## The monoidal Eilenberg-Moore construction and bialgebroids

Citations: | 12 - 3 self |

### BibTeX

@MISC{Szlachányi_themonoidal,

author = {Kornél Szlachányi},

title = {The monoidal Eilenberg-Moore construction and bialgebroids},

year = {}

}

### OpenURL

### Abstract

Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×R-bialgebras, appear as the special case when T has also a right adjoint. Street’s 2-category of monads then leads to a natural definition of the 2-category of bialgebroids. Contents

### Citations

134 |
Catégories tannakiennes, in The Grothendieck Festschrift
- Deligne
- 1990
(Show Context)
Citation Context ... The underlying functor T must have a right adjoint (see Theorem 5.4). Tannaka duality for bialgebroids has recently been proven by Phùng Hô Hái [19] following the tradition of Saavedra [20], Deligne =-=[6]-=- and generalizing the results of Ulbrich [30], Schauenburg [21] for Hopf algebras and of Hayashi [8] for weak Hopf algebras. For more about this theory we refer to [18] and [9] and the references ther... |

132 |
The Formal Theory of Monads
- Street
- 1972
(Show Context)
Citation Context ...bialgebras and weak Hopf algebras [2, 3, 17], has not been investigated in detail yet. Unfortunately, it offers two natural ways. At first, since bimonads are monads, one can take Street’s definition =-=[25]-=- of the 2-category Mnd(ComonCat) of monads in ComonCat. In this framework all functors are lax comonoidal so the monad morphisms 〈G, ϕ〉 involve lax comonoidal functors G. This approach was chosen by M... |

130 |
Categories for the Working Mathematician, 2nd ed
- Lane
- 1998
(Show Context)
Citation Context ...ebroids 35 5.4. Bialgebroid maps 36 5.5. Bimodule induced bialgebroid morphisms 37 5.6. An exotic example: Hom 38 References 38 12 KORNÉL SZLACHÁNYI 1. Introduction In the classical theory of monads =-=[14, 15]-=- or triples [1] one deals with the following construction. If a functor U : C → M has a left adjoint F with unit η: M → UF and counit ε: FU → C then there is a monad T = 〈UF, UεF, η〉 on M such that U ... |

109 |
A new duality theory for compact groups
- Doplicher, Roberts
- 1989
(Show Context)
Citation Context ...alence classes of objects in this 2-category may turn out to be the appropriate objects which characterize a class of monoidal categories uniquely, similarly to the Doplicher-Roberts characterization =-=[7]-=- of certain symmetric monoidal C ∗ -categories as representation categories of uniquely determined compact groups.4 KORNÉL SZLACHÁNYI 2. Monoidal adjunctions 2.1. Lax monoidal functors with left adjo... |

78 |
Hopf algebroids and quantum groupoids
- Lu
- 1996
(Show Context)
Citation Context ...lgebroid. Hopf algebroid could then be the name for a bimonad possessing some sort of antipode. The motivating example of bimonads is associated to a Takeuchi ×R-algebra [28], also called bialgebroid =-=[12, 5, 22, 26]-=-. For a bialgebroid A over R the algebra A is an extension of Re := R ⊗ Rop , so A is an Re-Re-bimodule. The underlying − : RMR → RMR where we identified functor T of the bimonad is T = A ⊗Re RMR with... |

65 |
Szlachányi: Weak Hopf algebras I
- Böhm, Nill, et al.
- 1999
(Show Context)
Citation Context ...of coalgebras [28]. At last but not least the bimonad description offers natural ways to define the category of bialgebroids which, even for the special case of weak bialgebras and weak Hopf algebras =-=[2, 3, 17]-=-, has not been investigated in detail yet. Unfortunately, it offers two natural ways. At first, since bimonads are monads, one can take Street’s definition [25] of the 2-category Mnd(ComonCat) of mona... |

58 |
A coassociative C⋆ quantum group with non-integral dimensions
- Böhm, Szlachányi
- 1996
(Show Context)
Citation Context ...of coalgebras [28]. At last but not least the bimonad description offers natural ways to define the category of bialgebroids which, even for the special case of weak bialgebras and weak Hopf algebras =-=[2, 3, 17]-=-, has not been investigated in detail yet. Unfortunately, it offers two natural ways. At first, since bimonads are monads, one can take Street’s definition [25] of the 2-category Mnd(ComonCat) of mona... |

50 |
An introduction to Tannaka duality and quantum groups, from: “Category theory
- Joyal, Street
- 1990
(Show Context)
Citation Context ... Saavedra [20], Deligne [6] and generalizing the results of Ulbrich [30], Schauenburg [21] for Hopf algebras and of Hayashi [8] for weak Hopf algebras. For more about this theory we refer to [18] and =-=[9]-=- and the references therein. The approach of the present paper does not fit into this series mathematically but perhaps ‘physically’. The categories we think about are module categories and therefore ... |

