The category of locally compact locales over any elementary topos is characterised by means of the axioms of abstract Stone duality (monadicity of the topology, considered as a selfadjoint exponential Σ(−), and Scott continuity, Fφ = ∃`. F (λn. n ∈ `)∧∀n ∈ `. φn), together with an “underlying set” functor that is right adjoint to the inclusion of the full subcategory of overt discrete objects (...

A metalanguage for denotational semantics is given a logical interpretation: types are interpreted as propositional theories; terms (in an extended typed λ-calculus, denoting elements of types) are embedded in an endogenous program logic, which characterises their behaviour in terms of which properties they satisfy; terms (in an extension of the algebraic metalanguage of cartesian closed catego...

We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming we develop a notion that is directly taken from concepts used ...

We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete self-dual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a two-dimensional space we call the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from −1 for sets to 1 for complete atom...

We study structures called d-frames which were developed by the last two authors for a bitopological treatment of Stone duality. These structures consist of a pair of frames thought of as the opens of two topologies, together with two relations which serve as abstractions of disjointness and covering of the space. With these relations, the topological separation axioms regularity and normality ...

1. Introduction Stone, in [8], developed for distributive lattices a representation theory generalizing that for Boolean algebras. This he achieved by topologizing the set X of prime ideals of a distributive lattice A (with a zero element) by taking as a base {P a : aeA} (where P a denotes the set of prime ideals of A not containing a), and by showing that the map a i-> P a is an isomorphism re...

In the topos approach to quantum physics, a functor known as the spectral presheaf of a von Neumann algebra plays the role of a generalized state space. Mathematically, the spectral presheaf also provides an interesting generalization of the Gelfand spectrum, which is only defined for abelian von Neumann algebras, to the nonabelian case. A partial duality result, analogous to Gelfand duality, e...

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categ...

Boolean powers were introduced by Foster [5]. It was noticed by Jónsson in the review of [6], and further elaborated by Banaschewski and Nelson [1], that the Boolean power of an algebra A by a Boolean algebra B can be described as the algebra of continuous functions from the Stone space of B to A, where A has the discrete topology. It follows that a Boolean power of the group Z is an `-group ge...