## BACHUKI MESABLISHVILI, TBILISI AND (710)

### BibTeX

@MISC{Wisbauer710bachukimesablishvili,,

author = {Robert Wisbauer and D Üsseldorf},

title = {BACHUKI MESABLISHVILI, TBILISI AND},

year = {710}

}

### OpenURL

### Abstract

Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal

### Citations

134 | G.: Towards a mathematical operational semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ... of algebras (monads) as well as of coalgebras (comonads) is well understood in various fields of mathematis as algebra (e.g. [8]), universal algebra (e.g. [13]), logic or operational semantics (e.g. =-=[31]-=-), theoretical computer science (e.g. [23]). The relationship between monads and comonads is controlled by distributive laws introduced in the seventies 12 BACHUKI MESABLISHVILI, TBILISI AND ROBERT W... |

66 |
Adjoint functors and triples
- Eilenberg, Moore
- 1965
(Show Context)
Citation Context ...joints of bimonads This section deals with the transfer of properties of monads and comonads to adjoint (endo-)functors. The relevance of this interplay was already observed by Eilenberg and Moore in =-=[11]-=-. An effective formalism to handle this was developed for adjunctions in 2categories and is nicely presented in Kelly and Street [16]. For our purpose we only need this for the 2-category of categorie... |

38 |
Distributive laws, in: Seminar on Triples and Categorical Homology Theory
- Beck
(Show Context)
Citation Context ....g. [23]). The relationship between monads and comonads is controlled by distributive laws introduced in the seventies 12 BACHUKI MESABLISHVILI, TBILISI AND ROBERT WISBAUER, D ÜSSELDORF by Beck (see =-=[2]-=-). In algebra one of the fundamental notions emerging in this context are the Hopf algebras. The definition is making heavy use of the tensor product and thus generalisations of this theory were mainl... |

34 |
Kan Extensions in Enriched Category Theory
- Dubuc
- 1970
(Show Context)
Citation Context ...mmutative diagram R B �� ��� ��� R �� UT ��������� A. Then R(b) = (R(b),βb) for some βb : TR(b) → R(b) and the collection {βb, b ∈ B} constitutes a natural transformation βR : TR → R. It is proved in =-=[10]-=- that the natural transformation tR : T Tη βL �� TRL �� RL is a morphism of monads. By the dual of [21, Theorem 4.4], we obtain: The functor R is an equivalence of categories iff the functor R is mona... |

30 |
Review of the elements of
- Kelly, Street
- 1974
(Show Context)
Citation Context ...e of this interplay was already observed by Eilenberg and Moore in [11]. An effective formalism to handle this was developed for adjunctions in 2categories and is nicely presented in Kelly and Street =-=[16]-=-. For our purpose we only need this for the 2-category of categories and for convenience we recall the basic facts of this situation here. 7.1. Adjunctions. Let L : A → B, R : B → A be an adjoint pair... |

25 | Monads on tensor categories
- Moerdijk
(Show Context)
Citation Context ...ads on a monoidal category as monads whose functor part is comonoidal by Bruguières and Virelizier in [7, 2.3] may be seen as going in this direction. Such functors are called Hopf monads in Moerdijk =-=[22]-=- and opmonoidal monads in McCrudden [18, Example 2.5]. In 2.2 we give more details of this notion. Another extension of the theory of corings are the generalised bialgebras in Loday in [17]. These are... |

19 | Frobenius monads and pseudomonoids
- Street
(Show Context)
Citation Context ... the endofunctor Map(G, −) is a Hopf monad on the catgeory of sets (7.9). Note that the pattern of our definition of bimonads resembles the definition of Frobenius monads on any category by Street in =-=[27]-=-. Those are monads T = (T,µ,η) with natural transformations ε : T → I and ρ : T → TT, subject to suitable conditions, which induce a comonad structure δ = Tµ · ρT : T → TT and product and coproduct on... |

18 | On Galois comodules - Wisbauer - 2006 |

17 | Integrals for Braided Hopf Algebras - Bespalov, Kerler, et al. |

15 | Hopf (bi-)modules and crossed modules in braided monoidal categories
- Bespalov
(Show Context)
Citation Context ...ons of this theory were mainly considered for monoidal categories. They allow readily to transfer formalisms from the category of vector spaces to the more general settings (e.g. Bespalov and Brabant =-=[3]-=- and [21]). A Hopf algebra is an algebra as well as a coalgebra. Thus one way of generalisation is to consider distinct algebras and coalgebras and some relationship between them. This leads to the th... |

