bimonads and hopf monads on categories

8/8/2019 BIMONADS AND HOPF MONADS ON CATEGORIES

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arXiv:0710

.1163v3

[math.QA

]11Jun2008

BIMONADS AND HOPF MONADS ON CATEGORIES

BACHUKI MESABLISHVILI, TBILISIANDROBERT WISBAUER, DUSSELDORF

Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monadson arbitrary categories thus providing the possibility to transfer the essentials of the theoryof Hopf algebras in vector spaces to more general settings. There are several extensionsof this theory to monoidal categories which in a certain sense follow the classical trace.Here we do not pose any conditions on our base category but we do refer to the monoidalstructure of the category of endofunctors on any category A and by this we retain someof the combinatorial complexity which makes the theory so interesting. As a basic toolwe use distributive laws between monads and comonads (entwinings) on A: we define abimonad on A as an endofunctor B which is a monad and a comonad with an entwining : BB BB satisfying certain conditions. This is also employed to define the categoryABB of (mixed) B-bimodules. In the classical situation, an entwining is derived from the

twist map for vector spaces. Here this need not be the case but there may exist specialdistributive laws : BB BB satisfying the Yang-Baxter equation (local prebraidings)which induce an entwining and lead to an extension of the theory ofbraided Hopf algebras.

An antipode is defined as a natural transformation S : B B with special propertiesand for categories A with limits or colimits and bimonads B preserving them, the existenceof an antipode is equivalent to B inducing an equivalence between A and the category ABBofB-bimodules. This is a general form of the Fundamental Theorem of Hopf algebras.

Finally we observe a nice symmetry: If B is an endofunctor with a right adjoint R,then B is a (Hopf) bimonad if and only ifR is a (Hopf) bimonad. Thus a k-vector spaceH is a Hopf algebra if and only if Homk(H,) is a Hopf bimonad. This provides a richsource for Hopf monads not defined by tensor products and generalises the well-knownfact that a finite dimensional k-vector space H is a Hopf algebra if and only if its dualH = Homk(H, k) is a Hopf algebra. Moreover, we obtain that any set G is a group if andonly if the functor Map(G,) is a Hopf monad on the category of sets.

Contents

1. Introduction 12. Distributive laws 33. Actions on functors and Galois functors 64. Bimonads 125. Antipode 156. Local prebraidings for Hopf monads 18

7. Adjoints of bimonads 28References 32

1. Introduction

The theory of algebras (monads) as well as of coalgebras (comonads) is well understoodin various fields of mathematis as algebra (e.g. [8]), universal algebra (e.g. [13]), logic oroperational semantics (e.g. [31]), theoretical computer science (e.g. [23]). The relationshipbetween monads and comonads is controlled by distributive laws introduced in the seventies

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2 BACHUKI MESABLISHVILI, TBILISI AND ROBERT WISBAUER, DUSSELDORF

by Beck (see [2]). In algebra one of the fundamental notions emerging in this context arethe Hopf algebras. The definition is making heavy use of the tensor product and thusgeneralisations of this theory were mainly considered for monoidal categories. They allowreadily to transfer formalisms from the category of vector spaces to the more general settings(e.g. Bespalov and Brabant [3] and [21]).

A Hopf algebra is an algebra as well as a coalgebra. Thus one way of generalisation is toconsider distinct algebras and coalgebras and some relationship between them. This leadsto the theory of entwining structures and corings over associative rings (e.g. [8]) and onemay ask how to formulate this in more general categories. The definition of bimonads on amonoidal category as monads whose functor part is comonoidal by Bruguieres and Virelizierin [7, 2.3] may be seen as going in this direction. Such functors are called Hopf monadsin Moerdijk [22] and opmonoidal monads in McCrudden [18, Example 2.5]. In 2.2 we givemore details of this notion.

Another extension of the theory of corings are the generalised bialgebras in Loday in[17]. These are Schur functors (on vector spaces) with a monad structure (operads) and aspecified coalgebra structure satisfying certain compatibility conditions [17, 2.2.1]. While in[17] use is made of the canonical twist map, it is stressed in [7] that the theory is built up

without reference to any braiding. More comments on these constructions are given in 2.3.The purpose of the present paper is to formulate the essentials of the classical theory of

Hopf algebras for any (not necessarily monoidal) category, thus making it accessible to awide field of applications. We also employ the fact that the category of endofunctors (withthe Godement product as composition) always has a tensor product given by compositionof natural transformations but no tensor product is required for the base category.

Compatibility between monads and comonads are formulated as distributive laws whoseproperties are recalled in Section 2. In Section 3, general categorical notions are presentedand Galois functors are defined and investigated, in particular equivalences induced forrelated categories (relative injectives).

As suggested in [33, 5.13], we define a bimonadH = (H,m,e,,) on any category A as anendofunctor H with a monad and a comonad structure satisfying compatibility conditions(entwining) (see 4.1). The latter do not refer to any braiding but in special cases they canbe derived from a local prebraiding : HH HH (see 6.3). In this case the bimonad showsthe characteristics of braided bialgebras (Section 6).

Related to a bimonad H there is the (Eilenberg-Moore) category AHH of bimodules witha comparison functor KH : A A

HH. An antipode is defined as a natural transformation

S : H H satisfying m SH = e = m HS . It exists if and only if the naturaltransformation := Hm H : HH HH is an isomorphism. If the category A is Cauchycomplete and H preserves limits or colimits, the existence of an antipode is equivalent to thecomparison functor being an equivalence (see 5.6). This is a general form of the FundamentalTheorem for Hopf algebras. Any generalisation of Hopf algebras should offer an extensionof this important result.

Of course, bialgebras and Hopf algebras over commutative rings R provide the prototypesfor this theory: on R-Mod, the category of R-modules, one considers the endofunctor B R : R-Mod R-Mod where B is an R-module with algebra and coalgebra structures, andan entwining derived from the twist map (braiding) M R N N R M (e.g. [5, Section8]).

More generally, for a comonad H, the entwining : HH HH may be derived froma local prebraiding : HH HH (see 6.7) and then results similar to those known forbraided Hopf algebras are obtained. In particular, the composition HH is again a bimonad(see 6.8) and, if 2 = 1, an opposite bimonad can be defined (see 6.10).


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BIMONADS AND HOPF MONADS ON CATEGORIES 3

In case a bimonad H on A has a right (or left) adjoint endofunctor R, then R is againa bimonad and has an antipode (or local prebraiding) if and only if so does H (see 7.5).In particular, for R-modules B, the functor HomR(B, ) is right adjoint to B R andhence B is a Hopf algebra if and only if HomR(B, ) is a Hopf monad. This provides arich source for examples of Hopf monads not defined by a tensor product and extends a

symmetry principle known for finite dimensional Hopf algebras (see 7.8). We close withthe observation that a set G is a group if and only if the endofunctor Map(G, ) is a Hopfmonad on the catgeory of sets (7.9).

Note that the pattern of our definition of bimonads resembles the definition of Frobeniusmonads on any category by Street in [27]. Those are monads T = (T , , ) with naturaltransformations : T I and : T T T, subject to suitable conditions, which inducea comonad structure = T T : T T T and product and coproduct on T satisfy thecompatibility condition T T = = T T .

2. Distributive laws

Distributive laws between endofunctors were studied by Beck [2], Barr [1] and others in

the seventies of the last century. They are a fundamental tool for us and we recall somefacts needed in the sequel. For more details and references we refer to [33].

