Cubical Type Theory: a constructive interpretation of the univalence axiom · HackerPapers discussions

Cubical Type Theory: a constructive interpretation of the univalence axiom — mentioned by mauricioc

Fri, 12 Jul 2019 18:42:35 GMT

One exciting new development is "Cubical Type Theory: a constructive interpretation of the univalence axiom" [0], a constructive version of homotopy type theory. The "axiom of choice" in this context is not really an axiom, just a provable theorem whose proof in this setting is given by Martin-Löf. The reason, I believe, is that constructive mathematics has a different meaning for the exists (∃) quantifier. In a constructive context, existence is "stronger" since you have to specify a witness of existence; this makes the hypothesis of the "axiom" of choice stronger, allowing for a proof.…

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Cubical Type Theory: a constructive interpretation of the univalence axiom — mentioned by hiker

Tue, 31 Oct 2017 09:44:10 GMT

Cubical https://github.com/mortberg/cubicaltt is a proof of concept programming language based on constructive (cubical type theory https://arxiv.org/abs/1611.02108) interpretation of HoTT. Agda also added recently support for cubical paths https://agda.readthedocs.io/en/latest/language/cubical.html . I'm also waiting on Lean https://leanprover.github.io to introduce cubical type theory for HoTT. Version 2 of the language had HoTT based on the univalence axiom which was dropped in version 3. With cubical type theory the univalence axiom can be constructively proved and is no longer an axiom.

Paper and discussion history. A mention is not an endorsement.