## Higher-Dimensional Algebra and Topological Quantum Field Theory (1995)

Venue: | Jour. Math. Phys |

Citations: | 138 - 14 self |

### BibTeX

@ARTICLE{Baez95higher-dimensionalalgebra,

author = {John C. Baez and James Dolan},

title = {Higher-Dimensional Algebra and Topological Quantum Field Theory},

journal = {Jour. Math. Phys},

year = {1995},

volume = {36},

pages = {6073--6105}

}

### Years of Citing Articles

### OpenURL

### Abstract

The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a `suspension' operation on n- categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k n + 2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a description of n- dimensional unitary extended TQFTs as weak n-functors from the `free stable weak n-category with duals on one object' to the n-category of `n-Hilbert spaces'. We conclude by describing n-categorical generalizations of deformation quantization and the quantum double construction. Introduction One important lesson we have learned from topological quantum field theory is that describ...

### Citations

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Citation Context ...important clues about the nature of higher-dimensional algebra. First, both nCob and Vect are ‘monoidal’ categories. For precise definitions of this and other terms from category theory, see Mac Lane =-=[57]-=-; roughly speaking, a category is monoidal if it has tensor products of objects and morphisms satisfying all the usual axioms, and an object 1 playing the role of identity for the tensor product. In n... |

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Citation Context ...tr = 10. S 1 as a commutative monoid object with nondegenerate trace Moving to higher dimensions, the best presentations of 3Cob for the purposes of constructing TQFTs are based on the Kirby calculus =-=[50, 66, 67]-=-. While very algebraic in flavor, these have not yet been distilled to a statement comparable to those for 1Cob and 2Cob. The Kirby calculus also gives a description of 4Cob which has yielded a few TQ... |

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Citation Context ...e invariants arising this way can be expanded as formal power series in �� h, and the coefficients, known as `Vassiliev invariants' or `invariants of finite type', have special topological propert=-=ies [9, 13]-=-. Their relation with the deformation quantization of commutative algebras is clarified by the manner in which they arise in Chern-Simons perturbation theory [2]. The operation of `taking the center' ... |

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68 |
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Citation Context ...strands of progress along these lines. First, the category of homotopy 2-types has been shown equivalent to a category whose objects are strict 2-categories having strict inverses for all k-morphisms =-=[17, 58]-=-. Moreover, the category of homotopy 3-types has been shown equivalent to a category whose objects are semistrict 3-categories having strict inverses [44, 55]. This naturally suggests the possibility ... |

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Citation Context ...theory, due to the h-cobordism theorem [61]. Also, while we will not go into it here, it is important to note the existence of a theory of piecewiselinear (PL) manifolds paralleling the smooth theory =-=[10, 21, 23, 36, 64]-=-. The smooth and PL versions of nCob are equivalent for n ≤ 6, but not in general for larger n. What we seek, however, is a unified algebraic framework for this entire collection of results, one that ... |

60 |
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Citation Context ...ing involving path integrals suggests that the TQFTs described in terms of local Lagrangians should be of this extended sort, and so far this has been borne out in rigorous work on important examples =-=[32, 33, 54, 65, 72]-=-. It is, in fact, the theory of extended TQFTs that provides the best information about the relationship between higher-dimensional algebra and TQFTs. We could at this point attempt to define `manifol... |

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Citation Context ...le semistrict monoidal and braided monoidal 2-categories have been defined by Kapranov and Voevodsky [46]. Semistrict weakly and strongly involutory monoidal 2-categories have been discussed by Breen =-=[14]-=-. Semistrict 3-categories, mentioned in the previous section, have been studied by Gordon, Power and Street [37] and Leroy [55]. n = 0 n = 1 n = 2 k = 0 sets categories 2-categories k = 1 monoids mono... |

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Citation Context ...in nature, such as nCob and Vect, are usually weak. People frequently ignore this fact, however (and the reader will note we did so in Section 1). The justification for doing so is Mac Lane’s theorem =-=[56]-=- that any weak monoidal category is equivalent to a strict one. However, the sense of ‘equivalence’ here is rather subtle and itself intimately connected with weakening. Following Kapranov and Voevods... |