2.1. Entwining from monad to comonad. Let T = (T , m , e) be a monad and G =(G,,) a comonad on a category A. A natural transformation : T G GT is called amixed distributive law or entwining from the monad T to the comonad G if the diagrams

GeG

~~|||||||| Ge

!3ggg

gggg

g

T G

/G GT,

T G

T !3ggg

gggg

g /G GT

T}}{{{{{{{{

T

T G

T/G T GGG/G GT G

G

and T T G

mG

T/G T GTT/G GT T

Gm

GTT

/G GGT T G

/G GT

are commutative.It is shown in [34] that for an arbitrary mixed distributive law : T G GT from a

monad T to a comonad G, the triple G = ( G, , ), is a comonad on the category AT ofT-modules (also called T-algebras), where for any object (a, ha) ofAT,

G(a, ha) = (G(a), G(ha) a);

()(a,ha) = a, and ()(a,ha) = a.

G is called the lifting of G corresponding to the mixed distributive law .

Furthermore, the triple T = ( T , m, e) is a monad on the category AG of G-comodules,where for any object (a, a) of the category A

G,

T(a, a) = (T(a), a T(a));

( m)(a,a) = ma, and

(e)(a,a) = ea.


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This monad is called the lifting of T corresponding to the mixed distributive law . Onehas an isomorphism of categories

(AG) bT (AT)bG,

and we write AGT() for this category. An object of AGT() is a triple (a, ha, a), where

(a, ha) AT and (a, a) AG with commuting diagram

(2.1) T(a)ha /G

T(a)

aa /G G(a)

T G(a)a

/G GT(a).

G(ha)

Oy

We consider two examples of entwinings which may (also) be considered as generalisationsof Hopf algebras. They are different from our approach and we will not refer to them lateron.

2.2. Opmonoidal functors. Let (V,

, I

) be a strict monoidal category. Following Mc-Crudden [18, Example 2.5], one may call a monad (T , , ) on V opmonoidal if there existmorphisms

: T(I) I and X,Y : T(X Y) T(X) T(Y),

the latter natural in X, Y V, which are compatible with the tensor structure ofV and themonad structure of T.

Such functors can also be characterised by the condition that the tensor product ofV canbe lifted to the category of T-modules (e.g. [33, 3.4]). They were introduced and namedHopf monads by Moerdijk in [22, Definition 1.1] and called bimonads by Bruguieres andVirelizier in [7, 2.3]. It is mentioned in [7, Example 2.8] that Szlachanyis bialgebroids in[29] may be interpreted in terms of such bimonads. It is preferable to use the terminologyfrom [18] since these functors are neither bimonads nor Hopf monads in a strict sense but

rather an entwining (as in 2.1) between the monad T and the comonad T(I) on V:Indeed, the compatibility conditions required in the definitions induce a coproduct I,I :

T(I) T(I) T(I) with counit : T(I) I. Moreover, the relation between and (e.g.(15) in [7, 2.3]) lead to the commutative diagram (using X I = X)

T T(X) /G

T(I,X)

T(X)I,X /G T(I) T(X)

T(T(I) T(X))T(I),T(X) /G T T(I) T T(X)

ITT(X) /G T(I) T T(X)

T(I)X

Oy

This shows that T(X) is a mixed (T, T(I) )-bimodule for the entwining map

= (I T()) T(I), : T(T(I) ) T(I) T().

The antipode of a classical Hopf algebra H is defined as a special endomorphism ofH. Sinceopmonoidal monads T relate two distinct functors it is not surprising that the notion of anantipode can not be transferred easily to this situation and the attempt to do so leads to anapparently complicated definition in [7, 3.3 and Remark 3.5]. Hereby the base categoryC is required to be autonomous.

2.3. Generalized bialgebras and Hopf operads. The generalised bialgebras over fieldsas defined in Loday [17, Section 2.1] are similar to the mixed bimodules (see 2.1): they arevector spaces which are modules over some operad A (Schur functors with multiplicationand unit) and comodules over some coalgebras Cc, which are linear duals of some operad C.


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Similar to the opmonoidal monads the coalgebraic structure is based on the tensor product(of vector spaces). The Hypothesis (H0) in [7] resembles the role of the entwining in2.1. The Hypothesis (H1) requires that the free A-algebra is a (CC, A)-bialgebra: this issimilar to the condition on an A-coring C, A an associative algebra, to have a C-comodulestructure (equivalently the existence of a group-like element, e.g. [8, 28.2]). The condition

(H2iso) plays the role of the canonical isomorphism defining Galois corings and the GaloisCoring Structure Theorem [8, 28.19] may be compared with the Rigidity Theorem [17, 2.3.7].The latter can be considered as a generalisation of the Hopf-Borel Theorem (see [17, 4.1.8])and of the Cartier-Milnor-Moore Theorem (see [17, 4.1.3]). In [17, 3.2], Hopf operads aredefined in the sense of Moerdijk [22] and thus the coalgebraic part is dependent on thetensor product. This is only a sketch of the similarities b etween Lodays setting and ourapproach here. It will be interesting to work out the relationship in more detail.

Similar to 2.1 we will also need the notion of mixed distributive laws from a comonad toa monad.

2.4. Entwining from comonad to monad. A natural transformation : GT T G is amixed distributive law from a comonad G to a monad T, also called an entwiningof G and

T, if the diagrams

GGe

~~|||||||| eG

!3hhh

hhhh

h GT

T 2fff

ffff

f /G T G

T~~||||||||

GT

/G T G , T

GT T

Gm

T/G T GTT/G T T G

mG

GGTG/G GT G

G/G T GG

GT

/G T G, GT

T

Oy

/G T G

T

Oy

are commutive.

For convenience we recall the distributive laws between two monads and between twocomonads (e.g. [2], [1], [33, 4.4 and 4.9]).

2.5. Monad distributive. Let F = (F,m,e) and T = (T, m, e) be monads on the categoryA. A natural transformation : F T T F is said to be monad distributive if it induces thecommutative diagrams

TeT

~~|||||||| Te

!3fff

ffff

f

F T

/G T F,

FFe

}}{{{{{{{{ eF

!3ggg

gggg

g

F T /G T F.

F F TmT /G

F

F T

F T FF /G T F F

Tm/G T F,

F T TFm /G

T

F T

T F TT/G T T F

mF /G T F.

In this case : F T T F induces a canonical monad structure on T F.

2.6. Comonad distributive. Let G = (G,,) and T = (T, , ) be comonads on thecategory A. A natural transformation : T G GT is said to be comonad distributive if itinduces the commutative diagrams


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T G

T !3ggg

gggg

g

/G GT

T}}{{{{{{{{

T ,

T G

G !3ggg

gggg

g

/G GT

G}}{{{{{{{{

G ,

T GT/G

T GGG /G GT G

G

GTT /G GGT,

T GG/G

T T GT/G T GT

T

GT

G /G GTT.

In this case : T G GT induces a canonical comonad structure on T G.

3. Actions on functors and Galois functors

The language of modules over rings can also be used to describe actions of monads on

functors. Doing this we define Galois functors and to characterise those we investigate therelationships between categories of relative injective objects.