37 |
Groups of algebras over A
- Takeuchi
- 1977
(Show Context)
Citation Context ... rhymes with bialgebra and bialgebroid. Hopf algebroid could then be the name for a bimonad possessing some sort of antipode. The motivating example of bimonads is associated to a Takeuchi ×R-algebra =-=[28]-=-, also called bialgebroid [12, 5, 22, 26]. For a bialgebroid A over R the algebra A is an extension of Re := R ⊗ Rop , so A is an Re-Re-bimodule. The underlying − : RMR → RMR where we identified funct... |

25 | Monads on tensor categories
- Moerdijk
(Show Context)
Citation Context ... structures. Such monads will be called bimonads. Bimonads are the abstract versions of bialgebras in the same spirit as monads are related to algebras. Bimonads first appeared in the work of Moerdijk=-=[16]-=- under the name Hopf monads. He was motivated by generalizing the notion of Hopf operad. In our context the name bimonad seems to be the more natural as it rhymes with bialgebra and bialgebroid. Hopf ... |

25 |
Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules
- Schauenburg
- 1998
(Show Context)
Citation Context ... monoidal forgetful functors U : AM → RMR. That bialgebroid structures on A over R are in one-to-one correspondence with the strict monoidal forgetful functors UA was first pointed out by Schauenburg =-=[23]-=-, see also [26]. Of course, the notion of bimonad is much more general than bialgebroids. But there is a simple criterion for a bimonad T on RMR to be a bialgebroid: The underlying functor T must have... |

25 |
On Hopf algebras and rigid monoidal categories
- Ulbrich
- 1990
(Show Context)
Citation Context ...adjoint (see Theorem 5.4). Tannaka duality for bialgebroids has recently been proven by Phùng Hô Hái [19] following the tradition of Saavedra [20], Deligne [6] and generalizing the results of Ulbrich =-=[30]-=-, Schauenburg [21] for Hopf algebras and of Hayashi [8] for weak Hopf algebras. For more about this theory we refer to [18] and [9] and the references therein. The approach of the present paper does n... |

23 | Finite quantum groupoids and inclusions of finite type
- Szlachányi
(Show Context)
Citation Context ...lgebroid. Hopf algebroid could then be the name for a bimonad possessing some sort of antipode. The motivating example of bimonads is associated to a Takeuchi ×R-algebra [28], also called bialgebroid =-=[12, 5, 22, 26]-=-. For a bialgebroid A over R the algebra A is an extension of Re := R ⊗ Rop , so A is an Re-Re-bimodule. The underlying − : RMR → RMR where we identified functor T of the bimonad is T = A ⊗Re RMR with... |

21 |
Tannaka duality for arbitrary Hopf algebras. Algebra Berichte 66
- Schauenburg
- 1992
(Show Context)
Citation Context ...em 5.4). Tannaka duality for bialgebroids has recently been proven by Phùng Hô Hái [19] following the tradition of Saavedra [20], Deligne [6] and generalizing the results of Ulbrich [30], Schauenburg =-=[21]-=- for Hopf algebras and of Hayashi [8] for weak Hopf algebras. For more about this theory we refer to [18] and [9] and the references therein. The approach of the present paper does not fit into this s... |

15 |
A canonical Tannaka duality for finite semisimple tensor categories, preprint
- Hayashi
- 1999
(Show Context)
Citation Context ...ids has recently been proven by Phùng Hô Hái [19] following the tradition of Saavedra [20], Deligne [6] and generalizing the results of Ulbrich [30], Schauenburg [21] for Hopf algebras and of Hayashi =-=[8]-=- for weak Hopf algebras. For more about this theory we refer to [18] and [9] and the references therein. The approach of the present paper does not fit into this series mathematically but perhaps ‘phy... |

11 | Galois actions by finite quantum groupoids in ”Locally Compact Quantum Groups and Groupoids
- Szlachányi
(Show Context)
Citation Context ...(2) = ϕ(b (1)) ⊗R ϕ(b (2)) εA(ϕ(b)) = ω(εB(b))THE MONOIDAL EILENBERG-MOORE CONSTRUCTION AND BIALGEBROIDS 37 for all b ∈ B. The equations (122-123- 124-125) define what is called a bialgebroid map in =-=[27]-=-. A bialgebroid map is completely determined by ϕ: B → A since ω = εA ◦ϕ◦sB. 5.5. Bimodule induced bialgebroid morphisms. Another class of bialgebroid morphisms are obtained if we take the functor G: ... |