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Citation Context ...ing involving path integrals suggests that the TQFTs described in terms of local Lagrangians should be of this extended sort, and so far this has been borne out in rigorous work on important examples =-=[32, 33, 54, 65, 72]-=-. It is, in fact, the theory of extended TQFTs that provides the best information about the relationship between higher-dimensional algebra and TQFTs. We could at this point attempt to define `manifol... |

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Quantum Groups, Quantum Categories and Quantum Field Theory
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Citation Context ...ing categories in which the existence of the balancing is a separate postulate [20]. In our approach it arises automatically. In fact, this idea is already implicit in the work of Frohlich and Kerler =-=[35]-=-. By a symmetric monoidal category `with duals' we mean just a braided monoidal category with duals which is also symmetric. The morphisms in C 1;3 correspond to isotopy classes of framed 1-tangles in... |

48 | Yetter D.: On algebraic structures implicit in topological quantum field theories, preprint
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Citation Context ...what extent this miracle can be generalized to higher dimensions. The search for algebraic structures appropriate for 4-dimensional TQFTs is already underway, with Donaldson theory as a powerful lure =-=[18, 22, 23, 24, 54]-=-. One would like higher-dimensional algebra to offer some guidance here, and eventually one would very much like a comprehensive picture of quantization for topological field theories in all dimension... |

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Citation Context ...which the level of one critical point passes another, two critical points coalesce, or a critical point splits in two. The study of these generic paths between Morse functions is known as Cerf theory =-=[19, 50]-=-. In the same sense as which handle attachments give generators for nCob, these paths between Morse functions give relations, known as handle slides and cancellations. We can visualize these as `movie... |

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H–Spaces from a Homotopy Point of View
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Citation Context ...ids it does so only up to a homotopy, which in turn satisfies a coherence condition up to homotopy, and so on. In fact, the whole tower of these ‘higher associativity laws’ was worked out by Stasheff =-=[69]-=- in 1963, and have an appealing geometrical description as faces of the ‘associahedron’. For example, the pentagon, being a 2-morphism, is a 2-dimensional face. One expects these higher associativity ... |

37 |
Lattice Topological Field Theory in two Dimensions, Commun.Math.Phys
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Citation Context ...heory, due to the h-cobordism theorem [61]. Also, while we will not go into it here, it is important to note the existence of a theory of piecewise-linear (PL) manifolds paralleling the smooth theory =-=[10, 21, 23, 36, 64]-=-. The smooth and PL versions of nCob are equivalent for ns6, but not in general for larger n. What we seek, however, is a unified algebraic framework for this entire collection of results, one that ap... |

34 |
On Witten’s 3-manifold invariants, preprint
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Citation Context ...ing involving path integrals suggests that the TQFTs described in terms of local Lagrangians should be of this extended sort, and so far this has been borne out in rigorous work on important examples =-=[32, 33, 54, 65, 72]-=-. It is, in fact, the theory of extended TQFTs that provides the best information about the relationship between higher-dimensional algebra and TQFTs. We could at this point attempt to define ‘manifol... |

32 |
Topological quantum field theories,” Inst. Hautes
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Citation Context ...he inner product must be determined in some other way. Many different approaches have been proposed to both these problems [3, 41], but here we only consider one, namely Atiyah's definition of a TQFT =-=[4]-=-. This is in fact a very radical approach! First, rather than attempting to describe the dynamics of fields on a single spacetime manifold, a TQFT describes the dynamics of fields in terms of a catego... |

31 |
The combinatorics of n-categorical pasting
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Citation Context .... In an n-category one has composition operations that allow one to paste together n-morphisms according to a wide variety of ‘pasting schemes’, much as one can glue together n-manifolds with corners =-=[42]-=-. Moreover, the primordial example of an (n + 1)-category is nCat. The reason is simply that nCat is closed, i.e., enriched over itself. That is, in addition to ‘n-functors’ between n-categories and ‘... |

29 |
A Guide to Quantum Groups (Cambridge
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(Show Context)
Citation Context ...sting, namely a `braided' monoidal category, in which the coherence law B y;x B x;y = 1 x\Omega y is dropped. As the name suggests, braided monoidal categories are important in 3-dimensional topology =-=[20, 34, 66]-=-. A further `categorification' of the notion of commutative monoid, namely a braided monoidal 2-category, appears to play a corresponding role in 4-dimensional topology [18, 46]. One goal of the n-cat... |

