Parametricity, automorphisms of the universe, and excluded middle
Auke Bart Booij, Martı́n Hötzel Escardó, Peter LeFanu Lumsdaine, Michael Shulman · arXiv (Cornell University) · 2017
It is known that one can construct non-parametric functions by assuming classical axioms. Our work is a converse to that: we prove classical axioms in dependent type theory assuming specific instances of non-parametricity. We also address the interaction between classical axioms and the existence of automorphisms of a type universe. We work over intensional Martin-Löf dependent type theory, and in some results assume further principles including function extensionality, propositional extensionality, propositional truncation, and the univalence axiom.