## Balanced Coalgebroids (2000)

Citations: | 2 - 0 self |

### BibTeX

@MISC{McCrudden00balancedcoalgebroids,

author = {Paddy McCrudden},

title = {Balanced Coalgebroids},

year = {2000}

}

### OpenURL

### Abstract

A balanced coalgebroid is a V op -category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of representations. This includes the reconstruction of dual quasi-bialgebras, quasi-triangular dual quasi-bialgebras, and balanced quasi-triangular dual quasi-bialgebras; the latter of which is a quantum group when equipped with a compatible antipode. As an application we construct a balanced coalgebroid whose category of representations is equivalent to the symmetric monoidal category of chain complexes. The appendix provides the definitions of a braided monoidal bicategory and sylleptic monoidal bicategory.

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Citation Context ...cs. This article studies the reconstruction of balanced coalgebroids. A balanced coalgebroid equipped with a compatible antipode is a quantum opgroupoid [DS97] which generalizes the quantum groups of =-=[Dri87]-=-. The reconstruction theorem presented in this article includes the reconstruction of dual quasi-bialgebras, quasi-triangular dual quasi-bialgebras, and balanced quasi-triangular dual quasi-bialgebras... |

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Citation Context ...he first defines braided monoidal bicategories and the second defines sylleptic monoidal bicategories. Ordinary and enriched category theory are assumed throughout this article; as usual, [Mac71] and =-=[Kel82]-=- are references for these subjects respectively. Familiarity with 2dimensional algebra is assumed and the reader may wish to consult [B'en67] or [KS74] for general theory on this subject; in particula... |

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Citation Context ...the forgetful functor is a strong monoidal functor. In detail, if M and N are finite dimensional C-comodules Theory and Applications of Categories, Vol. 7, No. 6 74 then, using the string calculus of =-=[JS93]-=-, the arrow /.-,()*+ ffi /.-,()*+ ffi M N /.-,()*+sM N C ? ? ? @ ? ? ? ? ? ? ? @ ? ? ? CD ? ? equips M\Omega N with the structure of a C-comodule. Similarly, the unit j : C ! C equips the vector space... |

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Citation Context ...s article; the first defines braided monoidal bicategories and the second defines sylleptic monoidal bicategories. Ordinary and enriched category theory are assumed throughout this article; as usual, =-=[Mac71]-=- and [Kel82] are references for these subjects respectively. Familiarity with 2dimensional algebra is assumed and the reader may wish to consult [B'en67] or [KS74] for general theory on this subject; ... |

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Citation Context ...ive ring is determined up to Morita equivalence by its category of modules; see for example [DI71]. More generally, an algebraic theory in the sense of Lawvere may be reconstructed from its semantics =-=[Law63]-=-, and this is the subject of the celebrated theory of structure and semantics. This article studies the reconstruction of balanced coalgebroids. A balanced coalgebroid equipped with a compatible antip... |

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Citation Context ...literature. For example, when the associativity isomorphism is the identity, they are called coquasi-triangular bialgebras in [Mon93, Sch92a], braided bialgebras in [LT91] and cobraided bialgebras in =-=[Kas95]-=-. When equipped with an antipode, they have been called quasi-quantum groups [Dri87]. At the risk of creating further confusion, the author prefers braided pseudomonoidal comonoids. Theory and Applica... |

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Citation Context ... throughout this article; as usual, [Mac71] and [Kel82] are references for these subjects respectively. Familiarity with 2dimensional algebra is assumed and the reader may wish to consult [B'en67] or =-=[KS74]-=- for general theory on this subject; in particular we shall use the theory of mates [KS74]. We use the string calculus of [JS93], and the definition of a monoidal bicategory [GPS95]. The approach take... |

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Citation Context ...nd whose tensor product is the Gray tensor product ; see for example [DS97]. A Gray-category is a Gray-enriched category in the sense of [Kel82], and may be considered to be a semi-strict tricategory =-=[GPS95]-=-. In particular, every 3-category may be considered to be a Gray-category. There is a factorization system (E; M) on the category Gray 0 . The class E consists of those 2functors that are bijective on... |

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Citation Context .... 7, No. 6 72 Pontrjagin duality. Tannaka duality shows that a compact group can be reconstructed from its continuous representations in the category of finite dimensional, complex vector spaces. See =-=[Che46]-=- and [JS91] for a good exposition of Pontrjagin and Tannaka duality. Morita theory shows that a commutative ring is determined up to Morita equivalence by its category of modules; see for example [DI7... |

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Citation Context ...efinition of a pseudomonoid in a monoidal 2-category is provided. This definition mildly generalizes that of a pseudomonoid in a Gray-monoid [DS97, Section 3] and may be considered a categorification =-=[BD98]-=- of the definition of a monoid in a monoidal category. Motivating examples are provided. Recall that a monoidal bicategory is a tricategory [GPS95, Section 2.2] with exactly one object and that a mono... |

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Citation Context ... the celebrated theory of structure and semantics. This article studies the reconstruction of balanced coalgebroids. A balanced coalgebroid equipped with a compatible antipode is a quantum opgroupoid =-=[DS97]-=- which generalizes the quantum groups of [Dri87]. The reconstruction theorem presented in this article includes the reconstruction of dual quasi-bialgebras, quasi-triangular dual quasi-bialgebras, and... |

