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2Categorical Poincaré representations and state sum applications, available as math.QA/0306440
"... This is intended as a selfcontained introduction to the representation theory developed in order to create a Poincaré 2category state sum model for Quantum Gravity in 4 dimensions. We review the structure of a new representation 2category appropriate to Lie 2group symmetries and discuss its appl ..."
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This is intended as a selfcontained introduction to the representation theory developed in order to create a Poincaré 2category state sum model for Quantum Gravity in 4 dimensions. We review the structure of a new representation 2category appropriate to Lie 2group symmetries and discuss its application to the problem of finding a state sum model for Quantum Gravity. There is a remarkable richness in its details, reflecting some desirable characteristics of physical 4dimensionality. We begin with a review of the method of orbits in Geometric Quantization, as an aid to the intuition that the geometric picture unfolded here may be seen as a categorification of this process. 1
Distributive Laws For Pseudomonads
 T. A. C
, 1999
"... . We define distributive laws between pseudomonads in a Graycategory A, as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the Graycategory PSM(A) of pseudomonads in A, and define a l ..."
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. We define distributive laws between pseudomonads in a Graycategory A, as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the Graycategory PSM(A) of pseudomonads in A, and define a lifting to be a pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure with respect to two given pseudomonads. We show how to obtain a pseudomonad with compatible structure from a distributive law, how to get a lifting from a pseudomonad with compatible structure, and how to obtain a distributive law from a lifting. We show that one triangle suffices to define a distributive law in case that one of the pseudomonads is a (co)KZdoctrine and the other a KZdoctrine. 1. Introduction Distributive laws for monads were introduced by J. Beck in [2]. As pointed out by G. M. Kelly in [7], strict distributive laws for higher dimensional monads are rare. We need then a study ...
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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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...
Polycategories via pseudodistributive laws
"... In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomo ..."
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In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘twosided Kleisli bicategory’ of this pseudodistributive law are precisely symmetric polycategories. 1
Functorial calculus in monoidal bicategories
 Applied Categorial Structures
, 2002
"... The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomp ..."
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The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomplete symmetric monoidal
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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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
Sums over graphs and integration over discrete groupoids
 Applied Categorical Structures
"... Abstract. We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pullback or pushforward formulas for integrals ov ..."
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Abstract. We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pullback or pushforward formulas for integrals over suitable groupoids.
Categorification
 Contemporary Mathematics 230. American Mathematical Society
, 1997
"... Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘c ..."
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Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘coherence laws’. Iterating this process requires a theory of ‘ncategories’, algebraic structures having objects, morphisms between objects, 2morphisms between morphisms and so on up to nmorphisms. After a brief introduction to ncategories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle ncategories, cobordism ncategories, and the homotopy ntypes of the loop spaces Ω k S k. We conclude by describing a definition of weak ncategories based on the theory of operads. 1
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
Lie 2algebras
, 2004
"... I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times ..."
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I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times during my graduate studies. I also thank my ‘mathematical brothers’: Miguel CarriónÁlvarez, Toby Bartels, Jeffrey Morton, and Derek Wise for their friendship and engaging, educational conversations. I am grateful for the assistance of Aaron Lauda in drawing various braid diagrams, and thank Ronnie Brown, Andrée Ehresmann, Thomas Larsson, James Stasheff, J. Scott Carter, and Masahico Saito for helpful discussions and correspondence. Finally, I am extremely appreciative of the love and support of my family, friends, and former professors during my time as a graduate student. I certainly could not have accomplished all that I have without them. iii ABSTRACT OF THE DISSERTATION