## Higher-Dimensional Algebra I: Braided Monoidal 2-Categories (1996)

Venue: | Adv. Math |

Citations: | 53 - 9 self |

### BibTeX

@ARTICLE{Baez96higher-dimensionalalgebra,

author = {John C. Baez and Martin Neuchl},

title = {Higher-Dimensional Algebra I: Braided Monoidal 2-Categories},

journal = {Adv. Math},

year = {1996},

volume = {121},

pages = {196--244}

}

### Years of Citing Articles

### OpenURL

### Abstract

We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of semistrict monoidal 2-categories and braided monoidal 2-categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the `center' of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2-categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `Higher-Dimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...

### Citations

138 | Higher-dimensional algebra and topological quantum field theory
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- 1995
(Show Context)
Citation Context ...raided monoidal 2-categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `Higher-Dimensional Algebra and Topological Quantum Field Theory' =-=[1]-=-, henceforth referred to as `HDA'. This program consists of generalizing algebraic concepts from the context of set theory to the context of n-category theory, and using the resulting language to unif... |

127 |
Tannakian categories
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- 1982
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Citation Context ...egant approach to quantum groups, which, as we shall see, makes their appearance in 3-dimensional topology much less mysterious. The class of theorems known as `Tannaka-Krein reconstruction theorems' =-=[9, 27, 32]-=- further clarifies the relation between the center construction and quantum doubles. Given a Hopf algebra H, the category Reps(H) is a C -linear abelian rigid monoidal category and equipped with a fai... |

116 | Topological gauge theories and group cohomology - Dijkgraaf, Witten - 1990 |

107 |
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- 1995
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Citation Context ...termining the correct coherence laws is a rather tricky business, so that weak n-categories have been defined so far only for ns3. They are usually called bicategories [2] for n = 2 and tricategories =-=[17]-=- for n = 3. A major challenge for higher-dimensional algebra is 2 to find a good theory of weak n-categories for all n. In any event, one expects quite generally that in either the strict or the weak ... |

101 |
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- 1994
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Citation Context ... fact, it is natural to conjecture a kind of `categorification' of the whole theory of quantum doubles. For example, one should be able to start with a `Hopf category' as defined by Crane and Frenkel =-=[7]-=- --- or, better, a `Hopf 2-algebra' --- and form the monoidal 2-category Reps(H) of its representations on `2-vector spaces' [21, 34]. The monoidal 2-category Reps(H) should be equipped with a monoida... |

82 | Formal category theory: adjointness for 2-categories - Gray - 1974 |

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- Breen
- 1994
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Citation Context ...dal category. However, they give the same 2-tangle. There is also a deep relationship between n-category theory and homotopy theory, described in HDA and the references therein, and using this, Breen =-=[4]-=- has deduced that the condition S + = S \Gamma should hold. 12 These facts constitute topological evidence that in the correct definition of a braided monoidal category, there should be an extra coher... |

48 | Yetter D.: On algebraic structures implicit in topological quantum field theories, preprint
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- 1994
(Show Context)
Citation Context ... not well developed. So far, the clearest description of hom(;; S 1 ) as a braided monoidal category in dimension 3 and a braided monoidal 2-category in dimension 4 has been given by Crane and Yetter =-=[8]-=-. There are many interesting projects left to do, however. For example, in dimension 3 it should be possible to use existing results of Kerler [24] and others to obtain a presentation of hom(;; S 1 ) ... |

21 |
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17 |
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- Eckmann, Hilton
- 1962
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- 1989
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10 |
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- 1994
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Citation Context ... in 4 dimensions: n = 2, k = 2. Topologists have already studied these 2-tangles, and the work of Carter and Saito [5] strongly suggests that they form a braided monoidal 2-category. In fact, Fischer =-=[13]-=- claims to have already shown this. His work is unfortunately rather unclear, but Kharlamov and Turaev [25] have begun to redo it more carefully. It should also 11 be re-evaluated in the light of our ... |

7 |
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(Show Context)
Citation Context ...ategory C n;1 as a limiting case. A very interesting example is the case of 2-tangles in 4 dimensions: n = 2, k = 2. Topologists have already studied these 2-tangles, and the work of Carter and Saito =-=[5]-=- strongly suggests that they form a braided monoidal 2-category. In fact, Fischer [13] claims to have already shown this. His work is unfortunately rather unclear, but Kharlamov and Turaev [25] have b... |

4 |
Introduction to bicategories, Springer
- Bénabou
- 1967
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Citation Context ...f n-morphisms. Unfortunately, determining the correct coherence laws is a rather tricky business, so that weak n-categories have been defined so far only for ns3. They are usually called bicategories =-=[2]-=- for n = 2 and tricategories [17] for n = 3. A major challenge for higher-dimensional algebra is 2 to find a good theory of weak n-categories for all n. In any event, one expects quite generally that ... |

1 |
Complete theories
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Citation Context ...at 2Cat can be regarded as a semistrict 3-category having small 2-categories as objects, 2-functors as morphisms, `pseudonatural transformations' as 2-morphisms, 18 and `modifications' as 3-morphisms =-=[3, 23]-=-. A pseudonatural transformation T between 2-functors F ; G: C ! D assigns to each object A 2 C a morphism TA : F(A) ! G(A) which satisfies the definition of a natural transformation only up to a spec... |