Canonicity and homotopy canonicity for cubical type theory
Journal article, 2022

Cubical type theory provides a constructive justification of homotopy type theory. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. We present in this article two canonicity results, both proved by a sconing argument: a homotopy canonicity result, every natural number is path equal to a numeral, even if we take away the equations defining the lifting operation on the type structure, and a canonicity result, which uses these equations in a crucial way. Both proofs are done internally in a presheaf model.

univalence

sconing

cubical type theory

canonicity

Artin glueing

Author

Thierry Coquand

University of Gothenburg

Simon Huber

University of Gothenburg

Christian Sattler

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

Logical Methods in Computer Science

18605974 (eISSN)

Vol. 18 1 35

Proof theory and higher categorical semantics of homotopy type theory

Swedish Research Council (VR) (2019-03765), 2020-01-01 -- 2023-12-31.

Subject Categories

Algebra and Logic

Computational Mathematics

Computer Science

DOI

10.46298/LMCS-18(1:28)2022

More information

Latest update

11/15/2023