Hopf algebroids, Hopf categories, and their Galois theories

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1 opf algebroids, opf categories, and their Galois theories Clarisson izzie Canlubo University of Copenhagen ariv: v1 [math.qa] 19 Dec 2016 December 20, 2016 Abstract opf algebroids are generalization of opf algebras over non-commutative base rings. It consists of a left- and a right-bialgebroid structure related by a map called the antipode. owever, if the base ring of a opf algebroid is commutative one does not necessarily have a opf algebra. Meanwhile, a opf category is the categorification of a opf algebra. It consists of a category enriched over a braided monoidal category such that every hom-set carries a coalgebra structure together with an antipode functor. In this article, we will introduce the notion of a topological opf category a small category whose set of objects carries a topology and whose categorical structure maps are sufficiently continuous. The main result of this paper is to describe the relation between finitely-generated projective opf algebroids over commutative unital C -algebras and topological coupled opf categories of finitetype whose space of objects is compact and ausdorff. We will accomplish this by using methods in algebraic geometry and spectral theory. astly, we will show that not only the two objects are tightly related, but so are their respective Galois theories. Mathematics Subject Classification (2010: 16T05, 14A20, 18F99, 18B40, 58B34 Keywords: opf algebroid, opf category, Galois theory. Contents 1 Introduction 2 2 opf algebroids Definitions Examples epresentation of opf algebroids opf categories Definitions and properties A good example of a opf category Galois extensions of opf categories

2 1 INTODUCTION 2 4 The category associated to a opf algebroid ocal eigenspace decomposition The geometry of C(-ring structures The geometry of C(-coring structures opf algebroids over C( The central case Correspondence of Galois extensions 33 6 eferences 35 1 Introduction opf algebras are robust generalization of groups. ecently, many authors have studied much more general opf-like structures: weak opf algebras, opf monads, -opf algebras, compact quantum groups to name a few. In this article, we will mainly be interested with opf algebroids and opf categories. In the literature, there are plenty of inequivalent notions of a opf algebroid. For the exposition on these notions, see Böhm [2]. Batista et al. [1] introduced the notion of a opf category which is the natural categorification of a opf algebra. Motivated by a fundamental related to his PhD thesis, the author tries to describe the geometry of opf algebroids over C(. This geometric description necessitates a structure closely related to a opf category, but which has not appeared in the literature as far as the author s knowledge. We will define such structures in section (3. We will recall in section (2 definitions and properties of opf algebroids. For completeness, we will also give a short exposition on the representation theoretic and Galois theoretic aspects of opf algebroids. Most of section (2 follows [2] except for the definition of morphisms of opf algebroids and the definition of a coupled opf algebra. In section (3, we will define what topological opf categories are and we will also define coupled opf categories and their topological version. We will end that section with a formulation of Galois theory for opf categories and all the related variant we will introduce in that section. One of the main result of this paper is theorem (1. It gives a bijective correspondence between finitely-generated projective opf algebroids over C( and topological coupled opf categories of finite type. Using algebraic geometric and spectral theoretic methods, spanning the entirety of section (4, we will prove this result. The second main result is theorem (2, which states that, not only is there a bijection between opf algebroids and topological opf categories, their Galois theories also matched in a bijective manner. Following David ilbert s statement: The art of doing mathematics consists in finding that special case which contains all the germs of generality. we will discuss a very important example in section (3.2 which completely illustrates the general situation. Acknowledgement. I would like to thank my PhD supervisor yszard Nest for guiding me through my studies in non-commutative geometry and for the valuable discussions that help me write this article. I would also like to thank DSF Grant, UP Diliman and the support of the Danish National esearch Foundation through the Centre for Symmetry and Deformation (DNF92.

3 2 OPF AGEBOIDS 3 2 opf algebroids 2.1 Definitions There are several inequivalent notions of a opf algebroid. We will briefly present here the one defined in Böhm [2]. An -ring is a monoid object in the category of -bimodules. Explicitly, an -ring is a triple (A, µ, η where A A µ A and η A are -bimodule maps satisfying the associativity and unit axioms similar for algebras over commutative rings. A morphism of -rings is a monoid morphism in category of -bimodules. It is important to note that there is a bijection between -rings (A, µ, η and k-algebra morphisms η A. Similar to the case of algebras over commutative rings, we can define modules over -rings. For an -ring (A, µ, η, a right (resp. left (A, µ, η-module is an algebra for the monad A (resp. A on the category M (resp. M of right (resp. left modules over. We can dualize all the objects we have defined in the previous paragraph. An -coring is a comonoid in the category of -bimodules, i.e a triple (C,, ɛ where C C C and C ɛ are -bimodule maps satisfying the coassociativity and counit axioms dual to those axioms satisfied by the structure maps of an -ring. A morphism of -corings is a morphism of comonoids. Given an -coring (C,, ɛ, similar to coalgebras over commutative rings, we define a right (resp. left (C,, ɛ-comodule as a coalgebra for the comonad C (resp. C on the category M (resp. M. Definition 1. A right (resp. left -bialgebroid B is an k op -ring (B, s, t and an -coring (B,, ɛ satisfying: (a s B and op t B are k-algebra maps with commuting images defining the k op -ring structure on B which is compatible to the -bimodule structure as an -coring thru the following relation: r b r := bs(r t(r, (resp. r b r := s(rt(r b, r, r, b B. (b With the above -bimodule structure on B one can form B B. The coproduct is required to corestrict to a k-algebra map to { B B := b i b i s(rb i b i = } b i t(rb i, r i i i respectively, { B B := b i b i b i t(r b i = i i i b i b is(r, r }. (c The counit B ɛ extends the right (resp. left regular -module structure on to a right (resp. left (B, s-module. A morphism of -bialgebroids is a morphism of op -rings and -corings. emark 1.

