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14
Structural Induction and Coinduction in a Fibrational Setting
 Information and Computation
, 1997
"... . We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for in ..."
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Cited by 67 (14 self)
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. We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions X = oe(X) where X may occur both positively and negatively in the type constructor oe. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunc...
Axiomatic Domain Theory
 in Categories of Partial Maps. Distinguished Dissertation Series
, 1995
"... The denotational semantics approach to the semantics of programming languages interprets the language constructions by assigning elements of mathematical structures to them. The structures form socalled categories of domains and the study of their closure properties is the subject of domain theory ..."
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Cited by 16 (2 self)
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The denotational semantics approach to the semantics of programming languages interprets the language constructions by assigning elements of mathematical structures to them. The structures form socalled categories of domains and the study of their closure properties is the subject of domain theory [Sco70, Sco82, Plo83, GS90, AJ94]. Typically, categories of domains consist of suitably complete partially ordered sets together with continuous maps. But, what is a category of domains? The main aim of axiomatic domain theory is to answer this question by axiomatising the structure needed on a mathematical universe so that it can be considered a category of domains. Criteria required from categories of domains can be of the most varied sort. For example, we could ask them to * have a rich collection of type constructors: sums, products, exponentials, powerdomains, dependent types, polymorphic types, etc; * have fixedpoint operators for programs and type constructors; * have only computable maps [Sco76, Smy77, Mul81, McC84, Ros86, Pho90, Lon95]; * have a Stone dual providing a logic of observable properties [Abr87, Vic89, Zha91]. An additional aim of the axiomatic approach is to relate these mathematical criteria with computational criteria. As we indicate below an axiomatic treatment of various of the above aspects is now available but much research remains to be done.
Inductive, Coinductive, and Pointed Types
 In Proceedings of the 1996 ACM SIGPLAN International Conference on Functional Programming
, 1996
"... An extension of the simplytyped lambda calculus is presented which contains both wellstructured inductive and coinductive types, and which also identifies a class of types for which general recursion is possible. The motivations for this work are certain natural constructions in category theory, i ..."
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Cited by 16 (0 self)
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An extension of the simplytyped lambda calculus is presented which contains both wellstructured inductive and coinductive types, and which also identifies a class of types for which general recursion is possible. The motivations for this work are certain natural constructions in category theory, in particular the notion of an algebraically bounded functor, due to Freyd. We propose that this is a particularly elegant core language in which to work with recursive objects, since the potential for general recursion is contained in a single operator which interacts well with the facilities for bounded iteration and coiteration. Keywords: type theory, language design, semantic foundations. 1 Introduction In designing typed languages that include recursion, there has long been a tension between the structure provided by types based on wellfounded induction and the freedom permitted by types based on general recursion. Very few languages outside of purely theoretical studies have chosen...
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 ..."
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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
Categorical aspects of polar decomposition
, 2010
"... Polar decomposition unquestionably provides a notion of factorization in the category of Hilbert spaces. But it does not fit existing categorical notions, mainly because its factors are not closed under composition. We observe that the factors are images of functors. This leads us to consider notion ..."
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Polar decomposition unquestionably provides a notion of factorization in the category of Hilbert spaces. But it does not fit existing categorical notions, mainly because its factors are not closed under composition. We observe that the factors are images of functors. This leads us to consider notions of factorization that emphasize reconstruction of the composite
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