26 |
The Poisson structure on the moduli space of flat connections and chord diagrams. Topology 35
- Andersen, Mattes, et al.
- 1996
(Show Context)
Citation Context ...ave special topological properties [9, 13]. Their relation with the deformation quantization of commutative algebras is clarified by the manner in which they arise in Chern-Simons perturbation theory =-=[2]-=-. The operation of `taking the center' can also be generalized, in a subtle and striking manner. We can think of a k-tuply monoidal n-category C --- strict, semistrict, or weak --- as an object in the... |

23 | Bridged links and tangle presentations of cobordism categories
- Kerler
(Show Context)
Citation Context ...which the level of one critical point passes another, two critical points coalesce, or a critical point splits in two. The study of these generic paths between Morse functions is known as Cerf theory =-=[19, 50]-=-. In the same sense as which handle attachments give generators for nCob, these paths between Morse functions give relations, known as handle slides and cancellations. We can visualize these as ‘movie... |

21 |
Coherence theorems for lax algebras and for distributive laws
- Kelly
- 1974
(Show Context)
Citation Context ... a ‘rig category’. These are often called ring categories, but there need be no additive inverses. Precise definitions and strictification theorems for these have been given by Laplaza [53] and Kelly =-=[48]-=-. The analogy between the commutative rig R and the symmetric rig category Vect suggests the existence of a recursive hierarchy of ‘n-vector spaces’. For example, the categorical analog of an Rmodule ... |

21 |
Coherence for distributivity
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(Show Context)
Citation Context ... one might call a ‘rig category’. These are often called ring categories, but there need be no additive inverses. Precise definitions and strictification theorems for these have been given by Laplaza =-=[53]-=- and Kelly [48]. The analogy between the commutative rig R and the symmetric rig category Vect suggests the existence of a recursive hierarchy of ‘n-vector spaces’. For example, the categorical analog... |

20 |
Structure of topological lattice field theories in three dimensions
- Chung, Fukuma, et al.
- 1994
(Show Context)
Citation Context ...heory, due to the h-cobordism theorem [61]. Also, while we will not go into it here, it is important to note the existence of a theory of piecewise-linear (PL) manifolds paralleling the smooth theory =-=[10, 21, 23, 36, 64]-=-. The smooth and PL versions of nCob are equivalent for ns6, but not in general for larger n. What we seek, however, is a unified algebraic framework for this entire collection of results, one that ap... |

18 |
Lectures on non-Perturbative Canonical Quantum Gravity (World Scientific
- Ashtekar
- 1991
(Show Context)
Citation Context ...ace of states, but in a formalism where states are diffeomorphism-invariant the inner product must be determined in some other way. Many different approaches have been proposed to both these problems =-=[3, 41]-=-, but here we only consider one, namely Atiyah's definition of a TQFT [4]. This is in fact a very radical approach! First, rather than attempting to describe the dynamics of fields on a single spaceti... |

18 | Knots and quantum gravity: progress and prospects
- Baez
- 1996
(Show Context)
Citation Context ...cally equivalent. One does not expect a realistic theory of quantum gravity to have this property. Indeed, while there are many clues indicating that quantum gravity is closely related to known TQFTs =-=[6]-=-, it may be an inherently more complex sort of theory. Ideas from higher-dimensional algebra, however, may still be very useful [5]. Cobordisms While it is customary to begin in field theory by writin... |

17 |
Group-like structures in categories
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(Show Context)
Citation Context ... the same as the monoid hom(x; x). Why does a 2-category C with only one object x and one morphism 1 x give a commutative monoid hom(1 x ; 1 x )? The argument goes back at least to Eckmann and Hilton =-=[27]-=-. The elements of hom(1 x ; 1 x ) are the 2-morphisms of C, and as described in our brief review of 220 categories, we can compose ff; fi: 1 x ) 1 x either vertically or horizontally to obtain a new 2... |

16 |
Pursuing stacks, unpublished manuscript
- Grothendieck
- 1983
(Show Context)
Citation Context ...ontext that Brown first coined the term ‘higher-dimensional algebra’. Here, however, we restrict our attention to n-categorical approaches. Ever since Grothendieck’s famous 600-page letter to Quillen =-=[39]-=-, it has been tempting to associate to a space X a ‘fundamental n-groupoid’ Πn(X), some sort of n-category whose objects are points, whose morphisms are paths, whose 2-morphisms are paths between path... |