4 2 OPF AGEBOIDS 4 (1 The k-algebra maps s and t define a k-algebra map η = s k t. As we have noted, such k-algebra uniquely determines an k op -ring structure on B. The maps s and t are called the source and target maps, respectively. (2 The k-submodule B B (resp. B B of B B is a k-algebra with factorwise multiplication. This is called the Takeuchi product. The map k op B B, r k r t(r s(r is easily seen to be a k-algebra morphism and hence, B B is an k op -ring. The corestriction of is an k op -bimodule map. ence, is an op -ring map. The same is true for B B. (3 The source map s is a k-algebra map and so it defines a unique -ring structure on B. The right version of condition (c explicitly means that r b := ɛ(s(rb, r, b B defines a right (B, s-action on. Definition 2. et k be a commutative, associative unital ring and let and be associative k-algebras. A opf algebroid is a triple = (,, S. and are bialgebroids having the same underlying k-algebra. Specifically, is a left - bialgebroid with (, s, t and (,, ɛ as its underlying k op -ring and -coring structures. Similarly, is a right -bialgebroid with (, s, t and (,, ɛ as its underlying k op -ring and -coring structures. et us denote by µ (resp. µ the multiplication on (, s (resp. (, s. S is a (bijective k-module map S, called the antipode. The compatibility conditions of these structures are as follows. (a the sources s, s, targets t, t and counits ɛ, ɛ satisfy s ɛ t = t, t ɛ s = s, s ɛ t = t, t ɛ s = s, (b the left- and right-regular comodule structures commute, i.e. id id id id (c for all l, r and for all h we have S(t (lht (r = s (rs(hs (l, (d S is the convolution inverse of the identity map i.e., the following diagram commute

5 2 OPF AGEBOIDS 5 S id ɛ s µ ɛ s id S µ emark 2. (1 In the constituent bialgebroids and, the counits ɛ and ɛ extend the regular module structures on the base rings and to the -ring (, s and to the -ring (, s, respectively. Equivalently, the counits extend the regular module structures on the base rings and to the op -ring (, t and to the op -ring (, t. This particularly implies that the maps s ɛ, t ɛ, s ɛ and t ɛ are idempotents. This means that the images of s and t coincides in. Same is true for the images of s and t. (2 Notice that for condition (b to make sense, apart from being an -bimodule map, has to be an -bimodule map. This is the case using remark (1. Similarly, is an -bimodule map. (3 We can equip with two (, -bimodule structures one using t and t and the other using s and s. Condition (c relates these two (, -bimodules structures via the antipode S which in turn makes the diagram in condition (d defined. (4 A most convenient way to summarize the property of the antipode of a opf algebra is to express it as the inverse of the identity map in the convolution algebra of endomorphisms of that opf algebra. For opf algebroids, the antipode is the inverse of the identity map in the appropriate category, called the convolution category of. As before, and are k-algebras. et and Y be k-modules such that has an -coring (,, ɛ and an -coring (,, ɛ structures and Y has an k - ring structure with multiplications µ : Y Y Y and µ : Y Y Y. Define the convolution category Conv(, Y to be the category with two objects labelled and. For I, J {, }, a morphism I J is a J-I-bimodule map Y. For I, J, K {, } and morphisms J f I and K g J, we define the composition f g to be the following convolution f g = µ J (f J g J. The antipode S of a opf algebroid is the inverse of the identity map viewed as an arrow in Conv(,. id (5 et us note that condition (c in the definition of a bialgebroid implies that ɛ s : is the identity. Similarly, ɛ s : is also the identity.

6 2 OPF AGEBOIDS 6 Using condition (a in the definition of a opf algebroid, we see that the following compositions define pairs of inverse k-algebra maps. ɛ s op ɛ t ɛ s op ɛ t This is particular implies that and are anti-isomorphic k-algebras. (6 Since there are two coproducts involved in a opf algebroid, namely and, we will use different Sweedler notations for their corresponding components. We will write (h = h [1] h [2] and (h = h [1] h [2] for h. (7 With a fixed bijective antipode S, the constituent left- and right-bialgebroids of a opf algebroid determine each other, see for example the article [3]. In view of this and the fact that and are anti-isomorphic, in the sequel where we will be mainly interested with opf algebroids with bijective antipodes we will simply call a opf algebroid over instead of explicitly mentioning. Definition 3. et (,, S and (,, S be opf algebroids over. An algebraic morphism (,, S (,, S of opf algebroids is a pair (ϕ, ϕ of a leftbialgebroid morphism ϕ and a right-bialgebroid morphism ϕ for which the following diagrams commute S S ϕ ϕ ϕ ϕ S and composition of such a pair is componentwise. et and be k-algebras and (,, S and (K, K, S be opf algebroids over and, respectively. In view of remark (2 (7 above, denote by = op and = ( op. A geometric morphism (,, S (K, K, S of opf algebroids is a pair (f, φ of k-algerba maps f and φ K, where, K denote the underlying k-algebra structures of the opf algebroids under consideration. These two maps satisfy the following compatibility conditions. (a f and φ intertwines the source, target and counit maps of the left-bialgebroid structures of and K, i.e. S ɛ t s φ f f φ f φ K ɛ K t K K s K K. Same goes for the source, target and counit maps of the right-bialgebroid structures.

7 2 OPF AGEBOIDS 7 (b In view of condition (a, the k-bimodule map φ k φ defines k-bimodule maps φ f φ K K, φ f φ K K. We then require that the following diagrams commute φ f φ K K φ f φ K K µ µ K µ µ K φ K φ K (c Also by of condition (a, the k-bimodule maps φ f φ and φ f φ of condition (b further define k-bimodule maps φ f φ K K, φ f φ K K. We then require that the following diagrams commute. φ K φ K K K φ f φ K K φ f φ K K (d φ intertwines the antipodes of and K, i.e. φ S = S K φ. emark 3. (1 For a k-algebra, let us denote by AG alg ( the category whose objects are opf algebroids over and morphisms are algebraic morphisms. For a fixed k, let us denote by AG geom (k the category whose objects are opf algebroids over k-algebras and morphisms are geometric morphisms. The existence of these two naturally defined categories reflect the fact that opf algebroids are generalization of both opf algebras and groupoids. (2 Equip e with the opf algebroid structure defined in example 5 of the next section. et (,, S be a opf algebroid over. Then the unit maps η, η together with the identity map on define geometric morphisms (id, η : e and (id, η : e.

