## Algebras versus coalgebras (2007)

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Venue: | Appl. Categorical Structures, DOI |

Citations: | 12 - 10 self |

### BibTeX

@ARTICLE{Wisbauer07algebrasversus,

author = {Robert Wisbauer},

title = {Algebras versus coalgebras},

journal = {Appl. Categorical Structures, DOI},

year = {2007},

pages = {10--1007}

}

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### Abstract

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 70’s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between

### Citations

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Citation Context ...aiding τ, the natural transformation − ⊗R τB,A : − ⊗R B ⊗R A → − ⊗R A ⊗R B is monad distributive for all pairs of R-algebras A, B. Similar constructions can be considered in monoidal categories (e.g. =-=[28]-=-, [36], [44]). In 4.4(2), conditions are given for the lifting of a monad to be a monad. More generally one may ask how the lifted functor T becomes a monad without T being required to be a monad. The... |

196 | Toposes, Triples and Theories
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(Show Context)
Citation Context ...his clear we will pay some attention to the so called mixed distributive laws between endofunctors introduced in Beck [6] and further studied in van Osdol [54] and Wolff [57] (see also Barr and Wells =-=[4]-=- and ˇ Skoda [46, Section 9]). These were rediscovered in Turi and Plotkin [52] in the context of operational semantics and by Brzeziński and Majid [14] in the form of entwining structures between the... |

177 |
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(Show Context)
Citation Context ...ory which applies to a large part of algebraic structures. Initially it started with the study of abstract algebras, that is, sets A with a collection of (n-ary) operations on A. In his thesis (1963, =-=[32]-=-) Lawvere suggested to formulate these general settings in the language of categories and functors. In particular it turned out that adjoint pairs of functors are of central importance. The study of m... |

132 |
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(Show Context)
Citation Context ...s considered by ˇ Skoda in [45] puts this in a wider context. We note that for some purposes it is natural to consider monads and comonads in 2-categories and we refer to Street [47], Lack and Street =-=[31]-=-, Power and Watanabe [43], Lenisa, Power and Watanabe [33] and Tanaka and Power [51] for a treatment in this direction. Szlachányi also considers 2-categories in [48] to understand (op)monoidal functo... |

87 | The structure of corings: Inductions functors, Maschke type theorem, and Frobenius and Galois-type properties - Brzeziński |

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Citation Context ...rned out that adjoint pairs of functors are of central importance. The study of monads (triples) and their modules and comonads (cotriples) and their comodules was initiated by Eilenberg and Moore in =-=[21]-=- and in Beck [5]. This approach covers an extremely wide range of applications including not only the classical notions from module theory and topology but also those from model theory and logic, the ... |

69 | A categorical programming language
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(Show Context)
Citation Context ...nding of this situation and in this case the fundamental theorem implies in turn the existence of an antipode (see 5.19, 5.20). Inspired by questions arising in computer science, Hagino introduced in =-=[25]-=- for two functors F, G : A → B between arbitrary categories the notion of (F, G)-algebras (we call them (F, G)-dimodules). These generalise modules as well as comodules of endofunctors and in Section ... |

64 |
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(Show Context)
Citation Context ... and Wang in [53] in braided monoidal categories by defining unifying braided bialgebras (and Hopf algebras). For further work in this direction the reader may consult [55], Böhm, Nill, and Szlachány =-=[8]-=-, and Caenepeel, Wang, and Yin [20]. Given the data for a braided bimonad it may be possible to adapt the arguments from [7] to derive for (braided) bimonads the fundamental theorem from the existence... |

41 |
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(Show Context)
Citation Context ...dling of functors as suggested by Eilenberg, Mac Lane and their schools. To make this clear we will pay some attention to the so called mixed distributive laws between endofunctors introduced in Beck =-=[6]-=- and further studied in van Osdol [54] and Wolff [57] (see also Barr and Wells [4] and ˇ Skoda [46, Section 9]). These were rediscovered in Turi and Plotkin [52] in the context of operational semantic... |

