Equilogical spaces

Bauer, Andrej; Birkedal, Lars; Scott, Dana

doi:10.1016/j.tcs.2003.11.012

Cited by 50 publications

(64 citation statements)

References 15 publications

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“…Recall from [Sco96,BBS04] that an equilogical space E = (S E , τ E , ∼ E ) consists of a T 0 -space (S E , τ E ) and an equivalence relation ∼ E ⊆ S E × S E on the points of the space.…”

Section: Equilogical Spacesmentioning

confidence: 99%

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The category of equilogical spaces and the effective topos as homotopical quotients

Rosolini¹

2016

J. Homotopy Relat. Struct.

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Section: Equilogical Spacesmentioning

confidence: 99%

“…The construction of the exponential F E is less direct and we refer the reader to the basic sources [Sco76,Sco96,BBS04] as well as [BR14,BCRS98].…”

Section: Equilogical Spacesmentioning

confidence: 99%

The category of equilogical spaces and the effective topos as homotopical quotients

Rosolini¹

2016

J. Homotopy Relat. Struct.

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“…α = n β =⇒ f (α) = f (β), where α = n β means ∀i < n. α i = β i . These models include Kleene-Kreisel functionals [19], compactly generated spaces [19], limit spaces [20], equilogical spaces [3], sequential spaces [12], and QCB spaces [12]. However, even though these models are introduced for the purposes of computability theory, they are developed within a classical meta-theory.…”

Section: Introductionmentioning

confidence: 99%

A Constructive Model of Uniform Continuity

Escardó

2013

Lecture Notes in Computer Science

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Abstract. We construct a continuous model of Gödel's system T and its logic HA ω in which all functions from the Cantor space 2 N to the natural numbers are uniformly continuous. Our development is constructive, and has been carried out in intensional type theory in Agda notation, so that, in particular, we can compute moduli of uniform continuity of T-definable functions 2 N → N. Moreover, the model has a continuous Fan functional of type (2 N → N) → N that calculates moduli of uniform continuity. We work with sheaves, and with a full subcategory of concrete sheaves that can be presented as sets with structure, which can be regarded as spaces, and whose natural transformations can be regarded as continuous maps.

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“…This is similar to a long-standing problem in recursion theory, concerning definable total maps of higher type such as N N → N or R R → R. Amongst many authors, Hyland and Bauer studied it in their settings of filter and equilogical spaces [Hyl79,BBS04].…”

Section: Beyond Local Compactnessmentioning

confidence: 79%

“…Curiously, one way of extending the traditional categories to have all exponentials is to add new colimits, specifically quotients of equivalence relations. The new objects, as proposed by Dana Scott, are called equilogical spaces [BBS04]. Giuseppe Rosolini put them in a sheaf-like categorical framework [Ros00] that also includes Martin Hyland's earlier filter spaces [Hyl79], and there also a cartesian closed extension for locales [Hec06].…”

Section: Beyond Local Compactnessmentioning

confidence: 99%

Foundations for Computable Topology

Taylor¹

2011

The Western Ontario Series in Philosophy of Science

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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.This method is applied to devising a new paradigm for general topology, called Abstract 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."In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl.)"Point-set topology is a disease from which the human race will soon recover." (Henri Poincaré, quoted in Comic Sections by Des MacHale.)

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Equilogical spaces

Cited by 50 publications

References 15 publications

The category of equilogical spaces and the effective topos as homotopical quotients

The category of equilogical spaces and the effective topos as homotopical quotients

A Constructive Model of Uniform Continuity

Foundations for Computable Topology

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