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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...
1 On Trasformations Between Belief Spaces
"... Abstract: Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and employing a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter, in particular, takes values in a space of Bayesian ..."
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Abstract: Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and employing a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter, in particular, takes values in a space of Bayesian states. The pignistic map is not a homomorphism between these same spaces. We demonstrate an impact this may have on robust decision making for frames of cardinality at least 3. We adapt the measure zero reflection property of some maps between probability spaces to define a category of belief states having plausibility zero reflecting functions as morphisms. Our definition encapsulates a generalization of the notion of absolute continuity to the context of belief spaces. We show that the Voorbraak map is a functor valued in this category. 1
Causal Theories: A Categorical Perspective on Bayesian Networks
"... It’s been an amazing year, and I’ve had a good time learning and thinking about the contents of this essay. A number of people have had significant causal influence on this. Foremost among these is my dissertation supervisor Jamie Vicary, who has been an excellent guide throughout, patient as I’ve j ..."
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It’s been an amazing year, and I’ve had a good time learning and thinking about the contents of this essay. A number of people have had significant causal influence on this. Foremost among these is my dissertation supervisor Jamie Vicary, who has been an excellent guide throughout, patient as I’ve jumped from idea to idea and with my vague questions, and yet careful to ensure I’ve stayed on track. We’ve had some great discussions too, and I thank him for them. John Baez got me started on this general topic, has responded enthusiastically and generously to probably too many questions, and, with the support of the Centre for Quantum Technologies, Singapore, let me come visit him to pester him with more. Bob Coecke has been a wonderful and generous general supervisor, always willing to talk and advise, and has provided many of the ideas that lurk in the background of those here. I thank both of them too. I also thank Rob Spekkens, Dusko Pavlovic, Prakash Panangaden, and Samson Abramsky for some interesting discussions