## Mackaay: Categorical representations of categorical groups (2004)

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Citations: | 15 - 0 self |

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@TECHREPORT{Barrett04mackaay:categorical,

author = {John W. Barrett and Marco Mackaay},

title = {Mackaay: Categorical representations of categorical groups},

institution = {},

year = {2004}

}

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### Abstract

The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1

### Citations

182 | State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31 - Turaev, Viro - 1992 |

140 | Introduction to bicategories - Bénabou - 1967 |

133 |
Review of the elements of 2-categories
- Kelly, Street
(Show Context)
Citation Context ...˜ h(f) ◦ 1G(g)). (b) ˜ h(1X) = 1h(X). This notion of natural 2-transformation is called a quasi-natural transformation in [G] (but with the 2-cell arrows reversed) and a lax natural transformation in =-=[KS]-=-; it is more general than the strictest possible notion in which the natural transformations in the definition given here are all identities. Definition 3.4 Let F, G, H : C → D be three 2-functors and... |

132 | Relativistic spin networks and quantum gravity - Barrett, Crane - 1998 |

107 |
Coherence for tricategories
- Gordon, Power, et al.
- 1995
(Show Context)
Citation Context ...n the definition of a monoidal bicategory will always be trivial in this paper. This entire structure, together with the axioms it obeys, is a special case of the definition of a tricategory given by =-=[GPS]-=-, in which the tricategory has only one object. 3.3 2-Vector spaces In this section we recall the definition, due to Kapranov and Voevodsky [KV], of the monoidal bicategory of 2-vector spaces in the c... |

101 |
Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases
- Crane, Frenkel
- 1994
(Show Context)
Citation Context ...ramatically in increasing dimension (though it might eventually stabilise). Formalisms exist for the application of categorical algebra to four-dimensional topology, for example using Hopf categories =-=[CF]-=-, categorical groups [Y-HT] or monoidal 2-categories [CS, BL, M-S]. Since braided monoidal categories are a special type of monoidal 2-category (ones with only one object), then there are examples of ... |

90 |
2-categories and Zamolodchikov tetrahedra equations,” in: Algebraic groups and their generalizations: quantum and infinite-dimensional methods
- Kapranov, Voevodsky
- 1994
(Show Context)
Citation Context ...ecial case of the definition of a tricategory given by [GPS], in which the tricategory has only one object. 3.3 2-Vector spaces In this section we recall the definition, due to Kapranov and Voevodsky =-=[KV]-=-, of the monoidal bicategory of 2-vector spaces in the completely coordinatized version. Definition 3.9 We define the monoidal bicategory 2Vect as follows: 1. 2Vect0 = N, the set of natural numbers in... |

82 |
Formal category theory: adjointness for 2-categories
- Gray
- 1974
(Show Context)
Citation Context ...ormation categorical group is K → AutK, where ∂ maps a group element to the corresponding inner automorphism. 3 Bicategories In this section we recall the definitions of 2-dimensional category theory =-=[G]-=-. First we define 2-categories, sometimes called strict 2-categories, and then indicate the changes required to give the weaker notion of bicategories. Finally we discuss monoidal structures on bicate... |

50 | Barrett-Crane model from a BoulatovOoguri eld theory over a homoge- neous space, Nucl.Phys. B574 - Pietri, Freidel, et al. - 2000 |

37 |
General relativity without coordinates
- Regge
- 1961
(Show Context)
Citation Context ...rom several problems, one of which is that there is no ‘data’ on the edges of a triangulation of the manifold, which is where one might expect to find the combi2natorial version of the metric tensor =-=[REG]-=-. Thus we arrived at the idea of constructing the monoidal 2-category of representations for the example of the categorical Lie group determined by the Lorentz group and its action on the translation ... |

32 | Spherical 2-categories and 4-manifold invariants, Adv
- Mackaay
- 1999
(Show Context)
Citation Context ... S2 as a representation of groups, one uses an intertwiner with negative and fractional entries. Consequently the monoidal 2-category of categorical representations of G(2, 3) is not semi-simple (see =-=[M-S]-=- for the precise definition of semi-simplicity). Therefore it is not clear if it can be used for the construction of topological state-sums, because semi-simplicity is an essential ingredient in the p... |

30 | State-Sum Invariants of 4-Manifolds - Crane, Kauffman, et al. - 1997 |

30 | Review of the elements of - Kelly, Street - 1974 |

27 | Skein theory and Turaev-Viro invariants, Topology - Roberts - 1995 |

22 |
G -groupoids, crossed modules and the fundamental groupoid of a topological group
- Brown, Spencer
(Show Context)
Citation Context ...two categorical groups is a strict monoidal functor. Categorical groups are equivalent to crossed modules of groups. This equivalence, and the basic properties of categorical groups, are explained in =-=[BS]-=-. Here we give a brief outline. In a categorical group G with hom-sets G(X, Y ), the categorical composition f · g and the group product ◦ are related by the interchange law (f ◦ g) · (h ◦ k) = (f · h... |

