J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum eld theory, Jour. Math. Phys. 36 (1995), 6073-6105.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... cobordisms between cobordisms , and so on up to the nth dimension. An n dimensional extended TQFT will then be a well behaved n functor from nCob to the n category nVect of n vector spaces . Baez and Dolan have conjectured a purely algebraic description of nCob in terms of a universal property [4], which if verified will greatly assist in constructing examples. At present we 11 expect to focus on developing the theory of n categories to the point where nCob and nVect have been given rigorous definitions for all n; currently this has only been done for low values of n [6, 84] 12. ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....Indeed this was the original motivation for their de nition. See [Lam69, Sza75] Categories in which one has a reasonable notion of tensor product 15 are called monoidal, and have recently gured prominently in several areas of mathematical physics, most notably topological quantum eld theory [Ati90, BD95]. The second well known application of polycategories is to logic. Typically logicians are interested in the analysis of sequents, written: Now A 1 ; A 2 ; A n ; B 1 ; B 2 ; Bm represent formulas in some logical system. We say that the above sequent holds if and only if the ....

J. Baez and J. Dolan. Higher-dimensional algebra and topological quantum eld theory. J. Math. Phys, 36:6073-6105, 1995.

.... cobordisms between cobordisms , and so on up to the nth dimension. An n dimensional extended TQFT will then be a well behaved n functor from nCob to the n category nVect of n vector spaces . Baez and Dolan have conjectured a purely algebraic description of nCob in terms of a universal property [4], which if verified will greatly assist in constructing examples. At present we expect to focus on developing the theory of n categories to the point where nCob and nVect have been given rigorous definitions for all n; currently this has only been done for low values of n [6, 79] 11. ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....Indeed this was the original motivation for their de nition. See [Lam69, Sza75] Categories in which one has a reasonable notion of tensor product are called monoidal, and have recently gured prominently in several areas of mathematical physics, most notably topological quantum eld theory [Ati90, BD95]. The second well known application of polycategories is to logic. Typically logicians are interested in the analysis of sequents, written: A 1 ; A 2 ; A n B 1 ; B 2 ; Bm Now A 1 ; A 2 ; A n ; B 1 ; B 2 ; Bm represent formulas in some logical system. We say ....

J. Baez and J. Dolan. Higher-dimensional algebra and topological quantum eld theory. J. Math. Phys, 36:6073-6105, 1995.

....conditions for iterative application of the axioms. For example, for a structure with associativity up to isomorphism one needs an additional axiom stating that even four not only three factors can be rebracketed which then suffices to gurantee that an arbitrary number of factors can be. See [8] for the details. There is especially the idea that categorification is linked to quantization (see again [8] and [9] 10] We will see that we can make this correspondence precise in our logic based approach. A more advanced goal is then to quantize more involved (non algebraic) ....

....one needs an additional axiom stating that even four not only three factors can be rebracketed which then suffices to gurantee that an arbitrary number of factors can be. See [8] for the details. There is especially the idea that categorification is linked to quantization (see again [8] and [9] 10] We will see that we can make this correspondence precise in our logic based approach. A more advanced goal is then to quantize more involved (non algebraic) mathematical structures like manifolds. Especially, we are interested in a quantization of Penrose s nonlinear gravitons. ....

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J. C. Baez, J. Dolan, Higher dimensional algebra and topological quantum field theory, J. Math. Phys. 36, 6073-6105 (1995).

....here we deal with the topological side. We directly start from the observation that categorifications are actually quantizations of discretized versions of the structures of classical mathematics, e.g. a finite dimensional module category over Hilb (or a 2 vector space in the terminology of [15] and [16] is not really a complex vector space in quantum set theory but the analog of a module over the natural numbers. So, if we try to categorify the manifold notion, it is natural to start with the triangulation of a classical manifold (and we will see that a natural candidate for the ....

.... connected, Hausdorff, paracompact) In this case, we have to study representations in a 2category, i.e. if we first consider representations in a one dimensional (higher) 12 quantized module, we have to consider functors to 2Hilb, the 2 category of finite dimensional 2 Hilbert spaces (see [6] [15]) Again, we get isomorphy of the 2 Hilbert spaces attached to the different U i as a consequence of connectedness, and the 1 cocycle condition implies that we deal with a bundle of 2 Hilbert spaces (where the transition functors should satisfy a unitarity condition) over M . Since the 1 cocycle ....

