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Boundedness And Complete Distributivity
 IV, Appl. Categ. Structures
"... . We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely bounded suprema. The KZdoctrine for bounded suprema is of some independent interest and a few results about it are given. The 2category of ordered sets admitting bounded ..."
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. We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely bounded suprema. The KZdoctrine for bounded suprema is of some independent interest and a few results about it are given. The 2category of ordered sets admitting bounded suprema over which nonempty infima distribute is shown to be biequivalent to a 2category defined in terms of idempotent relations. As a corollary we obtain a simple construction of the nonnegative reals. 1. Introduction 1.1. The main theorem of [RW1] exhibited a biequivalence between the 2category of (constructively) completely distributive lattices and suppreserving arrows, and the idempotent splitting completion of the 2category of relations  relative to any base topos. Somewhat in passing in [RW1], it was pointed out that this biequivalence provides a simple construction of the closed unit interval ([0; 1]; ), namely as the ordered set of downsets for the idempotent relat...
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 ..."
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Cited by 6 (0 self)
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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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Cited by 4 (3 self)
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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
An Elementary Theory of the Category of Locally Compact Locales
, 2003
"... 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 # , and Scott continuity, F# = ##. ..."
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Cited by 3 (3 self)
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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 # , and Scott continuity, F# = ##.
MONAD COMPOSITIONS I: GENERAL CONSTRUCTIONS AND RECURSIVE DISTRIBUTIVE LAWS
"... ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursivelydefined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad ..."
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ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursivelydefined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad approximations for compositions which fail to be a monad. 1.
MONADS AS EXTENSION SYSTEMS —NO ITERATION IS NECESSARY
"... Abstract. We introduce a description of the algebras for a monad in terms of extension systems, similar to the one for monads given in [Manes, 1976]. We rewrite distributive laws for monads and wreaths in terms of this description, avoiding the iteration of the functors involved. We give a profuncto ..."
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Cited by 1 (1 self)
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Abstract. We introduce a description of the algebras for a monad in terms of extension systems, similar to the one for monads given in [Manes, 1976]. We rewrite distributive laws for monads and wreaths in terms of this description, avoiding the iteration of the functors involved. We give a profunctorial explanation of why Manes’ description of monads in terms of extension systems works. 1.
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
Powersets of Terms and Composite Monads
"... Abstract. Composing various powerset functors with the term monad gives rise to the concept of generalised terms. This in turn provides a technique for handling manyvalued sets of terms in a framework of variable substitutions, thus being the prerequisite for categorical unification in manyvalued ..."
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Abstract. Composing various powerset functors with the term monad gives rise to the concept of generalised terms. This in turn provides a technique for handling manyvalued sets of terms in a framework of variable substitutions, thus being the prerequisite for categorical unification in manyvalued logic programming using an extended notion of terms. As constructions of monads involve complicated calculations with natural transformations, proofs are supported by a graphical approach that provides a useful tool for handling various conditions, such as those wellknown for distributive laws.
Continuous Categories Revisited
, 2003
"... Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the categor ..."
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Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the category of all precontinuous categories, i.e., categories in which limits and filtered colimits distribute. This concept is closely related to the continuous categories of P. T. Johnstone and A. Joyal.
Categories of Partial Frames
"... Abstract. This article discusses the basic categorical algebra for categories of partial frames. Categories of partial frames are labelled by subset selectors that indicate which joins exist. Constructions for limits, colimits, and free functors connecting various categories of partial frames are gi ..."
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Abstract. This article discusses the basic categorical algebra for categories of partial frames. Categories of partial frames are labelled by subset selectors that indicate which joins exist. Constructions for limits, colimits, and free functors connecting various categories of partial frames are given. Examples of partial frame categories are given. Subset selectors which preserve surjections are virtually the same as rules which select all subsets smaller than a given cardinal. Preface The article’s goal is to describe categories of partial frames. A partial frame is a meetsemilattice in which certain distinguished joins exist and finite meets distribute over distinguished joins. This is made precise in Subsection 2.1. Particular types of “partial frames ” have already appeared in frame theoretic literature. Madden [19] and Madden & Molitor [20] use κframes for any regular cardinal κ to draw useful frametheoretic conclusions: many monoreflections on the category of Tychonoff locales are produced, and epimorphisms of frames (monomorphisms of locales) are identified. Johnstone and Vickers [11] and Banaschewski [1]