35 |
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(Show Context)
Citation Context ...dol [54] and Wolff [57] (see also Barr and Wells [4] and ˇ Skoda [46, Section 9]). These were rediscovered in Turi and Plotkin [52] in the context of operational semantics and by Brzeziński and Majid =-=[14]-=- in the form of entwining structures between the tensor product of an algebra and a coalgebra over a commutative ring R. Some of these notions were more generally handled in monoidal categories by Mes... |

26 | Distributivity for endofunctors, pointed and copointed endofunctors, monads and comonads
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(Show Context)
Citation Context ...xt. We note that for some purposes it is natural to consider monads and comonads in 2-categories and we refer to Street [47], Lack and Street [31], Power and Watanabe [43], Lenisa, Power and Watanabe =-=[33]-=- and Tanaka and Power [51] for a treatment in this direction. Szlachányi also considers 2-categories in [48] to understand (op)monoidal functors for bialgebroids. His approach is similar in spirit to ... |

18 |
From varieties of algebras to covarieties of coalgebras, Electron
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(Show Context)
Citation Context ...m ←− L exists in A, then it belongs to A G . For more information about the behaviour of limits and colimits and examples of modules and comodules in the category of sets we refer to Adámek and Porst =-=[1]-=-. 2.11. Comonads. A comonad is a triple G = (G, δ, ε), where G : A → A is a functor and δ : G → GG, ε : G → IA, are natural transformations with commuting diagrams G δ �� δ �� GG Gδ �� GG δG �� GGG , ... |

16 | The factorisation problem and smash biproducts of algebras and coalgebras’, Algebr. Represent. Theory 3 - Caenepeel, Ion, et al. - 2000 |

15 | Hopf (bi-)modules and crossed modules in braided monoidal categories
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(Show Context)
Citation Context ...ili in [40]. To prove this for module categories, the twist map plays a central role. In monoidal categories this can be replaced by a more general braiding and it is shown in Bespalov and Drabant in =-=[7]-=- that in braided monoidal categories the existence of an antipode implies the fundamental theorem provided idempotents split in the base category. There is some hope to get a similar result in not nec... |

11 |
Elements of the general theory of coalgebras. LUATCS’99
- Gumm
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(Show Context)
Citation Context ...lassical) algebra still awaits exploration. For a deeper presentation of examples and usage of (co)algebras in universal algebra and computer science the interested reader may consult Gumm’s articles =-=[23, 24]-=-. To avoid confusion, we mention that, as usual, functor symbols are written on the left side of an object. When referring to examples in module categories over a commutative ring R, following the usa... |

8 | Composite cotriples and derived functors, Seminar on Triples and Categorical Homology Theory - Barr - 1969 |

7 | De Lombaerde, A categorical approach to Turaev’s Hopf groupcoalgebras
- Caenepeel, M
(Show Context)
Citation Context ... space categories) are investigated by Brzeziński and Nichita in [15]. As a special case we can look at a set G and the endofunctor G×− on the category of sets. As outlined in Caenepeel and Lombaerde =-=[19]-=-, the existence of an antipode for the related bimonad is then equivalent to G having a group structure and implies the fundamental theorem. The notions introduced here deepens the understanding of th... |

5 | Crossed products by a coalgebra - Brzeziński - 1997 |

5 | Yang-Baxter systems and entwining structures
- Brzezinski, Nichita
- 2005
(Show Context)
Citation Context ...ds to similar formulas and is developed by Baez in [2]. Relations between Yang-Baxter op3s4 erators and entwining structures (on vector space categories) are investigated by Brzeziński and Nichita in =-=[15]-=-. As a special case we can look at a set G and the endofunctor G×− on the category of sets. As outlined in Caenepeel and Lombaerde [19], the existence of an antipode for the related bimonad is then eq... |

5 |
Universelle Coalgebra, in: Allgemeine Algebra
- Gumm
- 2003
(Show Context)
Citation Context ...lassical) algebra still awaits exploration. For a deeper presentation of examples and usage of (co)algebras in universal algebra and computer science the interested reader may consult Gumm’s articles =-=[23, 24]-=-. To avoid confusion, we mention that, as usual, functor symbols are written on the left side of an object. When referring to examples in module categories over a commutative ring R, following the usa... |