3.1. T-actions on functors. Let A and B be categories. Given a monad T = (T , m , e) onA and any functor L : A B, we say that L is a (right) T-module if there exists a naturaltransformation L : LT L such that the diagrams

(3.1) L

eeee

eeee

eeee

eeeeLe /G LT

L

L,

LT TLm/G

LT

LT

L

LT L

/G L

commute. It is easy to see that (T, m) and (T T , T m) both are T-modules.Similarly, given a comonad G = (G,,) on A, a functor K : B A is a leftG-comoduleif there exists a natural transformation K : K GK for which the diagrams

K

gggg

gggg

gggg

gggg

K /G GK

K

K,

KK /G

K

GK

K

GKGK

/G GGK

commute.Given two T-modules (L, L), (L

, L), a natural transformation g : L L is called

T-linear if the diagram

(3.2) LTgT/G

L

LT

L

L g

/G L

commutes.

3.2. Lemma. Let (L, L) be aT-module. If f, f : T T L are T-linear morphisms from

the T-module (T T , T m) to the T-module (L, L) such that f T e = f T e, then f = f.


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Proof. Since f T e = f T e, we have L f T T eT = L fT TeT. Moreover, since f

and f are both T-linear, we have the commutative diagrams

T T T

Tm

fT /G LT

L

T T

f /G L,

T T T

Tm

fT/G LT

L

T T

f /G L.

Thus L f T = f T m and L fT = f T m, and we have f T m T eT = f T m T eT.

It follows - since T m T eT = 1 - that f = f.

3.3. Left G-comodule functors. Let G be a comonad on a category A, let UG : AG Abe the forgetful functor and write G : A AG for the cofree G-comodule functor. Fix afunctor F : B A, and consider a functor F : B AG making the diagram

(3.3) BF /G

F

1bb

bbbb

b AG

UG

~~}}}}}}}}

A

commutative. Then F(b) = (F(b), F(b)) for some F(b) : F(b) GF(b). Consider thenatural transformation

(3.4) F : F GF,

whose b-component is F(b). It should be pointed out that F makes F a left G-comodule,

and it is easy to see that there is a one to one correspondence between functors F : B AG

making the diagram (3.3) commute and natural transformations F : F GF making F aleft G-comodule.

The following is an immediate consequence of (the dual of) [10, Propositions II,1.1 and

II,1.4]:3.4. Theorem. Suppose that F has a right adjoint R : A B with unit : 1 RF andcounit : F R 1. Then the composite

tF : F RFR/G GF R

G /G G.

is a morphism from the comonad G = (F R , F R , ) generated by the adjunction , : F R : A B to the comonadG. Moreover, the assignment

F tF

yields a one to one correspondence between functors F : B AG making the diagram (3.3)commutative and morphisms of comonads tF : G

G.

3.5. Definition. We say that a left G-comodule F : B A with a right adjoint R : B A isG-Galois if the corresponding morphism tF : F R G of comonads on A is an isomorphism.

As an example, consider an A-coring C, A an associative ring, and any right C-comoduleP with S = EndC(P). Then there is a natural transformation

: HomA(P, ) SP A C

and P is called a Galois comodule provided X is an isomorphism for any right A-moduleX, that is, the functor SP : MS M

C is a A C-Galois comodule (see [32, Definiton4.1]).


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3.6. Right adjoint functor of F. When the category B has equalisers, the functor F hasa right adjoint, which can be described as follows: Writing R for the composite

RR /G RF R

RtF/G RG,

it is not hard to see that the equaliser (R, e) of the following diagram

RUGRUGG /G

RUG

/G RGUG = RUGGUG,

where G : 1 GUG is the unit of the adjunction UG G, is right adjoint to F.

3.7. Adjoints and monads. For categories A, B, let L : A B be a functor with rightadjoint R : B A. Let T = (T , m , e) be a monad on A and suppose there exists a functorR : B AT yielding the commutative diagram

BR /G

R 1ddddd

ddd AT

UT}}||||||||

A.

Then R(b) = (R(b), b) for some b : T R(b) R(b) and the collection {b, b B} con-stitutes a natural transformation R : T R R. It is proved in [10] that the naturaltransformation

tR : TT /G T RL

L /G RL

is a morphism of monads. By the dual of [21, Theorem 4.4], we obtain:

The functor R is an equivalence of categories iff the functor R is monadic and tR is anisomorphism of monads.

In view of the characterisation of Galois functors we have a closer look at some relatedclasses of relative injective objects.

Let F : B A be any functor. Recall (from [14]) that an object b B is said to beF-injective if for any diagram in B,

b1

g

f /G b2

hb

with F(f) a split monomorphism in A, there exists a morphism h : b2 b such that hf = g.We write Inj(F,B) for the full subcategory ofB with objects all F-injectives.

The following result from [26] will be needed.

3.8. Proposition. Let , : F R : A B be an adjunction. For any object b B, the following assertions are equivalent:

(a) b is F-injective;

(b) b is a coretract for some R(a), with a A;

(c) the b-component b : b RF(b) of is a split monomorphism.

3.9. Remark. For any a A, R(a) R(a) = 1 by one of the triangular identities for theadjunction F R. Thus, R(a) Inj(F,B) for all a A. Moreover, since the composite ofcoretracts is again a coretract, it follows from (b) that Inj(F,B) is closed under coretracts.


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3.10. Functor between injectives. Let KG : B AG be the comparison functor (nota-

tion as in 3.4). Ifb B is F-injective, then KG(b) = (F(b), F(b)) is UG-injective, since bythe fact that b is a split monomorphism in B, (G)G (b) = F(b) is a split monomorphism

in AG

(G as in 3.4). Thus the functor KG : B AG yields a functor

Inj(KG

) : Inj(F,B

) Inj(G

,AG

).When B has equalisers, this functor is an equivalence of categories (see [26]).

We shall henceforth assume that B has equalisers.

3.11. Proposition. The functor R : AG B restricts to a functor

R

: Inj(UG,AG) Inj(F,B).

Proof. Let (a, a) be an arbitrary object of Inj(UG,AG). Then, by Proposition 3.8,

there exists an object a0 A such that (a, a) is a coretraction of G(a0) = (G(a0), a0) in

AG, i.e., there exist morphisms

f : (a, a) (G(a0), a0) and g : (G(a0), a0) (a, a)

inAG

with gf = 1. Since f and g are morphisms inAG

, the diagram

G(a0)

g

(G)a0/G GG(a0)

G(g)

a

f

Oy

a

/G G(a)

G(f)

Oy

commutes. By naturality of R, the diagram

RG(a0)

R(g)

(R)G(a0) /G RGG(a0)

RG(g)

R(a)

R(f)

Oy

(R)a/G RG(a)

RG(f)

Oy

also commutes. Consider now the following commutative diagram

(3.5) R(a0)

a0/G RG(a0)

R(g)

(R)G(a0)/G

R((G)a0)/G RGG(a0)

RG(g)

R(a, a)

Oy

e(a,a)

/G R(a)

R(f)

Oy

(R)a /G

R(a)/G RG(a).

RG(f)

Oy

It is not hard to see that the top row of this diagram is a (split) equaliser (see, [12]),and since the bottom row is an equaliser by the very definition of e, it follows from thecommutativity of the diagram that R(a, a) is a coretract of R(a0), and thus is an object ofInj(F,B) (see Remark 3.9). It means that the functor R : AG B can be restricted to a

functor R

: Inj(UG,AG) Inj(F,B).

3.12. Proposition. Suppose that for any b B, (tF)F(b) is an isomorphism. Then the

functor F : B AG can be restricted to a functor

F

: Inj(F,B) Inj(UG,AG).


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Proof. Let denote the comultiplication in the comonad G (see 3.4), i.e., = FR.Then for any b B,

F(RF(b)) = AtF

(G

(U F(b))) = AtF

(F RF(b), F RF(b))

= AtF

(GF(b), F(b)) = (G

F(b), (tF)GF(b) F(b)).

