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122
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 28 (10 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
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...
A closed model structure for ncategories, internal Hom, nstacks and generalized SeifertVan Kampen
, 1997
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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
"... ..."
The Bekenstein bound, topological quantum field theory and pluralistic quantum cosmology
"... this paper a new approach to the problem of constructing a quantum theory of gravity in the cosmological context is proposed. It is founded on results from four separate directions of investigation, which are: 1) A new point of view towards the interpretation problem in quantum cosmology[1, 2, 3, 4] ..."
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Cited by 19 (12 self)
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this paper a new approach to the problem of constructing a quantum theory of gravity in the cosmological context is proposed. It is founded on results from four separate directions of investigation, which are: 1) A new point of view towards the interpretation problem in quantum cosmology[1, 2, 3, 4], which rejects the idea that a single quantum state, or a single Hilbert space, can provide a complete description of a closed system like the universe. Instead, the idea is to accept Bohr's original proposal that the quantum state requires for its interpretation a context in which we distinguish two subsystems of the universethe quantum system and observer. However, we seek to relativize this split, so that the boundary between the part of the universe that is considered the system and that which might be considered the observer may be chosen arbitrarily. The idea is then that a quantum theory of cosmology is specified by giving an assignment of a Hilbert space and algebra of observables to every possible boundary that can be considered to split the universe into two such subsystems. A quantum state of the universe is then an assignment of a statistical state to every one of these Hilbert spaces, subject to certain conditions of consistency. Each of these states is interpreted to contain the information that an observer on one side of each boundary might have about the system of the other side. This formulation then accepts the idea that each observer can only have incomplete information about the universe, so that the most complete description possible of the universe is given by the whole collection of incomplete, but mutually compatible quantum state descriptions of all the possible observers. At the same time, the information of different observers is, to some extent, ...
Descente pour les nchamps
"... We develop the theory of nstacks (or more generally Segal nstacks which are ∞stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of nstacks in terms of limits, which sh ..."
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Cited by 18 (5 self)
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We develop the theory of nstacks (or more generally Segal nstacks which are ∞stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of nstacks in terms of limits, which should be perfectly general for stacks of any type of objects; several other characterizations of nstacks in terms of “effectivity of descent data”; construction of the stack associated to an nprestack; a strictification result saying that any “weak ” nstack is equivalent to a (strict) nstack; and a descent result saying that the (n + 1)prestack of nstacks (on a site) is an (n + 1)stack. As for other examples, we start from a “left Quillen presheaf ” of cmc’s and introduce the associated Segal 1prestack. For this situation, we prove a general descent result, giving sufficient conditions for this prestack to be a stack. This applies to the case of complexes, saying how complexes of sheaves of Omodules can be glued together via quasiisomorphisms. This was the problem that originally motivated us. Résumé
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 18 (8 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Finite groups, spherical 2categories, and 4manifold invariants. arXiv:math.QA/9903003
"... In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], althou ..."
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Cited by 16 (5 self)
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In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the statesum invariants of Birmingham and Rakowski [11, 12, 13], who studied DijkgraafWitten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3types, such as [15], for example. 1 1
Homotopy quantum field theories and the homotopy cobordism category in dimension 1+1
"... Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor ..."
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Cited by 16 (0 self)
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Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor