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The fixed point property in synthetic domain theory
 Logic in Computer Science 6
, 1991
"... We present an elementary axiomatisation of synthetic domain theory and show that it is sufficient to deduce the fixed point property and solve domain equations. Models of these axioms based on partial equivalence relations have received much attention, but there are also very simple sheaf models bas ..."
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We present an elementary axiomatisation of synthetic domain theory and show that it is sufficient to deduce the fixed point property and solve domain equations. Models of these axioms based on partial equivalence relations have received much attention, but there are also very simple sheaf models based on classical domain theory. In any case the aim of this paper is to show that an important theorem can be derived from an abstract axiomatisation, rather than from a particular model. Also, by providing a common framework in which both PER and classical models can be expressed, this work builds a bridge between the two. 1
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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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.
The HasCasl prologue: categorical syntax and semantics of the partial λcalculus
 COMPUT. SCI
, 2006
"... We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we ..."
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We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we establish an equivalence between partial cartesian closed categories (pccc’s) and partial λtheories. Building on these results, we define (settheoretic) notions of intensional Henkin model and syntactic λalgebra for Moggi’s partial λcalculus. These models are shown to be equivalent to the originally described categorical models in pccc’s via the global element construction. The semantics of HasCasl is defined in terms of syntactic λalgebras. Correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side; as an application, we pinpoint unique choice as the distinctive feature of topos logic (in comparison to intuitionistic higherorder logic of partial functions, which by our results is the logic of pccc’s with equality). Finally, we give some applications of the modeltheoretic equivalence result to the semantics of HasCasl and its relation to firstorder Casl.
STABLE MEET SEMILATTICE FIBRATIONS AND FREE RESTRICTION CATEGORIES
"... Abstract. The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are “unitary”, in a sen ..."
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Abstract. The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are “unitary”, in a sense which generalizes that from the theory of inverse semigroups. Characterization theorems for unitary restriction categories are derived. The paper ends with an explicit description of the free restriction category on a directed graph. 1.
The Logic of the Partial λCalculus With Equality
"... We investigate the logical aspects of the partial λcalculus with equality, exploiting an equivalence between partial λtheories and partial cartesian closed categories (pcccs) established here. The partial λcalculus with equality provides a fullblown intuitionistic higher order logic, which in a ..."
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We investigate the logical aspects of the partial λcalculus with equality, exploiting an equivalence between partial λtheories and partial cartesian closed categories (pcccs) established here. The partial λcalculus with equality provides a fullblown intuitionistic higher order logic, which in a precise sense turns out to be almost the logic of toposes, the distinctive feature of the latter being unique choice. We give a linguistic proof of the generalization of the fundamental theorem of toposes to pcccs with equality; type theoretically, one thus obtains that the partial λcalculus with equality encompasses a MartinLöfstyle dependent type theory. This work forms part of the semantical foundations for the higher order algebraic specification language HasCasl.
The synthetic Plotkin powerdomain
, 1990
"... Plotkin [1976] introduced a powerdomain construction on domains in order to give semantics to a nondeterministic binary choice constructor, and later [1979] characterised it as the free semilattice. Smyth [1983] and Winskel [1985] showed that it could be interpreted in terms of modal predicate tran ..."
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Plotkin [1976] introduced a powerdomain construction on domains in order to give semantics to a nondeterministic binary choice constructor, and later [1979] characterised it as the free semilattice. Smyth [1983] and Winskel [1985] showed that it could be interpreted in terms of modal predicate transformers and Robinson [1986] recognised it as a special case of Johnstone’s [1982] Vietoris construction, which itself generalises the Hausdorff metric on the set of closed subsets of a metric space. The domain construction involves a curious order relation known as the EgliMilner order. In this paper we relate the powerdomain directly to the free semilattice, which in a topos is simply the finite powerset, i.e. the object of (Kuratowski)finite subobjects of an object. We show that the EgliMilner order coincides (up to “¬¬”) with the intrinsic order induced by a family of “observable predicates.” This problem originally arose in the context of the Effective topos, in which the observable predicates are the recursively enumerable subsets. However we find that the results of this paper hold for any elementary topos, and so by considering a (pre)sheaf topos (which the Effective topos is not) we may compare them with the classical approach. Important Note: Much of the credit for the work in this paper is due to Wesley Phoa and Martin Hyland, but I take the blame for its presentation. Comments on it are most welcome. When it is finished it will be submitted as a joint paper with Wesley Phoa, and an announcement will be made on types.
An Elementary Theory of Various Categories of Spaces in Topology
, 2005
"... In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclid ..."
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In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclidean principle Fσ ∧ σ = F> ∧ σ makes Σ the classifier for open subspaces. Computably based locally compact locales provide the leading model for these axioms, although the methods are also applicable to CCDop (constructively completely distributive lattices). In this paper we recover the textbook theories, using the additional axiom that the subcategory of overt discrete objects have a coreflection, the “underlying set ” functor. This subcategory is then a topos, and the whole category is characterised in the minimal situation as that of locally compact locales over that topos. However, by adding further axioms regarding the existence of equalisers and injectivity of Σ, we find the category of sober spaces or of locales over the topos as a reflective subcategory, whilst the whole category is cartesian closed and has all finite limits and colimits.
Foundations for Computable Topology
, 2009
"... Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between al ..."
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Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between algebra and geometry as an abstract monadic adjunction that we turn into a new type theory. To this we add an equation that is satisfied by the Sierpiński space, which plays a key role as the classifier for both open and closed subspaces. In the resulting theory there is a duality between open and closed concepts. This captures many basic properties of compact and closed subspaces, despite the absence of any explicitly infinitary axiom. It offers dual results that link general topology to recursion theory. The extensions and applications of ASD elsewhere that this paper survey include a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0, 1] in ASD is compact.