Consider now the diagram

GF(b)(tF

)F(b) /G

F(b)

GF(b)

F(b)

GGF(b)

(1)

(tF)F(b).(tF)F(b)

'9xxxx

xxxx

xxxx

xxxx

xxxx

xxxx

(tF

)GF(b)

GG

F(b) G((tF)F(b)) /G GGF(b) ,

in which the triangle commutes by the definition of the composite (tF)F(b).(tF)F(b), whilethe diagram (1) commutes since tF is a morphism of comonads. The commutativity of

the outer diagram shows that (tF)F(b) is a morphism from the G-coalgebra F(RF(b)) =(GF(b), (tF)GF(b)

F(b)) to the G-coalgebra (GF(b), F(b)). Moreover, (tF)F(b) is an isomor-

phism by our assumption. Thus, for any b B, F(RF(b)) is isomorphic to the G-coalgebra(GF(b), F(b)), which is of course an object of the category Inj(U

G,AG). Now, since anyb Inj(F,B) is a coretract ofRF(b) (see Remark 3.9), and since any functor takes coretractsto coretracts, it follows that, for any b Inj(F,B), F(b) is a coretract of the G-coalgebra(GF(b), F(b)) Inj(U

G,AG), and thus is an object of the category Inj(UG,AG) again byRemark 3.9. This completes the proof.

The following technical observation is needed for the next proposition.

3.13. Lemma. Let , : W W : Y X be an adjunction of any categories. If i : x xand j : x x are morphisms inX such that ji = 1 and if x is an isomorphism, then xis also an isomorphism.

Proof. Since ji = 1, the diagram

xi /G x

1 /G

ij/G x

is a split equaliser. Then the diagram

WW(x) W

W(i) /G WW(x) 1 /GWW(ij)

/G WW(x)

is also a split equaliser. Now considering the following commutative diagram

x

x

i /G x

x

1 /G

ij/G x

x

WW(x)WW(i)

/G WW(x)1 /G

WW(ij)/G WW(x)


11/33


and recalling that the vertical two morphisms are both isomorphisms by assumption, we getthat the morphism x is also an isomorphism.

3.14. Proposition. In the situation of Proposition 3.12, Inj(F,B) is (isomorphic to) acoreflective subcategory of the category Inj(UG,AG).

Proof. By Proposition 3.11, the functor R restricts to a functor

R

: Inj(UG,AG) Inj(F,B),

while according to Proposition 3.12, the functor F restricts to a functor

F

: Inj(F,B) Inj(UG,AG).

Since

F is a left adjoint to R, Inj(F,B) is a full subcategory ofB, and Inj(UG,AG) is a full subcategory ofAG,

the functor F

is left adjoint to the functor R, and the unit : 1 R

F

of the adjunction

F

R

is the restriction of : F R to the subcategory Inj(F,B

), while the counit : F

R

1 of this adjunction is the restriction of : F R 1 to the subcategoryInj(UG,AG).

Next, since the top of the diagram 3.5 is a (split) equaliser, R(G(a0), a0) R(a0). Inparticular, taking (GF(b), F(b)), we see that

RF(b) R(GF(b), F(b)) = R F(U F(b)).

Thus, the RF(b)-component RF(b) of the unit : 1 R

F

of the adjunction F

R

is

an isomorphism. It now follows from Lemma 3.13 - since any b Inj(F,B) is a coretractionof RF(b) - that b is an isomorphism for all b Inj(F,B) proving that the unit

of the

adjunction F

R

is an isomorphism. Thus Inj(F,B) is (isomorphic to) a coreflectivesubcategory of the category Inj(UG,AG).

3.15. Corollary. In the situation of Proposition 3.12, suppose that each component of theunit : 1 RF is a split monomorphism. Then the category B is (isomorphic to) acoreflective subcategory of Inj(UG,AG).

Proof. When each component of the unit : 1 RF is a split monomorphism, it followsfrom Proposition 3.8 that every b B is F-injective; i.e. B = Inj(F,B). The assertion nowfollows from Proposition 3.14.

3.16. Characterisation of G-Galois comodules. Assume B to admit equalisers, let Gbe a comonad onA, and F : B A a functor with right adjoint R : A B. If there existsa functor F : A AG with UGF = F, then the following are equivalent:

(a) F is G-Galois, i.e. tF : G G is an isomorphism;(b) the following composite is an isomorphism:

F RGFR/G GUGF R = GF R

G/G G ;

(c) the functor F : B AG restricts to an equivalence of categories

Inj(F,B) Inj(UG,AG);

(d) for any (a, a) Inj(UG,AG), the (a, a)-component (a,a) of the counit of the

adjunction F R, is an isomorphism;


12/33


13/33


14/33


15/33


5. Antipode

We consider a bimonad H = (H,m,e,,,) on any catgeory A.

5.1. Canonical maps. Define the composites

(5.1) : HHH/G HHH Hm/G HH,

: HHH /G HHH

mH/G HH.

In the diagram

HHHHH /G

Hm

HHHHHmH /G

HHm

HHH

Hm

HH

H/G HHH

Hm/G HH,

the left square commutes by naturality of , while the right square commutes by associativityof m. From this we see that is left H-linear as a morphism from (HH,Hm) to itself. Asimilar diagram shows that is right H-linear as a morphism from (HH,mH) to itself.Moreover, in the diagram

HHe /G

HHH /G HHH

Hm

HH

HHe

6TllllllllllllllllllllllllllllllHH

the top triangle commutes by functoriality of composition, while the bottom triangle com-mutes because m He = 1. Drawing a similar diagram for H and mH, we obtain

(5.2) He = , eH = .

5.2. Definition. A natural transformation S : H H is said to be

a left antipode if m (SH) = e ;

a right antipode if m (HS) = e ;

an antipode if it is a left and a right antipode.

A bimonad H is said to be a Hopf monad provided it has an antipode.

Following the pattern of the proof of [8, 15.2] we obtain:

5.3. Proposition. We refer to the notation in 5.1.

(1) If has an H-linear left inverse, then H has a left antipode.

(2) If has an H-linear left inverse, then H has a right antipode.

Proof. (1) Suppose there exists an H-linear morphism : HH HH with = 1.Consider the composite

S : HHe /G HH

/G HHH /G H.


16/33


We claim that S is a left antipode of H. Indeed, in the diagram

H /G HH

HeH /G

HHH

(1)

H /G

Hm

HHH(2)

Hm

HH /G HH

m

HH /G HH H /G H ,

the triangle commutes since e is the unit for the monad H, rectangle (1) commutes byH-linearity of , and rectangle (2) commutes by naturality of . Thus

m SH = m HH H HeH = H ,

and using (5.2), we have

H = H He = H He = e .

Therefore S is a left antipode of H.

(2) Denoting the left inverse of by , it is shown along the same lines that S =H eH is a right antipode.

5.4. Lemma. Suppose that is an epimorphism. If f, g : H H are two natural transfor-mations such that

m f H = m gH or m Hf = m Hg ,

then f = g.

Proof. Assume m f H = m gH . Since He = by (5.2), we have

m f H He = m gH He,

and, since is also H-linear, it follows by Lemma 3.2 that

m f H = m gH .But is an epimorphism by our assumption, thus

m f H = m gH.

By naturality of e : 1 H, we have the commutative diagrams

H

He

f /G H

He

HHfH/G HH,

H

He

g /G H

He

HHgH/G HH.