J. C. Baez, J. Dolan, "Higher-dimensional algebra and topological quantum field theory", J. Math. Phys. 36, 6073-6105 (1995).

....When we do this from the top horizontal line to the bottom one, we get a sequence of words (a sentence) Then the LHS and the RHS of each chart move gives each of the above equivalences between sentences. Higher categories including the one defined above were defined and studied by Baez and Dolan [2]. Here we included explicit relations among 2 morphisms. 3.4.1 Remark. In this setting we see that singularities naturally give rise to a 2 category where the 1 morphisms, 2 morphisms, and relations among the 2 morphisms correspond to singularities of plane curves, surfaces, and the projections ....

Baez, John, and Dolan, James, Higher-dimensional Algebra and Topological Quantum Field Theories, J. Math. Phys. 36 (1995) 6073-6105.

....some functions can be written for all types. Map and fold are examples for polytypic functions. The results of this paper can be seen as first examples of higher order polytypic programming. On the mathematical side the paper was inspired by Baez and Dolan s program of higher dimensional algebra [BD95]. The foundation of this work is a hierarchy of weak n categories, the definition of which is current research [BD98, HMP98, Bat98] The free theories used in this paper are instances of the topological motivated notion of operads which are central structures in above work. The formal connection ....

J. Baez and J. Dolan. Higher-dimensional algebra and topological quantum field theory. Jour. Math. Phys., 36:6073--6105, 1995.

.... is inspired by the ongoing research discovering higher dimensional categorical structures (termed post modern algebra by John Baez) The notion of Lawvere theories and its inductive definition is present in Wagners work [Wag94] The notion of combing a operad tensor is inspired by Baez, Dolan [BD95, BD98] and Schmitt [Sch93] The notion of left globularity is inspired by the Batanin, Street globular approach [Bat98, Str98] The work on the multicategory approach by Hermida, Makkai, Power [HMP98, Her97] and Leinster [Lei98a, Lei98b, Lei99] bears a striking resemblance. One might be tempted to call ....

J. Baez and J. Dolan. Higher-dimensional algebra and topological quantum field theory. Jour. Math. Phys., 36:6073--6105, 1995.

....braided compact closed categories [28] In this paper, we will only consider a commutative monoid, which gives a symmetric monoidal category. We hope to explore the nonsymmetric and braided versions of this construction in future work, as well as the connections to topological quantum field theory [10]. Definition 5.12 Let M be a commutative monoid with identity e. Define a crossed M set to be a (left) M set X, together with a function j j : X M such that jmxj = jxj. This formula is more complicated in the nonabelian case. With a nonabelian group, we would require that jgxj = g Gamma1 ....

....A conformal field theory would then be a nuclear functor to the tensored category Hilb. A related issue is the extension of our work to higher dimensional categories. The theory of n Hilbert spaces [11] a higher dimensional analogue of Hilbert space, has become quite important in TQFT [10]. Baez has developed the theory of 2 Hilbert spaces with this in mind, and extended some of the work of Doplicher and Roberts to this setting [23] Finally, the category DRel suggests several further topics of investigation. One possible extension of DRel is to the theory of noncommutative ....

J. Baez, J. Dolan. Higher-dimensional algebra and topological quantum field theory. Journal of Mathematical Physics, 36:6073-6105, 1995.

....should also say that Verity has announced a similar construction, based on parity complexes. It is useful to have the various results, because different approaches may be appropriate for the various applications, which include homotopy theory [5, 9] non abelian homology [14, 7] quantum physics [2, 4, 6] and computer science [13, 12] We now outline the approach of the paper. The most important operations in an category are the compositions in various directions. In practice, one often gives a composition by means of a pasting diagram; see for example [10, 3] It is therefore desirable to ....

J. C. Baez and J. Dolan. Higher dimensional algebra and topological quantum field theory. Jour. Math. Phys., 36:6073--6105, 1995.

