Categorical univalence of a universe does not entail function extensionality, as shown by polynomial models of type theory that refute the latter while satisfying the former.
Cubical assemblies, a univalent and impredicative universe and a failure of propositional resizing
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Extends intensional type theory with large sizes and parametric quantifiers to construct inductive and coinductive types, justified by a realisability model interpreting sizes as an uncountable ordinal.
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Univalence without function extensionality
Categorical univalence of a universe does not entail function extensionality, as shown by polynomial models of type theory that refute the latter while satisfying the former.
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Constructing (Co)inductive Types via Large Sizes
Extends intensional type theory with large sizes and parametric quantifiers to construct inductive and coinductive types, justified by a realisability model interpreting sizes as an uncountable ordinal.