Thus, since m He = 1,

f = m He f = m f H He = m gH He = m He g = g.

If m Hf = m Hg similar arguments apply.

5.5. Characterising Hopf monads. LetH = (H,m,e,,,) be a bimonad.

(1) The following are equivalent:

(a) = Hm H : HH HH is an isomorphism;

(b) = mH H : HH HH is an isomorphism;

(c) H has an antipode.


17/33


(2) If H has an antipode andA admits equalisers, then the comparison functor (see 4.3)

KH : A AHH()

makes A (isomorphic to) a coreflective subcategory of the categoryAHH().

Proof. (1) (c)(a) The proof for [21, Proposition 6.10] applies almost literally.(a)(c) Write : HH HH for the inverse of . Since is H-linear, it follows that

also is H-linear. Then, by Proposition 5.3, S = H He is a left antipode of H. Weshow that S is also a right antipode of H. In the diagram

H /G

!3fff

ffff

ffff

ffff

ffHH

(1)

H /G HHH

(2)

HSH /G HHH

(3)

mH /G

Hm

HH

m

HH

H


18/33


19/33


and (ii) is equivalent to the identities

(6.5) eH = He

(6.6) H = H

(6.7) H = H H H

(6.8) Hm = mH H H

6.2. -bimonad. Let : HH HH be a double entwining. Then H is called a -bimonadprovided the diagram

(6.9) HH

m /G H /G HH

HHHHHH

/G HHHH

mm

Oy

is commutative, that is

m = mm H H = Hm mHH H H HH H,and also the following diagrams commute

(6.10) HHH /G

m

H

H

/G 1,

1e /G

e

H

HeH

/G HH,

1e /G

=1ccc

cccc

c H

1.

6.3. Proposition. LetH be a -bimonad. Then the composite

: HHH/G HHH

H/G HHHmH/G HH

is a mixed distributive law from the monad H to the comonad H. Thus H is a bimonad (asin 4.1) with mixed distributive law .

Proof. We have to show that satisfies

(6.11) He = eH

(6.12) H = H

(6.13) H = H H H

(6.14) mH = Hm H H

Consider the diagram

H

(1)

eH/G

eH

HH

(2)

/G

eHH

HH

eH

ppp

pppp

pppp

pppp

ppp

pppp

pppp

pppp

pppp

pp

HHH

/G HHHH /G HHH

mH /G HH ,

which is commutative since square (1) commutes by (6.10); square (2) commutes by functo-riality of composition; the triangle commutes since e is the identity of the monad H. Thus eH = mH H H eH = eH, and (6.1) implies eH = He, showing (6.11).


20/33



HHH/G HHH

H/G

HH %7uuuu

uuuu

uu

HH

HHH(1)

mH/G

HH

HH

H

HH

H

/G H

HH

(2)

H

4Riiiiiiiiiiiiiiiiiiii

in which square (1) commutes because is a morphism of monads and thus m = H;the triangle commutes because of (6.2), diagram (2) commutes because of functoriality ofcomposition.

Thus H = H mH H H = H HH H = H, showing (6.12).Constructing suitable commutative diagram we can show

mH = mH H H mH= mH HHm HmHH HHH HHH HHH H,

Hm H H = Hm mHH H H H H HmH HH HH= mH HHm HmHH HHH HHH HHH H.

Comparing this two identities we get the condition (6.14).To show that (6.13) also holds, consider the diagram

HHHHH /G

HH

%QQQQ

QQQQQQQQQQQQQQ

QQQQ

(1)HHHH

HH/G

HHH

HHHHmHH /G

HHH

(3)HHH

HH

HHHHH(2)

HHH

HHHHH

HHH

mHHH/G

(4)

HHHH

HH

HH

/G

H

Oy

HHHH

HHH

>b|||||||||||||||||

HHHHH

HHH

=azzzzzzzzzzzzzzzzz

HHHHH

mmH )A

mHHH /G HHHH

HmH

HHH,

in which the triangles and diagrams (1) and (3) commute by functoriality of composition;diagram (2) commutes by (6.7); diagram (4) commutes by naturality of m.

Finally we construct the diagram

HH

H /G HHH(1)

HHttjjjjjjjjjj

jjjjjj

H /G HHH

(2)

HH

mH /G HH

H

HHHH

(3)

HH/G

HHH

HHHH

(4)HH

HH /G HHHH

HHH

HHH

HHHHHHHH

/G HHHHH

HHH *B

HHH /G HHHHH

HHH /G

(5)

HHHHH

mmH

Oy

HHHHH

HHH

4Rjjjjjjjjjjjjjjj


21/33


in which diagram (1) commutes by (6.3); diagram (2) commutes by (6.9) because HHHHH = H; the triangle and diagrams (3), (4) and (5) commute by functoriality of com-position.

It now follows from the commutativity of these diagrams that

H = H mH H H

= mmH HHH HHH HHH HHH

= (HmH HH HH) (mHH H H H H) H

= H H H.

Therefore satisfies the conditions (6.11)-(6.14) and hence is a mixed distributive law fromthe monad H to the comonad H.

6.4. Corollary. In the situation of the previous proposition, if(a, a) AH, then(H(a), H(a))

AH, where H(a) is the composite

H(a)H(a)/G HH(a)

H(a)/G HHH(a)Ha/G HHH(a)

mH(a)/G HH(a) .

Proof. Write H for the monad on the category AH that is the lifting ofH correspondingto the mixed distributive law . Since H(a) = a H(a), it follows that (H(a), H(a)) =H(a, a), and thus (H(a), H(a)) is an object of the category A

H.

6.5. -Bimodules. Given the conditions of Proposition 6.3, we have the commutativediagram (see (4.1))

HHm /G

H

H /G HH

HHHH

/G HHH,

Hm

Oy

and thus H is a bimonad by the entwining and the mixed bimodules are objects a in Awith a module structure ha : H(a) a and a comodule structure a : a H(A) with acommutative diagram

H(a)

H(a)

ha /G aa/G H(a)

HH(a)a /G HH(a).

H(ha)

Oy

By definition of , commutativity of this diagram is equivalent to the commtativity of

(6.15) H(a)H(a)

yrrrrrrr

rrr

ha /G aa /G H(a)

HH(a)

H(a) &8vvvv

vvvv

vvHH(a)

H(ha)fvvvv

vvvvvv

HHH(a)H(a)

/G HHH(a)

mH(a)

9Wrrrrrrrrrr.

A morphism f : (a, ha, a) (a, ha , a) is a morphism f : a a

such that f AH andf AH.

We denote the category AHH() by AHH.


22/33


6.6. Antipode of a -bimonad. LetH = (H,m,e,,) be a -bimonad with an antipodeS where : HH HH is a double entwining. Then

(6.16) S m = m SS and S = SS .

If HS = SH and SH = HS , then S : H H is a monad as well as a comonad

morphism.Proof. Since (H H , H H ,) is a comonad and (H,m,e) is a monad, the collection

Nat(HH,H) of all natural transformations from HH to H forms a semigroup with unite and with product

f g : HH /G HHHH

HH/G HHHHfg /G HH

m /G H .


HH

m

H

wwoooooo

ooooooo

/G HHHH

(2)

HH /G HHHH

mHH

H (1)

1ccc

cccc

cccc

cccc

ccc H

(@

HHHSHH

Hm

uukkkkkkk

kkkkkk

kk

HH

(3)

(4)

SH)A

HHH

Hm

I e/G H HHm

oo

in which the diagrams (1),(2) and (3) commute because H is a bimonad, while diagram (4)commutes by naturality. It follows that

m Hm SH H mHH H H = e H = e.

