Galois Theory for Corings and Comodules

Transcription

1 FACULTEIT WETENSCHAPPEN DEPARTEMENT WISKUNDE Galois Theory for Corings and Comodules Proefschrift voorgelegd aan de Faculteit Wetenschappen, voor het verkrijgen van de graad van Doctor in de Wetenschappen door Joost Vercruysse Promotor: Prof. S. Caenepeel Academiejaar

2

3 Whenever there is existence, There shall be co-existence...

4

5 Preface This work is written as a PhD thesis under the supervision of Professor Stefaan Caenepeel. The thesis was defended on March 9, 2007 at the Vrije Universiteit Brussel (Free University of Brussels, VUB). The members of the jury were Tomasz Brzeziński, Stefaan Caenepeel, Eric Jespers, Rudger Kieboom (secretary), Claudia Menini, Michel Van den Bergh (president) and Yinhuo Zhang. I would like all of them for the effort they have put into this job. The present version of this thesis is slightly different from the one orriginally presented on March 9. Some typograffical and mathematical inconsistencies have been cured. In Chapter 4, some of the structure theorems have been reformulated in a way that makes the flatness condition on the coring less prominent. Let me thank the members of the jury of their useful suggestions and as well Jawad Abuhlail for his remark on local projectivity for modules generated by countable sets (see Remark 2.55). Since without any doubt many mistakes and misprints did still survive endless proofreading, all corrections and remarks are welcomed at jvercruy@vub.ac.be. Brussel/Gent, April iii

6

7 Ik ben blij dat jij in mijn team zit. Guido Pallemans, Eilandbewoner Acknowledgements A thousand words of gratitude for... Jan Van Geel, my supervisor at Universiteit Gent during my licentiaatsthesis. He was the first who learnt me about (classical) Galois theory and in this way guided me into a fascinating algebraic world. He also informed me about an open position at the VUB for which I applied. So, without him, this thesis would have never existed for different reasons. Stef Caenepeel, my current supervisor. Beside the mathematics he learnt me, it was a great pleasure to join him during conferences and foreign visits, especially when the bottles at the conference dinner turned out to be too small. Gabriella (Gabi) Böhm and José (Pépé) Gómez-Torrecillas, with whom I had and have the chance to collaborate. I want to thank them for inviting me twice to their home institute in Budapest and Granada, where I had very pleasing stays. I learnt a lot from these collaborations and their contribution to this work is of great value. Laiachi El Kaoutit, Miodrag Iovanov and Shuanhong Wang, my other collaborators. Beside their contribution to this work I am grateful to them for the pleasant moments when they visited Brussels, and to Mio for the hospitality in Bucharest. Erwin De Groot and Kris Janssen, my friends and roommates during the working days at VUB and even longer during conferences. I enjoyed a lot to talk with them about mathematics, play frozen bubble or laugh with things that should not be mentioned here. Philippe Cara, one of my other nice colleagues, who is always willing to help with any computer- or L A TEX-problem. My dear parents, for their continuous support. Finally..., special thanks to my best friends Annelies, Ilse, Stijn, Jarno, Thomas and of course most especially to Carolien, for all the fine moments when mathematics was far away. v

8

9 Table Of Contents Acknowledgement v Notational conventions xiii Introduction 1 Part I : Algebraic and Categorical Constructions 1. Algebras in Monoidal Categories and Bicategories 9 2. Local Projectivity versus local algebraic Structures Corings and Comodules 73 Part II : Galois Theory 5. Galois Comodules Morita Theory for Corings Cleft Bicomodules 167 Part III : Frobenius and Separable Functors 8. Separable Functors and relative Cohomology Co-Frobenius Corings and related Functors Applications to Galois Theory 235 Appendix A. Nederlandse samenvatting 249 vii

10

11 Contents Preface Acknowledgements Table Of Contents Notational Conventions iii v vii xiii Introduction 1 Chapter 1. Algebras in Monoidal Categories and Bicategories Bicategories and monoidal categories Bicategories Bicategories versus 2-categories Monoidal categories Monads and algebras Monads, algebras and modules Comonads, coalgebras and comodules Bicategories of bimodules Adjoint pairs and Morita contexts The bicategories of Eilenberg-Moore objects Locally finite duality The bicategory of bi(co)modules Enriched bicategories 31 References 34 Chapter 2. Local Projectivity versus local algebraic Structures Colimits and split direct systems Non-unital rings The Dorroh extension Firm rings Rings with local units Rings with idempotent local units The M-adic and finite topology Projectivity Firmly projective modules Weakly locally projective modules Strongly locally projective modules Local projectivity versus local units Representations of rings with local units Local structure maps Corings with local comultiplications Corings with local counits Rings with local multiplication 70 References 72 Chapter 3. Corings and Comodules Basic properties of corings and comodules Definitions 73 ix

12 x CONTENTS Adjunctions The Dorroh coring Corings and entwining Structures Entwining structures and entwined modules Factorizable enwining structures Rational modules R-rational modules The rational functor Comatrix corings Comatrix corings over firm rings Locally finite duality Comatrix corings and cotriples coming from adjunctions Corings from colimits Corings from colimits Colimit comatrix corings Factorizing split direct systems 107 References 110 Chapter 4. Galois Comodules Introduction : Motivating problems The descent problem Hopf-Galois theory Galois comodules Comonadic-Galois comodules The canonical cotriple morphism Firm Galois comodules Comonadic-Galois versus firm Galois comodules Structure theorems Applications Galois theory in bicategories: a unifying approach Push-out and pull-back functors Weak and strong structure theorems examples 142 References 144 Chapter 5. Morita Theory for Corings Finite Galois theory and Morita theory Finite Galois theory Morita theory in the bicategory of bimodules The dual of the canonical map Morita contexts associated to a comodule The *-Morita context associated to a comodule The Morita context associated to a comodule Morita contexts associated to a bimodule Structure theorems Application: Morita contexts associated to a grouplike element Grouplike characters Grouplike elements 164 References 166 Chapter 6. Cleft Bicomodules Some remarks on notation and coring extensions Morita theory for coring extensions Weak and strong structure theorems Cleft bicomodules Examples 179

