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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 comp ..."
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Cited by 30 (9 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...
Constructive complete distributivity IV
 Appl. Cat. Struct
, 1994
"... A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and suppreserving arrows. There is a restrict ..."
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Cited by 9 (5 self)
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A complete lattice L is constructively completely distributive, (CCD), when the sup arrow from downclosed subobjects of L to L has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and suppreserving arrows. There is a restriction to order ideals and "totally algebraic" lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos. Research partially supported by grants from NSERC Canada. Diagrams typeset using Michael Barr's diagram package. AMS Subject Classification Primary: 06D10 Secondary 18B35, 03G10. Keywords: completely distributive, adjunction, projective, nuclear Introduction Idempotents do not split in the category of rel...
Dagger categories and formal distributions
"... Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a cat ..."
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Summary. Monoidal dagger categories play a central role in the abstract quantum mechanics of Abramsky and Coecke. The authors show that a great deal of elementary quantum mechanics can be carried out in these categories; for example, the Born rule emerges naturally. In this paper, we construct a category of tame formal distributions with coefficients in a commutative associative algebra and show that it is a dagger category. This gives access to a broad new class of models, with the abstract scalars in the sense of Abramsky being the elements of the algebra. We will also consider a subcategory of local formal distributions, based on the ideas of Kac. Locality has been of fundamental significance in various formulations of quantum field theory. Thus our work may provide the possibility of extending the abstract framework to QFT. We also show that these categories of formal distributions are monoidal and contain a nuclear ideal, a weak form of adjunction appropriate for analyzing categories
QF FUNCTORS AND (CO)MONADS
"... Abstract. One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K: A → B between categories is Frobenius if there exists a functor G: B → A which is at the same time a right and left adjoint of K; a monad F on ..."
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Cited by 3 (1 self)
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Abstract. One reason for the universal interest in Frobenius algebras is that their characterisation can be formulated in arbitrary categories: a functor K: A → B between categories is Frobenius if there exists a functor G: B → A which is at the same time a right and left adjoint of K; a monad F on A is a Frobenius monad provided the forgetful functor AF → A is a Frobenius functor, where AF denotes the category of Fmodules. With these notions, an algebra A over a field k is a Frobenius algebra if and only if A ⊗k − is a Frobenius monad on the category of kvector spaces. The purpose of this paper is to find characterisations of quasiFrobenius algebras by just referring to constructions available in any categories. To achieve this we define QF functors between two categories by requiring conditions on pairings of functors which weaken the axioms for adjoint pairs of functors. QF monads on a category A are those monads F for which the forgetful functor UF: AF → A is a QF functor. Applied to module categories (or Grothendieck categories), our notions coincide with definitions first given K. Morita (and others). Further applications
Edinburgh, Scotland
, 1998
"... Most tensored \Lambdacategories 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 ..."
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Most tensored \Lambdacategories 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 \Lambdacategories, all morphisms are nuclear, and in the tensored \Lambdacategory of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored \Lambdacategories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations.
A complete lattice admitting no comonoid structure
, 2006
"... In this paper we show that M3, the fiveelement modular nondistributive lattice, does not admit a comonoid structure in the category of complete lattices and joinpreserving maps, equipped with the natural tensor product; this is surprising because it does admit several comonoid structures when t ..."
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In this paper we show that M3, the fiveelement modular nondistributive lattice, does not admit a comonoid structure in the category of complete lattices and joinpreserving maps, equipped with the natural tensor product; this is surprising because it does admit several comonoid structures when the same category is equipped with a seemingly less natural tensor product.