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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 140 (14 self)
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For a copy with the handdrawn figures please email
HigherDimensional Algebra I: Braided Monoidal 2Categories
 Adv. Math
, 1996
"... We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give lon ..."
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Cited by 53 (9 self)
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We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2category Z(C) as the `center' of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `HigherDimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...
2Tangles as a Free Braided Monoidal 2Category with Duals
, 1997
"... The algebraic characterization of tangles by Freyd, Turaev and Yetter has led to the discovery of new invariants for links. In this dissertation, we prove an analogous result one dimension higher: that the 2category of unframed, unoriented 2tangles is the free semistrict braided monoidal 2catego ..."
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Cited by 9 (3 self)
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The algebraic characterization of tangles by Freyd, Turaev and Yetter has led to the discovery of new invariants for links. In this dissertation, we prove an analogous result one dimension higher: that the 2category of unframed, unoriented 2tangles is the free semistrict braided monoidal 2category with duals on one unframed self dual object. We give appropriate definitions of the 2category of 2tangles, and of duality for monoidal and braided monoidal 2categories. We use the movie moves of Carter, Rieger and Saito, to show that there is a 2functor from this 2category to any braided monoidal 2category with duals containing an unframed self dual object. Knotted surfaces in 4space are naturally included in this characterization, sinc...
2Tangles
, 1997
"... Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R 4 may be described as certain 2morphisms in the 2category of `2tangles in 4 dimensions'. In this announcement we give a purely algebraic characterization of the ..."
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Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R 4 may be described as certain 2morphisms in the 2category of `2tangles in 4 dimensions'. In this announcement we give a purely algebraic characterization of the 2category of unframed unoriented 2tangles in 4 dimensions as the `free semistrict braided monoidal 2category with duals on one unframed selfdual object'. A forthcoming paper will contain a proof of this result using the movie moves of Carter, Rieger and Saito. We comment on how one might use this result to construct invariants of 2tangles. 1 Introduction Recent work on `quantum invariants' of knots, links, tangles, and 3manifolds depends crucially on a purely algebraic characterization of tangles in 3dimensional space. It follows from work of Freyd and Yetter, Turaev, and Shum [13, 18, 19, 20] that isotopy classes of framed oriented tangles in 3 dimensions are the morphisms of a cert...
BaxterBazhanov Model, FrenkelMoore Equation and the Braid Group
, 1995
"... In this paper the threedimensional vertex model is given, which is the duality of the threedimensional BaxterBazhanov (BB) model. The braid group corresponding to FrenkelMoore equation is constructed and the transformations R, I are found. These maps act on the group and denote the rotations of ..."
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In this paper the threedimensional vertex model is given, which is the duality of the threedimensional BaxterBazhanov (BB) model. The braid group corresponding to FrenkelMoore equation is constructed and the transformations R, I are found. These maps act on the group and denote the rotations of the braids through the angles π about some special axes. The weight function of another threedimensional vertex model related the 3D lattice integrable model proposed by Boos, Mangazeev, Sergeev and Stroganov is presented also, which can be interpreted as the deformation of the vertex model corresponding to the BB model.
Solutions of nsimplex Equation from Solutions of Braid Group Representation 1
, 1994
"... It is shown that a kind of solutions of nsimplex equation can be obtained from representations of braid group. The symmetries in its solution space are also discussed. ..."
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It is shown that a kind of solutions of nsimplex equation can be obtained from representations of braid group. The symmetries in its solution space are also discussed.
Preprint DM/IST 9/97 REFLECTIONS ON TOPOLOGICAL QUANTUM FIELD THEORY 1
, 1997
"... The aim of this article is to introduce some basic notions of Topological Quantum Field Theory (TQFT) and to consider a modification of TQFT, applicable to embedded manifolds. After an introduction based around a simple example (Section 1) the notion of a ddimensional TQFT is defined in categoryth ..."
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The aim of this article is to introduce some basic notions of Topological Quantum Field Theory (TQFT) and to consider a modification of TQFT, applicable to embedded manifolds. After an introduction based around a simple example (Section 1) the notion of a ddimensional TQFT is defined in categorytheoretical terms, as a certain type of functor from a category of ddimensional cobordisms to the category of vector spaces (Section 2). A construction due to Turaev, an operatorvalued invariant of tangles, is discussed in Section 3. It bears a strong resemblance to 1dimensional TQFTs, but carries much richer structure due to the fact that the 1dimensional manifolds involved are embedded in a 3dimensional space. This leads us, in Section 4, to propose a class of TQFTlike theories, appropriate to embedded, rather than pure, manifolds. 1.