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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...
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
On Yetter’s invariant and an extension of the DijkgraafWitten invariant to categorical groups
 Theory Appl. Categ
"... We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the ..."
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Cited by 10 (0 self)
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We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined
Formal Homotopy Quantum Field Theories
 II : Simplicial Formal Maps
"... Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQ ..."
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Cited by 5 (2 self)
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Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G,1) and also when it is simply connected. Here we study general HQFTs with d = 1 and target a general 2type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2types that we use are crossed modules, C, and we introduce a notion of formal Cmap, which extends the usual latticetype constructions to this setting. This leads to a classification of ‘formal ’ 2dimensional HQFTs with target C,
TQFTs from Homotopy ntypes
, 1995
"... : Using simplicial methods developed in [22], we construct topological quantum field theories using an algebraic model of a homotopy ntype as initial data, generalising a construction of Yetter in [23] for n=1 and in [24] for n=2 Introduction In [23], Yetter showed how to construct a topological q ..."
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Cited by 4 (2 self)
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: Using simplicial methods developed in [22], we construct topological quantum field theories using an algebraic model of a homotopy ntype as initial data, generalising a construction of Yetter in [23] for n=1 and in [24] for n=2 Introduction In [23], Yetter showed how to construct a topological quantum field theory with coefficients in a finite group. In [24], he showed that his construction could be extended to handle coefficients in a finite categorical group, or cat 1 group. These objects are algebraic models for certain homotopy 2types. The topological quantum field theories thus constructed are (2+1) TQFTs, but the methods used do not depend on the manifolds being surfaces, except to avoid possible irregularities related to problems of triangulations in low dimensions. Yetter ended that second note with some open questions, the third of which was: can one carry out the same sort of construction for algebraic models of higher homotopy types? In this note we will show that a ...
THE FUNDAMENTAL CROSSED MODULE OF THE COMPLEMENT Of A Knotted Surface
, 2009
"... We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, the ..."
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Cited by 3 (0 self)
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We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π2(M,M 1) →Gcan be rescaled to a homotopy invariant IG(M), depending only on the algebraic 2type of M. We describe an algorithm for calculating π2(M,M (1) ) as a crossed module over π1(M (1)), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0 and 1handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant IG yields a nontrivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.
Categorical Groups, Knots and Knotted Surfaces
, 2008
"... We define a knot invariant and a 2knot invariant from any finite categorical group. ..."
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We define a knot invariant and a 2knot invariant from any finite categorical group.
Contents
, 1998
"... Abstract. To characterize categorical constraints associativity, commutativity and monoidality in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups gives nonabelian cohomology. The categorification func ..."
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Abstract. To characterize categorical constraints associativity, commutativity and monoidality in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups gives nonabelian cohomology. The categorification functor from groups to monoidal categories provides the correspondence between the respective parity (quasi)complexes and allows to interpret 1cochains as functors, 2cocycles monoidal structures, 3cocycles associators. The cohomology spaces H 3, H 2, H 1, H 0 correspond as usual to quasiextensions, extensions, split extensions and invariants, as in the abelian case. A larger class of commutativity constraints for monoidal categories is identified. It is
Contemporary Mathematics Formal Homotopy Quantum Field Theories, II: Simplicial Formal Maps
, 2005
"... Abstract. Simplicial formal maps were introduced in the first paper of this series as a tool for studying Homotopy Quantum Field Theories with background a general homotopy 2type. Here we continue their study, showing how a natural generalisation can handle much more general backgrounds. The questi ..."
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Abstract. Simplicial formal maps were introduced in the first paper of this series as a tool for studying Homotopy Quantum Field Theories with background a general homotopy 2type. Here we continue their study, showing how a natural generalisation can handle much more general backgrounds. The question of the geometric interpretation of these formal maps is partially answered in terms of combinatorial bundles. This suggests new interpretations of HQFTs.