Thus S m = m1 in Nat(HH,H). Furthermore, by (a somewhat tedious) computation we

can show

m Hm HHS HSH H mHH H H = e H = e .

This shows that m SS = m1 in Nat(HH,H). Thus m SS = S m.

To prove the formula for the coproduct consider Nat(H,HH) as a monoid with unit ee and the convolution product for f, g Nat(H,HH) given by

f g : H /G HH

fH/G HHHHHg/G HHHH

mm /G HH .

By computation we get

( S) = eH e = ee , ( SS ) = He e = ee .

Thus ( S) = 1 and ( SS ) = 1, and hence S = SS .

Now assume HS = SH and SH = HS . Then we have

SS = SH HS = SH SH = HS SH = SS, thus

S m = m SS = m SS = m SS.

Moreover, since m He = 1, we have

S e = m He S enat= m SH He e

(6.10)= m SH e

antip.= e e

(6.10)= e .

Hence S is a monad morphism from (H,m,e) to (H, m , e).


23/33


For the coproduct, SS = SS implies

S = SS = SS = SS .

Furthermore,

S = S H

nat

= H SH

(6.10)

= m SH

antip.

= e

(6.10)

= .This shows that S is a comonad morphism from (H,,) to (H, , ).

It is readily checked that for a bimonad H, the composite HH is again a comonad as wellas a monad. However, the compatibility between these two structures needs an additionalproperty of the double entwining . This will also help to construct a bimonad oppositeto H.

6.7. Local prebraiding. Let : HH HH be a natural transformation. is said tosatisfy the Yang-Baxter equation (YB) if it induces commutativity of the diagram

HHHH/G

H

HHHH /G HHH

H

HHHH

/G HHHH

/G HHH .

is called a local prebraiding provided it is a double entwining (see 6.1) and satisfies theYang-Baxter equation.

6.8. Doubling a bimonad. Let H = (H,m,e,,) be a -bimonad where : HH HHis a local prebraiding. Then HH = (HH, m, e, , ) is a -bimonad with e = ee, = ,

m : HHHHHH/G HHHH

mm/G HH ,

: HH /G HHHH

HH/G HHHH

and double entwining

: HHHHHH/G HHHH

HH/G HHHHHH/G HHHH

HH/G HHHH .

Proof. We already know that (HH, m, e) is a monad and that (HH, , ) is a comonad.First we have to show that is a mixed distributive law from the monad (HH, m, e) to thecomonad(HH, , ), that is

HHe = eH H, H H = H H ,

HHm HH HH = mHH,

HH HH HH = H H .

The first two equalities can be verified by placing the composites in suitable commutative

diagrams. The second two identities are obtained by lengthy standard computations (asknown for classical Hopf algebras).

It remains to show that (HH, m, e, , ) satisfies the conditions of Definition 4.1 withrespect to . Again

m = H HH = HH, and

e = HHee He e = HHee

are shown by standard computations and

e = H eH e(4.2)

= e = 1.


24/33


To show that (HH, m, e, , , ) satisfies (4.1), consider the diagram

H4

(1)

H2H

HH/G H4

HH2

'9xxxxxxx

xxxx

xxmH

2

/G H3

HH '9xxxxxxx

xxxx

xxHm /G H2

(3)

H /G H3

H2

H5

(4)

(2)

H3

'9xxxx

xxxx

xxxx

x H4H

2

m

7Uooooooooooooo

H3

H5

(5)

H4

HH2

/G H5

(6)

H2H

7U

H4

H3

/G H5

(7)H

4

H2H

/G H5(8)

HmH2

7U

H4

H5(9)

(10)

H2H

2

'9yyyy

yyyy

yyyy

y H4HH /G H4

H5(11)

(12)

HH2

/G

H3m

Oy

H5

H3m

Oy

H6

H3H

HH3

/G H6

(14)H

3H

(13)

H4

/G H6

(15)H

3H

H2H

2

/G H6

HmH3

@d

H3H

/G H6HmH

3

7Uooooooooooooo

H2H

2

/G H6

(16)

HH3

/G H6

H2mH

2

Oy

H6HH

3

/G H6H

4

/G H6H

2H

2

/G H6

H3H

>b}}}}}}}}}}}}}}}}}

HH3

/G H6,

H3H

>b}}}}}}}}}}}}}}}}}

in which diagram (1) commutes because is a mixed distributive law and thus

H H H = H ;

the diagrams (2) and (9) commute by (4.1); the diagrams (3)-(8), (10), (11), (13), (14) and(16) commute by naturality; diagram (12) commutes because is a mixed distributive law(hence Hm H H = mH); diagram (15) commutes by 6.7. By commutativity of thewhole diagram,

m = H H H2 H Hm mH2 H H

= H2m H2mH2 H3 H H H3 H2 H2 H4 H H3 H3 H H4 H2H

= HH HH HHm,

and hence HH = (HH, m, e, , ) is a -bimonad.

6.9. Opposite monad and comonad. Let : HH HH be a natural transformationsatisfying the Yang-Baxter equation.

(1) If (H,m,e) is a monad and is monad distributive, then (H, m , e) is also a monadand is monad distributive for it.

(2) If (H,,) is a comonad and is comonad distributive, then (H, , ) is also acomonad and is comonad distributive for it.


25/33


Proof. (1) To show that m is associative construct the diagram

HHHH /G

H

(1)

HHHmH /G

H (2)

HH

HHHH

HHH

H /G

Hm

(3)

HHHH/G HHH

Hm /G

mH

(4)

HH

m

HH

/G HHm /G H,

where the rectangle (1) is commutative by the YB-condition, (2) and (3) are commutativeby the monad distributivity of , and the square (4) is commutative by associativity of m.Now commutativity of the outer diagram shows associativity of m .

From 2.5 we know that eH = He and He = eH and this implies that e is also theunit for (H, m , e).

The two pentagons for monad distributivity of for (H, m m, e) can be read from theabove diagram by combining the two top rectangles as well as the two left hand rectangles.

(2) The proof is dual to the proof of (1).

6.10. Opposite bimonad. Let : HH HH be a local prebraiding with 2 = I and letH = (H,m,e,,) be a -bimonad onA. Then:

(1) H = (H, m , e , , ) is also a -bimonad.

(2) If H has an antipode S with HS = SH and SH = HS , then S is a-bimonad morphism between the -bimonads H and H.

In this case S is an antipode for H.

Proof. (1) By (1), (2) in 6.9, is a (co)monad distributive law from the (co)monad Hto the (co)monad H, and e = e = 1 by (6.10). Moreover,

m = m (6.10)

= H 2.4= H = H = H, and

e = e(6.10)

= eH e2.1= He e = eH e = eH e.

To prove compatibility for H we have to show the commutativity of the diagram

(6.17) HHm /G

H /G HH

HHHHHH

/G HHHH.

mm

Oy

For this standard computations (from Hopf algebras) apply.(2) By 6.6, S is a -bimonad morphism from the -bimonad H to the -bimonad H. To

show that S is an antipode for H we need the equalities

m SH = e = e and m HS = e = e .

Since SH = HS , we have

m SH = m SH = m HS 2=1

= m HS = e .

Since HS = SH , we have

m HS = m HS = m SH 2=1

= m SH = e .