13 CONTENTS xi Cleft entwining structures Cleft extensions of algebras by a coalgebra Cleft extensions of algebras by a Hopf algebra and the fundamental theorem Cleft weak entwining structures Cleft extensions by partial group actions Cleft entwining structures over arbitrary base Cleft extensions of algebras by a Hopf algebroid Cleft factorization structures 187 References 189 Chapter 7. Separable Functors and relative Cohomology Separability and relative injectivity Bicomodules and separability Coderivations and cointegrations Cohomology for bicomodules Applications Coseparable corings Coseparable coalgebra co-extensions 207 References 210 Chapter 8. Co-Frobenius Corings and related Functors Elementary results Direct sums and direct products Frobenius corings Morita contexts Locally adjoint functors Action of a set of natural transformations on a category Locally adjoint functors The induction functor Adjunctions Description of sets of natural transformations The Yoneda-approach The coproduct functor Characterizations of co-frobenius and quasi-co-frobenius corings Locally Frobenius corings Characterization of Frobenius corings Quasi-co-Frobenius corings and related functors 228 References 233 Chapter 9. Applications to Galois Theory Separable corings and Galois comodules Frobenius properties and Morita theory for comodules Frobenius corings Co-Frobenius corings The coring as a Galois comodule The coring as a finite Galois comodule The coring as an infinite Galois comodule: rationality properties The coring as an infinite Galois comodule: quasi co-frobenius corings 246 References 247 Appendix A. Nederlandse Samenvatting Ringen en coringen Comatrix coringen en Galois comodulen Separabiliteitseigenschappen en Frobenius coringen 251 Bibliography 253

14 xii CONTENTS Index 256

15 Willen we dat afspreken? Lydia Protut Notational Conventions Rings and corings. We will usually denote by k a commutative ring, it will only be a field if it is explicitly mentioned. Capitals A, B, A, B,.. or R, R, S, S will be used for associative rings (k-algebras). Mostly we will preserve capitals A, B for rings with unit (i.e. an element 1 such that 1a = a1 for all a in the algebra) and R, S for firm rings. However we mention explicitly what is the case as much as possible. Capitals C, D are also used to denote coalgebras (with commutative basering). This is in contrast to corings, which are denoted by gothic capitals C, D,... Modules and comodules, whether over algebras, coalgebras or corings, are always denoted by capitals M, N, P,.... By Σ we will denote a comodule whose Galois theory is studied. Categories and functors. We denote categories with capitals in calligraphic font, A, B, C, M. In particular, the category of firm left A-modules, firm right B-modules and firm A-B bimodules over rings A and B are respecively denoted by A M, M B and A M B. The category of arbitrary left A-modules, right B-modules and A-B bimodules over rings A and B are respecively denoteted by A M, MB, A MB. In the same way, the categories of left C-comodules, right D-comodules and C-D bicomodules over corings C and D are denoted by C M, M D and C M D. Let C and D be two categories. A functor F : C D will be the name for a covariant functor, it will only be a contravariant functor if it is explicitly mentioned. Consider two functors F, G : C D. The class of all natural transformations between F and G is denoted by Nat(F, G). Finally, 1 denotes the discrete one-object category, i.e. 1 is the category with one object that we will denote by and with only one morphism, the identity on. Morphisms. Let X and Y be two objects of a category C then Hom C (X, Y ) is the notation for all morphisms in C from X to Y. In particular, for rings A and B, we denote by A Hom(X, Y ), Hom B (X, Y ) and A Hom B (X, Y ) the sets of left A-linear, right B-linear and A-B bilinear maps between modules X and Y. Similarly, for corings C and D, we denote by C Hom(X, Y ), Hom D (X, Y ), C Hom D (X, Y ) are sets of left C, right D and C-D colinear morphisms between X and Y. For an object X in a category C, X will also be the name for the identity morphism on X. In some cases, where this would cause too much confusion or to not overload the notation, we will denote the identity on X by 1 X or just by 1. xiii

16

17 Get on with it! Monthy Python, Search for the Holy Grail Introduction Recall the sentence that is stated on the first page of this book : Whenever there is existence, there shall be co-existence (this quote is due to Raimundas Vidunas, a former visitor at the Vrije Universiteit Brussel). In mathematics, especially for category theorists, this expression is an evidence, since every mathematical idea, can be dualized, i.e. every construction in a category A can be performed in the opposite category A op, where arrows are reversed. This principle is standing at the origin of the notion of a coalgebra (co-algebra). A k-algebra over a commutative base ring k is a monoid in the monoidal category M k. Hence a coalgebra is a monoid in the category M op k. In the same way, A-rings (being monoids in the monoidal category of bimodules over a not necessarily commutative base ring A) can be dualized to corings and modules dualize to comodules. Although corings are just the dual notion of the well-understood rings, the whole theory of corings cannot be directly obtained from the theory of rings, by use of a general duality principle. The reason is simple: by dualizing rings to corings, we dualize in the first place the category of modules and this category is not self-dual. A first difference between rings and corings that can be easily understood from the definition is the following. For every ring R, the ring of integers Z can play the role of base ring for R, since Z R is a ring morphism. For corings there does not exist such a universal base ring, nor does there exist a universal coring Ω which allows a coring morphism C Ω, for all corings C. The main goal of this work is to study the relation between the world and the co-world. Explicitly, we want to examine functors between categories of modules and categories of comodules (over corings). Module categories have been studied for decades, therefore they are considered interesting for numerous reasons. Coring theory has not such a long standing tradition, but still corings and comodules are known to be relevant in several subfields of abstract algebra and even non-commutative geometry. Corings were defined by Sweedler [109] in 1975, who introduced these structures in order to formulate a theorem, which in a dual form gives rise to the Jacobson- Bourbaki Theorem. However, because of a lack of examples during the next 25 years, only about five articles appeared that really dealt with corings (e.g. [76], [83], [91]). Corings surfaced for a moment in the beginning of the eighties, under the name bocs, in the work of Kleiner [82] and Rojter [101]. Corings became almost forgotten, but this situation came to an abrupt end just before the end of the second millenium, when Takeuchi observed that new examples of corings can be constructed out of entwining structures. Takeuchi did not publish this observation, but we can read in a mathematical review by Masuoka [110] the following. The following observation due to Takeuchi helps our understanding of an entwining structure (A, C) ψ : ψ makes A C into a right A-module such that given the obvious left A-module structure and the comultiplication and the counit arising from those of C, it forms an A-coring, and a right (A, C) ψ -module is identified naturally with a right A C-comodule. In his well-known article Structure of Corings: Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Brzeziński explores this idea further and in this way the observation of Takeuchi, together with Brzeziński s paper gave rise to a renewed interest in corings 1

