J. C. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2-categories, Adv. Math. 121 (1996), 196-244.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1, 1999 Abstract We show that, for any braided compact closed bicategory B, the bicategory Ladj(B) of left adjoints in B also admits a braided compact closed structure. 1 Introduction Over the last few years monoidal bicategories have been studied in connection with physics and geometry ([BN], C] FY] JS] KV] and of particular interest to these applications have been the notions of a braiding for a tensor and a dual for an object. In this paper, we prove a theorem which demonstrates how an old construction (the bicategory of squares [G] can be used to define braided compact ....

Baez J and Neuchl M, Higher-dimensional algebra I: braided monoidal 2categories, in: Adv. in Math. Vol. 121 (1996) 196-244.

.... this section, the definition of a braided pseudomonoid in a braided monoidal 2 category is provided, generalizing that of a braided pseudomonoid in a braided Gray monoid [DS97, Section 4] First, the definition of a braided monoidal 2 category is given, generalizing that of a braided Gray monoid [KV94, BN96, DS97, Cra98]. Recall that an equivalence in a 2 category K may be replaced with an adjoint equivalence. That is, if f is an equivalence in K then f has a right adjoint g with invertible counit and unit and j respectively. Of course, g is also the left adjoint of f with invertible unit and counit Gamma1 ....

....is a monoidal 2 category equipped with a braiding. When K is in fact a Gray monoid, the associativity equivalence a is the identity and so we may take a to be the identity also. In this case the definition of a braided monoidal 2 category agrees with that of a braided Gray monoid as given in [BN96, DS97]. It differs from that given in [Cra98] in that we require no further axioms on the braiding of the unit, as [Cra98] does. If K is a braided monoidal category, then the monoidal 2 categories K op , K co and K rev , obtained by reversing 1 cells, 2 cells and the tensor product respectively, ....

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J. Baez and M. Neuchl. Higher dimensional algebra I: Braided monoidal 2-categories, Adv. in Math., 121 (1996), 196--244.

....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 as the quantum doubles of monoidal 2 categories [7, 15]. It seems plausible that applying this construction to a monoidal 2 category with duals will give a braided monoidal 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 ....

J. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math. 121 (1996), 196-244.

....categories is to give definitions which are easier to work with, but which are sufficiently general to be useful in most problems. The first definitions of monoidal and braided monoidal 2 categories were given by Kapranov and Voevodsky [11] More recently Day and Street [8] and Baez and Neuchl [4] have given equivalent definitions of monoidal and braided monoidal 2 categories, which are slightly modified from those given by Kapranov and Voevodsky. In the following we use the formulations given by Baez and Neuchl with the addition of the relations noted in (1) of the Conclusions of that ....

....smooth tangle diagram with an associated height function, and the source and target of the morphism are represented by the point set at the top and bottom of the tangle. We define T in terms of smooth manifolds, and show that this forms a braided monoidal 2 category in the sense of Baez and Neuchl [4]. We define duality for monoidal and braided monoidal 2 categories, and define an unframed self dual object. Finally, we show that the 2 category of 2 tangles is a braided monoidal 2 category with duals with one unframed self dual object. Throughout, 2 categories (monoidal, braided, or with duals) ....

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J. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math. 121(1996), 196-244.

....Piergiulio Katis and R.F.C. Walters School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia e mail: giuliok maths.usyd.edu.au 23 February 1999 1 Introduction Over the last few years monoidal bicategories have been studied in connection with physics and geometry ([BN], C] FY] JS] KV] and of particular interest to these applications have been the notions of a braiding for a tensor and a dual for an object. In this paper we prove a theorem which deepens the connection between duals and adjoints, and which demonstrates how an old construction (the ....

Baez J and Neuchl M, Higher-dimensional algebra I: braided monoidal 2categories, in: Adv. in Math. Vol. 121 (1996) 196-244.

....n dimensional cobordisms may be represented as n morphisms, and important equations such as the Yang Baxter and Zamolodchikov equations can also be seen as n categorical coherence laws. Here we sketch some of the main ideas involved, illustrating them with many examples. For references, see [5, 6]. Complexity of trajectories in rectangular billiards Yuliy Baryshnikov, School of Mathematics, University of Hull y.baryshnikov maths.hull.ac.uk Take a cube as a billiard domain and a generic trajectory in it. Apparently, nothing more simple can be imagined in billiards but even here some ....

J. Baez and M. Neuchl. Higher-dimensional algebra I: braided monoidal 2-categories. Available as ftp://math.ucr.edu/pub/baez/bm2cat.ps.Z.

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J. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math.121 (1996), 196-244.

....here, allowing 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 ....

....homotopy theoretic flavor. In particular, for any compact 60 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 as the quantum doubles of monoidal 2 categories [7, 15]. It seems plausible that applying this construction to a monoidal 2 category with duals will give a braided monoidal 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 ....

J. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math. 121 (1996), 196-244.

....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 ....

....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 M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math. 121 (1996), 196-244.

....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 Operads It ....

J. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math. 121 (1996), 196-244.

....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, with C as the ....

J. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math.121 (1996), 196-244.

....specified 2 isomorphism, the tensorator : O f;g : A Omega g) f Omega B 0 ) f Omega B) A 0 Omega g) Again there are various axioms that must hold. These were first explicitly listed by Kapranov and Voevodsky [14] and later expressed more tersely in the language of 2 category theory [5, 11]. While the details are rather lengthy, we can equip T with unit object, tensor products, and tensorator, and check that T becomes a semistrict monoidal 2 category. As noted by Kharlamov and Turaev, Fischer s paper on 2tangles [12] has serious flaws, such as not discussing the tensorator. One ....

....check that T becomes a semistrict monoidal 2 category. As noted by Kharlamov and Turaev, Fischer s paper on 2tangles [12] has serious flaws, such as not discussing the tensorator. One can also consider gluing 2 tangles together along a hyperplane of constant x. By the Eckmann Hilton argument [2, 5], this form of gluing makes T into a semistrict braided monoidal 2 category. This means, first of all, that for any objects A and B there is a morphism R A;B : A Omega B B Omega A, called the braiding . This morphism must be a invertible up to a 2 isomorphism. In addition, for any morphisms ....

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J. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2categories, Adv. Math. 121(1996), 196-244.

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J. C. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2-categories, Adv. Math. 121 (1996), 196-244.

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J. C. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2-categories, Adv. Math. 121 (1996), 196-244.

No context found.

J. C. Baez and M. Neuchl, Higher-dimensional algebra I: Braided monoidal 2-categories, Adv. Math. 121 (1996), 196-244.

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Baez, John and Neuchl, M., Higher-Dimensional Algebra I : Braided Monoidal 2- Categories, to appear in Adv. Math.

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