8 2 OPF AGEBOIDS Examples Example 1. opf algebras. A opf algebra over the commutative unital ring k gives an example of a opf algebroid. ere, we take = = k as k-algebras, take s = t = s = t = η to be the source and target maps, set ɛ = ɛ = ɛ to be the counits, and = = to be the coproducts. Example 2. Coupled opf algebras. It might be tempting to think that opf algebroids for which = = k must be opf algebras. This is not entirely the case. We will give a general set of examples for which this is not true. Two opf algebra structures 1 = (, m, η, 1, ɛ 1, S 1 and 2 = (, m, η, 2, ɛ 2, S 2 over the same k-algebra are said to be coupled if (a there exists a k-module map C :, called the coupling map such that C id 1 m ɛ 2 ɛ 1 k η 2 m id C commutes, and (b the coproducts 1 and 2 in commutes. Coupled opf algebras give rise to opf algebroids over k. The left k-bialgebroid is the underlying bialgerba of 1 while the right k-bialgebroid is the underlying bialgebra of 2. The coupling map plays the role of the antipode. et us give examples of coupled opf algebras. Connes and Moscovici constructed twisted antipodes in [4]. et us show that such a twisted antipode is a coupling map for some coupled opf algebras. et = (, m, 1,, ɛ, S be a opf algebra. Take 1 = as opf algebras. et σ : k be a character. Define 2 : by h h (1 σ(s(h (2 h (3. Take ɛ 2 = σ. Define S 2 : by h σ(h (1 S(h (2 σ(h (3. Note the Sweedler-legs of h appearing in the definition of S 2 is the one provided by and not by 2. Then, 2 = (, m, 1, 2, ɛ 2, S 2 is a opf algebra coupled with 1 by the coupling map S σ : defined by h σ(h (1 S(h (2. Example 3. Groupoid algebras. Given a small groupoid G with finitely many objects and a commutative unital ring k, we can construct what is called the groupoid algebra of G over k, denoted by kg. For such a groupoid G, let us denote by G (0 its set of objects, G (1 its set of morphisms, s, t : G (1 G (0 the source and target maps, ι : G (0 G (1 the unit map, ν : G (1 G (1 the inversion map, G (2 = G (1 t s G (1 the set of composable pairs of morphisms, and m : G (2 G (1 the partial composition. The groupoid algebra kg is the k-algebra generated by G (1 subject to the relation

9 2 OPF AGEBOIDS 9 f f, if f, f are composable ff = 0, otherwise for f, f G (1. The groupoid algebra kg is a opf algebroid as folows. The base algebras and are both equal to kg (0 and the two bialgebroids and are isomorphic as bialgebroids with underlying k-module kg (1. The partial groupoid composition m dualizes and extends to a multiplication m : kg (1 kg (1 kg (1 which then factors through the canonical surjection kg (1 kg (1 kg (1 kg (0 kg (1 to give the product kg (1 kg (0 kg (1 kg (1. The source and target maps s, t of the groupoid give the source and target maps s, t : kg (0 kg (1, respectively. The unit map gives the counit map ɛ : kg (1 kg (0. Finally, the inversion map gives the antipode map S : kg (1 kg (1. Note that the underlying bimodule structures of the right and the left bialgerboid is related by the antipode map. Example 4. Weak opf algebras. Another structure that generalize opf algebras, called weak opf algebras, also are opf algebroids. Explicitly, a weak opf algebra over a commutative unital ring k is a unitary associative algebra together with k-linear maps : (weak coproduct, ɛ : k (weak counit and S : (weak antipode satisfying the following axioms: (i is multiplicative, coassociative, and weak-unital, i.e. ( (1 1(1 (1 = (2 (1 = (1 (1( (1 1, (iii ɛ is counital, and weak-multiplicative, i.e. for any x, y, z ɛ(xy (1 ɛ(y (2 z = ɛ(xyz = ɛ(xy (2 ɛ(y (1 z, (v for any h, S(h (1 h (2 S(h (3 = S(h and h (1 S(h (2 = ɛ(1 (1 h1 (2, S(h (1 h (2 = 1 (1 ɛ(h1 (2 et us sketch a proof why a weak opf algebra is a opf algebroid. Consider the maps p :, h 1 (1 ɛ(h1 (2 and p :, h ɛ(1 (1 h1 (2. By k-linearity and weak-multiplicativity of ɛ, p and p are idempotents. Multiplicativity and coassiociativity of and counitality of ɛ implies that for any h, h (1 p (h (2 = 1 (1 h 1 (2 p (h (1 h (2 = 1 (1 h1 (2. Now, using these relations and coassiociativity of we get 1 (1 1 (1 1 (2 1 (2 = 1 (1 (1 p (1 (1 (2 1 (2 = 1 (1 p (1 (2 1 (3 1 (1 1 (1 1 (2 1 (2 = 1 (1 p (1 (2(1 1 (2(2 = 1 (1(1 p (1 (1(2 1 (2 Thus, the first tensor factor of the left-hand side of the first equation above is in the image of p. Similarly, the last tensor factor of the left-hand side of the second equation above

10 2 OPF AGEBOIDS 10 is in the image of p. Clearly, p (1 = p (1 = 1. ence, the images of p and p are unitary subalgebras of. Denote these subalgebras by and, respectively. By the weak-unitality of we see that these subalgebras are commuting subalgebras of. Taking the source map s as the inclusion and the target map as t : op, r ɛ(r1 (1 1 (2 equips with an k op -ring structure. Taking ɛ = p and as the composition k equips with an -coring structure (,, ɛ. The ring and coring structures just constructed gives a structure of right -bialgebroid. Using op in place of in the above construction, we get a left op -bialgebroid op. Together with the right -bialgebroid constructed and the existing weak antipode S, we get a opf algebroid ( op,, S. 2.3 epresentation of opf algebroids In this section, we will look at representations of opf algebroids. Towards the end of the section, we will look at the descent theoretic aspect of a special class of modules over opf algebroids, the so called relative opf modules. et = (,, S be a opf algebroid with underlying k-module. carries both a left -module sctructure and a left - module structure via the maps s and t, respectively. A right -comodule M is a right -module and a right -module together with a right -coaction ρ : M M and a right -coaction ρ : M M such that ρ is an -comodule map and ρ is an -comodule map. For the coaction ρ, let us use the following Sweedler notation: ρ (m = m [0] m [1] and for the coaction ρ, let us use the following Sweedler notation: ρ (m = m [0] m [1]. With these notations, the conditions above explicitly means that for all m M, l and r we have (m l [0] (m l [1] = ρ (m l = m [0] t (lm [1] (m r [0] (m r [1] = ρ (m r = m [0] m [1] s (r. We further require that the two coactions satify the following commutative diagrams M ρ M M ρ M (1 ρ ρ id ρ ρ id M id M M id M