18 2 INTRODUCTION that is still expanding today. The main reason why the connection with entwining structures makes corings so attractive, is that it turned out that entwining structures and entwined modules provide a lucid framework to study all types of Hopf modules that were constructed during the last decades of the 20th century, such as Yetter-Drinfeld modules, Long dimodules, graded modules, Doi-Hopf modules. In fact, by passing from entwining structures and entwined modules to corings and comodules, although it turned out that most structural theorems could be preserved, calculations became easier and more transparent and many results could be clarified as they were presented in a broader perspective. The thesis is devided into three parts, each part consists of three chapters. In the first part, we create the algebraic and categorical tools that are needed in the two other parts. Part II, is the heart of the thesis and deals with Galois theory. In the last part, we study some special types of functors between module and comodule categories and their implications to Galois theory. We start by recalling in Chapter 1 the definitions of algebras, coalgebras and their representations. We prefer to give a general treatment in bicategories, since this turns out to be the right perspective for Galois theory as well (see below). Chapter 2, is devoted to some ring theoretic aspects. In particular, we are interested in modules that are not finitely generated, but that still satisfy a certain projectivity condition. There is a close connection between these notions of local projectivity and rings with local units (remark that we use the term (local) unit for elements e in a ring R that satisfy the relation r e = r for a set of elements r R, and not for invertible elements e in R). In the literature, several notions of local projectivity have appeared. A first notion is due to Zimmermann-Huisgen [119], and is equivalent to the so-called α-condition. We will call modules that satisfy this kind of local projectivity weakly locally projective modules. A second notion (which we will call strongly local projectivity) is due to Abrams [1], and is related to rings with idempotent local units by Morita theory (see [7]). In Chapter 2, we further investigate the relations between local projectivity and local units. We introduce an even more general notion of projectivity wich we call firm projectivity and which is related to firm rings. Our main observation is that firmly projective modules are in bijective correspondence with pairs of adjoint functors between categories of modules over firm rings. If we specify the firm base ring to a ring with local units, we obtain a module that satisfies the corresponding type of projecitivity. If the ring has a unit, then the module is finitely generated and projective. In Chapter 3, the last Chapter of Part I, we make a basic study of corings and comodules. We develop in this chapter all tools from the general theory of corings that we will need in Parts II and III. We explain in more detail the relation between corings and entwining structures, that we already mentioned before. We develop the theory of rational modules, which will be important if we study Galois theory of corings by means of Morita theory in Chapter 5. The main part of Chapter 3 is devoted to the construction of comatrix corings. The first type of comatrix corings was introduced in [63]. Starting from a B-A bimodule that is finitely generated and projective as right A-module El Kaoutit and Gómez-Torrecillas constructed an A-coring Σ B Σ that generalizes the canonical coring that appeared in Sweedler s original paper [109]. Although it seemed from their construction that it is necessary that Σ is finitely generated and projective, several attempts were made to drop the finiteness condition in this construction. In [64] El Kaoutit and Gómez-Torrecillas construct a type of infinite comatrix corings that are built from a direct sum of B-A bimodules that are finitely generated and projective as right A-modules. In [113], the author constructed corings with local comultiplications, starting from a B-A bimodule that is locally projective as right A-module. Combining the techniques of both approaches, Gómez-Torrecillas and the author constructed corings out of firmly projective modules. This approach generalizes the construction of [64] and has been specified in [41] to a construction of corings out of colimits in a joint work with Caenepeel and De Groot. Part II deals with the main subject of this work: Galois theory for Corings and Comodules. As mentioned at the beginning, Galois theory describes properties of functors between categories of modules and categories of comodules, i.e. it relates the objects of Chapter 2 with the objects of Chapter 3, using the language we introduced in Chapter 1. In Chapter 4, we give a general treatment of the theory of Galois comodules. This theory originated from the classical work on Galois extensions of commutative fields. This theory has been extended to a Galois theory for

