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Topological and Limitspace subcategories of Countablybased Equilogical Spaces
, 2001
"... this paper we show that the two approaches are equivalent for a ..."
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Cited by 22 (4 self)
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this paper we show that the two approaches are equivalent for a
Developing Theories of Types and Computability via Realizability
, 2000
"... We investigate the development of theories of types and computability via realizability. ..."
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Cited by 20 (6 self)
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We investigate the development of theories of types and computability via realizability.
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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Cited by 13 (4 self)
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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
A universal characterization of the closed euclidean interval (Extended Abstract)
 PROC. OF 16TH ANN. IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE, LICS'01
, 2001
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basi ..."
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Cited by 10 (0 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the
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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Cited by 8 (7 self)
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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
Categories of spaces may not be generalized spaces as exemplified by directed graphs
 Matematicas XX, 179–86. Republished in: Reprints in Theory and Applications of Categories
, 1986
"... to his continued generosity in granting copyright permission, it can now be reprinted in ..."
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Cited by 7 (0 self)
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to his continued generosity in granting copyright permission, it can now be reprinted in
Two constructive embeddingextension theorems with applications to continuity principles and to BanachMazur computability
 Mathematical Logic Quarterly
"... We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor ..."
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Cited by 6 (1 self)
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We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to Z extends to a sequentially continuous function from X to R. The second asserts an analogous property for Baire space relative to any inhabited locally noncompact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between “continuity principles ” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to R are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally noncompact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally noncompact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursiontheoretic setting, then, for any such space X, there exists a BanachMazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers.
The Largest Topological Subcategory of Countablybased Equilogical Spaces
, 1998
"... There are two main approaches to obtaining "topological" cartesianclosed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed  for example, the category of sequential spaces. Under the other, one generalises the notion of s ..."
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Cited by 4 (1 self)
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There are two main approaches to obtaining "topological" cartesianclosed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed  for example, the category of sequential spaces. Under the other, one generalises the notion of space  for example, to Scott's notion of equilogical space. In this paper we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countablybased equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. This category consists of certain "!projecting" topological quotients of countablybased topological spaces, and contains, in particular, all countablybased spaces. We show that this category is cartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the cate...
Parametrized spaces model locally constant homotopy sheaves
 Topology Appl
, 2008
"... Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification ..."
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Abstract. We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopytheoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopytheoretic version of the correspondence between covering spaces and π1sets. We then use these two equivalences to study base change functors for parametrized spaces. Contents
A Universal Characterisation of the Closed Euclidean Interval
 in: Proceedings of 16th Annual IEEE Symposium on Logic in Computer Science
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the ca ..."
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Cited by 3 (3 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. We test the notion in categories of interest. In the category of sets, any closed and bounded interval of real numbers is an interval object. In the category of topological spaces, the interval objects are closed and bounded intervals with the Euclidean topology. We also prove that an interval object exists in any elementary topos with natural numbers object. The universal property of an interval object provides a mechanism for defining functions on the interval. We use this to define basic arithmetic operations, and to verify equations between them. It also allows us to develop an analogue of the primitive recursive functions, yielding a natural class of computable functions on the interval. Contents 1