11 2 OPF AGEBOIDS 11 We will denote by M the category of right -comodules. Symmetrically, we can define left -comodules and we denote the category of a such by M. Comodules over opf algebroids are comodules over the constituent bialgebroids. Thus, one can speak of two different coinvariants, one for each bialgebroid. For a given right -comodule M, they are defined as follows: M co = M co = { m M { m M ρ (m = m ρ (m = m } 1, } 1. In the general case, we have M co M co. But in our case, where we assume S is bijective these two spaces coincide. This will be important in the formulation of Galois theory for opf algebroids. To see that these coinvariants coincide, consider the following map Φ M : M M m h ρ (m S(h ere, acts on the right of M through the second factor. If m M co, then we have ρ (m = ρ (m S(h = Φ M (m 1 = Φ M (ρ (m = Φ M (m [0] m [1] = ρ (m [0] S(m [1] = (m [0] [0] m [0] [1] S(m[1] = m [0] [0] m [0] [1] S(m[1] = m [0] m [0] [1] S(m[1] [1] = m [0] s (ɛ (m [1] 1 = m [0] s (ɛ (m [1] 1 = m This shows the inclusion M co M co. To show the other inclusion, one can run the same computation but using the inverse of Φ M which is the following map Φ 1 M : M M m h S 1 (h ρ (m. In this case, we can simply write M co for M co = M co and refer to it as the - coinvariants of M instead of distinguishing the - from the -coinvariants, unless it is necessary to do so. et us now discuss monoid objects in M. They are called -comodule algebras. A right -comodule algebra is an -ring (M, µ, η such that M is a right -comodule and η : M and µ : M M M are -comodule maps. Using Sweedler notation for coactions, this explicitly means that for any m, n M we have (mn [0] (mn [1] = ρ (mn = m [0] n [0] m [1] n [1], (2

12 3 OPF CATEGOIES 12 (mn [0] (mn [1] = ρ (mn = m [0] n [0] m [1] n [1], (3 1 [0] M 1 [1] M = ρ (1 M = 1 M 1, (4 (1 M [0] (1 M [1] = ρ (1 M = 1 M 1. (5 et = (,, S be a opf algebroid with underlying k-module. A k-algebra extension A B is said to be (right -Galois if B is a right -comodule algebra with B co = A and the map B B A gal B a b ab [0] b [1] A is a bijection. The map gal is called the Galois map associated to the bialgebroid extension A B. Symmetrically, the extension A B is (right -Galois if B is a right -comodule algebra with B co = A and the map B B A gal B a b a [0] b a [1] A is a bijection. We say that a k-algebra extension A B is -Galois if it is both -Galois and -Galois. It is not known in general if the bijectivity of gal and gal are equivalent. owever, if the antipode S is bijective (which is part of our standing assumption then gal is bijective if and only if gal. To see this, note that gal = Φ B gal where Φ B is the map defined in the previous section for M = B. Since S is bijective, Φ B is an isomorphism which gives the desired equivalence of bijectivity of gal and gal. Thus, the extension A B is -Galois if it is a bialgebroid Galois extension for any of its constituent bialgebroids. 3 opf categories 3.1 Definitions and properties Batista et al. [1] introduced the notion of a opf category over an arbitrary strict braided monoidal V. In this section, we will introduce its topological version. For this purpose, we specialize V as the category of complex vector spaces whose braiding is the usual flip of tensor factors. Also, we will assume that the underlying categories of such opf categories are small. We will be primarily interested with finite-type V-enriched categories, by which we mean the hom-sets are finite-dimensional vector spaces. Before giving the definition of a opf category, let is introduce some notation first. For two V-enriched categories A and B with the same set of objects, we define A B to the the V-enriched category with as the set of objects and for x, y, the hom-set of arrows from x to y is the vector space (A B x,y := A x,y B x,y. (6

13 3 OPF CATEGOIES 13 We call A B the tensor product of A and B. With this, the category of V-enriched categories over becomes a strict monoidal category whose monoidal unit, denoted by 1, is the category over such that for any x, y we have 1 x,y = C. Definition 4. A opf category over is a V-enriched category satisfying the following conditions. (a There are functors, ɛ 1 called the coproduct and counit, respectively, such that is coassociative and counital with respect to ɛ, i.e. the diagram of functors 1 id is ɛ id 1 ɛ id commute. (b There is a functor S : op, called the antipode, satisfying S id op ɛ 1 η id S op ere, denotes the bifunctor induced by the categorical composition in and η is the functor that send 1 1 x,y to the identity element of x,y. emark 4. Functoriality of and ɛ means that for any x, y, we have linear maps x,y x,y x,y x,y x,y ɛ x,y C where x,y is coassociative and counital with respect to ɛ x,y in the usual sense. This implies that x,y is a coalgebra. If we denote by CV the category of coalgebras on V, another way to package part (a of definition (4 is to say that is enriched over C(V. For the main results of this paper, we will be mostly interested with the case is a topological space. In such a case, it makes sense to reflect continuity on the functors, ɛ and S along with the categorical structure maps. This calls for the following definition.

