### 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 ..."

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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 and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres. 1

### 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 ..."

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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 and-then connectives are obtained, as adjoints, via the existential-pullback 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

### Expressive Logics for Coalgebras via Terminal Sequence Induction

, 2005

"... Abstract This paper presents a logical characterisation of coalgebraic behavioural equivalence. The characterisation is given in terms of coalgebraic modal logic, an abstract framework for reasoning about, and specifying properties of, coalgebras, for an endofunctor on the category of sets. Its main ..."

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Abstract This paper presents a logical characterisation of coalgebraic behavioural equivalence. The characterisation is given in terms of coalgebraic modal logic, an abstract framework for reasoning about, and specifying properties of, coalgebras, for an endofunctor on the category of sets. Its main feature is the use of predicate liftings, which give rise to the interpretation of modal operators on coalgebras. We show that coalgebraic modal logic is adequate for reasoning about coalgebras, that is, behaviourally equivalent states cannot be distinguished by formulas of the logic. Subsequently, we isolate properties which also ensure expressiveness of the logic, that is, logical and behavioural equivalence coincide. 1 Introduction Coalgebras for an endofunctor on the category of sets can be used to model a large class of state based systems, including Kripke models, labelled transition systems, Moore and Mealy machines and deterministic automata (see [18] for an overview). This raises the question of a uniform logical framework, which can be used to reason about, and specify properties of, coalgebraically modelled systems.

### Inductive, Coinductive, and Pointed Types

"... An extension of the simply-typed lambda calculus is presented which contains both well-structured 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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An extension of the simply-typed lambda calculus is presented which contains both well-structured 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. 1

### *-autonomous categories, Unique decomposition categories.

"... We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of ..."

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We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of Abramsky’s GoI situations–ones based on Unique Decomposition Categories (UDC’s)–exactly captures Girard’s functional analytic models in his first GoI paper, including Girard’s original Execution formula in Hilbert spaces, his notions of orthogonality, types, datum, algorithm, etc. Here we associate to a UDC-based GoI Situation a denotational model (a ∗-autonomous category (without units) with additional exponential structure). We then relate this model to some of the standard GoI models via a fully-faithful embedding into a double-gluing category, thus connecting up GoI with earlier Full Completeness

### A Categorical Model for the Geometry of Interaction Abstract

"... We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via ..."

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We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.

### Another iteration on “A synthesis of several sorting algorithms”

, 1994

"... In “A synthesis of several sorting algorithms”, Darlington showed how to use program transformation techniques to develop versions of six well-known sorting algorithms. We provide more evidence for the naturalness of the resulting taxonomy of algorithms by showing how it follows almost immediately f ..."

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In “A synthesis of several sorting algorithms”, Darlington showed how to use program transformation techniques to develop versions of six well-known sorting algorithms. We provide more evidence for the naturalness of the resulting taxonomy of algorithms by showing how it follows almost immediately from a consideration of the types of the objects involved. By exploiting the natural operations of iteration and coiteration over recursively defined data types, we may automatically derive the structure of each algorithm. 1