19 INTRODUCTION 3 commutative rings by Auslander and Goldman [10] and by Chase, Harrison and Rosenberg [50]. A group action can be generalized to a Hopf algebra (co)action. This leads to the Hopf-Galois theory, developed first for finitely generated and projective Hopf algebras (see [51] and [85]) and later for arbitrary Hopf algebras (see [58] and [104]). As we explained, corings provide a general framework to explain many results of Hopf algebra theory in a simple and clarifying way. In this repect, it is no surprise that Hopf-Galois theory has a formulation in terms of corings. This was shown in [28], where a Galois theory is developed for corings with a grouplike element. To a ring morphism i : B A, we can associate an A-coring, the so-called canonical Sweedler coring. A morphism from this coring to another A-coring C is completely determined by a grouplike element g C. If this morphism is an isomorphism, we say that (C, g) is a Galois coring. We can construct a pair of adjoint functors between the categories M B and M C and formulate sufficient and necessary conditions for this pair to be an equivalence of categories. The development of Galois theory in terms of corings provides as well an elegant formulation of descent theory. El Kaoutit and Gómez- Torrecillas [63] introduced a yet more general version of Galois theory, replacing the grouplike element by a right C-comodule Σ that is finitely generated and projective as right A-module. They construct an A-coring out of Σ, called the comatrix coring. We call Σ a Galois comodule if the canonical coring morphism from the comatrix coring to C is bijective (see also [29], [42]). There exists an adjunction between the categories M T and M C, where T = End C (Σ). Several attempts were made to drop or weaken the finiteness condition on Σ and construct infinite versions of Galois theory for comodules. El Kaoutit and Gómez-Torrecillas introduced Galois comodules that are (infinite) direct sums of finitely generated and projective right A-modules [64], Caenepeel, De Groot and the author generalized this to a method to construct corings out of colimits and Gómez- Torrecillas and the author developed a theory of Galois comodules over firm rings and Wisbauer introduced a functorial definition for a Galois comodule [117]. In Chapter 4, we discuss these different theories and their relation. During the last years, other versions and generalizations of Hopf-Galois theory have been formulated (see [31], [118], [35]) and it became clear that the proper framework to study all these theories in a general way, is the framework of bicategories. Therefore, we develop in the second section of Chapter 4, a Galois theory in general bicategories and reduce other Galois theories as special cases. This general formulation allows us as well to clarify the relation between Galois theory for corings and the theory of (co)tripleability of functors developed by Beck. In Chapter 5 we treat some special cases of Galois theory. If the coring satisfies a finiteness condition (whether the coring is finitely generated and projective over its base ring or the coring is locally projective over its base ring such that the rational part of the dual ring is dense with respect to the finite topology), the category of comodules can be described as a category of modules. Hence, Galois theory, handling about functors between the category of comodules and a category of modules, reduces to Morita theory, which studies functors between two module categories. The Morita theory that we develop generalizes a long standing tradition of applying Morita theory in Hopf-Galois theory (see [53], [57]). We extend the Morita theory of Chapter 5 further in Chapter 6, where we associate a Morita context to any coring extension (D : L) of (C : A) and an L-C bicomodule. If D = L = k is the trival k-coring, then this Morita context reduces to the one from Chapter 5. The new Morita context allows us to develop the theory of cleft bicomodules, which unifies all previously known notions of cleft extensions. Cleft bicomodules provide Galois comodules for which the associated pair of adjoint functors between the category of comodules and the category of modules over the endomorphism ring of the Galois comodule is always an equivalence. This generalizes the fundamental theorem for Hopf algebras, which states that the category of Hopf modules over a Hopf k-algebra H is equivalent to the category of modules over the base ring k. Part III is devoted to the study of separable and Frobenius functors. Both separablity and Frobenius properties have interesing implications for Galois theory. In the first two chapters of Part III we develop the theory of separable Frobenius, co-frobenius and quasi co-frobenius corings in a functorial way. In the last chapter we discuss their implications on the Galois theory of comodules. In [80] Jonah studied the second and the third cohomology groups of coalgebras defined in a, not necessarily Abelian, multiplicative category (see also [9]). Kleiner gave in [84] a cohomological

20 4 INTRODUCTION characterization of separable algebras using integrations. Another approach via derivations was given by Barr and Rinehart in [12]. This last one has been dualized to the case of coseparable coalgebras by Doi [56]. Nakajima [94] showed that Doi s result can be extended to coalgebra extensions (or coextensions) with a cocommutative base coalgebra. In [76], Guzman used Jonah s methods to generalize Doi s characterization for corings over an arbitrary base ring and unified this with a dualization of Kleiner s approach of cointegrations. This gives rise to a nice characterization of coseparable corings in terms of cohomology, derived functors and both cointegrations and coderivations. Unfortunately this last characterization cannot be applied to coalgebra coextensions, and Nakajima s results are not recovered. The common framework behind Guzman s and Nakajima s approach is the fact that both coseparable corings and coseparable coalgebra coextensions can be interpreted as cotriples (comonads) with a separable forgetful functor (in the sense of [95]). In all situations discussed before, the multiplicative base category was additive with cokernels and arbitrary direct sums, and the (co)triple functor preserved cokernels and direct sums. In Chapter 7 we will approach the problem from this cotriple point of view. We work with a cotriple over a Grothendieck category (not necessarily multiplicative) whose underlying functor fits in the above mentioned class of functors. These functors were studied in relation with corings in [69]. We will present a generalization of Guzman s characterization in this situation, and as a particular application we also give, under different assumptions, Nakajima s result. Frobenius and co-frobenius coalgebras and Hopf algebras, Frobenius ring extensions and Frobenius bimodules have been intensively studied over the last decades. In [32], [33] the close relation between Frobenius extensions, Frobenius bimodules and Frobenius corings is discussed. Although the name indicates differently, the co-frobenius property of a coring is a weakening and not a dualization of the Frobenius property. In particular, although the Frobenius property is left-right symmetric, the co-frobenius property is not. Nevertheless, coalgebras over a base field which are at the same time left and right co-frobenius can be understood as a dual version of Frobenius algebras. Indeed, for a Frobenius k-algebra A, the functors Hom A (, A) and Hom k (, k) from M A to A M are naturally isomorphic (see [54]), for a left and right co-frobenius k-coalgebra C, the functors Hom C (, C ) and Hom k (, k) from M C to C M are naturally isomorphic (see [78]). Frobenius corings have a very nice characterization in terms of Frobenius functors. This result says that a coring C is Frobenius if and only if the forgetful functor F : M C M A is at the same time a left and right adjoint of the induction functor A C. An overview of most results with regard to this subject can be found in [46]. A similar categorical interpretation of (quasi )co-frobenius coalgebras and corings remained somewhat mysterious. In Chapter 8, we will provide this categorical description of quasi co-frobenius corings and we will generalize some results of [78]. For this reason, we construct several Morita contexts. Starting from the observation (see [21, Remark 3.2]) that a Morita context can be identified with a (k-linear) category with two objects, we construct a Morita context relating a coring C with its dual C. This context describes the Frobenius property of the coring. More precisely, if there exists a pair of invertible elements in this Morita context, then the coring is exactly a Frobenius coring. A similar Morita context relates representable functors, such as those used in [78] and [54] to describe (co-)frobenius properties. A last type of Morita contexts, that is constructed in a different way, describes the adjuction property of a pair of functors. More precisely, if there exists a pair of invertible elements in this Morita context, then the pair of functors is exactly an adjoint pair. By relating these Morita contexts with (iso)morphisms of Morita contexts, we recover the result that a coring is Frobenius if and only if the forgetful functor and the induction functor make up a Frobenius pair if and only if certain representable functors are isomorphic. The advantage of our presentation is that it clarifies underlying relations between the different equivalent descriptions of the Frobenius property, and that these relations can be described even if the coring is not Frobenius. In particular, using these Morita contexts, we can formulate a categorical interpretation of (quasi )co-frobenius corings. In Chapter 9, the final chapter of this thesis, we study how Galois theory of a coring is affected by separibility and Frobenius properties of corings. In Hopf-Galois theory, which can be generalized to corings as we already discussed, there exists a stronger version of the usual structure

