Dear all,
Taichi Uemura and I would like to announce our new paper on the axiom of Church's thesis in cubical assemblies at
https://arxiv.org/abs/1905.03014 . Our main results are that Church's thesis (which states that all functions N -> N are computable) is false in cubical assemblies but that it does hold in a reflective subuniverse of cubical assemblies. We use the reflective subuniverse to show that Church's thesis is consistent with univalent type theory.
Along the way we show a couple of other results that might also be of interest. The technique we use is fairly general - it takes statements of a certain form that hold in the internal language of a locally cartesian closed category satisfying the Orton-Pitts axioms, and then constructs a reflective subcategory of the Orton-Pitts model of cubical type theory where the same statement holds. We use this to give a new proof of Thierry Coquand's result that Brouwer's principle (all functions from Baire space to N are continuous) is consistent with univalent type theory. To construct propositional truncation in the reflective subuniverse we needed first to construct certain higher inductive types in cubical assemblies (suspensions, propositional truncation and nullification). To do this we first construct the underlying objects using W types with reductions and then apply the Coquand-Huber-Mortberg method. That method applies not just to cubical assemblies but any lcc satisfying the Orton-Pitts axioms together with (a split version of) W types with reductions.
Comments, questions, etc are welcome.
Best,
Andrew