14 3 OPF CATEGOIES 14 Definition 5. et be a topological space and let O be the sheaf of continuous complex-valued functions on. A topological opf category over is a opf category together with a sheaf sh over (with the product topology of O -bimodules satisfying the following conditions. (a Denote by π 1, π 2 : the projection onto the first and second factor, respectively. Over an open set U, for any σ sh (U, f O (π 1 U and g O (π 2 U we have for any (x, y U. (f σ g (x, y = f(xσ(x, yg(y (b x,y is the fiber of sh at (x, y. (c, η,, ɛ and S are the induced maps on global sections of the following map of sheaves sh O sh sh sh, O η sh sh, sh sh sh sh sh ɛsh, O O, sh S ( sh sh op respectively. ere, ( sh op is the pullback of the sheaf sh along the map flipping the factors. emark 5. The bimodule tensor product O used in part (c for sh of definition (5 is the tensor product of the appropriately modified O -bimodule sh, one in which we have ( f σ τ g = (σ g (f τ O O for any f, g O. For the bimodule tensor product O used for sh is the one with ( f σ τ g = (f σ (τ g O O for any f, g O. The following, which will play an important role in our formulation of the main result, is the categorification of a coupled opf algebra. Definition 6. A coupled opf category is a V-enriched category with two C(V- enrichments, denoted by and, with coproducts, and counits ɛ, ɛ, respectively; and a functor S : op, called the coupling functor, such that the following conditions are satisfied:

15 3 OPF CATEGOIES 15 (a The following diagrams, indicating the coupling condition, commute. S id op ɛ 1 η ɛ 1 η id S op (b The coproducts and commute, i.e. id id id id emark 6. (1 Coupled opf categories are almost the categorification of coupled opf algebras. While the constituent bialgebras of a coupled opf algebra is a opf algebras in itself, the constituent categories and of a coupled opf category need not be opf categories. (2 Just like opf categories, we can also topologize coupled opf categories. We can take definition (5: assert the existence of a sheaf sh over of O -bimodules, take conditions (a and (b as they are, and replace condition (c by (c,, ɛ, ɛ and S are the induced maps on global sections of the following map of sheaves sh ( sh sh O sh, sh (ɛ sh O, sh ( sh sh O sh, sh (ɛ sh O, sh S ( sh sh op respectively, making the following diagram

16 3 OPF CATEGOIES 16 ( sh (U sh (U sh (U O (U S U id O (U sh (U op sh (U O (U sh (U (ɛ sh (U O (U η U µ U sh (π diag 2 U sh (U (ɛ sh (U O (U η U sh (π diag 1 U ( sh (U sh (U sh (U O (U id O (U S U sh (U O (U ( sh (U op µ U commute for any U. ere, µ U and η U denote the maps induced by the composition and unit maps of C. The maps π diag 1 and π diag 2 denote, (x, y (x, x and, (x, y (y, y, respectively. 3.2 A good example of a opf category In this section, we will look at a very important example of a opf category. This example will also be an example of our main result. This is a special case of proposition 7.1 of [1]. Consider a finite set whose elements are conveniently labelled as 1, 2,..., n. Equipped with the discrete topology. Consider the category C whose set of objects is and define C x,y = C. The category C is obviously a opf category. By proposition 7.1 of [1], = x,y C x,y is a weak opf algebra. Using the arguments in example 4 of section (2.2, is a opf algebroid over A = C n = O (. The opf algebroid has a more familiar form. It is isomorphic, as a opf algebroid, the algebra M n (C over its diagonal D n = Diag n (C. With the D n -bimodule structure on M n (C defined as P M Q := MP Q, the coproduct and the counit ɛ are given as (E ij = E ij Dn E ij, ɛ (M = P, Q D n, M M n (C, n E ii φ(me ii where φ is the linear functional defined by φ(e ij = 1 for all i, j. With the usual matrix multiplication and unit, and ɛ constitutes a right D n -bialgebroid structure on M n (C. For completeness, let us define the structure maps of the left D n -bialgebroid structure of M n (C. Consider the D n -bimodule structure on M n (C defined as P M Q := P QM, The coproduct and the counit ɛ are defined as i=1 P, Q D n, M M n (C. (E ij = E ij Dn E ij, ɛ (M = n φ(e ii ME ii i=1

17 3 OPF CATEGOIES 17 where φ is the same linear functional used to defined ɛ. The antipode S of this opf algebroid is defined as S(E ij = E ji. As a weak opf algebra, φ is the counit of. The coproducts and are the extension of the weak coproduct to M n (C Dn M n (C relative to the D n -bimodule structure used. As we will see in section (4, this is not a coincidence. This is in fact a special case of a more general result which we shall prove at the end of that section. 3.3 Galois extensions of opf categories Formulation of Galois theory for opf category is straightforward. ecall that in the case of opf algebras, only the underlying bialgebra structure is relevant. In the coaction picture, the coalgebra is used to make sense of a coaction while the algebra structure is used to make sense of the Galois map. All these ingredients are already present in the case of a opf category. We will discuss the situation for topological opf categories. The case for opf categories follow almost immediately by dropping any manifestation of topology. Before giving the definition of the categorical analogue of a comodule algebra, let us first discuss what a topological category is, at least for our purpose. A V-enriched category M over a space is a topological category if there is a sheaf M sh of O -bimodules such that conditions (a, (b and the relevant part of condition (c of definition (5 hold. Definition 7. et be a topological opf category with space of objects, coproduct, counit ɛ and antipode S with associated sheaf sh. (1 A topological category M over enriched over V, with associated sheaf M sh, is a right -comodule if there is a functor ρ : M M such that the following conditions hold. (a ρ is coassociative with respect to and counital with respect to ɛ, i.e. the diagrams of functors M ρ M M M 1 ρ id ρ id ɛ M ρ id M M commute, and (b the functor ρ is the map induced by the map of sheaves M sh M sh O sh where the tensor product is the same as the first one we described in remark (5. A left -comodule can be symmetrically defined. (2 A morphism M φ N of right -comodules is a functor that commutes with the right coactions, i.e. one which makes the following diagram commute