4 | R-commutative geometry and quantization of Poisson algebras
- Baez
- 1991
(Show Context)
Citation Context ...noncommutative geometry the role of the twist map can be replaced by an arbitrary YangBaxter operator (on vector space categories). This approach leads to similar formulas and is developed by Baez in =-=[2]-=-. Relations between Yang-Baxter op3s4 erators and entwining structures (on vector space categories) are investigated by Brzeziński and Nichita in [15]. As a special case we can look at a set G and the... |

4 |
Theory of braided Hopf crossed products
- Guccione, Guccione
(Show Context)
Citation Context ...lisations of braided Hopf algebras, he did not refer to the braiding on the base category (of vector spaces). Similar situations were studied by Menini and Stefan in [38] (see 5.18), the Gucciones in =-=[22]-=-, and Kharchenko in [30]. Another view on the relations between Yang-Baxter operators and entwining structures is given by Brzeziński and Nichita in [15]. The conditions for a braided bialgebra given ... |

4 |
Adjoint lifting theorems for categories of modules
- Johnstone
- 1975
(Show Context)
Citation Context ...R : LR → LRLR and counit ε : LR → IB. 3 Relations between functors To study the relationship between various module categories, the following definition is of interest. It was formulated in Johnstone =-=[27]-=- for monads but we also consider it for arbitrary endofunctors. 3.1. Lifting of functors. Let F and G be endofunctors of the categories A and B respectively. Given functors T : A → B, T : AF → BG, and... |

3 |
Connected braided Hopf algebras
- Kharchenko
(Show Context)
Citation Context ...f algebras, he did not refer to the braiding on the base category (of vector spaces). Similar situations were studied by Menini and Stefan in [38] (see 5.18), the Gucciones in [22], and Kharchenko in =-=[30]-=-. Another view on the relations between Yang-Baxter operators and entwining structures is given by Brzeziński and Nichita in [15]. The conditions for a braided bialgebra given in [49, Definition 5.1] ... |

2 | Brzeziński’s crossed products and Braided Hopf crossed products - Luigi, Guccione, et al. |

2 |
Double quantum groups
- Hobst, Pareigis
- 2001
(Show Context)
Citation Context ...or a mixed distributive law for an algebra A and a coalgebra C over a commutative ring R in [14, Definition 2.1] (see 5.8). The connection between this notions is also mentioned in Hobst and Pareigis =-=[26]-=-. It was observed by Takeuchi that these structures are closely related to corings (see [13, Proposition 2], [16, 32.6]). This is a special case of 5.4(b) since the coring A⊗R C is just a comonad on t... |

1 |
Triples, modules and cohomology
- Beck
- 1967
(Show Context)
Citation Context ...oint pairs of functors are of central importance. The study of monads (triples) and their modules and comonads (cotriples) and their comodules was initiated by Eilenberg and Moore in [21] and in Beck =-=[5]-=-. This approach covers an extremely wide range of applications including not only the classical notions from module theory and topology but also those from model theory and logic, the latter being of ... |

1 |
Dingguo and Yin Yanmin, Yetter-Drinfeld modules over weak Hopf algebras and the center construction
- Caenepeel, Wang
- 2005
(Show Context)
Citation Context ...dal categories by defining unifying braided bialgebras (and Hopf algebras). For further work in this direction the reader may consult [55], Böhm, Nill, and Szlachány [8], and Caenepeel, Wang, and Yin =-=[20]-=-. Given the data for a braided bimonad it may be possible to adapt the arguments from [7] to derive for (braided) bimonads the fundamental theorem from the existence of an antipode S : B → B (see 5.13... |

1 | Coalgebras, braidings, and distributive laws
- Kasangian, Lack, et al.
- 2004
(Show Context)
Citation Context ... Yang-Baxter equation are also used by Menini and Stefan in [38] to define compatible flip morphisms for monads on arbitrary categories and their concept was extended by Kasangian, Lack and Vitale in =-=[29]-=- (see 5.18). By a suggestion of Manin, in noncommutative geometry the role of the twist map can be replaced by an arbitrary YangBaxter operator (on vector space categories). This approach leads to sim... |