* [HoTT] Paper on Church's Thesis
@ 2019-05-09 8:18 Andrew Swan
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From: Andrew Swan @ 2019-05-09 8:18 UTC (permalink / raw)
To: Homotopy Type Theory
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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
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.
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2019-05-09 8:18 [HoTT] Paper on Church's Thesis Andrew Swan
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