18 3 OPF CATEGOIES 18 M ρ M M φ φ id N ρ N N. ere, ρ M and ρ N are the coactions of on M and N, respectively. (3 A right -comodule M is a right -comodule-category if in addition, the composition map M M M is a map of right -comodules, where M M is equipped with the diagonal coaction. (4 The coinvariants of a right -comodule-category M is the subcategory M co whose space of objects is and whose hom-sets are defined as ( M co C x,y := {α M x,y ρ(α = α id y } for any x, y. emark 7. A opf category is the categorification of a opf algebra with categorical composition corresponding to the algebra product. A right -comodule M is in particular a category, it already has a composition. This means that we only need to impose requirement (3 in definition (7 to get a categorification of the notion of a comodulealgebra. In the classical set-up, one has to require the existence of a product and assert its compatibility with the comodule structures. In the set-up of opf-galois theory with respect to opf algebras, there is a wellunderstood notion for extensions of k-algebras A B to be -Galois for a opf algebra even if A k. This is because B A B makes sense as a k-module. All that is left to do is require A = B co and that the map B A B B, a b (a 1ρ(b is bijective. On the other hand, in the situation of a opf category and extensions of comodule-categories A M with A = (M co, we can only make sense of the product M A M in the case A is the subcategory of M whose hom-sets A x,y are all zero except when x = y, in which case A x,x = C. In this case, we identify M A M with M M. et us call such a category the trivial linear category over, and denote by I. There might be a way to consider Galois extensions by opf categories in which the subcategory of coinvariants is strictly larger than I, but at present it is not clear to the author how to make sense of it. Fortunately, for our purpose of proving theorem (2 it is enough to have I as the subcategory of coinvariants. Definition 8. A right -comodule-category M is a -Galois extension of I provided (a M co = I, and (b the functor M M gal M, α β ( α β [0] β[1]

19 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 19 called the Galois morphism, is fully faithful. emark 8. (1 We are using Sweedler notation for the legs of the coaction ρ : M M. In other words, for any x, y and α M x,y, we have ρ(α = α [0] α [1], where α [0] M x,z and α [1] z,y for some z. This, in particular, tells us that the map gal above make sense. (2 Galois extension by a coupled opf category = (,, S means simultaneous Galois extensions of the constituent C(V-enriched categories and. 4 The category associated to a opf algebroid In this section, we will consider opf algebroids over a commutative unital C -algebra A. We will restrict to the case where is finitely-generated and projective as a left and a right A-module. With the underlying assumption that the antipode is bijective, by [3], finitely-generated projectivity of any of the A-module structures of coming from the source and target maps are all equivalent. Note that even though A is commutative, its image under the source or the target map need not be central in. We will deal with this general situation and specialize in the case when we have centrality. 4.1 ocal eigenspace decomposition et = (,, S be a opf algebroid over A, a commutative unital C -algebra. Assume that is finitely-generated and projective as a left- and a right-a-module via the source and target maps. With our standing assumption, has the same properties. et us first consider the left bialgebroid. The Gelfand duality implies that A = C( for some compact ausdorff space. The Serre-Swan theorem applied to the left A-module gives us a finite-rank vector bundle E p such that = Γ(, E as left modules, where the left C(-module structure on Γ(, E is by pointwise multiplication, i.e. (f σ(x = f(xσ(x for all x, f C( and σ Γ(, E. By the bimodule nature of, the right A-module structure of Γ(, E commutes with the left ρ A-module which implies that we have a representation C( End(E of C( into the endomorphism bundle of E p. Since C( is abelian and ρ is a -morphism, ρ(c( lands in a maximal abelian subalgebra D(n of End(E. Choose a finite collection of open sets {U i i = 1, 2,..., m} that cover over which E is trivializable. Choose a system of coordinates such that E trivial over each U i, i.e. E Ui = U i V, where V is a finite-dimensional vector space.choosing a basis v 1, v 2,..., v n V one has End(E Ui = C(U i, M n (C where n is the rank of E. Commutativity of C( implies that up to unitaries V i U(n, we have C( ρ C(U i, Diag(n where Diag(n denotes the subalgebra of diagonal matrices on M n (C and V i C(U i, Diag(n V i = D(n Ui.

20 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 20 For each i = 1, 2,..., m, choosing a set of central orthogonal idempotents {e j j = 1,..., n} gives n projections p i j given by the following composition C( ρ i C(Ui, Diag(n = n C(U i k=1 proj j C(Ui These projections are in particular continuous C -morphisms. ence, they give, for each ϕ i j i = 1, 2,..., m, (possibly non-distinct n continuous injective maps U i, j = 1,..., n. Geometrically, the situation is depicted figure (1. Figure 1: ocal eigenspace decomposition of E. et us describe the nature of the set Z = i,j ϕi j(u i over the intersections U α U β. Over U α U β U α we get a unitary V α which gives n central orthogonal idempotents and up to ordering of such idempotents, one gets the sets ϕ i j(u i. The union j ϕi j(u i does not depend on the ordering of these idempotents. Thus, over U α U β one gets unitaries V α and V β which simultaneously diagonalize ρ(c(. Thus, we have from which we get that ( ϕ α j (U α j ϕ α j (U α U β = j j ϕ β j (U α U β ( ϕ β j (U β = j that is, the sets j ϕi j(u i agree on the intersections. A subset T is called transverse if j ϕ α j (U α U β

