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Geometric and higher order logic in terms of abstract Stone duality
 THEORY AND APPLICATIONS OF CATEGORIES
, 2000
"... The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this ..."
Abstract

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The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.
Subspaces in abstract Stone duality
 Theory and Applications of Categories
, 2002
"... ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idemp ..."
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ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory. Paré showed that any elementary topos has this duality, and we prove it intuitionistically for the category of locally compact locales. The paper is largely concerned with the construction of such a category out of one that merely has powers of some fixed object Σ. It builds on Sober Spaces and Continuations, where the related but weaker notion of abstract sobriety was considered. The construction is done first by formally adjoining certain equalisers that Σ (−) takes to coequalisers, then using Eilenberg–Moore algebras, and finally presented as a lambda calculus similar to the axiom of comprehension in set theory. The comprehension calculus has a normalisation theorem, by which every type can
1 Summary Abstract Stone Duality
"... Computer science has enjoyed topological interpretations for thirty years, but arbitrary infinite joins have precluded the converse, a computational interpretation of general topology. Abstract Stone Duality (ASD) is a type theory in which the topology on a space is an exponential with a λcalculus, ..."
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Computer science has enjoyed topological interpretations for thirty years, but arbitrary infinite joins have precluded the converse, a computational interpretation of general topology. Abstract Stone Duality (ASD) is a type theory in which the topology on a space is an exponential with a λcalculus, not an infinitary lattice. But instead of rewriting old proofs in a preconceived logic, it exploits a deep mathematical theme, Stone duality, reconciling conceptual and computational traditions in mathematics. ASD gives a computational interpretation to continuous functions, not only for domains but between all locally compact spaces, including those from geometry. Published work develops the necessary infrastructure, defining notions such as compact Hausdorff spaces very naturally, with lattice duality between open and closed phenomena. Recent work generalises interval analysis from R to other spaces, but the intervals themselves are only mentioned during compilation. The categorical structure allows a conceptual development, whilst the λcalculus handles higher types. This will be used to investigate differential and integral calculus. ASD also throws new light on discrete mathematics, giving a computational status to the powerset and other constructions, following Stone’s dictum that they carry topologies. ASD can be implemented by compilation by continuationpassing into a language that combines functional and logic programming. 1 2 A bridge from mathematics to computation