16 |
quantum groups and TQFTs
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- 1996
(Show Context)
Citation Context ...resenting them, but it is important to keep in mind the distinction: for example, composition is not strictly associative, but only associative up to equivalence, unless we treat cobordisms carefully =-=[67]-=-. f ◦ g = fg 1. Composition in nCob 3A representation of nCob is thus a functor Z: nCob → Vect. The fact that Z assigns to the cylindrical spacetime [0, 1] × M the identity on Z(M), that is, the triv... |

15 |
Operator Invariants of Tangles
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- 1990
(Show Context)
Citation Context ... and the ‘balancing’ shown in Figure 31 be unitary, where a morphism f is said to be unitary if ff † = f † f = 1. 25• �� 31. The balancing in a braided monoidal category with duals Turaev and Yetter =-=[71, 73]-=- have shown that the morphisms in C1,2 correspond to isotopy classes of framed 1-tangles in 3 dimensions. Here a couple of remarks are in order. First, in this dimension, our sort of framing is equiva... |

14 |
G-groupoids, crossed modules, and the classifying space of a topological group
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- 1976
(Show Context)
Citation Context ...strands of progress along these lines. First, the category of homotopy 2-types has been shown equivalent to a category whose objects are strict 2-categories having strict inverses for all k-morphisms =-=[17, 58]-=-. Moreover, the category of homotopy 3-types has been shown equivalent to a category whose objects are semistrict 3-categories having strict inverses [44, 55]. This naturally suggests the possibility ... |

14 |
Braided compact monoidal categories with applications to low dimensional topology
- Freyd, Yetter
- 1989
(Show Context)
Citation Context ...sting, namely a `braided' monoidal category, in which the coherence law B y;x B x;y = 1 x\Omega y is dropped. As the name suggests, braided monoidal categories are important in 3-dimensional topology =-=[20, 34, 66]-=-. A further `categorification' of the notion of commutative monoid, namely a braided monoidal 2-category, appears to play a corresponding role in 4-dimensional topology [18, 46]. One goal of the n-cat... |

12 |
Markob algebras
- YETTER
- 1986
(Show Context)
Citation Context ... and the ‘balancing’ shown in Figure 31 be unitary, where a morphism f is said to be unitary if ff † = f † f = 1. 25• �� 31. The balancing in a braided monoidal category with duals Turaev and Yetter =-=[71, 73]-=- have shown that the morphisms in C1,2 correspond to isotopy classes of framed 1-tangles in 3 dimensions. Here a couple of remarks are in order. First, in this dimension, our sort of framing is equiva... |

11 |
The Foundation of the General Theory of Relativity in The Principle of Relativity
- Einstein
- 1923
(Show Context)
Citation Context ...ating gravity. However, it has traditionally been a bit unclear what we really are asking for when we say we want a theory to be generally covariant. In Einstein's original work on general relativity =-=[30], he -=-emphasized that the equations should preserve their form under arbitrary coordinate transformations: "The general laws of nature are to be expressed by equations which hold good for all systems o... |

10 |
2-categories and 2-knots
- Fischer
- 1994
(Show Context)
Citation Context ... description of such 2-tangles as movies in which each frame is a 1-tangle in 3 dimensions, giving explicit `movie moves' which go between any two movies representing isotopic 2-tangles [18]. Fischer =-=[31]-=- has used this information to describe a 2-category of 2-tangles in 4 dimensions, and came close to proving the tangle hypothesis in this case. There are a number of loose ends, however, and recently ... |

8 |
Higher dimensional group theory’, Low dimensional topology
- BROWN
- 1982
(Show Context)
Citation Context ... popular involves Kan complexes [60], which model a space by an algebraic analog of a simplicial complex. Alternative approaches based on cubes have been developed by Brown, Higgins, Loday and others =-=[16]-=-. Indeed, it was in this context that Brown first coined the term `higher-dimensional algebra'. Here, however, we restrict our attention to n-categorical approaches. Ever since Grothendieck's famous 6... |