21 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 21 proj 1 T :, proj 2 T : are homeomorphisms, where proj 1 and proj 2 denotes the projection onto the first and second factor, respectively. In particular, T is homeomorphic to. Using the above argument, we have the following proposition. Proposition 1. For every i = 1, 2,..., m, j = 1, 2,..., n the set ϕ i j(u i extends to a transverse subset of completely contained in Z. In particular, Z is the union of n (possibly overlapping transverse subsets of. This means that the curves in figure (1 overlap. emark 9. Another way to see why the closed subset Z is the union of transverse subsets of is by the fact the we can run the construction of the sets ϕ i j(u i described in the beginning of this section in a symmetric fashion, one for each factor of. The whole picture (1 is a decomposition of into U i, i I. The graphs of ϕ i j are labelled accordingly. Note that each f(x End(E x, f C( are diagonalizable since they commute with their adjoint f(x C(. And since such operators commute with each other, the collection {f(x End(E x f C(} is simultaneously diagonalizable. Over a point x U 1, the fiber E x decomposes into joint eigenspaces of {f(x End(E x f C(}. The dimension of these eigenspaces are determined by the number of intersections of the vertical dotted line through x U 1 with the graphs of ϕ 1 j. Using this eigenspace decomposition, we have the following proposition which describes geometrically the right C(-module structure of. Proposition 2. Given σ Γ(, E and f C( the section σ f Γ(, E is given as (σ f (x = n f(ϕ i j(xe j σ(x. (7 j=1 where x U i and σ(x = n e j σ(x. emark 10. j=1 (1 In case C( is central in, the above picture reduce to {U i i I} the trivial cover and ϕ : is the identity, i.e. the graph in the above picture is the diagonal of. The action defined by equation (7 then reduces to pointwise multiplication which then coincides with the left C(-module structure of = Γ(, E. (2 One can understand the right action above as pointwise-eigenvalue-scaled action. Compared to the central case, every f C( acts on a σ Γ(, E in a way that f(x acts diagonally on σ(x, i.e. E x constitutes a single eigenspace for the operator f(x corresponding to the eigenvalue f(x C. In the noncentral case, the action is still pointwise. owever, the operator f(x no longer has a single eigenspace. The eigenspaces are labelled by the points ϕ i j(x where x U i and the eigenvalues of f(x are f ( ϕ i j(x, j = 1,..., n.

22 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 22 Proposition 3. As a C(-bimodule, = Γ(Z, E where E is a sheaf of complex vector spaces over supported on a closed subset Z. The C(-bimodule structure on Γ(Z, E is defined as for f, g C( and σ Γ(Z, E. (f σ g(x, y = f(xσ(x, yg(y The C( C( op is dense in C( thus we can extend the C( C( op - module structure of to a C( -module structure. Consider the annihilator of, Ann( = {f C( f σ = 0, for all σ B }. Then, there is an open set U such that Ann( = C(U. Then Z = ( U, the support of the bimodule. Proposition 4. The subset Z is completely determined by the C(-bimodule structure of. Moreover, Z is the support of = Γ(, E. By proposition (1, Z is the union of transverse subsets of which is individually are unions of graphs of ϕ i j. et E (x,y = ϕ i j (x=y (E x ϕ i j (x be the fiber of E over (x, y Z, where (E x ϕ i j (x denotes the eigensubspace of E x over the point ϕ i j(x. This defines a sheaf of vector spaces on supported on Z. A section of τ Γ(, E defines a section ˆτ Γ(Z, E whose value at a point (x, y is proj ij τ(x, if y = ϕ i j(x for some i, j ˆτ(x, y = 0, otherwise, where proj ij denotes the projection E x (E x ϕ i j (x. Conversely, any section τ Γ(Z, E defines a section ˇτ Γ(, E by ˇτ(x = y τ(x, y. Now, given h C( C( we have h(x, y = k f k (xg k (y for some f k, g k C(. For any σ B = Γ(, E we have (h ˆσ (x, y = k = k f k (xˆσ(x, yg k (y f k (xg k (ϕ i j(xe j (σ(x = proj ij ( k f k σ g k (x, y

23 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 23 which shows that : Γ(, E Γ(Z, F, τ ˆτ is a bimodule map whose inverse is the map : Γ(Z, F Γ(, E, τ ˇτ. Using proposition (1, we can relate the vector bundles the Serre-Swan theorem gives when applied to the left and right C(-module structure of as follows. p 1 p 2 Proposition 5. et E 1 and E2 be the vector bundles given by the Serre- Swan theorem applied to the finitely-generated projective left and right C(-module, respectively. Then E 1 and E 2 are the direct-images of the sheaf E along π 1 and π 2, respectively. First, the direct-image of E along π 1 is easily seen to be a vector bundle and the space of sections Γ(, (π 1 E is easily seen to be isomorphic as left C(-modules to the left C(-module Γ(Z, E. By proposition (3, Γ(Z, E = Γ(, E 1 as left C(-modules. Thus, by corollary 2.8 of [5] we see that E 1 and (π 1 E are isomorphic as vector bundles. Similar argument works for E 2. et us say more about the nature of the eigenspaces E (x,y, x, y in relation to the subset Z. Proposition 6. (i E x = E (x,y y (ii dim ( E (x,y is the number of transverse subsets of contained in Z passing through (x, y, with multiplicities. ( (iii dim E (x,y = n for any y. x 4.2 The geometry of C(-ring structures The previous section describes the geometry of using its bimodule structure over C(. But has more structure than just being a bimodule. In particular, it is a s C(-ring via the left source map C(. In this section, we will look at what this additional structure contributes to the geometry of. We will keep the notations of the previous section. The C(-ring structure on = Γ(, E via the source map s consists of a pair of C(-bimodule maps µ Γ(Z, E Γ(Z, E Γ(Z, E C( C( η Γ(Z, E satisfying the associativity and unitality conditions. For brevity we will write η = s. The unit map η gives an element 1 Γ(Z, E satisfying f 1 = 1 f for all f C(. Since is ausdorff, if x y then we can find an f C( such that f(x = 1 and f(y = 0. Thus, for x y we have 1(x, y = f(x1(x, y = (f 1(x, y = (1 f(x, y = 1(x, yf(y = 0. Thus, the source map A s is implemented by C( Γ(Z, E, f f 1. This means that s f(x = f(x1(x, x and choosing f such that f(x 0 and s f(x 0

24 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 24 Figure 2: The geometry of the product and unit maps. Figure 3: Support Z of the bimodule B. we see that 1(x, x E (x,x is a nonzero element. Thus, the diagonal / of is in Z. See figure (3. Note that Γ(Z, E C( Γ(Z, E = Γ(Z, E (2 where E (2 is the sheaf of vector spaces whose fiber at a point (x, z Z is the vector space ( E(x,y E (y,z y due to the balancing condition σ f C( τ = σ C( f τ for σ, τ Γ(Z, E and f C(. Notice that all but finitely many summands above are zero. Specifically, only

