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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...
Higherdimensional algebra IV: 2Tangles
"... Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we p ..."
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Cited by 35 (10 self)
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Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2category is the ‘free semistrict braided monoidal 2category with duals on one unframed selfdual object’. By this universal property, any unframed selfdual object in a braided monoidal 2category with duals determines an invariant of 2tangles in 4 dimensions. 1
An invariant of tangle cobordisms
"... In [9] to a plane diagram D of an oriented tangle T with 2n bottom and 2m top endpoints we associated a complex F(D) of (H m, H n)bimodules, ..."
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Cited by 30 (3 self)
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In [9] to a plane diagram D of an oriented tangle T with 2n bottom and 2m top endpoints we associated a complex F(D) of (H m, H n)bimodules,
Diagrammatics, Singularities, and Their Algebraic Interpretations
 in ``10th Brazilian Topology Meeting, Sa~ o Carlos, July 22 26, 1996,'' Mathematica Contempora^ nea
, 1996
"... This series of lectures reviews the remarkable feature of quantum topology: There are unexpected direct relations among algebraic structures and the combinatorics of knots and manifolds. The 6j symbols, Hopf algebras, triangulations of 3manifolds, TemperleyLieb algebra, and braid groups are rev ..."
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Cited by 21 (2 self)
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This series of lectures reviews the remarkable feature of quantum topology: There are unexpected direct relations among algebraic structures and the combinatorics of knots and manifolds. The 6j symbols, Hopf algebras, triangulations of 3manifolds, TemperleyLieb algebra, and braid groups are reviewed in the first three lectures. In the second lecture, we discuss parentheses structures and 2categories of surfaces in 3space in relation to the TemperleyLieb algebras. In the fourth lecture, we give diagrammatics of 4 dimensional triangulations and their relations to the associahedron, a higher associativity condition. We prove that the 4dimensional Pachner moves can be decomposed in terms of singular moves, and lower dimensional relations. In our last lecture, we give a combinatorial description of knotted surfaces in 4space and their isotopies. MRCN: 57Q45 Key words: Reidemeister Moves, 2categories, Movie Moves, Knotted Surfaces 1 1 Introduction In this series of tal...
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...
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Cited by 6 (0 self)
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
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...