Discussion of Homotopy Type Theory and Univalent Foundations
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From: Andrew Swan <wakelin.swan@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: [HoTT] Paper on Church's Thesis
Date: Thu, 9 May 2019 01:18:38 -0700 (PDT)
Message-ID: <426a5bdc-35e2-4cdf-8b1d-70068fd7b3f5@googlegroups.com> (raw)

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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

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