## A CATEGORY OF QUANTUM CATEGORIES

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@MISC{Chikhladze_acategory,

author = {Dimitri Chikhladze},

title = {A CATEGORY OF QUANTUM CATEGORIES},

year = {}

}

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### Abstract

Abstract. Quantum categories were introduced in [5] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We introduce notions of functor and natural transformation for quantum categories and consider various constructions on quantum structures.

### Citations

379 | Basic concepts of enriched category theory, volume 64 - Kelly - 1982 |

185 |
Topos Theory
- Johnstone
- 1977
(Show Context)
Citation Context ...old coproduct of an object V of V. There is a coreflexive-equalizer-preserving braided strong-monoidal functor − · − : Setf × V → V. The preservation of coreflexive equalizers is due to Lemma 0.17 in =-=[6]-=-. We have qCatSetf = Catf. Thus we obtain a functor Catf × qCatV → qCatV (28) Let i : ∆ → Catf be the canonical embedding of the simplicial category into the category of finite categories. Precomposin... |

144 |
The geometry of tensor calculus i
- Joyal, Street
- 1991
(Show Context)
Citation Context ...Appendix: Computations for quantum categories in string diagrams We introduce framed string diagrams to represent morphisms in a braided monoidal bicategory as an enrichment of the string diagrams of =-=[7]-=-. These diagrams are designed to ease presentation of quantum structures. A string diagram of [7] has edges labeled by objects of V and nodes labeled by morphisms of V and represents a morphism in V. ... |

141 |
Introduction to bicategories
- B'enabou
- 1967
(Show Context)
Citation Context ...2 DIMITRI CHIKHLADZE 2. Monoidal comonads Let B be a bicategory. We write as if B were a 2-category, regarding associativity and unitivity isomorphisms as identities. Recall that a comonad in B [14], =-=[1]-=- is a pair (B, g), where B is an object of B and g = (g, δ : g ⇒ gg, ϵ : g ⇒ 1g) is a comonoid in the homcategory B(B, B). A map of comonads (k, κ) : (B, g) → (A, g ′ ) consists of a morphism k : B → ... |

133 |
Review of the elements of 2-categories
- Kelly, Street
- 1974
(Show Context)
Citation Context ...hism consists of a morphism w : E → F and 2-cells ψ2 : wp ⇒ p(w ⊗ w), ψ0 : hj ⇒ j in B satisfying three axioms. Monoidal morphisms and opmonoidal morphisms lead us to the setting of a double category =-=[9]-=-, [19]. Recall briefly, that a double category has objects and two types of arrows, called horizontal morphisms and vertical morphisms, forming bicategories in the two directions. Also, there is a set... |

125 |
The formal theory of monads
- Street
- 1972
(Show Context)
Citation Context ... and ordinary categories, by taking the monoidal category V to be the category of sets. In this paper we approach quantum categories using the bicategorical version of the formal theory of (co)monads =-=[14]-=-. One benefit of this approach is that it makes clear the connection between quantum categories and ordinary categories, which enables us to reproduce ordinary category theory for quantum categories. ... |

77 |
Hopf algebroids and quantum groupoids
- Lu
- 1996
(Show Context)
Citation Context ...ught of as “several object” generalisations of bialgebras. They were considered for the first time in M. Takeuchi’s paper [17] and appeared later in the work of many authors in different fields, e.g. =-=[10]-=-, [13], [20], [3]. A quantum category in a general monoidal category was defined by B. Day and R. Street [5] incorporating both bialgebroids, in the way mentioned above, and ordinary categories, by ta... |

61 |
Monoidal bicategories and Hopf algebroids
- Day, Street
(Show Context)
Citation Context ...re is an isomorphism: B g ˆk �� A g′ (1) u A ∼ = k � B u By an equivalence between suitable categories, comonad structures on k : B → A correspond to diagrams (2) in B. Let B be a monoidal bicategory =-=[4]-=-. We specify n-ary tensor product pseudofunctors B n ⊗n �� BA CATEGORY OF QUANTUM CATEGORIES 3 by choosing bracketing for the tensor product to be from the left. So, the expression B1 ⊗ . . . ⊗ Bn re... |

30 |
Homotopy field theory in dimension 3 and crossed group-categories, arXiv:math/0005291
- Turaev
(Show Context)
Citation Context ...FamV → Set taking (S, {As}) to the set S determines a functor qCatFamV → Cat. (31) In this way each object in qCatFamV has an underlying category. Next we show how Hopf group coalgebras introduced in =-=[18]-=- are quantum categories (groupoids [5]).28 DIMITRI CHIKHLADZE Let G denote a group. A Hopf G-coalgebra consists of a family of algebras {Ag} indexed by elements g of G together with a family of linea... |

26 |
Hopf algebroids with bijective antipodes: axioms, integrals, and duals
- Böhm, Szlachányi
(Show Context)
Citation Context ...l object” generalisations of bialgebras. They were considered for the first time in M. Takeuchi’s paper [17] and appeared later in the work of many authors in different fields, e.g. [10], [13], [20], =-=[3]-=-. A quantum category in a general monoidal category was defined by B. Day and R. Street [5] incorporating both bialgebroids, in the way mentioned above, and ordinary categories, by taking the monoidal... |

