) )ˆ T : A ⊗ E G → Aa ⊗ g π ( T ( a ) g ) . The connection between the opmonoidal monad and the canonical monad R ⊛ on Set A is theisomorphism of their Eilenberg-Moore categories over the forgetful functor A Set A → Set A . But R ⊛ is lacking any (op)monoidal structure and such properties as a bialgebroid being a Hopfalgebroid can only be formulated in terms of ˆ T . The left and right fusion morphisms associated toˆ T are H ℓX,Y = ( ˆ T X ⊗ A ˆ µ Y ) ◦ ˆ T X, ˆ T Y : ˆ T ( X ⊗ A ˆ T Y ) → ˆ T X ⊗ A ˆ T Y (99) : ( x ⊗ A ( y ⊗ E g )) ⊗ E h ( x ⊗ E h (1) ) ⊗ A ( y ⊗ E gh (2) )(100) H rX,Y = (ˆ µ X ⊗ A ˆ T Y ) ◦ ˆ T ˆ T X,Y : ˆ T ( ˆ T X ⊗ A Y ) → ˆ T X ⊗ A ˆ T Y (101) : (( x ⊗ E g ) ⊗ A y ) ⊗ E h ( x ⊗ E gh (1) ) ⊗ A ( y ⊗ E h (2) )(102)and we know from [1, Theorem 3.6] that ˆ T is left (right) Hopf, i.e., H ℓ (resp. H r ) is invertible,precisely when ( A Set A ) ˆ T is left (right) closed and the forgetful functor to A Set A preserves the left(right) internal hom. If we try to translate this Theorem to the skew monoidal language we againrun into the problem that left and right closed structures behave differently. Lemma 5.6.
For a skew monoidal monoid A = h A, T, Q, γ, η, ε i the following conditions are equiv-alent: (i) G , the right A -bialgebroid associated to A , is left Hopf, i.e., the opmonoidal monad ˆ T = ⊗ E G is a left Hopf monad, (ii) the canonical map can G : G ⊗ T ( A ) G → G ⊗ A G, g ⊗ T ( A ) h h (1) ⊗ A gh (2) is a bijection, (iii) γ is invertible. (iv) the canonical map can F : F ⊗ Q ( A ) F → F ⊗ A F, f ⊗ T ( A ) f ′ f (1) f ′ ⊗ A f (2) is a bijection, (v) F , the left A -bialgebroid associated to A , is right Hopf, i.e., the opmonoidal monad F ⊗ A × A op is a right Hopf monad.Proof. (i) ⇔ (ii) is obvious from formula (99).(ii) ⇔ (iii) Composing the canonical map with the isomorphisms τ : G ⊗ A G ∼ → G, g ⊗ A g g S ( g η ) σ : G ⊗ T ( A ) G ∼ → G, g ⊗ T ( A ) g T ( g η ) g we obtain a map c G : G → G, g µT ( µδgη )such that τ ◦ can G = c G ◦ σ . Using the bijection K : A ∼ → G , a µT ( a ), we can write c G ( g ) = K ( γK − ( g )) from which the statement follows.By dualizing the above arguments one obtains (iii) ⇔ (iv) and (iv) ⇔ (v). (cid:3) SMM-s with the above five properties could be called one-sided Hopf SMM although we cannotsay whether left or right due to the rival conditions (i) and (v). Notice also that these one-sidedHopf properties do not require the skew monoidal structure of A to be closed.The other Hopf property, i.e., invertibility of H r for G or invertibility of H ℓ for F , is equivalentto the maps can ′ G : G ⊗ S ( A ) G → G ⊗ A G, g ⊗ S ( A ) h gh (1) ⊗ A h (2) can ′ F : F ⊗ S ( A ) F → F ⊗ A F, f ⊗ S ( A ) f ′ f (1) ⊗ A f (2) f ′ being bijective. Unfortunately these conditions are not equivalent to invertibility of some simplecombination of the skew monoidal structure maps. However, if we assume the A is left coclosed,or, just for symmetry, that it is biclosed, then there is a simple characterization in terms of a left skew monoidal structure. (So far, in this paper we used ‘skew monoidal’ for a right skew monoidalstructure.) If Q has a left adjoint, let’s say P with unit i : id A → QP and counit e : P Q → id A , KEW MONOIDAL MONOIDS 17 then a left skew monoidal structure on the monoid A can be defined by a ∗ ′ b := eP ( S ( b ) ia ) , a, b ∈ Aγ ′ := eP ( µT ( i ) eP ( δi )) η ′ := eP ( η ) ε ′ := εi . This structure can be obtained via a contravariant embedding (similarly to Theorem 5.4) into theright skew monoidal category A Set induced by the right A op -bialgebroid G coop . Lemma 5.7.
For a biclosed skew monoidal monoid A = h A, T, Q, γ, η, ε i the following conditionsare equivalent: (i) G , the right A -bialgebroid associated to A , is right Hopf. (ii) can ′ G is a bijection. (iii) γ ′ is invertible. Concluding the paper we mention that the existence of non-trivial skew monoidal monoids is anopen problem. It seems that the SMM structure is very sensitive to the addition of new axiomsand can easily collapse into something trivial. Let us make this precise.
Definition 5.8.
A skew monoidal monoid h A, T, Q, µ, η, δ, ε i is called trivial if A is commutative, T = Q = id A , ε is invertible and µ = ε , δ = η = ε − . Lemma 5.9.
For a skew monoidal monoid A any one of the following conditions alone impliesthat A is a trivial skew monoidal monoid: (i) A is finite. (ii) A is a cancellative monoid. (iii) A is commutative. (iv) T is an automorphism. (v) Q is an automorphism. (vi) η is invertible. (vii) µ is invertible. (viii) δ is invertible. (ix) ε is invertible. (x) µ • µ ∼ = ε as objects of A T . Notice that invertibility of γ does not occur among these fatal conditions, thereby granting Hopfskew monoidal monoids the chance of being non-trivial. References [1] A. Bruguieres, S. Lack, A. Virelizier, Hopf monads on monoidal categories,
Advances in Mathematics (2011) 745-800[2] S. Lack, R. Street, On monads and warpings,
Cahiers de Topologie et Geometrie Differentielle Categoriques ,to appear; arXiv:1408.4953 [3] P. Schauenburg, Duals and doubles of quantum groupoids ( × R -Hopf algebras), in ”New trends in Hopf algebratheory”, Contemporary Mathematics (2000) 273-299[4] K. Szlach´anyi, Skew-monoidal categories and bialgebroids,
Advances in Mathematics , 1694-1730 (2012)[5] K. Szlach´anyi, Skew monoidal categories beyond bialgebroids, http://maths-temp.swan.ac.uk/staff/tb/LMS-Workshop , talk at the LMS workshop ”Categorical and Homological Methods in Hopf Algebras”, Swansea,UK (2013)
Wigner Research Centre for Physics, Budapest
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