26/33


As we have seen in Theorem 5.6, the existence of an antipode for a bimonad H on acategory A is equivalent to the comparison functor being an equivalence provided A isCauchy complete and H preserves colimits. Given a local prebraiding the latter conditionon H can be replaced by conditions on the antipode (compare [3, Theorem 3.4], [4, Lemma4.2] for the situation in braided monoidal category).

6.11. Antipode and equivalence. Let : HH HH be a local prebraiding and letH = (H,m,e,,) be a -bimonad on a categoryA in which idempotents split. Consider thecategory of bimodules

AHH = A

HH(),

where = mH H H (see 6.3, 6.5).IfH has an antipode S such that SH = HS andHS = SH, then the comparison

functor KH : A AHH is an equivalence of categories.

Proof. We know that the functor KH has a right adjoint if for each (a, ha, a) AHH,

the equaliser of the (a, ha, a)component of the pair of functors

(6.18) UHUbH

UHUbHe b

H /G

UHUbH

/G UHHU

bH = UHU

bH

bHU

bH

exists. Here e bH

: 1 bHU

bH is the unit of the adjunction U

bH

bH and UH is the composite

UHeHUH /G UHHUH

UH (tKH ) /G UHH.

Using the fact that for any (a, ha) AH,

(tKH )(a,ha) = H(ha) a and

H(ha) a ea = H(ha) H(ea) ea = ea,

it is not hard to show that the (a, ha, a)-component of Diagram 6.18 is the pair

aa

/Gea /G

H(a).

Thus, KH has a right adjoint if for each (a, Ha, a) AHH, the equaliser of the pair of

morphisms (ea, a) exists.Suppose now that H has an antipode S : H H. For each (a, ha, a) A

HH, consider the

composite qa = ha Sa a : a a. By a (tedious) standard computation - applying 6.15,

6.6, 2.4 - one can showea qa = a qa and qa qa = qa.

6.12. Remark. Dually, one can prove that for each (a, ha, a) AHH, qa a = qa ha, thus

ia qa a = ia qa ha, and since ia is a (split) monomorphism, it follows that qa a = qa ha.

Since idempotents split in A, there exist morphisms ia : a a and qa : a a such thatqa ia = 1a and ia qa = qa. Since qa is a (split) epimorphism and since ea ia qa = ea qa =a qa = ia qa, it follows that

(6.19) ea ia = a ia.


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Using this equality it is straightforward to show that the diagram

(6.20) aia

/G aqa

ttea

/G

a

/G H(a)

haSass

is a split equaliser. Hence for any (a, ha, a) AHH, the equaliser of the pair of morphisms(ea, a) exists, which implies that the functor KH has a right adjoint RH : A

HH A which

is given byRH(a, Ha, a) = a.

Since for any (a, ha, a) AHH,

a ea = eH(a) ea and a ea = 1 by 6.2; H(a) a = 1, since (H,,) is a comonad; ea a = H(a) eH(a) by naturality,

we get a split equaliser diagram

a ea /G H(a)

a

ss eH(a) /Ga

/G H2(a)

H(a)rr

.

This is preserved by any functor, and since RH(H(a), ma, a) is the equaliser of the pair ofmorphisms (eH(a), a), in particular a RH(H(a), ma, a) = RH(KH(a)). Thus RHKH 1.

For any (a, ha, a) AHH, write a for the composite ha H(ia) : H(a) a. We claim that

a is a morphism in AHH from KH(a) = (H(a), ma, a) to (a, ha, a). Indeed, we have

a ma = ha H(ia) ma

naturality = ha ma H2(ia)

(a, ha) AH = ha H(ha) H2(ia) = ha H(H(ha) ia) = ha H(a),

and this just means that a is a morphism in AH from (H(a), ma) to (a, ha).

Next - using 6.15, 6.19 - we computea a = H(a) a.

Thus, a is a morphism in AH from (H(a), a) to (a, a), and hence a is a morphism in

AHH from KH(a) = (a, ma, a) to (a, ha, a).Similarly it is proved that the composite a = H(qa) a : a H(a) is a morphism in

AH from (a, ha, a) to (H(a), ma, a) and a further calculation yields

a a = 1a and a a = 1H(a).

Hence we have proved that for any (a, ha, a) AHH, a is an isomorphism in A

HH, and

using the fact that the same argument as in Remark 2.4 in [12] shows that a is the counitof the adjunction KH RH, one concludes that KHRH 1. Thus the functor KH is an

equivalence of categories. This completes the proof. For an example, let V = (V, , I , ) be a braided monoidal category and H = (H,m,e,,)

a bialgebra in V. Then

(H , m , e , , , = H,H )

is a bimonad on V, and it is easy to see that the category VHH of Hopf modules is just the

category VHH() = VHH.

6.13. Theorem. LetV = (V, , I , ) be a braided monoidal category such that idempotentssplit inV. Then for any bialgebra H = (H,m,e,,) in V, the following are equivalent:


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(a) H has an antipode;

(b) the comparison functor

KH : V VHH, V (H V, m V, V), f H f,

is an equivalence of categories.

7. Adjoints of bimonads

This section deals with the transfer of properties of monads and comonads to adjoint(endo-)functors. The relevance of this interplay was already observed by Eilenberg andMoore in [11]. An effective formalism to handle this was developed for adjunctions in 2-categories and is nicely presented in Kelly and Street [16]. For our purpose we only needthis for the 2-category of categories and for convenience we recall the basic facts of thissituation here.

7.1. Adjunctions. Let L : A B, R : B A be an adjoint pair of functors with unitand counit , , and L : A B, R : B A be an adjoint pair of functors with unit andcounit , . Given any functors F : A A and G : B B, there is a bijection betweennatural transformations

: LF GL and : F R RG

where is obtained as the composite

F RFR RLF R

RR RGLR

RG RG,

and is given as the composite

LFLF LF RL

LL LRGL

GF GL.

In this situation, and are called mates under the given adjunction and this is denotedby a . It is nicely displayed in the diagram

AL /G

F

BR /G

G

A

Fz||

|||||

|||||||

A

:f}}}}}}}

}}}}}}}

L/GB

R/GA.

Given further(i) adjunctions L : C A, R : A C and L : C A, R : A C and a functor

H : C C, or(ii) an adjunction L : A B, R : B A and functors F : A A and

G : B B, we get the diagram

CL /G

H

AL /G

F

BR /G

G

A

Fy{{

{{{{{{

{{{{{{{{

R /G C

y{{{{{{{

{{{{{{{

H

C

R/G

:f|||||||

|||||||

A

9e||||||||

||||||||

L/G

F

BR

/G

G

A

F

y||||||||

|||||||| R

/GC

AL

/G

:f}}}}}}}

}}}}}}}

BR

/G A,


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yielding the mates

(M1) LFFF

GLFG GGL FF G

F FRG

G RGG,

(M2) LLHL LFL

L LGL HRR

G RRG

R RRG.

7.2. Properties of mates. Let L, L : A B be functors with right adjoints R, R,respectively, and : L L a natural transformation.

(i) If L : A B is a functor with right adjoint R and : L L a natural transfor-mation, then

.

(ii) If L : C A is a functor with right adjoint R, then

(L : LL LL) (R : RR RR).

(iii) If Lo : B C is a functor with right adjoint Ro, then

(Lo : LoL LoL) (Ro : RRo RRo).

