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Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 62 (10 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
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
StateSum Invariants of 4Manifolds
 J. Knot Theory Ram
, 1997
"... Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery ..."
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Cited by 30 (6 self)
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Abstract: We provide, with proofs, a complete description of the authors ’ construction of statesum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda’s surgery invariants [Br1,Br2] using techniques developed in the case of the semisimple subquotient of Rep(Uq(sl2)) (q a principal 4r th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4manifolds equipped with 2dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. 1 1
FRAMED DISCS OPERADS AND BATALIN–VILKOVISKY ALGEBRAS
, 2001
"... The framed ndiscs operad f Dn is studied as semidirect product of SO(n) and the little ndiscs operad. Our equivariant recognition principle says that a grouplike space acted on by f Dn is equivalent to the nfold loop space on an SO(n)space. Examples of f D2spaces are nerves of ribbon braided mo ..."
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Cited by 19 (4 self)
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The framed ndiscs operad f Dn is studied as semidirect product of SO(n) and the little ndiscs operad. Our equivariant recognition principle says that a grouplike space acted on by f Dn is equivalent to the nfold loop space on an SO(n)space. Examples of f D2spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of f Dn, which produces higher Batalin–Vilkovisky algebra structures for n even. We study quadratic duality for semidirect product operads and compute the double loop space homology of a manifold as BValgebra. 1.
Generalized BarrettCrane vertices and invariants of embedded graphs
 Journal of Knot Theory and its Ramifications 8
, 1999
"... ..."
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
Bottom tangles and universal invariants
, 2006
"... A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite ..."
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Cited by 5 (2 self)
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A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of “braided Hopf algebra action ” on the set of bottom tangles. Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H, we define a braided functor J from B to the category ModH of left H–modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in ModH. Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants.