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Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
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.
HigherOrder Categorical Grammars
 Proceedings of Categorial Grammars 04
"... into two principal paradigms: modeltheoretic syntax (MTS), which ..."
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into two principal paradigms: modeltheoretic syntax (MTS), which
Ambiguity, Neutrality, and Coordination in Higherorder Grammar
, 2003
"... We show that the standard account of neutrality and coordination in typelogical grammar is untenable. However, when using as our framework a version of Lambek’s categorical grammar with a type theory based on Lambek and Scott’s higher order intuitionistic logic (the internal language of a topos) ra ..."
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Cited by 3 (2 self)
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We show that the standard account of neutrality and coordination in typelogical grammar is untenable. However, when using as our framework a version of Lambek’s categorical grammar with a type theory based on Lambek and Scott’s higher order intuitionistic logic (the internal language of a topos) rather than the Lambek calculus, the account can largely be salvaged. Because of the difficulty of phonologically interpreting coordinated functors of differing directionality we need to handle both phonology and syntax within a single polymorphically typed lambda calculus.
Cover semantics for quantified lax logic
 Journal of Logic and Computation
"... Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of LawvereTierneyGrothendieck topologies on topoi. This paper provides a complete semantics for quantified la ..."
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Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of LawvereTierneyGrothendieck topologies on topoi. This paper provides a complete semantics for quantified lax logic by combining the BethKripkeJoyal cover semantics for firstorder intuitionistic logic with the classical relational semantics for a “diamond ” modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of “propositions ” of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4. An alternative completeness proof is given for (nonmodal) firstorder intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkinstyle construction of a characteristic model whose points are principal theories rather than prime saturated ones. The paper provides further evidence that there is more to intuitionistic modal logic than the generalisation of properties of boxes and diamonds from Boolean modal logic.
1 Reflections on a categorical foundations of mathematics
"... Summary. We examine Gödel’s completeness and incompleteness theorems for higher order arithmetic from a categorical point of view. The former says that a proposition is provable if and only if it is true in all models, which we take to be local toposes, i.e. Lawvere’s elementary toposes in which the ..."
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Summary. We examine Gödel’s completeness and incompleteness theorems for higher order arithmetic from a categorical point of view. The former says that a proposition is provable if and only if it is true in all models, which we take to be local toposes, i.e. Lawvere’s elementary toposes in which the terminal object is a nontrivial indecomposable projective. The incompleteness theorem showed that, in the classical case, it is not enough to look only at those local toposes in which all the numerals are standard. Thus, for a classical mathematician, Hilbert’s formalist program is not compatible with the belief in a Platonic standard model. However, for pure intuitionistic type theory, a single model suffices, the linguistically constructed free topos, which is the initial object in the category of all elementary toposes and logical functors. Hence, for a moderate intuitionist, formalism and Platonism can be reconciled after all. The completeness theorem can be sharpened to represent any topos by continuous sections of a sheaf of local toposes. 1.1
QUANTUM MECHANICS AS A SPACETIME THEORY
, 2005
"... Abstract. We show how quantum mechanics can be understood as a spacetime theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects ..."
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Abstract. We show how quantum mechanics can be understood as a spacetime theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects differs from that of classical objects. The systems that are nonlocal when measured in the classical spacetime continuum may be localized in the quantum continuum. We compare this new description of spacetime with the Bohmian picture of quantum mechanics. 1. What is quantum spacetime? Both modern mathematics and modern physics underwent serious foundational crises during the 20th century. The crisis in mathematics occured at the beginning of the century and the main problem was to deal with certain infinities that are directly related to the concept of real number. Poincaré [31] explained this crisis in terms of different attitudes to infinity, related to Aristotle’s actual infinity and the potential infinity (the first attitude believes that the actual infinity exists, we begin with the collection in which we find the preexisting objects, the second holds that a collection is formed by successively adding new members, it is infinite because we can see no reason why this process should stop). It led finally to the emergence of new, nonstandard definitions of real numbers. The crisis in physics concerns the interpretation of the quantum theory, the measurement problem and the question of nonlocality. In previous works we showed how in principle certain paradoxes of the quantum theory can be explained provided we enlarge our conception of number [10].Our goal was to show how the basic axioms of quantum mechanics can be reformulated in terms of nonstandard real numbers that we call qrumbers. It is our goal in the present paper to analyze nonlocality and the concept of spacetime at the light of the new conceptual tools that we developed in the past.
TYPES, SETS AND CATEGORIES
"... This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category t ..."
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This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, I have elected to offer detailed technical presentations of just a few important instances. 1 THE ORIGINS OF TYPE THEORY The roots of type theory lie in set theory, to be precise, in Bertrand Russell’s efforts to resolve the paradoxes besetting set theory at the end of the 19 th century. In analyzing these paradoxes Russell had come to find the set, or class, concept itself philosophically perplexing, and the theory of types can be seen as the outcome of his struggle to resolve these perplexities. But at first he seems to have regarded type theory as little more than a faute de mieux.
Topological Completeness of FirstOrder Modal Logic
"... As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topo ..."
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As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity ” operation is modeled by taking the interior of an arbitrary subset of a topological space. This topological interpretation was recently extended in a natural way to arbitrary theories of full firstorder logic by Awodey and Kishida [3], using topological sheaves to interpret domains of quantification. This paper proves the system of full firstorder S4 modal logic to be deductively complete with respect to such extended topological semantics. The techniques employed are related to recent work in topos theory, but are new to systems of modal logic. They are general enough to also apply to other modal systems. Keywords: Firstorder modal logic, topological semantics, completeness.