6 | Opmonoidal monads
- McCrudden
(Show Context)
Citation Context ...monCat) of monads in ComonCat. In this framework all functors are lax comonoidal so the monad morphisms 〈G, ϕ〉 involve lax comonoidal functors G. This approach was chosen by McCrudden in the preprint =-=[13]-=- many results of which overlap ours. But the present paper uses another method to define the category of bimonads and therefore of bialgebroids. We insist on having lax monoidal functors in monad morp... |

6 |
Axioms for weak bialgebras (preprint mathQA/9805104
- Nill
- 1998
(Show Context)
Citation Context ...of coalgebras [28]. At last but not least the bimonad description offers natural ways to define the category of bialgebroids which, even for the special case of weak bialgebras and weak Hopf algebras =-=[2, 3, 17]-=-, has not been investigated in detail yet. Unfortunately, it offers two natural ways. At first, since bimonads are monads, one can take Street’s definition [25] of the 2-category Mnd(ComonCat) of mona... |

4 |
Doctrinal adjunction, in Category Seminar (Proc
- Kelly
- 1974
(Show Context)
Citation Context ...x monoidal functors with left adjoint. The whole content of this paper rests on the following result which can be found in various forms in the categorical literature, in most general form perhaps in =-=[11]-=-. Nevertheless we provide an explicit proof in order for the paper to be selfcontained even for non-specialists. Theorem 2.1. Let 〈U, τ, ι〉 be a lax monoidal functor from 〈C, ✷ , e〉 to 〈M, ⊗, i〉. If U... |

4 |
Duals and doubles of quantum groupoids (×R-bialgebras), in: New trends in Hopf algebra theory (La Falda
- Schauenburg
- 1999
(Show Context)
Citation Context ...lgebroid. Hopf algebroid could then be the name for a bimonad possessing some sort of antipode. The motivating example of bimonads is associated to a Takeuchi ×R-algebra [28], also called bialgebroid =-=[12, 5, 22, 26]-=-. For a bialgebroid A over R the algebra A is an extension of Re := R ⊗ Rop , so A is an Re-Re-bimodule. The underlying − : RMR → RMR where we identified functor T of the bimonad is T = A ⊗Re RMR with... |

4 |
Morita theory – formal ring laws and monoidal equivalences of categories of bimodules
- Takeuchi
- 1987
(Show Context)
Citation Context ... the isomorphisms ReM ≡ RMR and SeM ≡ SMS the G has to be naturally isomorphic to a functor G ⊗Re − with a Morita equivalence bimodule SeGRe. That is to say, the rings R and S are √ Morita-equivalent =-=[29]-=-. Ordinary Morita equivalence R ∼ S arises under the further assumption that SeGRe is the k-tensor product of equivalence bimodules RHS and SH ′ R . This latter situation is the Morita base change pro... |

3 |
Dual bialgebroids for depth 2 ring extensions
- Kadison, Szlachányi
(Show Context)
Citation Context ... = ξ ′. 5. Bialgebroids 5.1. From bialgebroids to bimonads. Let k be a commutative ring, R a (possibly non-commutative) k-algebra. A Takeuchi ×R bialgebra or a left bialgebroid over R in the sense of =-=[10]-=- consists of such that • a k-algebra A with a k-algebra map s ⊗k t: R ⊗k R op → A making A into an R-R bimodule via r · a · r ′ := s(r)t(r ′ )a and • a comonoid structure 〈A, ∆, ε〉 on A in RMR BGD 1.a... |

3 |
Quantum groups and non-commutative geometry, http://www.mathematik.uni-muenchen.de/∼pareigis/Vorlesungen/02SS/QGandNCG.pdf
- Pareigis
(Show Context)
Citation Context ...dition of Saavedra [20], Deligne [6] and generalizing the results of Ulbrich [30], Schauenburg [21] for Hopf algebras and of Hayashi [8] for weak Hopf algebras. For more about this theory we refer to =-=[18]-=- and [9] and the references therein. The approach of the present paper does not fit into this series mathematically but perhaps ‘physically’. The categories we think about are module categories and th... |

2 |
Bialgebroids, ×A-bialgebras and duality, preprint QA/0012164
- Brzeziński, Militaru
(Show Context)
Citation Context |

2 | Morita base change in quantum groupoids
- Schauenburg
(Show Context)
Citation Context ...ivalence R ∼ S arises under the further assumption that SeGRe is the k-tensor product of equivalence bimodules RHS and SH ′ R . This latter situation is the Morita base change proposed by Schauenburg =-=[24]-=- while the former was named as √ Morita base change.38 KORNÉL SZLACHÁNYI 5.6. An exotic example: Hom. For the tired Reader’s sake let stand here an example of a bimonad that is not a bialgebroid. It ... |