2016
DOI: 10.1016/j.apal.2016.04.011
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A constructive manifestation of the Kleene–Kreisel continuous functionals

Abstract: We identify yet another category equivalent to that of Kleene-Kreisel continuous functionals. Reasoning constructively and predicatively, all functions from the Cantor space to the natural numbers are uniformly continuous in this category. We do not need to assume Brouwerian continuity axioms to prove this, but, if we do, then we can show that the full type hierarchy is equivalent to our manifestation of the continuous functionals. We construct this manifestation within a category of concrete sheaves, called C… Show more

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Cited by 11 publications
(6 citation statements)
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“…This paper does not focus on homotopy theory, thus the methodology is different. Escardó and Xu (2016) use a similar big topos, but restricted to compact spaces to model the fan-theorem in a simple type theory. Coquand et al (2017) provide a stack model over Cantor space for univalent type theory.…”
Section: Discussionmentioning
confidence: 99%
“…This paper does not focus on homotopy theory, thus the methodology is different. Escardó and Xu (2016) use a similar big topos, but restricted to compact spaces to model the fan-theorem in a simple type theory. Coquand et al (2017) provide a stack model over Cantor space for univalent type theory.…”
Section: Discussionmentioning
confidence: 99%
“…In addition, they often do a better job of faithfully encoding topological notions such as unions of closed sets, constructions of cell complexes (including geometric realization of simplicial sets), and cohomology. In [EX16] such a topos was used to represent the Kleene-Kreisel functionals and model principles of intutionism.…”
Section: Examplesmentioning
confidence: 99%
“…There are several related categories. For example, C-spaces [12] arise in a similar way but replacing (N ∪ {∞}) with the Cantor space 2 N . In Section 6.3 we will replace (N ∪ {∞}) with the vertical natural numbers equipped with the Scott topology, following [18].…”
Section: Categories Of Concrete Sheavesmentioning
confidence: 99%