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When is the Kleene--Kreisel hierarchy full?

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If you have a question about this talk, please contact Neel Krishnaswami.

In the category of Kleene-Kreisel continuous functionals, all maps from the Cantor space to the natural numbers are uniformly continuous. However, its traditional treatment available in the literature relies on either classical logic or constructively contentious principles. In a weak constructive setting, we develop the theory of Kleene-Kreisel spaces within a category of concrete sheaves, called C-spaces, which form a locally cartesian closed category. This category has a fan functional and validates the uniform continuity axiom in System T and dependent types. We do not need to assume Brouwerian continuity axioms, but, if we do, then we can show constructively that the full type hierarchy is equivalent to our manifestation of the continuous functionals.

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