A note on split extensions of bialgebras

García-Martínez, Xabier; Linden, Tim Van der

doi:10.1515/forum-2017-0016

Cited by 6 publications

(3 citation statements)

References 13 publications

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“…Let C be any pointed non-unital category. (For instance, the category of Hopf algebras over a field is such [23].) Necessarily then, certain product inclusions are not jointly strongly epimorphic.…”

Section: 8mentioning

confidence: 99%

“…The present paper is the starting point of an exploration of this new object-wise approach, which is being further developed in ongoing work. For instance, the article [22] provides a simple direct proof of a result which implies our Theorem 7.7, and in [23] cocommutative Hopf algebras over an algebraically closed field are characterised as the protomodular objects in the category of cocommutative bialgebras.…”

Section: Introductionmentioning

confidence: 98%

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Two characterisations of groups amongst monoids

Montoli

Rodelo

Linden

2018

Journal of Pure and Applied Algebra

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Abstract. The aim of this paper is to solve a problem proposed by Dominique Bourn: to provide a categorical-algebraic characterisation of groups amongst monoids and of rings amongst semirings. In the case of monoids, our solution is given by the following equivalent conditions:(i) G is a group; (ii) G is a Mal'tsev object, i.e., the category Pt G pMonq of points over G in the category of monoids is unital; (iii) G is a protomodular object, i.e., all points over G are stably strong, which means that any pullback of such a point along a morphism of monoids Y Ñ G determines a split extensionin which k and s are jointly strongly epimorphic. We similarly characterise rings in the category of semirings.On the way we develop a local or object-wise approach to certain important conditions occurring in categorical algebra. This leads to a basic theory involving what we call unital and strongly unital objects, subtractive objects, Mal'tsev objects and protomodular objects. We explore some of the connections between these new notions and give examples and counterexamples.

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Section: 8mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Two characterisations of groups amongst monoids

Montoli

Rodelo

Linden

2018

Journal of Pure and Applied Algebra

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“…This fact opens the way to many new applications of a wide range of results obtained in that general context, results having consequences in non-abelian homological algebra, radical theory and commutator theory, for instance. We refer the reader to [23] for some other categorical properties of Hopf K,coc , and to [13] for a conceptual characterization of cocommutative Hopf algebras among cocommutative bialgebras.…”

Section: Introductionmentioning

confidence: 99%

Split extension classifiers in the category of cocommutative Hopf algebras

Gran¹,

Kadjo²,

Vercruysse³

2018

Bull. Belg. Math. Soc. Simon Stevin

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We describe the split extension classifiers in the semi-abelian category of cocommutative Hopf algebras over an algebraically closed field of characteristic zero. The categorical notions of centralizer and of center in the category of cocommutative Hopf algebras is then explored. We show that the categorical notion of center coincides with the one that is considered in the theory of general Hopf algebras.3.4.1, that uses an explicit description of split exact sequences in Hopf K,coc in terms of semi-direct products (also called smash products) of cocommutative Hopf algebras. We then analyse the abstract notions of center and of centralizer, introduced by D. Bourn and G. Janelidze in [8], in the category Hopf K,coc . This part is also based on some nice results concerning centralizers due to A. Cigoli and S. Mantovani in [11]. We finally compare our description of the center in Hopf K,coc with the defintion given by A. Chirvasitu and P. Kasprzak in [10] for arbitrary (not necessarily cocommutative) Hopf algebras, and observe that they coincide in the cocommutative case. Since our results heavily rely on the Cartier-Gabriel-Konstant-Milnor-Moore decomposition theorem for cocommutative Hopf algebras which is only valid over an algebraically closed field of characteristic zero, our description only holds in this case. It remains an open question if this description can be extended to the general case. Acknowledgement. The authors are grateful to Tim Van der Linden for an important remark on a preliminary version of this paper, and to the referee for some useful suggestions.

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Crossed Modules of Monoids I: Relative Categories

Böhm

2019

Appl Categor Struct

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This is the first part of a series of three strongly related papers in which three equivalent structures are studied:-internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -crossed modules of monoids relative to this class of spans -simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this first part the theory of relative pullbacks is worked out leading to the definition of a relative category.Date: March 2018.

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A note on split extensions of bialgebras

Cited by 6 publications

References 13 publications

Two characterisations of groups amongst monoids

Two characterisations of groups amongst monoids

Split extension classifiers in the category of cocommutative Hopf algebras

Crossed Modules of Monoids I: Relative Categories

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