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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
Constructive set theories and their categorytheoretic models
 IN: FROM SETS AND TYPES TO TOPOLOGY AND ANALYSIS
, 2005
"... We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicat ..."
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Cited by 10 (0 self)
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We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of categorytheoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar categorytheoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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Cited by 4 (2 self)
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
Computability structures, simulations and realizability
, 2011
"... We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of si ..."
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We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of simulation between such computability structures, and show that such simulations correspond precisely to certain functors between the realizability models. Furthermore, we show that our class of computability structures has good closure properties — in particular, it is ‘cartesian closed ’ in a slightly relaxed sense. We also investigate some important subclasses of computability structures and of simulations between them. We suggest that our 2category of computability structures and simulations may offer a framework for a general investigation of questions of computational power, abstraction and simulability for a wide range of computation models from across computer science.
Constructive Toposes with Countable Sums as Models of Constructive Set Theory
"... We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This pa ..."
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We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive set theory. We show that any constructive topos with countable coproducts provides a model of a standard constructive set theory, CZFExp (that is, the variant of Aczel’s Constructive ZermeloFraenkel set theory obtained by weakening Subset Collection to the Exponentiation axiom). The model is constructed as a category of classes, using ideas derived from Joyal and Moerdijk’s programme of algebraic set theory. A curiosity is that our model always validates the axiom V = Vω1 (in an appropriate formulation). Hence the full Separation schema is always refuted. 1.
Notions of Computability for General Datatypes
, 2001
"... ncratic details of the programming language). We would place our work near the latter end of this spectrum, our intention being to uncover deep mathematical structure that is inherent in the nature of certain kinds of computation, independent of details of language designand whose relevance will ..."
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ncratic details of the programming language). We would place our work near the latter end of this spectrum, our intention being to uncover deep mathematical structure that is inherent in the nature of certain kinds of computation, independent of details of language designand whose relevance will therefore persist as particular languages come and go. The distinctive contribution of our project has been to focus on various natural notions of computability that programming languages may embody. It is well known that all reasonable programming languages give the same computable functions on natural numbers, but there are interesting senses in which this is not true for all other commonly occurring datatypes. Our philosophy is that an understanding of these notions of computability gives insight into those semantic models that most closely match the languages in question, which in turn supports the design of program logics with a clear operational meaning. Particular col
The Bulletin of Symbolic Logic
 Bull. Symbolic Logic
, 1996
"... The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods. ..."
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The Loeb space construction in nonstandard analysis is applied to the theory of processes to reveal basic phenomena which cannot be treated using classical methods.
An Abstract Look At Realizability
, 2000
"... This paper is about purely categorical approaches to realizability, and contrasts with recent work particularly by Longley [14] and Lietz and Streicher [13], in which the basis is taken as a typed generalisation of a partial combinatory algebra. We, like they, will be interested in when the construc ..."
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This paper is about purely categorical approaches to realizability, and contrasts with recent work particularly by Longley [14] and Lietz and Streicher [13], in which the basis is taken as a typed generalisation of a partial combinatory algebra. We, like they, will be interested in when the construction yields a topos, and hence gives a full interpretation of higherorder logic. This is also a theme of Birkedal's work, see [1, 2], and his joint work in [3]. Birkedal makes considerable use of the construction we study. We present realizability toposes as the product of two constructions. First one takes a category (which corresponds to the typed partial combinatory algebra), and then one glues Set to it in a variant of the comma construction. This, as we shall see, has the eect of improving the categorical properties of the algebra category. Then one takes an exact completion of the result. This also has the eect of improving the categorical properties. Formally the main result of the paper is that the result is a topos just (modulo some technical conditions) when the original category has a universal object. Early work on realizability (e.g.[12, 22], or see [23]) is characterised by its largely syntactic nature. The core denition is when a sentence of some formal logic is realised, and the main interest is in when certain deductive principles (such as Markov's rule) are validated. Martin Hyland's invention y The authors wish to acknowledge the support of the EPSRC, EU Working Group 26142 APPSEM, and MURST 1 2 of realizability toposes [10] advances on this, not only in the simplicity of the construction, but by providing a semantic framework in which the formal logics can naturally be interpreted. Hyland was strongly motivated in his work by a then recent approach...