Partial Univalence in n-truncated Type Theory
Paper in 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

univalence

Author

Christian Sattler

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

Andrea Vezzosi

IT University of Copenhagen

ACM International Conference Proceeding Series

807-819 3394759
978-145037104-9 (ISBN)

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

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1145/3373718.3394759

More information

Latest update

8/4/2020 1