....require further work in higher dimensional algebra. There are a number of promising strategies. First, there is plenty of evidence that a certain class of braided monoidal 2categories with duals, the braided monoidal 2 groupoids, are essentially the same as homotopy 2 types of double loop spaces [3, 5]. These should give 2 tangle invariants with a purely homotopy theoretic flavor. In particular, for any compact 2 dimensional submanifold Sigma ae R 4 , these invariants should depend only on the homotopy type of R 4 Gamma Sigma. Second, one can construct braided monoidal 2 categories ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....monoidal category with duals, and, in particular, to the categories of representations of quantum groups. Any such functor naturally gives an invariant of framed oriented tangles. In higher dimensions we look for analogous algebraic structures. The tangle hypothesis proposed by Baez and Dolan [2] gives a characterization of n dimensional surfaces embedded in (n k) dimensional space as a k tuply monoidal n category for all n and k . An n category is a generalization of a category for which there are objects, morphisms between objects, 2 morphisms between morphisms, and so on up to ....

.... Crane and Frenkel give a method for constructing Hopf categories from Lusztig s canonical bases for quantum groups [5] Another possibility for finding appropriate 2 categories is homotopy 2 types of double loop spaces, which Baez and Dolan suggest are likely to be braided monoidal 2 categories [2, 3]. Further, Baez suggests that braided monoidal 3 Hilbert spaces are likely to be braided monoidal 2 categories with duals [1] Finally, there are several examples of braided monoidal 2 categories which have been constructed from solutions of the Zamolodchikov tetrahedron equations [11, 7] and it ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....5, 6, etc. because ultimately the general, n dimensional, notion will be needed anyway. Instead, I use the indications above to investigate 3 and 4 monoidal structures on 2D teisi. This will be more immediately useful, for example in the theory of 2 tangles [4] and in quantum field theory [2], and gives feedback which should be valuable for the further development of a theory of higher dimensional teisi. So far there have been two attempts at defining higher monoidal structures on braided monoidal 2 categories. The first attempt was by Breen [7] who defined weakly and strongly ....

J. C. Baez and J. Dolan, Higher dimensional algebra and Topological Quantum Field Theory, Jour. Math. Phys. 36 (1995), 6073--6105.

....of braided monoidal categories. These are expected to arise as algebraic homotopy types of particular connected spaces, and as weak n categories [2, 4] which have only one element in low dimensions. And there should be applications to higher dimensional TQFTs and to n tangles, see Baez et al. [1, 3], Day and Street [10] and Crane and Yetter [7, 8] Rather than dealing with weak higher dimensional categories in their full generality, I will restrict myself to the semistrict case here. Firstly, because currently semistrict 1991 Mathematics Subject Classification: 18D05 (18A05, 18D10, ....

....(k 1) monoidal n categories to semistrict k monoidal n categories is an isomorphism for k n 2 (compare [1, p. 6089] In section 4 I will give Day and Street s definition of sylleptic and symmetric monoidal 2 categories [10] also called weakly and strongly involutory 2 categories [1, 5]. The syllepsis corresponds to the 3 tensor, with the axioms for the syllepsis exactly being naturality and functoriality axioms. Day and Street s symmetry condition corresponds to the symmetry axiom above. An important class of examples of braided monoidal categories, which also comes up in ....

J. C. Baez and J. Dolan. Higher dimensional algebra and topological quantum field theory. Jour. Math. Phys., 36:6073--6105, 1995.

No context found.

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum eld theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....and applications. To complete our definitions formally, we have to prove some properties on algorithms proposed here. In general infinite cases, these are so complex that computers can provide good help. And many mathematical areas need the notion of weak n categories, or even weak categories [1, 4, 5, 7, 8, 14]. In author s feeling, it could provide a real tool to construct mathematical structures. ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995) 6073--6105.

....3 dimensions is especially important, because it has a beautiful algebraic characterization in terms of a universal property. This was initially developed by Turaev [28] Freyd Yetter [18, 29] and Joyal Street [19] and it reached a highly polished form in the work of Shum [25] In our language [1, 3], her result is that isotopy classes of framed oriented tangles in 3 dimensions are the morphisms of the free braided monoidal category with duals on one object . Using this universal property, we can easily obtain functors from this category to other braided monoidal categories with duals, such ....

....to other braided monoidal categories with duals, such as categories of representations of quantum groups. Any such functor gives an invariant of tangles, and in particular, a knot invariant. This is the easiest way to understand the Jones polynomial and its relatives [24] The tangle hypothesis [3] suggests a vast generalization of this result, applicable to n dimensional surfaces embedded in (n k) dimensional space for all n and k. This generalization involves the notion of a k tuply monoidal n category . A k tuply monoidal n category is an (n k) category that has only trivial ....