21 INTRODUCTION 5 theorem, known as Scheider s Theorem I. It says that under certain conditions, the surjectivity of the canonical map is sufficient to obtain its bijectivity and even an equivalence of categories. We discuss some generalizations of these theorems in the context of firm Galois comodules in Section 9.1, where we make use of the characterization of separable corings. The Morita context that we introduce in Chapter 5 is a generalization of a Morita context introduced by Doi [57]. Morita contexts similar to the one of Doi were studied by Cohen, Fischman and Montgomery in [52] and [53]. These are different from the one of Doi, in the sense that the two connecting modules in the context are equal to the underlying algebra A. On the other hand, they are more restrictive, in the sense that they only work for finite dimensional Hopf algebras over a field (see [52]) or Frobenius Hopf algebras over a commutative ring (see [53]). In Section 9.2, we study the Morita context associated to a Frobenius coring with a fixed comodule Σ. It turns out that the connecting modules in the context are then precisely Σ and its right dual Σ ; in the case where Σ = A, the two connecting modules are isomorphic to A. This clarifies the relationship between the Morita contexts of [57] on the one hand and [52] and [53] on the other hand. Weaker results are obtained in the situation where C is co-frobenius. If the coring satisfies some finiteness conditions, then we can study its Galois properties as comodule over itself. This allows us to obtain a formulation, by means of Galois theory, of the equivalence between the category of comodules over a coring and the category of modules over the dual of the coring, if the coring is finitely generated and projective over its base ring and between the category of comodules over the coring and the category of modules over the rational part of the dual if the coring is locally projective over its base ring. Another very interesting result is a characterization of quasi co-frobenius corings by means of Galois theory.

22

23 The Question is: What is a Mannahmannah?. - The Question is: Who cares?. Statler & Waldorf IPart I : Categorical and Algebraic Constructions

24

25 Chapter 1 Algebras in Monoidal Categories and Bicategories In this Chapter we introduce the basic algebraic framework that will be used throughout this book. In the first Section we study bicategories, monoidal categories and the way their structure is related. We give a proof for the well-known result among category theorists that any bicategory is equivalent to a 2-category. In Section 1.2, we show how the notion of an algebra arises naturally in this general setting and coalgebras are defined in a dual way. In Section 1.3 we compare several possible definitions of a bicategory of bimodules associated to a given bicategory. In the final Section, we discuss some special features of enriched bicategories, such as the construction of the dual monad of a comonad and an interesting relation between Morita contexts and adjoint pairs. For a general introduction to the theory of categories and functors we refer to the classics [25] and [88] Bicategories and monoidal categories Bicategories. Let us first recall the following terminology. A Hom-Class category is a category A with a class of objects and such that for any two objects A, B A, all morphisms between A and B, constitute a class Hom A (A, B). In a Hom-Set category we make the restriction that Hom A (A, B) is a set for all A, B A. Finally, a small is a Hom-Set category that contains only a set of objects. To avoid set-theoretical problems we will try to omit Hom-Class categories as most as possible. For this reason, if we speak about a category, then we will mean a Hom-Set category, unless it is mentioned differently. A bicategory B consists of the following data. (i) A class of objects A, B,... which are called 0-cells (or objects). (ii) For every two objects A and B, there exists a category Hom B (A, B) = Hom(A, B), whose class of objects we denote by Hom 1 (A, B) and which are called 1-cells. We denote f : A B for a 1-cell f Hom 1 (A, B). Take two 1-cells f, g Hom 1 (A, B). The set of morphisms from f to g in the category Hom(A, B) is denoted by A Hom B 2 (f, g). We call these morphisms 2-cells and denote them as α : f g. We will denote the composition of morphisms in the category Hom(A, B) by, i.e. for all f, g, h Hom 1 (A, B) such that α : f g and β : g h, we have β α : f h. This composition will now be called the vertical composition of 2-cells. (iii) For any three objects A, B, C B, there exist a functor c ABC : Hom(A, B) Hom(B, C) Hom(A, C). 9

