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Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
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Cited by 74 (6 self)
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
Higherdimensional algebra VI: Lie 2algebras,
, 2004
"... The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We ..."
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Cited by 44 (12 self)
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The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We define a ‘semistrict Lie 2algebra ’ to be a 2vector space L equipped with a skewsymmetric bilinear functor [·, ·]: L × L → L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the ‘Jacobiator’, which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang–Baxter equation. We construct a 2category of semistrict Lie 2algebras and prove that it is 2equivalent to the 2category of 2term L∞algebras in the sense of Stasheff. We also study strict and skeletal Lie 2algebras, obtaining the former from strict Lie 2groups and using the latter to classify Lie 2algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finitedimensional Lie algebra g a canonical 1parameter family of Lie 2algebras g � which reduces to g at � = 0. These are closely related to the 2groups G � constructed in a companion paper.
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 43 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
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
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Generalized Centers of Braided and Sylleptic Monoidal 2Categories
, 1997
"... Recent developments in higherdimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2categories, and a center construction for monoidal 2categories which gives a braided monoidal 2category. I give ge ..."
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Cited by 25 (3 self)
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Recent developments in higherdimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2categories, and a center construction for monoidal 2categories which gives a braided monoidal 2category. I give generalized center constructions for braided and sylleptic monoidal 2categories which give sylleptic and symmetric monoidal 2categories respectively, and I correct some errors in the original center construction for monoidal 2categories. 1 Introduction The initial motivation for the study of braided monoidal categories was twofold: from homotopy theory, where braided monoidal categories of a particular kind arise as algebraic 3types of arcconnected, simply connected spaces, and from higherdimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16]. These motivations have subsequently been brought together by the definition of tricategori...
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...
An introduction to ncategories
 In 7th Conference on Category Theory and Computer Science
, 1997
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Structures and Diagrammatics of Four Dimensional Topological Lattice Field Theories
 Advances in Math. 146
, 1998
"... Crane and Frenkel proposed a state sum invariant for triangulated 4manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double ..."
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Cited by 20 (5 self)
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Crane and Frenkel proposed a state sum invariant for triangulated 4manifolds. They defined and used new algebraic structures called Hopf categories for their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum invariant of triangulated 4manifolds using CraneYetter cocycles as Boltzmann weights. Our invariant generalizes the 3dimensional invariants defined by Dijkgraaf and Witten and the invariants that are defined via Hopf algebras. We present diagrammatic methods for the study of such invariants that illustrate connections between Hopf categories and moves to triangulations. 1 Contents 1 Introduction 3 2 Quantum 2 and 3 manifold invariants 4 Topological lattice field theories in dimension 2 . . . . . . . . . . . . . . . . . . . 4 Pachner moves in dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 TuraevViro inv...