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Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
Abstract

Cited by 9 (9 self)
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 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 4 (1 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.
Proofs as Polynomials
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Cited by 2 (1 self)
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
Quantum Logic in Dagger Categories with Kernels
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
Abstract
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, and orthomodularity. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
Abstract Quantum Logic in Dagger Categories with Kernels
, 902
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
Abstract
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, and orthomodularity. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
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.