26 10 CHAPTER 1. ALGEBRAS IN MONOIDAL CATEGORIES AND BICATEGORIES For all f Hom 1 (A, B) and g Hom 1 (B, C), we denote c ABC (f, g) = f B g Hom 1 (A, C). For all α A Hom B 2 (f, g) and β B Hom C 2 (h, k), we denote c ABC (α, β) = α B β : f B h g B k. This composition will be called the horizontal composition of 2-cells. (iv) For any object A B, there exists a functor 11 A : 1 Hom(A, A), where 1 denotes the discrete category with 1 object. We will denote 11 A ( ) just by 11 A. (v) For any four objects A, B, C, D B, there exists a natural isomorphism (1) α ABCD : c ACD (c ABC Hom(C, D)) c ABD (Hom(A, B) c BCD ). (2) For any two objects A, B B, there exist two natural isomorphisms λ AB : Hom(A, B) c AAB (11 A Hom(A, B)), ρ AB : Hom(A, B) c ABB (Hom(A, B) 11 B ). For all compatibility conditions we refer to e.g. [25, section 7.7] or [16], where the notion of a bicategory was introduced. For all objects A, B, C B, we obtain from the functorality of c ABC the interchange law, i.e. (3) (α B β) (γ B δ) = (α γ) B (β δ), for α A Hom B 2 (a, c), β B Hom C 2 (b, d), γ A Hom B 2 (c, e) and δ B Hom C 2 (d, f). From (3) one immediately deduces that for all α A Hom B 2 (a, c) and β B Hom C 2 (b, d), (4) (α B b) (c B β) = α β = (a B β) (α B d). A 2-category is a bicategory such that the isomorphisms α ABCD, λ AB and ρ AB are identities for all choices of A, B, C, D. In particular, 11 A = A for all objects A of a 2-category. To any bicategory B one can associate new bicategories denoted by B op, B co and B coop. These are constructed by taking respectively opposite composition for the 1-cells, for vertical composition of 2-cells and for both. Examples 1.1. (1) Let A be an ordinary category. Then A is also a bicategory, if we consider Hom A (A, B) as a discrete category for any two objects A, B A. In this way A is even a 2-category. (2) The second basic example is the bicategory Cat of small categories, functors and natural transformations. This is again a 2-category. Remark our convention to write the composition of 1-cells in a covariant way. This has the important implication if we compute the compostion of functors and the horizontal composition of natural transformations in Cat. Let A, B and C be categories and F : A B and G : B C functors. Then we will denote (5) F B G = GF : A C for the composite functor. In the same way, for categories A, B and C, functors F, G : A B and H, K : B C and natural transformations α : F G and β : H K, we will denote (6) α B β = βα : HF KG, where the right hand side is the Godement product of natural transformations. (3) If F and G are two functors between the Hom-Set categories A and B, then Nat(F, G) is in general no longer a set but a class. Consequently, if we consider a 2-category whose 0-cells are all Hom-Set categories, 1-cells are functors and 2-cells are natural transformations, then the Hom-category between two 0-cells in this 2-category is a Hom- Class category. We will denote this 2-category by CAT. (4) A particulary interesting sub-2-category of CAT is constructed as follows. The 0-cells consist of all Grothendieck categories and the 1-cells are so-called right continuous functors, i.e. functors that preserve colimits. By [49, Lemma 5.1] the Hom-categories of this 2-category are Hom-Set categories.

27 1.1. BICATEGORIES AND MONOIDAL CATEGORIES 11 We will give more examples of bicategories in the following sections. Let A and B be two bicategories. A lax Functor F : A B consists of the following data: (i) for every 0-cell A A, a 0-cell F A B; (ii) for every pair of 0-cells A, B A a functor F AB : Hom A (A, B) Hom B (F A, F B); (iii) for every triple of 0-cells A, B, C A, a natural transformation (7) γ ABC : c F A,F B,F C (F AB F BC ) F AC c ABC ; (iv) for every 0-cell A A, a natural transformation (8) δ A : 11 F A F AA 11 A ; such that the following diagram commutes for all objects A, B, C, D in B, (9) (F AB (f) F B F BC (g)) F C F CD (h) α F A,F B,F C,F D F AB (f) F B (F BC (g) F C F CD (h)) γ 2 γ 1 F AD ((f B g) C h) F AD (α ABCD ) F AD (f B (g C h)) where γ 1 = γ ACD (γ ABC F C F CD (h)) and γ 2 = γ ABD (F AB (f) F B γ BCD ). For the other coherence axioms that γ and δ have to satisfy, we refer to [25, section 7.5]. If all γ ABC and δ A are natural isomorphisms, then F is called a pseudo-functor. If F : A B is a lax functor, and P a property of functors, then we say that F is locally P, if F AB is P for all A, B A. In this way we can speak about locally an equivalence, locally faithful,.... Let us introduce the following notation. For a 1-cell f : A B in a bicategory A, we find a functor f : Hom A (C, A) Hom(C, B). Consider two lax functors F, G : A B. A lax natural transformation α : F G consists of the following data: (i) for every 0-cell A A, a 1-cell α A : F A GA; (ii) for every pair of objects A, B A, a natural transformation τ AB : (α A ) G AB α B F AB ; where α and τ satisfy coherence axioms for which we refer to [25, section 7.5]. When F and G are pseudo functors, and each τ AB is a natural isomorphism, then α is called a pseudo natural transformation. Consider two lax natural transformations α, β : F G between the lax functor F, G : A B. A modification Ξ : α β is a family Ξ A : α A β A of 2-cells in B, indexed by 0-cells of A. We require this family to satisfy the following property: for every pair of 1-cells f, g : A A in A and every 2-cell σ : f g the equality Ξ A GA Gσ = F σ F A Ξ A holds in B. With the above definitions, we can construct new bicategories. Let A and B be two bicategories, then we denote by Lax(A, B) the bicategory whose 0-cells are lax functors, 1-cells are lax natural transformations and 2-cells are modifications. Considering pseudo functors, pseudo natural transformations and modifications we obtain a sub-bicategory of Lax(A, B), that is denoted as [A, B]. If B is a 2-category, then Lax(A, B) and [A, B] are 2-categories as well. In general, the Hom-categories of Lax(A, B) and [A, B] can be Hom-Class categories, even if the Hom-categories of A and B are Hom-Set categories. Examples 1.2. pseudo functor (i) Let B be any bicategory and Ω a 0-cell in B. We can construct the (10) Rep Ω : B CAT, as follows.

