Partial Univalence in n-truncated Type Theory
Paper i proceeding, 2020

It is well known that univalence is incompatible with uniqueness of identity proofs (UIP), the axiom that all types are h-sets. This is due to finite h-sets having non-trivial automorphisms as soon as they are not h-propositions. A natural question is then whether univalence restricted to h-propositions is compatible with UIP. We answer this affirmatively by constructing a model where types are elements of a closed universe defined as a higher inductive type in homotopy type theory. This universe has a path constructor for simultaneous "partial" univalent completion, i.e., restricted to h-propositions. More generally, we show that univalence restricted to (n-1)-types is consistent with the assumption that all types are n-truncated. Moreover we parametrize our construction by a suitably well-behaved container, to abstract from a concrete choice of type formers for the universe.

cubical type theory

homotopy type theory



Christian Sattler

Chalmers, Data- och informationsteknik, Datavetenskap

Andrea Vezzosi

IT-Universitetet i Kobenhavn

ACM International Conference Proceeding Series

807-819 3394759

35th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2020
Saarbrucken, Germany,


Algebra och logik


Matematisk analys



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