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J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105. 62

....Section 4 we describe some algebraic structures that amount to iterated categorifications of the natural numbers and the integers. A large amount of interesting mathematics emerges from the study of these structures. In Section 5 we summarize our own approach to n categories. In a previous paper [4] we sketched a program of using n categories to clarify the relationships between topological quantum field theory and more traditional approaches to algebraic topology. The present paper covers some aspects of this program in more detail, taking advantage of work that has been done in the ....

....j morphisms are the (j k) morphisms of C. In doing so we obtain a particular sort of n category with extra structure and properties, which we call a k tuply monoidal n category. Table 2 shows what we expect these to be like for low values of n and k. For example, the Eckmann Hilton argument [4, 8, 27] shows that a 2 category with one object and one morphism is a commutative monoid. Categorifying this argument, one can show that a 3 category with one object and one morphism is a braided monoidal category. Similarly, we expect that a 4 category with one object, one morphism and one 2 morphism is ....

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J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....of categori cation to quantum theory. By now it is clear that categori cation is necessary for understanding the connections between quantum eld theory and topology. It has even played a role in some attempts to nd a quantum theory of gravity. But having reviewed these subjects elsewhere [2, 3, 4], we restrict ourselves here to some of the simplest aspects of quantum physics. One of the rst steps in developing quantum theory was Planck s new treatment of electromagnetic radiation. Classically, electromagnetic radiation in a box can be described as a collection of harmonic oscillators, ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum eld theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....and applications. To complete our definitions formally, we have to prove some properties on algorithms proposed here. In general infinite cases, these are so complex that computers can provide good help. And many mathematical areas need the notion of weak n categories, or even weak categories [1, 4, 5, 7, 8, 14]. In author s feeling, it could provide a real tool to construct mathematical structures. ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995) 6073--6105.

....question of when two approaches can be considered equivalent. Henceforth by n category we always mean weak n category , as defined in this paper. For more background on n category theory and why it should be interesting, see the previous papers in this series, which we refer to as HDA0 [4], HDA1 [6] and HDA2 [2] As in those papers, we use the ordering in which the composite of morphisms f : x y and g: y z is written as fg, but when dealing with operads we write the composite of a k ary operation f with the operations g 1 ; g k as f Delta (g 1 ; g k ) 2 ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

....put these ingredients together and give a precise definition of 2 Hilbert spaces in Section 3. Why bother categorifying the notion of Hilbert space As already noted, one motivation comes from the study of topological quantum field theories, or TQFTs. In the introduction to this series of papers [2], we proposed that n dimensional unitary extended TQFTs should be treated as n functors from a certain n category nCob to a certain n category nHilb. Roughly speaking, the n category nCob should have 0 dimensional manifolds as objects, 1 dimensional cobordisms between these as morphisms, ....

....composition we use the ordering in which, for example, the composite of f : x y and g: y z is denoted f ffi g. We denote the identity morphism of an object x either as 1 x or, if there is no danger of confusion, simply as x. We refer to our earlier papers on higher dimensional algebra as HDA0 [2] and HDA1 [3] 2 H Categories Let Hilb denote the category whose objects are finite dimensional Hilbert spaces, and whose morphisms are arbitrary linear maps. Henceforth, all Hilbert spaces will taken as finite dimensional unless otherwise specified. The category Hilb is symmetric monoidal, ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

.... for granted, such as the notion of equality [8] and the emphasis on doing mathematics using 1 dimensional strings of symbols [12, 19] Starting in the late 1980s, it became apparent that higher dimensional algebra is the correct language to formulate so called topological quantum field theories [7, 20, 30]. More recently, various people have begun to formulate theories of quantum gravity using ideas from higher dimensional algebra [6, 11, 16, 22, 23] While they have tantalizing connections to string theory, these theories are best seen as an outgrowth of loop quantum gravity [24] The plan of the ....

....are most naturally done using n dimensional diagrams. But this link between n categories and n dimensional topology is precisely why there may be a nice description of nCob in the language of n categories. Dolan and I have proposed such a description, which we call the cobordism hypothesis [7]. Much work remains to be done to make this hypothesis precise and prove or disprove it. Proving it would lay the groundwork for understanding topological quantum field theories in a systematic way. But beyond this, it would help us towards a purely algebraic understanding of space and ....

J. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, Jour. Math. Phys. 36 (1995), 6073-6105.

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J. Baez and J. Dolan. Higher-dimensional algebra and topological quantum eld theory. Jour. Math. Phys., 36:6073-6105, 1995.

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