S. E. Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183-223.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in HDA1. These modifications are necessary for the proper treatment of 2 tangles, and especially for an unambiguous statement of the Zamolodchikov tetrahedron equation, as had been noted by Breen [8] Day and Street [16] later gave a more terse formulation of the definition in HDA1. Then Crans [15] noted an error in the proof of Theorem 18 of HDA1, and explained how to fix it by adding some conditions concerning the unit object to the definition of a braided monoidal 2 category. In what follows, by a braided monoidal 2 category we mean a semistrict braided monoidal 2 category as defined ....

....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 ....

S. Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183-223.

....to work out what these are in concrete terms. More recently, Kapranov and Voevodsky [40] have considered the case n = k = 1. We gave a detailed discussion of Table 2 in an earlier paper [4] and subsequent work by various authors has improved our understanding of some of the higher entries [8, 24, 26]. There are many interesting processes going from each entry in this table to its neighbors. We list some of the main ones below. Most of these have only been thoroughly studied for low values of n and k, often in the framework of semistrict n categories, which are a kind of halfway house ....

....categories [38, 40, 48] In particular, when C is the monoidal category of representations of a Hopf algebra H, ZC is the braided monoidal category of representations of the quantum double D(H) Categorifying still further, Baez and Neuchl [8] treated the case n = 2, k = 1. Subsequently Crans [24] corrected some errors in their work and dealt with the cases n = 2, k 1. 3 Lessons from Homotopy Theory In Grothendieck s famous 600 page letter to Quillen [35] he proposed developing n category theory as a vast generalization of homotopy theory, with a special class of n categories the ....

S. E. Crans, Generalized centers of braided and sylleptic monoidal 2categories, to appear in Adv. Math., preprint available at http://wwwmath. mpce.mq.edu.au/crans/papers/papers.html

.... 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 ....

....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, are equipped with a braiding in a ....

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S. Crans. Generalized centers of braided and sylleptic monoidal 2-categories, Adv. in Math., 136 (1998), 183--223.

....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 ....

S. Crans, Generalized centers of braided and sylleptic monoidal 2categories, to appear in Adv. Math., preprint available at http://wwwmath. mpce.mq.edu.au/¸crans/papers/papers.html .

.... These modifications are necessary for the proper treatment of 2 tangles, and especially for an unambiguous statement of the Zamolodchikov tetrahedron equation, as had been noted by Breen [6] Subsequently Day and Street [11] re expressed the Baez Neuchl definition in a more compact way, and Crans [10] added some axioms governing the braiding of the unit object. In what follows we use the definition given by Crans. It turns out that one can equip T with braiding morphisms and 2 isomorphisms and check that it is a semistrict braided monoidal 2 category in this sense. There is a very special ....

....characterization of T , and thus of these knotted surfaces . 3 Statement of Theorem In what follows, by monoidal and braided monoidal 2 categories we mean semistrict ones as defined in reference [5] but with the braided monoidal 2 categories satisfying the extra axioms introduced by Crans [10], which say that R Delta; Delta , R ( Deltaj Delta; Delta) and R ( Delta; Deltaj Delta) are the identity whenever one of the arguments is the unit object I. Definition 1. A monoidal 2 category with duals is, to begin with, a monoidal 2category equipped with the following structures: 1. ....

S. Crans, Generalized centers of braided and sylleptic monoidal 2categories, to appear in Adv. Math., preprint available at http://wwwmath. mpce.mq.edu.au/¸crans/papers/papers.html

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S. E. Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183-223.

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S. E. Crans, Generalized centers of braided and sylleptic monoidal 2categories, Adv. Math. 136 (1998), 183--223.

....attempt was by Kapranov and Voevodsky [24, 23] who gave a long list of data and axioms. However, their definition contains several inaccuracies and errors, which was noted by Baez and Neuchl [5] who also gave an improved definition. I further improved the definition by adding axioms for the unit [10]. Day and Street [13] using different terminology, also gave a definition of braided monoidal 2 categories, but with fewer unit axioms. Kapranov and Voevodsky motivated their definition by referring to their MAIN PRINCIPLE OF CATEGORY THEORY [24, p. 179] that in any category it is unnatural and ....

....which hence are required to be identity 2 arrows. Also, in the weak view the braiding is an equivalence, whereas in the teisi view it is an isomorphism. The first result of this paper is that these are the only differences between (semistrict) braided monoidal 2 categories (as defined in [10]) and braided 2D teisi. The interpretation of this is that the main obstacles for proving the conjecture above will be the weakness of functoriality and the weakness of invertibility. I should mention here that Baez and Neuchl [5, p. 242] as corrected by me [10, p. 206] have shown that either ....

[Article contains additional citation context not shown here]

S. E. Crans, Generalized centers of braided and sylleptic monoidal 2categories, Adv. Math. 136 (1998), 183--223.

....unit axiom for composition in Tricat(C ; D ) does not hold, and hence that in this case Tricat(C ; D ) is not a Gray category. In fact, this error, and the underlying misconception about transformations, is exactly the same one as made by Baez and Neuchl in [2] which I observed and corrected in [12]. Nonetheless, the general result, that for tricategories C and D , Tricat(C ; D ) is a tricategory, still holds, with a virtually identical proof, the only difference being that, for D a Gray category, one shows that Tricat(C ; D ) is a (particularly simple sort of) tricategory, which is all ....

S. E. Crans, Generalized centers of braided and sylleptic monoidal 2-categories, Adv. Math. 136 (1998), 183--223.

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