28 12 CHAPTER 1. ALGEBRAS IN MONOIDAL CATEGORIES AND BICATEGORIES - For a 0-cell A B, we define Rep Ω (A) = Hom(Ω, A); - for a 1-cell a : X Y in B, define Rep Ω (a) = X a : Hom(Ω, X) Hom(Ω, Y ) (x : Ω X) (x X a : Ω Y ) (ϕ : x x ) (ϕ X a : x X a x X a) - for a 2-cell α X Hom Y 2 (a, a ), where a, a Hom 1 (X, Y ), define a Rep Ω (α) = α Nat( X a, X a ) as follows α x = x X α : x X a x X a, α ϕ = ϕ X α : f X a g X a ; for all x Hom 1 (Ω, X) and ϕ Ω Hom A 2 (f, g). Dually, we can define a pseudofunctor ΩRep : B op CAT, by Ω Rep(A) = Hom(A, Ω), Ω Rep(a) = a Y and Ω Rep(α) = α Y. (ii) Given a 1-cell f : Ω Ω we can construct a pseudo natural transformations Rep f = f : Rep Ω Rep Ω ; f Rep = f : Ω Rep Ω Rep. (iii) Starting from a 2-cell α Ω Hom Ω 2 (f, g), we obtain modifications Rep α = α : Rep f Rep g ; αrep = α : f Rep g Rep. Let A and B be two 0-cells in a bicategory B. We call A and B internally equivalent in B, if there exist 1-cells f Hom 1 (A, B) and g Hom 1 (B, A) and two isomorphisms η A Hom A 2 (A, f B g) and ε B Hom B 2 (g A f, B). Let A and B be two bicategories. A bi-equivalence from A to B consists of a pair of pseudo functors F : A B and G : B A together with an internal equivalence between 1 and G F in [A, A] and an internal equivalence between F G and 1 in [B, B]. In other words, there must exist pseudo natural transformations α : 1 G F, β : G F 1, γ : F G 1 and δ : 1 F G and modifications 1 = β α, α β = 1, 1 = δ γ and γ δ = 1. An equivalent condition for a bi-equivalence is the following. A pseudo functor F : A B is a bi-equivalence if and only if F is locally an equivalence and surjective up-to-equivalence on objects. This latter condition means that for any 0-cell B B we can find a 0-cell A A such that F A and B are internally equivalent in B Bicategories versus 2-categories. Computations in a bicategory B can become quite hard because of the presence of the associativity isomorphisms α ABCD and the unit isomorphism λ AB and ρ AB. Since in a 2-category A these isomorphisms are just the identities, computations in A are much simpler. For this reason we would like to find a way to tranfer the more complicated computations in B to the easier setting of A. A well-known result that belongs to the folklore among category-theorists tells that this is always possible (see e.g. [108], [89]). For sake of completeness, we provide now a full prove of this construction. Let F : B A be a locally faithful pseudo functor. Suppose we want to verify an equation of 2-cells in B of the form (11) α 1 α ABCD = α 2, where α 1 A Hom D 2 ((f B g) C h, p), α 2 A Hom D 2 (f B (g C h), p), with f Hom 1 (A, B), g Hom 1 (B, C), h Hom 1 (C, D) and p Hom 1 (A, D). Apply the pseudo functor F on this equation, then we obtain the following diagram in the 2-category A, (12) (F AB (f) F B F BC (g)) F C F CD (h) F AB (f) F B (F BC (g) F C F CD (h)) F AD (α 2 ) γ 2 F AD ((f B g) C h) F AD (p) F AD (α ABCD ) γ F AD (α 1 ) 1 F AD (f B (g C h))