13 | Adjointable monoidal functors and quantum groupoids”, in Hopf algebras in Noncommutative Geometry and
- Szlachányi
- 2005
(Show Context)
Citation Context ...ntroduced and named Hopf monads by Moerdijk in [22, Definition 1.1] and called bimonads by Bruguières and Virelizier in [7, 2.3]. It is mentioned in [7, Example 2.8] that Szlachányi’s bialgebroids in =-=[29]-=- may be interpreted in terms of such ”bimonads”. It is preferable to use the terminology from [18] since these functors are neither bimonads nor Hopf monads in a strict sense but rather an entwining (... |

12 | Monads and comonads in module categories
- Böhm, Brzeziński, et al.
- 804
(Show Context)
Citation Context ...R-modules is a Hopf monad. By 7.5, B ⊗R − is a bimonad (with antipode) if and only if its right adjoint functor HomR(B, −) is a bimonad (with antipode). This situation is considered in more detail in =-=[5]-=-. If B is finitely generated and projective as an R-module and B ∗ = HomR(B,R), then HomR(B, −) ≃ B ∗ ⊗R − and we obtain the familiar result that B is a Hopf algebra if and only if B ∗ is. 7.9. Charac... |

12 |
Generalized bialgebras and triples of operads, arXiv:math/0611885v2 [math.QA
- Loday
(Show Context)
Citation Context ...n Moerdijk [22] and opmonoidal monads in McCrudden [18, Example 2.5]. In 2.2 we give more details of this notion. Another extension of the theory of corings are the generalised bialgebras in Loday in =-=[17]-=-. These are Schur functors (on vector spaces) with a monad structure (operads) and a specified coalgebra structure satisfying certain compatibility conditions [17, 2.2.1]. While in [17] use is made of... |

12 | The monoidal Eilenberg-Moore construction and bialgebroids - Szlachányi - 2003 |

12 | Survey of braided Hopf algebras, in: New Trends in Hopf Algebra Theory - Takeuchi |

12 | Algebras versus coalgebras
- Wisbauer
(Show Context)
Citation Context ...ed by Beck [2], Barr [1] and others in the seventies of the last century. They are a fundamental tool for us and we recall some facts needed in the sequel. For more details and references we refer to =-=[33]-=-. 2.1. Entwining from monad to comonad. Let T = (T,m,e) be a monad and G = (G,δ,ε) a comonad on a category A. A natural transformation λ : TG → GT is called a mixed distributive law or entwining from ... |

9 |
Opmonoidal monads, Theory and
- McCrudden
(Show Context)
Citation Context ...res and Virelizier in [7, 2.3]. It is mentioned in [7, Example 2.8] that Szlachányi’s bialgebroids in [29] may be interpreted in terms of such ”bimonads”. It is preferable to use the terminology from =-=[18]-=- since these functors are neither bimonads nor Hopf monads in a strict sense but rather an entwining (as in 2.1) between the monad T and the comonad T(I) ⊗ − on V: Indeed, the compatibility conditions... |

8 |
Composite cotriples and derived functors, Seminar on Triples and Categorical Homology Theory
- Barr
- 1969
(Show Context)
Citation Context ... · ρT : T → TT and product and coproduct on T satisfy the compatibility condition Tµ · δT = δ · µ = µT · Tδ. 2. Distributive laws Distributive laws between endofunctors were studied by Beck [2], Barr =-=[1]-=- and others in the seventies of the last century. They are a fundamental tool for us and we recall some facts needed in the sequel. For more details and references we refer to [33]. 2.1. Entwining fro... |

7 | Monads of effective descent type and comonadicity
- Mesablishvili
(Show Context)
Citation Context ...ction U H ⊣ φ H are given by the formulas: η (a, θa) = θa : (a,θa) → φ H U H (a,θa) = (H(a),δa) and σa = εa : H(a) = U H φ H (a) → a. Since ε is a split epimorphism, it follows from Corollary 3.17 of =-=[20]-=- that, when A is Cauchy complete, the functor φ H is monadic. Since K H (a) = ((H(a),δa),ma), it is easy to see that the a-component of αK H : ̂ HK H → K H is just the morphism ma : HH(a) → H(a), and ... |

