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The Uniformity Principle on Traced Monoidal Categories
 In Proceedings of CTCS’02, volume 69 of ENTCS
, 2003
"... The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin's principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones ..."
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The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin's principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits. 1
Towards a typed geometry of interaction
, 2005
"... We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a v ..."
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Cited by 8 (2 self)
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We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of (the original) untyped GoI in a unique decomposition category.
Partially traced categories
, 1107
"... This paper deals with questions relating to Haghverdiand Scott’s notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced ..."
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This paper deals with questions relating to Haghverdiand Scott’s notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced
TrU (f) as
"... Street and Verity [4] to give a categorical account of a situation occurring in very different settings: linear algebra, knot theory, proof theory... The basic idea is that a trace is an operation that to any morphism f: A ⊗ U → B ⊗ U in a monoidal category associates a new morphism ..."
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Street and Verity [4] to give a categorical account of a situation occurring in very different settings: linear algebra, knot theory, proof theory... The basic idea is that a trace is an operation that to any morphism f: A ⊗ U → B ⊗ U in a monoidal category associates a new morphism
Partially traced categories
"... This paper deals with questions relating to Haghverdi and Scott’s notion of partially traced categories. The main result is a representationtheorem for such categories: we provethat everypartiallytraced categorycan be faithfully embedded in a totally traced category. Also conversely, every symmetric ..."
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This paper deals with questions relating to Haghverdi and Scott’s notion of partially traced categories. The main result is a representationtheorem for such categories: we provethat everypartiallytraced categorycan be faithfully embedded in a totally traced category. Also conversely, every symmetric monoidal subcategory ofatotallytracedcategoryispartiallytraced, sothis characterizesthe partiallytracedcategoriescompletely. The main technique we use is based on Freyd’s paracategories, along with a partial version of Joyal, Street, and Verity’s Intconstruction.
Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory
, 2002
"... In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction betwee ..."
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In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by the paracategory of bivariant functors and dinatural transformations. We introduce partial multicategories to account for partial tensor products. We also consider fibrations for paracategories and their indexedparacategory version. Finally, we instantiate all these concepts in the context of probabilistic automata.