(Hopf) bimonoids in duoidal categories

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

This chapter deals with unspecified duoidal —so in particular braided monoidal –categories; at which level of generality there is no bijection between the morphisms in the category and all natural transformations between the induced functors. Those monads are identified which correspond to monoids; and those opmonoidal functors are identified which correspond to comonoids. This leads to an equivalence between bimonoids in a duoidal category and bimonads on it of a certain kind. A characterization of those bimonoids is given whose induced bimonad is a Hopf monad. Examples include Hopf monoids in braided monoidal categories—such as classical Hopf algebras and Hopf group algebras—small categories, Hopf algebroids over commutative (but not arbitrary) base algebras, weak Hopf algebras of Chap. 6, Hopf monads of Chap. 3 and certain coalgebra-enriched categories called Hopf categories.

Original languageEnglish
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Verlag
Pages99-123
Number of pages25
DOIs
Publication statusPublished - Jan 1 2018

Publication series

NameLecture Notes in Mathematics
Volume2226
ISSN (Print)0075-8434

Fingerprint

Monads
Monoids
Functor
Weak Hopf Algebra
Enriched Category
Monoidal Category
Coalgebra
Morphisms
Bijection
Hopf Algebra
Equivalence
Algebra
Arbitrary

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Böhm, G. (2018). (Hopf) bimonoids in duoidal categories. In Lecture Notes in Mathematics (pp. 99-123). (Lecture Notes in Mathematics; Vol. 2226). Springer Verlag. https://doi.org/10.1007/978-3-319-98137-6_7

(Hopf) bimonoids in duoidal categories. / Böhm, G.

Lecture Notes in Mathematics. Springer Verlag, 2018. p. 99-123 (Lecture Notes in Mathematics; Vol. 2226).

Research output: Chapter in Book/Report/Conference proceedingChapter

Böhm, G 2018, (Hopf) bimonoids in duoidal categories. in Lecture Notes in Mathematics. Lecture Notes in Mathematics, vol. 2226, Springer Verlag, pp. 99-123. https://doi.org/10.1007/978-3-319-98137-6_7
Böhm G. (Hopf) bimonoids in duoidal categories. In Lecture Notes in Mathematics. Springer Verlag. 2018. p. 99-123. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-319-98137-6_7
Böhm, G. / (Hopf) bimonoids in duoidal categories. Lecture Notes in Mathematics. Springer Verlag, 2018. pp. 99-123 (Lecture Notes in Mathematics).
@inbook{eafe708d02c24a099e06cf94f4a104b3,
title = "(Hopf) bimonoids in duoidal categories",
abstract = "This chapter deals with unspecified duoidal —so in particular braided monoidal –categories; at which level of generality there is no bijection between the morphisms in the category and all natural transformations between the induced functors. Those monads are identified which correspond to monoids; and those opmonoidal functors are identified which correspond to comonoids. This leads to an equivalence between bimonoids in a duoidal category and bimonads on it of a certain kind. A characterization of those bimonoids is given whose induced bimonad is a Hopf monad. Examples include Hopf monoids in braided monoidal categories—such as classical Hopf algebras and Hopf group algebras—small categories, Hopf algebroids over commutative (but not arbitrary) base algebras, weak Hopf algebras of Chap. 6, Hopf monads of Chap. 3 and certain coalgebra-enriched categories called Hopf categories.",
author = "G. B{\"o}hm",
year = "2018",
month = "1",
day = "1",
doi = "10.1007/978-3-319-98137-6_7",
language = "English",
series = "Lecture Notes in Mathematics",
publisher = "Springer Verlag",
pages = "99--123",
booktitle = "Lecture Notes in Mathematics",

}

TY - CHAP

T1 - (Hopf) bimonoids in duoidal categories

AU - Böhm, G.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - This chapter deals with unspecified duoidal —so in particular braided monoidal –categories; at which level of generality there is no bijection between the morphisms in the category and all natural transformations between the induced functors. Those monads are identified which correspond to monoids; and those opmonoidal functors are identified which correspond to comonoids. This leads to an equivalence between bimonoids in a duoidal category and bimonads on it of a certain kind. A characterization of those bimonoids is given whose induced bimonad is a Hopf monad. Examples include Hopf monoids in braided monoidal categories—such as classical Hopf algebras and Hopf group algebras—small categories, Hopf algebroids over commutative (but not arbitrary) base algebras, weak Hopf algebras of Chap. 6, Hopf monads of Chap. 3 and certain coalgebra-enriched categories called Hopf categories.

AB - This chapter deals with unspecified duoidal —so in particular braided monoidal –categories; at which level of generality there is no bijection between the morphisms in the category and all natural transformations between the induced functors. Those monads are identified which correspond to monoids; and those opmonoidal functors are identified which correspond to comonoids. This leads to an equivalence between bimonoids in a duoidal category and bimonads on it of a certain kind. A characterization of those bimonoids is given whose induced bimonad is a Hopf monad. Examples include Hopf monoids in braided monoidal categories—such as classical Hopf algebras and Hopf group algebras—small categories, Hopf algebroids over commutative (but not arbitrary) base algebras, weak Hopf algebras of Chap. 6, Hopf monads of Chap. 3 and certain coalgebra-enriched categories called Hopf categories.

UR - http://www.scopus.com/inward/record.url?scp=85056255189&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056255189&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-98137-6_7

DO - 10.1007/978-3-319-98137-6_7

M3 - Chapter

AN - SCOPUS:85056255189

T3 - Lecture Notes in Mathematics

SP - 99

EP - 123

BT - Lecture Notes in Mathematics

PB - Springer Verlag

ER -