J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with programming semantics is also relevant to a rigorous formulation of TQFT (see below) But for that one will also need a good theory of higher dimensional duality. At the level of monoidal bicategories this is treated in [42] and the related theory of 2 Hilbert spaces is developed in [6]. 11. Applications to TQFT There are many applications of n categories to mathematical physics; let us mention just two. The most well established application is to topological quantum field theory. The connections between 3 dimensional TQFTs and category theory have been intensively studied ....

....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. Applications to string theory A second, newer, application of n categories to mathematical physics arises from string theory. In traditional quantum field theory particles are treated as pointlike, and the parallel transport of a particle along a path is described using gauge fields, that ....

J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.

....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. Applications to string theory A second, newer, application of n categories to mathematical physics arises from string theory. In traditional quantum field theory particles are treated as pointlike, and the parallel transport of a particle along a path is described using gauge fields: that ....

J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.

.... categories this scheme has to be enlarged: We have to allow for representations in a module category (Hilb is the prototype of a module category) or even higher categorification thereof (the question of giving a Hilbert space structure on such higher modules is dealt with in [8] 19] and [22]) Quantum set theory sheds new light on this enlargement of the scheme: We have to allow for modules in quantum set theory instead of simple vector space structures only, i.e. for the quantized version of the structures. We will now see that this view carries through even to the higher levels of ....

....(Universality of quantum mechanics) Quantum mechanics is universal in the sense of category theory, i.e. we conceive of it as an abstract structure formulated in arrow language which can be concretely realized in every category C with monoidal structure. Remark 8 The work in [8] 19] and [22] on the categorification of the full Hilbert space structure shows also a way how to formulate this structure in the abstract setting. The postulate above may need some additional restriction in the sense that one has to be more specific about the ring (or rig) of scalars allowed. 21 Certainly ....

J. C. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, preprint, q-alg/9609018v2. 24

.... then a weak (n 1) functor from a weak (n 1) groupoid to the weak (n 1) category (n 1)V of weak module n categories over nVect (the finite dimensional analog of nV, including Hilbert space structures we should actually take nHilb, the category of finite dimensional n Hilbert spaces instead, see [6] for an approach to n Hilbert spaces) again satisfying 8B 2 Obj (B) F (B) M On the level we then have representations of weak groupoids. Remark 1 In Part I of this work we have seen that we get the cubic versions of higher categories instead of the spherical ones if we use quantum ....

.... (again 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 ....

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J. C. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, preprint, q-alg/9609018v2.

.... the case of trialgebroids, one gets a functorial counterpart of a Hopf algebroid in the first case but a 2 Gamma C category structure is also reached in the second step (remember that the Doplicher Roberts theorem also generalizes straightforward from compact groups to compact groupoids see [Bae 1996] without changing the general algebraic structure of the representation categories occuring) From the structural requirements in the definition of a trialgebra and from the nature of the examples it suggests itself that maybe the trialgebroid generalization is just the natural setting of ....

J. C. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, preprint, q-alg/9609018v2.

....ambient tensored category. One of our goals in future work will be to find such a category. 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 ....

J. Baez. Higher-dimensional algebra II:2-Hilbert spaces. preprint, 1996.

....2 category with duals. This reduces the question to obtaining monoidal 2 categories with duals. The 2 category of unitary representations of a 2 groupoid should be a monoidal 2 category with duals, just as the category of unitary representations of a groupoid is a monoidal category with duals [1]. Moreover, the 2 category of representations of any Hopf category is a monoidal 2 category [23] and when unitary representations can be be defined, the 2 category of unitary representations should be a monoidal 2 category with duals. Some examples of Hopf categories and related structures have ....

....solutions of the Zamolodchikov tetrahedron equations [20] Many such solutions are known [11] so one may hope that some give braided monoidal 2 categories with duals. Finally, one expects braided monoidal 3 Hilbert spaces to be interesting examples of braided monoidal 2 categories with duals [1]. However, to obtain these we will probably need to use some of the constructions sketched above. Our result and its proof can probably be improved in various ways. First, we expect similar algebraic characterizations of the 2 category of framed and or oriented 2 tangles in 4 dimensions, where we ....

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J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.

.... 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 is possible that one of these may give a braided monoidal 2 category with duals. ....

J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, to appear in Adv. Math., preprint available as q-alg/9609018 and at http://math.ucr.edu/home/ baez/

....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 ....

....us to omit the conditions R (AjA;A) 1 and R (AjA;A) 1 which previously appeared in the definition of an unframed self dual object A. We refer to the paper in which the tangle hypothesis was first stated as HDA0 [3] and refer to the earlier papers in this series as HDA1 [7] HDA2 [1], and HDA3 [4] 2 A Topological Description of 2 Tangles In this section we describe the 2 category T of 2 tangles using the language of differential topology, and prove that T is a braided monoidal 2 category with duals. First we carefully describe the objects, 1 morphisms, and 2 morphisms of T ....

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J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.

....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 meantime. Various other aspects are treated in a series of papers entitled Higher Dimensional Algebra [2, 5, 6, 8]. 2 n Categories One philosophical reason for categorification is that it refines our concept of sameness by allowing us to distinguish between isomorphism and equality. In a set, two elements are either the same or different. In a category, two objects can be the same in a way while still ....

....frame for 0 j k. 7 Conclusions In this paper we have discussed iterated categorifications and stabilizations of some of the very simplest algebraic structures: the natural numbers and the integers. However, one can also categorify many other concepts: vector spaces [40] and Hilbert spaces [2], group algebras [21] algebras of formal 45 power series [5, 37] and other Hopf algebras [20, 22] sheaves [15, 17] and so on. Interesting results about these familiar structures typically have interesting categorified analogs. It is clear, therefore, that the set based mathematics we know and ....

J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.

....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, Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.

....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 Operads It turns out to be ....

J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, to appear in Adv. Math..

....of framings (or algebraically speaking, balancings) We can use this definition here because we are mainly interested in T , which is generated by an unframed selfdual object in the sense defined below. Geometrically, this object is simply a point embedded in the unit square. In previous work [1], it was shown how the balancing arises naturally in any braided monoidal category with duals. The same idea applies to braided monoidal 2 categories with duals. Explicitly, for any object A in a braided monoidal 2 category with duals, the balancing b A : A A is given by: b A = e A Omega ....

....reduces the question to obtaining monoidal 2 categories with duals. A good example of one of these should be the monoidal 2 category of unitary representations of a 2 groupoid, just as the monoidal category of unitary representations of a groupoid is an example of a monoidal category with duals [1]. Third, Crane and Frenkel have sketched a way to construct Hopf categories from Kashiwara and Lusztig s canonical bases for quantum groups [9] There is reason to hope that the representation 2categories of these Hopf categories are monoidal 2 categories with duals. Fourth, just as one can ....

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J. Baez, Higher-dimensional algebra II: 2-Hilbert spaces, to appear in Adv. Math., preprint available as q-alg/9609018 and at http://math.ucr.edu/home/ baez/

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J.C. Baez. Higher-dimensional algebra II: 2-Hilbert spaces. Research report, Department of Mathematics, University of California at Riverside, California, USA, 1996.

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