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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 &quot;good &quot; 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 &quot;best &quot; regular category (called its regular completion) that embeds it. The second assigns to
Programming Metalogics with a Fixpoint Type
, 1992
"... A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category th ..."
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Cited by 12 (6 self)
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A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way. The notion of a category with fixpoint object is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically. The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the same properties. PCFstyle languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...
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 9 (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.
Axioms and (Counter)examples in Synthetic Domain Theory
 Annals of Pure and Applied Logic
, 1998
"... this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the p ..."
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this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the purposes of the axiomatic part of this paper, we believe that it would also be
Enrichment and Representation Theorems for Categories of Domains and Continuous Functions
, 1996
"... This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported b ..."
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Cited by 8 (5 self)
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This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported by SERC grant RR30735 and EC project Programming Language Semantics and Program Logics grant SC1000 795 they explain the ordertheoretic structure). Our aim is 1. to provide a justification of Scott's original consideration of ordered structures, and 2. to deepen our understanding of the notion of passage to the limit
Synthetic Domain Theory in Type Theory: Another Logic of Computable Functions
 In Proceedings of TPHOL
, 1996
"... Abstract. We will present a Logic of Computable Functions based on the idea of Synthetic Domain Theory such that all functions are automatically continuous. Its implementation in the Lego proofchecker – the logic is formalized on top of the Extended Calculus of Constructions – has two main advantag ..."
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Abstract. We will present a Logic of Computable Functions based on the idea of Synthetic Domain Theory such that all functions are automatically continuous. Its implementation in the Lego proofchecker – the logic is formalized on top of the Extended Calculus of Constructions – has two main advantages. First, one gets machine checked proofs verifying that the chosen logical presentation of Synthetic Domain Theory is correct. Second, it gives rise to a LCFlike theory for verification of functional programs where continuity proofs are obsolete. Because of the powerful type theory even modular programs and specifications can be coded such that one gets a prototype setting for modular software verification and development. 1
Lifting as a KZdoctrine
 Proceedings of the 6 th International Conference, CTCS'95, volume 953 of Lecture Notes in Computer Science
, 1995
"... this paper, is the analysis of notions of approximation aiming at explaining and justifying (ordertheoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and orderenrichment we considered contextual approximation which, in ..."
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this paper, is the analysis of notions of approximation aiming at explaining and justifying (ordertheoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and orderenrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see [Fio94c]) and, second, we do not know of nonordertheoretic axioms making the specialisation preorder !complete. To overcome these drawbacks another notion of approximation was to be considered. And, it was the second problem that motivated the intensional notion of approximation provided by the path relation. In fact, it is shown in [Fio94b] that under suitable axioms the path relation can be equipped with a canonical passagetothelimit operator appropriate for fixedpoint computations; stronger axioms make this operator be given by lubs of !chains
Realizability Models for Sequential Computation
, 1998
"... We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of seq ..."
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We give an overview of some recently discovered realizability models that embody notions of sequential computation, due mainly to Abramsky, Nickau, Ong, Streicher, van Oosten and the author. Some of these models give rise to fully abstract models of PCF; others give rise to the type structure of sequentially realizable functionals, also known as the strongly stable functionals of Bucciarelli and Ehrhard. Our purpose is to give an accessible introduction to this area of research, and to collect together in one place the definitions of these new models. We give some precise definitions, examples and statements of results, but no full proofs. Preface Over the last two years, researchers in various places (principally Abramsky, Nickau, Ong, Streicher, van Oosten and the present author) have come up with a number of new realizability models that embody some notion of "sequential" computation. Many of these give rise to fully abstract and universal models for PCF and related languages. Alth...
A proposed categorical semantics for ML modules
, 1995
"... We present a simple categorical semantics for ML signatures, structures and functors. Our approach relies on realizablity semantics in the category of assemblies. Signatures and structures are modelled as objects in slices of the category of assemblies. Instantiation of signatures to structures and ..."
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We present a simple categorical semantics for ML signatures, structures and functors. Our approach relies on realizablity semantics in the category of assemblies. Signatures and structures are modelled as objects in slices of the category of assemblies. Instantiation of signatures to structures and hence functor application is modelled by pullback. 1 Introduction Building on work on the semantics of programming languages in realizability models, in particular that of Wesley Phoa [Pho90] and John Longley [Lon95], we sketch a simple approach to elements of the ML modules system, such as signatures, structures and functors. Once the basic machinery is set up, we will need only quite basic category theory. This paper is an updated and completely revised version of an earlier paper by Michael Fourman and Wesley Phoa [PF92]. The construction of "generic" (in a sense to be defined below) elements and types presented here is essentially the same as in that paper. However, our presentation is ...