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Citation Context ...e46] and [JS91] for a good exposition of Pontrjagin and Tannaka duality. Morita theory shows that a commutative ring is determined up to Morita equivalence by its category of modules; see for example =-=[DI71]-=-. More generally, an algebraic theory in the sense of Lawvere may be reconstructed from its semantics [Law63], and this is the subject of the celebrated theory of structure and semantics. This article... |

49 |
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Citation Context ...2 Pontrjagin duality. Tannaka duality shows that a compact group can be reconstructed from its continuous representations in the category of finite dimensional, complex vector spaces. See [Che46] and =-=[JS91]-=- for a good exposition of Pontrjagin and Tannaka duality. Morita theory shows that a commutative ring is determined up to Morita equivalence by its category of modules; see for example [DI71]. More ge... |

33 |
Two dual classes of bialgebras related to the concept of “quantum groups” and “quantum Lie algebra
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Citation Context ...e appeared under many names in the literature. For example, when the associativity isomorphism is the identity, they are called coquasi-triangular bialgebras in [Mon93, Sch92a], braided bialgebras in =-=[LT91]-=- and cobraided bialgebras in [Kas95]. When equipped with an antipode, they have been called quasi-quantum groups [Dri87]. At the risk of creating further confusion, the author prefers braided pseudomo... |

32 |
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Citation Context ...? ? ? ? ? ? G C CD ? ? @ ? ? ? Again the converse is true, and there is a bijection between cobraidings on C and braidings on the monoidal category Comod f (C) [JS91, Section 10, Propostion 3]. Majid =-=[Maj92]-=- considers a variant of the above reconstruction of extra structure motivated by the theory of quantum groups. A dual quasi-bialgebra is a coalgebra C in the category of vector spaces, equipped with a... |

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Citation Context ...o we may take a to be the identity also. In this case the definition of a braided monoidal 2-category agrees with that of a braided Gray-monoid as given in [BN96, DS97]. It differs from that given in =-=[Cra98]-=- in that we require no further axioms on the braiding of the unit, as [Cra98] does. If K is a braided monoidal category, then the monoidal 2-categories K op , K co and K rev , obtained by reversing 1-... |

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A non-commutative non-cocommutative Hopf algebra in ‘Nature
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Citation Context ... allows us to construct a balanced coalgebra in Section 11 whose category of representations is equivalent to the symmetric monoidal category of chain complexes. This coalgebra was first described in =-=[Par81]-=-. There are two appendices to this article; the first defines braided monoidal bicategories and the second defines sylleptic monoidal bicategories. Ordinary and enriched category theory are assumed th... |

7 |
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Citation Context ...al 2-functor Comod f : Comon(V) ! Cat=Vect f whose value on a comonoid C is its category of finite dimensional representations Comod f (C) equipped with the forgetful functor ! : Comod f (C) ! Vect f =-=[Str89]-=-. The commutativity of (1) then states that if a comonoid C is equipped with the structure of a dual quasibialgebra, then Comod f (C) is canonically a monoidal category and the forgetful functor is mu... |

6 |
Reconstruction of hidden symmetries
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Citation Context ... general case, a coalgebra cannot be reconstructed from its ordinary category of representations equipped with its forgetful functor, and so a more sensitive theory must be employed. Indeed, Pareigis =-=[Par96]-=- uses the theory of V-actegories to facilitate reconstruction. If C is a coalgebra in V, M is a C-comodule and V is an object of V, then the arrow V\Omega ffi : V\Omega M ! V\Omega M\Omega C equips V\... |

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4 |
Categories of representations of balanced coalgebroids
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(Show Context)
Citation Context ...example 129 A Braided monoidal bicategories 133 B Sylleptic monoidal bicategories 143 1. Introduction This article is the sequel to [McC99b] and reports on the results of the author's doctoral thesis =-=[McC99a]-=-. The classification and reconstruction of mathematical objects from their representations is a broad and significant field of mathematics. For example, a locally compact abelian group can be reconstr... |

4 |
Categories of representations of coalgebroids
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(Show Context)
Citation Context ...ctor Comod 119 10 Reconstruction of balanced coalgebroids 127 11 An example 129 A Braided monoidal bicategories 133 B Sylleptic monoidal bicategories 143 1. Introduction This article is the sequel to =-=[McC99b]-=- and reports on the results of the author's doctoral thesis [McC99a]. The classification and reconstruction of mathematical objects from their representations is a broad and significant field of mathe... |

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Citation Context ...y on the graph 1 ! 2 / 3. Let J : D ! Cat be the constant 2-functor at the terminal category. Suppose that R : D ! K is a homomorphism of bicategories. Then a bi-pullback of R is a J-weighted bilimit =-=[Str80]-=- of R, and it amounts to an object fJ; Rg of K equipped with a pseudonatural equivalence K(A; fJ; Rg) ' hom(D; Cat)(J; K(A;R)): Any homomorphism R : D ! K is isomorphic to a 2-functor R 0 : D ! K and ... |