Proof. (i) is a special case of 7.1(M1).(ii) follows from 7.1(M2) by putting A = A, B = B, C = C and H = H.(iii) is derived by applying 7.1 to the diagram

AL /G B

Lo /G CRo /G B

I{

R /G A

z~~~~~~~

~~~~~~~

AL/G

;g

B

I

;g

Lo/G C

Ro/G B

R/G A.

As observed by Eilenberg and Moore in [11, Section 3], for a left adjoint endofunctor which

is a monad, the right adjoint (if it exists) is a comonad (and vice versa). The techniquesoutlined above provide a convenient and effective way to describe this transition and toprove related properties. Recall that for any endofunctor L : A A with right adjoint R,for a positive integer n, the powers Ln have the right adjoints Rn.

7.3. Adjoints of monads and comonads. Let L : A A be an endofunctor with rightadjoint R.

(1) If L = (L, mL, eL) is a monad, then R = (R, R, R) is a comonad, where R, R arethe mates of mL, eL in the diagrams

AL /G A

R /G A

R

{~~~~~~~

~

~~~~~~

A

eL

;g

I/G A

I/G A,

AL /G A

R /G A

Rz~~~~~~~

~

~~~~~~

A

mL

;g

HH

/G ARR

/G A.

(2) If L = (L, L, L) is a comonad, then R = (R, mR, eR) is a monad where mR, eR arethe mates of L, L in the diagrams

AI /G A

I /G A

eR{~~~~~~~

~~~~~~~

A

L

;g

L/G A

R/G A,

ALL/G A

RR/G A

mRz~~~~~~~

~~~~~~~

A

L

;g

L/G A

R/G A.


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Proof. (1) Since eL R and mL R, it follows from 7.2 (ii) and (iii) that

LeL RR, eLL RR, mLL RR, LmL RR.

Applying 7.2 (i) now yields

mL LeL RR R, mL eLL RR R,mL mLL RR R, mL LmL RR R.

Since L is a monad we have mL eLL = mL LeL = I and mL mLL = mL LmL, implying

RR R = RR R = I and RR R = RR R.

This shows that R = (R, R, R) is a comonad.

The proof of (2) is similar.

The methods under consideration also apply to the natural transformations LL LLwhich were basic for the definition and investigation of bimonads in previous sections. Thefollowing results were obtained in cooperation with Gabriella Bohm and Tomasz Brzezinski.

7.4. Adjointness and distributive laws. Let L : A A be an endofunctor with rightadjoint R and a natural transformation L : LL LL. This yields a mate R : RR RRin the diagram

ALL/G A

RR/G A

R{

A

L

;g

LL/G A

RR/G A

with the following properties:

(1) LL RR and LL RR.

(2) L satisfies the Yang-Baxter equation if and only if R does.

(3) 2L = I if and only if 2R = I.(4) If L = (L, mL, eL) is a monad and L is monad distributive, then R is comonad

distributive for the comonad R = (R, R, R).

(5) If L = (L, L, L) is a comonad and L is comonad distributive, then R is monaddistributive for the comonad R = (R, mR, eR).

Proof. (1) follows from 7.2, (ii) and (iii). The remaining assertions follow by (1) andthe identities in the proof of 7.3.

Recall from Definition 4.1 that a bimonadH is a monad and a comonad with compatibilityconditions involving an entwining H : HH HH.

7.5. Adjoints of bimonads. Let H be a monad H = (H, mH, eH) and a comonadH = (H, H, H) on the category A. Then a right adjoint R of H induces a monadR = (R, mR, eR) and a comonad R = (R, R, R) (see 7.3) and

(1) H = (H, H) is a bimonad with entwining H : HH HH if and only if R = (R, R)is a bimonad with entwining R : RR RR.

(2) H = (H, H) is a bimonad with entwining H : HH HH if and only if R = (R, R)is a bimonad with entwining R : RR RR.

(3) If H = (H , H , H) is a bimonad with antipode, then R = (R,R,R) is a bimonadwith antipode (Hopf monad).


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Proof. (1) With arguments similar to those in the proof of 7.4 we get that R is anentwining from R to R. It remains to show the properties required in Definition 4.1. From7.2(i) we know that

H HH eRR eR, H mH R eR,H eH R mR, eHH eH R RR, H eH R eR.

Thus the equalities

H HH = H mH, H eH = HH eH, H eH = I

hold if and only if

eRR eR = R eR, R mR = R RR, R eR = I.

The transfer of the compatibility between product and coproduct 4.1 is seen from thecorresponding diagrams

HHmH /G

HH

HH /G HH

HHHHH

/G HHH,

HmH

Oy RR RRoo RR

mRoo

RR

RRR

mRR

Oy

RRR.RR

oo

The proof of (2) is similar.(3) By 5.5, the existence of an antipode is equivalent to the bijectivity of the morphism

H = HmH HH : HH HH.

Since HH RmR and HmH RR, H is an isomorphism if and only if R = RmR RRis an isomorphism.

Functors with right (resp. left) adjoints preserve colimits (resp. limits) and thus 5.6 and7.5 imply:

7.6. Hopf monads with adjoints. Assume the category A to admit limits or colimits.

Let H = (H, mH, eH, H, H, H) be a bimonad on A with a right adjoint bimonad R =(R, mR, eR, R, R, R). Then the following are equivalent:

(a) the comparison functor KH : A AHH(H) is an equivalence;

(b) the comparison functor KR : A ARR(R) is an equivalence;

(c) H has an antipode;

(d) R has an antipode.

Finally we observe that local prebraidings are also tranferred to the adjoint functor.

7.7. Adjointness of -bimonads. Let H be a monad H = (H, mH, eH) and a comonadH = (H, H, H) on the categoryA with a right adjoint R.

If H = (H, H) is a bimonad with double entwining H : HH HH, then R = (R, R) isa bimonad with double entwining R : RR RR.

Moreover, H satisfies the Yang-Baxter equation if and only if so does R.

Proof. Most of the assertions follow immediately from 7.4 and 7.5.It remains to verify the compatibility condition 6.9. For this observe that from 7.2(i) we

getHH mHmH, HHH RRR, mHmH RR,

and hence

mHmH HH HH mRmR RRR RR and H mM R mR.


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It follows that H satifies 6.9 if and only if so does R.

7.8. Dual Hopf algebras. Let B be a module over a commutative ring R. B is a Hopfalgebra if and only if the endofunctor B R on the catgeory of R-modules is a Hopfmonad. By 7.5, B R is a bimonad (with antipode) if and only if its right adjoint functor

HomR(B, ) is a bimonad (with antipode). This situation is considered in more detail in[5].If B is finitely generated and projective as an R-module and B = HomR(B, R), then

HomR(B, ) B R and we obtain the familiar result that B is a Hopf algebra if and

only if B is.

7.9. Characterisations of groups. For any set G, the endofunctor G : Set Setis a Hopf bimonad on the category of sets if and only if G has a group structure (e.g. [33,5.20]). Since the functor Map(G, ) is right adjoint to G , it follows from 7.6 that a setG is a group if and only if the functor Map(G, ) : Set Set is a Hopf monad.

Acknowledgements. The authors want to express their thanks to Gabriella Bohm

and Tomasz Brzezinski for inspiring discussions and helpful comments. The research wasstarted during a visit of the first author at the Department of Mathematics at the HeinrichHeine University of Dusseldorf supported by the German Research Foundation (DFG). Heis grateful to his hosts for the warm hospitality and to the DFG for the financial help.

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Addresses:Razmadze Mathematical Institute, Tbilisi 0193, Republic of Georgia

[email protected]

Department of Mathematics of HHU, 40225 Dusseldorf, [email protected]
http://arxiv.org/abs/math/0406310http://arxiv.org/abs/math/0406310

bimonads and hopf monads on categories

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