29 1.1. BICATEGORIES AND MONOIDAL CATEGORIES 13 where γ 1 and γ 2 are isomorphisms given by compositions of the natural isomorphisms γ from (7). The inner quadrangle is commutative by the coherence axiom on γ, see (9). Since A is a 2-category, γ1 1 F AD (α ABCD ) γ 2 is just the identity. Denote β 1 = F AD (α 1 ) γ 1 and β 2 = F AD (α 2 ) γ 2. Then, the diagram (12) commutes if and only if β 1 = β 2 if and only if F AD (α 1 ) = F AD (α ABCD ) F AD (α 2 ). Since F is locally faithful, in particular F AD is a faithful functor. This implies that if F AD (α 1 ) = F AD (α ABCD ) F AD (α 2 ), if and only if α 1 = α ABCD α 2. So (11) is satisfied in B if and only if β 1 = β 2 commutes in A, i.e. the equivalent condition in the 2-category holds true. We can follow the same reasoning for any diagram of 2-cells that is constructed in B. This justifies the following statement. Proposition 1.3. Every calculation that holds in 2-categories can be repeated in a bicategory for which there exists a locally faithful pseudo functor to a 2-category. Consider the following sub-2-category Cat(B) of CAT - The 0-cells are of the form Hom(A, B) with A and B 0-cells in B; - the 1-cells are functors of the form Rep Ω (a) = a : Hom(Ω, A) Hom(Ω, B), with a Hom 1 (A, B) as in Example 1.2(i). - the 2-cells Rep Ω (a) Rep Ω (a ) are natural transformations in Nat( a, a ) of the form Rep Ω (α) where α X Hom Y 2 (a, a ). In particular, we see that if the 2-cells between two given 1-cells determine a set in B, the same holds in Cat(B), i.e. the Hom-categories in Cat(B) are Hom-Set categories if and only if Homcategories in B are Hom-Set categories. Now we can construct the bicategory [B, Cat(B)] which consists of pseudo-functors, pseudo natural transformations and modifications from B to Cat(B). From the construction of Cat(B) it is clear that the representable pseudo functors, pseudo natural transformations and modifications as defined in Example 1.2 are in [B, Cat(B)]. Now define Rep : B [B, Cat(B)] A Rep(A) = Rep A a Rep(a) = Rep a α Rep(α) = Rep α for any 0-cell A, 1-cell a and 2-cell α in B, where we used again the notation of Example 1.2. Lemma 1.4. With notation as above, Rep : B op [B, Cat(B)] is a locally faithful pseudo functor. Proof. Let us first check that Rep is a pseudo functor. By definition of the representables in Example 1.2, we see that Rep is well-defined as a pseudo functor B op [B, CatB] once we have found natural isomorphisms γ ABC and δ A as in (7) and (8). Take any A, B, C B, f Hom 1 (A, B) and g Hom 1 (B, C). Consider Rep f, Rep g and Rep f B g. For any X B and h Hom 1 (X, A) we obtain from (1) a natural isomorphism α XABC : Rep f g (h) = (f g) h f (g h) = Rep f (Rep g (h)). In the same way, for any A B we find for all X B and all f Hom 1 (X, A) that λ AB (2) induces a natural isomorphism λ AB : Rep 11A (f) = 11 A A f f. We conclude that Rep is a pseudo functor. Recall that a pseudo functor F : B A is faithful if and only if the usual functor F AB : Hom B (A, B) Hom A (F A, F B) is faithful for all objects A and B in B. Thus we have to check whether the map A Hom B 2 (f, g) Mod(Rep f, Rep g ) is injective, where Mod denotes the class of all modifications from Rep f to Rep g. Take any 2-cell α : f g in B. If we apply the pseudo functor Rep, we obtain the modification Rep α : Rep f Rep g. By defintion Rep α consists of a family of natural transformations α X : f X g X

30 14 CHAPTER 1. ALGEBRAS IN MONOIDAL CATEGORIES AND BICATEGORIES indexed by the 0-cells X B, where f X and g X denote functors from the category Hom(X, B) to Hom(X, A). Consider now α, β A Hom B 2 (f, g) and suppose that Rep α = Rep β. Then we find in particular by taking X = B above two times the same natural transformation α B = β B : f B g B. Evaluating this equality in the identity morphism on B, we obtain α B B = β B B. Consider now the natural isomorphism ρ (2) from the definition of a bicategory. The naturality of ρ implies the commutativity of the following diagram α f g ρ AB (f) ρ AB (g) f B B α B B g B B Since ρ is an isomorphism, we thus obtain We can conlude that Rep is locally faithful. α = ρ 1 AB (g) (α BB) ρ AB (f) = ρ 1 AB (g) (β BB) ρ AB (f) = β As a final step in our construction, we consider the sub-2-category Rep(B) of [B, Cat(B)] consisting of the full image of the pseudo functor Rep. This means Rep(B) can be describes as follows - The 0-cells are pseudo functors of the form Rep A, where A is a 0-cell in B; - the 1-cells are pseudo natural transformations of the form Rep a : Rep B Rep A, where a Hom(A, B) is a 1-cell in B - the 2-cells are modifications of the form Rep α : Rep a Rep a, where α A Hom B 2 (a, a ) is a 2-cell in B. Theorem 1.5. Let B be any bicategory and Rep(B) the 2-category associated to B constructed as above. Then B and Rep(B) op are bi-equivalent. Proof. By the construction of Rep(B), the pseudofunctor Rep can be corestricted to a pseudo functor B op Rep(B). Since Rep is locally faithful, the corestriction is locally faithful as well. Moreover, by construction, the corestricted functor is locally full and surjective on 0-cells. We conclude that B op and Rep(B) are bi-equivalent, and thus B and Rep(B) op are bi-equivalent as well. Corollary 1.6. All calculations that hold in 2-categories can be repeated in bicategories. In particular, all diagrams that are constructed out of the associativity and identity isomorphisms commute in any bicategory. (This last statement is known as the Coherence Theorem.) Proof. The first statement is a consequence of Proposition 1.3 and Theorem 1.5. The second statement follows from the first one and the fact that diagrams built out of associativity and identity isomorphisms commute obviously in a 2-category. Consequently, from now on, we will rely on the 2-categorical calculus when we are dealing with bicategories. Remark 1.7. The associativity of the composition of 1-cells and the horizontal composition of 2-cells in Rep(B) op should be understood as follows. We relate to any 1-cell f Hom 1 (A, B) in B a functor Rep f = f and functors associate in a trivial way. This trivial associativity can be thought of as pre-ordening all brackets in B. Indeed : ((Rep h Rep g ) Rep f )(a) = (Rep h (Rep g Rep f ))(a) = h C (g B (f A a)) for all 1-cells f Hom 1 (A, B), g Hom 1 (B, C), h Hom 1 (C, D) and a Hom 1 (Ω, A) in B.