19 | Quantum categories, star autonomy, and quantum groupoids
- Day, Street
(Show Context)
Citation Context ...uchi’s paper [17] and appeared later in the work of many authors in different fields, e.g. [10], [13], [20], [3]. A quantum category in a general monoidal category was defined by B. Day and R. Street =-=[5]-=- incorporating both bialgebroids, in the way mentioned above, and ordinary categories, by taking the monoidal category V to be the category of sets. In this paper we approach quantum categories using ... |

16 | Enriched categories, internal categories, and change of base
- Verity
- 1992
(Show Context)
Citation Context ...consists of a morphism w : E → F and 2-cells ψ2 : wp ⇒ p(w ⊗ w), ψ0 : hj ⇒ j in B satisfying three axioms. Monoidal morphisms and opmonoidal morphisms lead us to the setting of a double category [9], =-=[19]-=-. Recall briefly, that a double category has objects and two types of arrows, called horizontal morphisms and vertical morphisms, forming bicategories in the two directions. Also, there is a set of sq... |

13 |
Quantum Groups: A Path to Current Algebra
- Street
- 2007
(Show Context)
Citation Context ...ntity comodule on C is C itself. As it is a convention to name such bicategories after arrows, ComodV is called the bicategory of comodules. For more on the theory of comodules we refer the reader to =-=[15]-=-. Each comonoid morphism f : C → D determines an adjoint pair in C: f∗ ⊣ f ∗ C � �� D The comodules f ∗ : C � �� �� D and f∗ : D � C are both C as objects of V with coactions respectively C δ3 �� C ⊗ ... |

11 |
Duals and doubles of quantum groupoids (×R-Hopf algebras
- Schauenburg
- 2000
(Show Context)
Citation Context ...f as “several object” generalisations of bialgebras. They were considered for the first time in M. Takeuchi’s paper [17] and appeared later in the work of many authors in different fields, e.g. [10], =-=[13]-=-, [20], [3]. A quantum category in a general monoidal category was defined by B. Day and R. Street [5] incorporating both bialgebroids, in the way mentioned above, and ordinary categories, by taking t... |

11 | Galois actions by finite quantum groupoids in ”Locally Compact Quantum Groups and Groupoids
- Szlachányi
(Show Context)
Citation Context ...via f = φν0.�� � � �� �� A CATEGORY OF QUANTUM CATEGORIES 23 The notion of the quantum functor includes the notion of functor between small categories and the notion of weak morphism of bialgebroids =-=[16]-=-. By Section 2 an opmorphism between monoidal comonads is determined by an opmonoidal morphism between the Eilenberg-Moore objects. Thus given a quantum functor (f, φ) : (C, A) → (C′, A ′) the comodul... |

6 | Opmonoidal monads
- McCrudden
(Show Context)
Citation Context ...ht biadjoint, then Mon(i) : MonB → MonComndB has a right biadjoint too. Using the the biequivalence (2) we infer that the canonical inclusion MonB → ComndMonB has a right biadjoint. This proves [12], =-=[11]-=-:�� � �� 4 DIMITRI CHIKHLADZE 2.2. Proposition. If B admits the Eilenberg-Moore construction for comonads, then so does Mon(B). Explicitly an Eilenberg-Moore object of a monoidal comonad (E, g) is ob... |

6 |
Groups of algebras over A⊗A
- Takeuchi
(Show Context)
Citation Context ...ring, a quantum category is the same as a bialgebroid. Bialgebroids can be thought of as “several object” generalisations of bialgebras. They were considered for the first time in M. Takeuchi’s paper =-=[17]-=- and appeared later in the work of many authors in different fields, e.g. [10], [13], [20], [3]. A quantum category in a general monoidal category was defined by B. Day and R. Street [5] incorporating... |

3 |
Monads on tensor categories. Category theory 1999
- Moerdijk
(Show Context)
Citation Context ... a right biadjoint, then Mon(i) : MonB → MonComndB has a right biadjoint too. Using the the biequivalence (2) we infer that the canonical inclusion MonB → ComndMonB has a right biadjoint. This proves =-=[12]-=-, [11]:�� � �� 4 DIMITRI CHIKHLADZE 2.2. Proposition. If B admits the Eilenberg-Moore construction for comonads, then so does Mon(B). Explicitly an Eilenberg-Moore object of a monoidal comonad (E, g)... |

1 |
Email: d.chikhladze@gmail.com This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/25/1/25-01.{dvi,ps,pdf
- Phys
(Show Context)
Citation Context ...several object” generalisations of bialgebras. They were considered for the first time in M. Takeuchi’s paper [17] and appeared later in the work of many authors in different fields, e.g. [10], [13], =-=[20]-=-, [3]. A quantum category in a general monoidal category was defined by B. Day and R. Street [5] incorporating both bialgebroids, in the way mentioned above, and ordinary categories, by taking the mon... |