25 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 25 those y for which (x, y and (y, z are both in Z contribute nontrivially. et us denote these y as y 1, y 2,..., y n. By proposition (3, Γ(Z, E = Γ(, E as C(-bimodules. Since Γ(, is a fully faithful functor by corollary 2.8 of [5], we can convert the global ring structures µ and η into something fiber-wise. In particular, the product map µ induces a map E (x,y1 E (y1,z... E (x,yn E (yn,z µ E(x,z (8 illustrated in figure (2. By the universal property of direct sums, there are maps E (x,yi E (yi,z µ y i E (x,z one for each y i. The collection of these maps satisfy a set of conditions which, though derivable from associativity, is complicated to write down. See (3 of the remark below for these conditions. owever, for the maps E (x,x E (x,x E (x,x these conditions are precisely the associativity condition. ikewise, the map η induces maps η x,y : C E (x,y which is nonzero when x = y and zero otherwise. The map µ x together with η x = η x,x makes the vector space E (x,x a unital algebra, whose dimension depend on the multiplicity of the associated eigenvalue. The following proposition is then immediate from these arguments. Proposition 7. et A be the C(-sub-bimodule of Γ(Z, E supported on the diagonal /. Then A is an A-subring of where the multiplication is pointwise. Moreover, A is the centralizer of A in. emark 11. (1 Using abuse of notation, let us identify A with its image in. In case A is central in, the fibers of the vector bundle E are algebras. These algebras correspond to E (x,x together with the maps E (x,x E (x,x E (x,x and C E (x,x since in the central case, E (x,x = E x. Thus, A = in the central case which is not surprising at all knowing that A is the centralizer of A. µ (2 The maps E (x,yi E i (y1,z E (x,z are only restricted by the associativity of µ. Since Γ(Z, E = Γ(, E and Γ(, is known to be a fully faithful functor by corollary 2.8 of [5], we have µ x µ x ηx ( E(x,yi E (yi,y j E (yj,z y i,y j ( µ i id i y j ( E(x,yj E (yj,z ( id µ j j ( E(x,yi E (yi,z y i Universal property of direct sums gives us µ i i µ j j E (x,z.

26 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 26 E (x,yi E (yi,y j E (yj,z µ i id E (x,yj E (yj,z id µ j E (x,yi E (yi,z µ i µ j E (x,z. This justifies the argument before proposition (7. We can also use this to say more about the fibers of E which we state in the next proposition. Proposition 8. E (x,y is a left E (x,x E (y,y bimodule for every x, y. emark 12. Using remark (11 above, we can construct a small category enriched over the category of complex vector spaces. The set of objects of is. For every x, y, we define E (x,y, if y = ϕ i j(x for some i, j om(x, y := {0}, otherwise. We will call the associated category of the left A-bialgebroid. In the next section, we will see the additional properties of coming from the A-coring structure of. On a different note, let us give a complete geometric description of the A-ring structure of. Proposition 9. Denote by a i b := µ y i (a, b, a E (x,yi and b E (yi,z. The product of σ, τ Γ(Z, E takes the form (στ(x, z = i σ(x, y i i τ(y i, z for all (x, z Z. This follows immediately from equation (8. Notice the resemblance of this formula to the one for matrix multiplication. This should remind the reader of an example we discussed in section (3.2. One can view a C(-ring to be a matrix of vector spaces whose entries are indexed by and what sits in entry (x, y is the vector space E (x,y. As we have defined after proposition (4, the vector space E (x,y is the zero vector space if (x, y / Z. For matrix algebras M n (C, would be an n-element set and the vector spaces E (x,y would all be C. There are a plethora of algebraic structures package into a bialgebroid let alone in a opf algebroid. Before we end this section, let us take a detour to describe the relationships among the structures of : being a C-algebra, the A-ring and the A e -ring structures being a left-bialgebroid over A = C(. For the purpose of this discussion, let us denote by (µ C, η C the C-algebra structure of and recall that (µ, s and (µ A e, η denote the relevant A-ring and A e -ring structures of, respectively. As we mentioned in section (2.1, for a k-algebra, -ring structures are in bijection with k-algebra maps η : k. Thus, the complex algebra structure of is uniquely determined by the unit map η C : C. Similarly, the A-ring and the A e -ring structures are determined by the C-linear maps s and η. These maps satisfy the following commutativity relations.

27 4 TE CATEGOY ASSOCIATED TO A OPF AGEBOID 27 C ηc A A e. A s η µc µ µ A e A e In terms of the local eigenspace decomposition, the map µ C induces maps E (x,w E (z,y E (x,y while, by (8, we have maps E (x,z E (z,y E (x,y. On the other hand, because the C( -bimodule structure of is given as follows, (f σ(x, y = f(x, yσ(x, y, (σ f(x, y = f(y, xσ(x, y, for any f C(, σ, and x, y, the product µ A e E (x,z E (z,x E (x,x. induces maps Another way of seeing this is by noting that the product µ uses the tensor product A which kills products E (x,w E (z,y E (x,y for which w z. ikewise, the tensor product A e kills products E (x,z E (z,y E (x,y for which x y. 4.3 The geometry of C(-coring structures In this section, using the techniques and results we have developed in sections (4.1 and (4.2 we will describe what the coring structure of contributes to the geometry of E. We will keep the notations of the previous two sections. The C(-bimodule structure of the underlying A-coring structure of is related to the C(-bimodule structure of the underlying A-ring via (f σ g(x, y = f(xg(xσ(x, y (9 for σ Γ(Z, E, f, g C(, and x, y. The left-hand side of equation (9 concerns the bimodule structure one has for the underlying A-coring of while the right-hand side concerns its A-ring structure. This, in particular, implies that if we run the construction we have in section (4.1 for the bimodule structure of the A-coring of, we will get the same sheaf E supported over the same closed subset Z. A, uses a different A-bimodule structure The coproduct of, from the A-bimodule structure involved in the A-ring structure. Thus, C( means different from the C( we have in the product µ. With this, let us denote by A this new tensor product. thus, we have Γ(Z, E Γ(Z, E C( Γ(Z, E. (10 owever, using the relation (9 the codomain of can be expressed as