7 |
Combining a monad and a comonad. Theor
- Power, Watanabe
- 2002
(Show Context)
Citation Context ...bras (comonads) is well understood in various fields of mathematis as algebra (e.g. [8]), universal algebra (e.g. [13]), logic or operational semantics (e.g. [31]), theoretical computer science (e.g. =-=[23]-=-). The relationship between monads and comonads is controlled by distributive laws introduced in the seventies 12 BACHUKI MESABLISHVILI, TBILISI AND ROBERT WISBAUER, D ÜSSELDORF by Beck (see [2]). In... |

6 | Facets of Descent, III : Monadic Descent for - Janelidze, Tholen |

6 | Entwining Structures in Monoidal Categories
- Mesablishvili
(Show Context)
Citation Context ...his theory were mainly considered for monoidal categories. They allow readily to transfer formalisms from the category of vector spaces to the more general settings (e.g. Bespalov and Brabant [3] and =-=[21]-=-). A Hopf algebra is an algebra as well as a coalgebra. Thus one way of generalisation is to consider distinct algebras and coalgebras and some relationship between them. This leads to the theory of e... |

6 |
Restricting the comparison functor of an adjunction to projective objects, Quaest
- Sobral
- 1983
(Show Context)
Citation Context ... �� with F(f) a split monomorphism in A, there exists a morphism h : b2 → b such that hf = g. We write Inj(F, B) for the full subcategory of B with objects all F-injectives. The following result from =-=[26]-=- will be needed. 3.8. Proposition. Let η,ε : F ⊣ R : A → B be an adjunction. For any object b ∈ B, the following assertions are equivalent: (a) b is F-injective; (b) b is a coretract for some R(a), wi... |

5 |
Universelle Coalgebra, in: Allgemeine Algebra
- Gumm
- 2003
(Show Context)
Citation Context ... 28 References 32 1. Introduction The theory of algebras (monads) as well as of coalgebras (comonads) is well understood in various fields of mathematis as algebra (e.g. [8]), universal algebra (e.g. =-=[13]-=-), logic or operational semantics (e.g. [31]), theoretical computer science (e.g. [23]). The relationship between monads and comonads is controlled by distributive laws introduced in the seventies 12... |

4 | Cauchy completion in category theory - Borceux, Dejean - 1986 |

4 |
Adjoint triangles, Rep. Midwest Category Semin
- Dubuc
- 1968
(Show Context)
Citation Context ... �� �� �� � �� �� �� 12 BACHUKI MESABLISHVILI, TBILISI AND ROBERT WISBAUER, D ÜSSELDORF (e) for any a ∈ A, ε φG(a) = ε (G(a),δa) is an isomorphism. Proof. That (a) and (b) are equivalent is proved in =-=[9]-=-. By the proof of [12, Theorem of 2.6], for any a ∈ A, ε φ G (a) = ε (G(a),δa) = (t F )a, thus (a) and (e) are equivalent. By Remark 3.9, (d) implies (e). Since B admits equalisers by our assumption o... |

4 | Descent in categories of (co)algebras - Mesablishvili - 2005 |

4 |
Algebra 24
- unknown authors
- 1973
(Show Context)
Citation Context ...d G if the diagrams G eG ��������� ��� ��Ge ��� TG � GT, λ λ TG�� ��� � Tε �� T �� εT ������� �� GT TG Tδ �� λG TGG �� GTG and TTG Tλ �� TGT λT �� GTT λ GT δT Gλ � GGT are commutative. It is shown in =-=[34]-=- that for an arbitrary mixed distributive law λ : TG → GT from a monad T to a comonad G, the triple ̂ G = ( ̂ G, ̂ δ, ̂ε), is a comonad on the category AT of T-modules (also called T-algebras), where ... |

3 |
Projectivity and injectivity relative to a functor
- Hardie
- 1957
(Show Context)
Citation Context ...is an isomorphism of monads. In view of the characterisation of Galois functors we have a closer look at some related classes of relative injective objects. Let F : B → A be any functor. Recall (from =-=[14]-=-) that an object b ∈ B is said to be F-injective if for any diagram in B, AT b1 f �� b2 g b �� with F(f) a split monomorphism in A, there exists a morphism h : b2 → b such that hf = g. We write Inj(F,... |

3 | Distributive laws for actions of monoidal categories - Skoda - 406 |