20 | Geometrical measurements in three-dimensional quantum gravity - Barrett |

20 | Fibrations in bicategories. Cahiers Topologie Géom - Street - 1980 |

16 | Finite groups, spherical 2-categories, and 4-manifold invariants
- Mackaay
(Show Context)
Citation Context ...2-category of representations for the example of the categorical Lie group determined by the Lorentz group and its action on the translation group of Minkowski space, generalising the construction of =-=[M-FG]-=-. An early draft of this paper is the reference cited by Crane and Yetter [CY-2G, CY-MC, Y-MC, CSH] who developed the particular example, and the machinery of measurable categories to handle the Lie a... |

16 |
TQFTs from homotopy 2-types
- Yetter
- 1993
(Show Context)
Citation Context ... dimension (though it might eventually stabilise). Formalisms exist for the application of categorical algebra to four-dimensional topology, for example using Hopf categories [CF], categorical groups =-=[Y-HT]-=- or monoidal 2-categories [CS, BL, M-S]. Since braided monoidal categories are a special type of monoidal 2-category (ones with only one object), then there are examples of the latter construction giv... |

11 | Measurable categories and 2-groups - Crane, Yetter |

9 |
Bicatégories monoïdales et extensions de gr-catégories”, Homology Homotopy Appl
- Rousseau
(Show Context)
Citation Context ...ties. We call this sort of bicategory a strictly unital bicategory 8The notions of 2-functor, natural 2-transformation and modification have suitable generalisations to the case of bicategories, see =-=[ROU]-=-, for example. One new phenomenon which occurs is that, for the 2-functors, the horizontal composition is no longer preserved exactly, but only up to a family of natural isomorphisms, defined as follo... |

6 | crossed modules and the fundamental groupoid of a topological group - G-groupoids - 1976 |

5 | Homologically Twisted Invariants Related to (2+1)- and (3+1)-Dimensional State-Sum Topological Quantum Field Theories. Eprint hep-th/9311082 - Yetter - 1993 |

4 | Measurable categories - Yetter |

4 |
Introduction to bicategories. 1967 Reports of the Midwest Category Seminar
- Benabou
(Show Context)
Citation Context ...associator, αX,Y,Z : f ◦ (g ◦ h) ⇒ (f ◦ g) ◦ h, which satisfies a certain coherence law. Similarly, the unital nature of the horizontal composition can be weakened by introducing natural isomorphisms =-=[BEN]-=-. However, in all the examples in this paper the unital isomorphisms are all identities. We call this sort of bicategory a strictly unital bicategory 8The notions of 2-functor, natural 2-transformati... |

3 | Knotted surfaces, braid moves, and beyond, in Knots and Quantum Gravity - Carter, Saito - 1993 |

3 |
Refined state-sum invariants of 3- and 4-manifolds. Geometric topology
- Roberts
- 1993
(Show Context)
Citation Context ...e used for the construction of topological state-sums of 3and 4-dimensional manifolds. We have not worked out what these invariants are, but possibly they are connected to Yetter’s [Y-EX] and Roberts =-=[R-EX]-=- refined invariants. One idea for a generalization would be to replace Vect by a more interesting braided monoidal category, such as the ones appearing in the representation theory of quantum groups. ... |

3 | Yetter: A more sensitive Lorentzian state sum - Crane, N - 2003 |

2 | A more sensitive lorentzian state sum. gr-qc/0301017 - Yetter |

2 |
Homologically Twisted Invariants Related to (2+1)and (3+1)-Dimensional State-Sum Topological Quantum Field Theories. Eprint archive hep-th/9311082
- Yetter
(Show Context)
Citation Context ...ategory and could be used for the construction of topological state-sums of 3and 4-dimensional manifolds. We have not worked out what these invariants are, but possibly they are connected to Yetter’s =-=[Y-EX]-=- and Roberts [R-EX] refined invariants. One idea for a generalization would be to replace Vect by a more interesting braided monoidal category, such as the ones appearing in the representation theory ... |

1 | Clebsh-Gordan decomposition for transitive representations - Lulek, Lulek, et al. - 1985 |

1 | Racah-Wigner approach to standardization of permutation representations for finite groups - Lulek, Lulek, et al. - 1985 |

1 | NG7 2RD, UK Departamento de Matemática Universidade do Algarve 8005-139 - Categ |