From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: X-Spam-Checker-Version: SpamAssassin 3.4.2 (2018-09-13) on inbox.vuxu.org X-Spam-Level: X-Spam-Status: No, score=-1.1 required=5.0 tests=DKIM_SIGNED,DKIM_VALID, DKIM_VALID_AU,DKIM_VALID_EF,FREEMAIL_FORGED_FROMDOMAIN,FREEMAIL_FROM, HEADER_FROM_DIFFERENT_DOMAINS,HTML_MESSAGE,MAILING_LIST_MULTI, RCVD_IN_DNSWL_NONE autolearn=ham autolearn_force=no version=3.4.2 Received: from mail-ot1-x33c.google.com (mail-ot1-x33c.google.com [IPv6:2607:f8b0:4864:20::33c]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id 47a2e73e for ; Thu, 9 May 2019 08:18:43 +0000 (UTC) Received: by mail-ot1-x33c.google.com with SMTP id e3sf240425otk.1 for ; Thu, 09 May 2019 01:18:42 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:date:from:to:message-id:subject:mime-version :x-original-sender:precedence:mailing-list:list-id:list-post :list-help:list-archive:list-unsubscribe; bh=mKrh8bU+Hu22N/LP3oHWJxYTNtO6FBGuToqLrxGdwiI=; b=W8qbZ9SeclD0W6KJ5bg5YeDNtQfdaFY4F1MbbbIK8utJqg8E9pL/vZUfrnXEb6jTQA dAHvf8FPMZ3srbnXcwzHu6uFze+TjteHazR7S4hP5LcMbjZEiBCOn9ucbGVOJeN6ZYn7 qgVKSQLRY1zFaCrLu0GMPIu5kO1gEjpA0wPj5tYpyzbSlfYlnzzpOIfd2GC3NrXP0UG2 rpmR0gh3SAGfXtjAqJ+ayuzqgrOfMVhkgaHeMRycwDragGuTp/7agMHNVN4GnCpt93HB 2uDzVcG76iAaHFGBspSbxhvZPsoQNofOVU0Zhj6P++/4VZpjc5tBBvgsk/1hs7j+A+d7 fHgw== DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=date:from:to:message-id:subject:mime-version:x-original-sender :precedence:mailing-list:list-id:list-post:list-help:list-archive :list-unsubscribe; bh=mKrh8bU+Hu22N/LP3oHWJxYTNtO6FBGuToqLrxGdwiI=; b=gMLri7MWaULLRc1oNUFdTAMwL+o+J1QVE+DbrcIV5iRObdhMYj1B6hqg+zND32AK3o Y1x6FMRZ029oUPiHl3N4q3BstiqBUzd4qeVvjy93cixRDrBsaSGLeabl8jJ+WJ5U8bAi /8bAd0tPq+0PQouMd5FbLuoxEwEFO8mcAQYX9Qdvc9NJrKkONjvHmdiOhuP8QTpCUn+T /uJXhuVYKRSCD4Dcyui7pZNIyqKYY8dnbF/yxl5wPzXpzRXZdxzIQ8+BwA9zq7eQbSI9 5IvSDteJC2Qvc+d3PawXDW0uiE8gSUNodDVgrFAgpuidlg9xq1UNKLHS849LwyNopr7l /auA== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=sender:x-gm-message-state:date:from:to:message-id:subject :mime-version:x-original-sender:precedence:mailing-list:list-id :x-spam-checked-in-group:list-post:list-help:list-archive :list-unsubscribe; bh=mKrh8bU+Hu22N/LP3oHWJxYTNtO6FBGuToqLrxGdwiI=; b=dyDK/BPoXVMA15Tv4vNMCpYEq8a1p+md+95RuPEPVdLJTwuxM04edpKil9OBIoZ/Tn J33RtYodjd9Ru18UlVNl9Ix9n4ffpX2mQjkKIN+iN5XxWQz/Xy1gdyjzunmSLU/bP4Fb ILNXICVoPiQwghDAWVDPtl0zJPdYoHVJgdPMGfydrKTj+YvvMlHVvO4rDju/BxpvjiG/ UJtr9cR4k2QAcpZ8ejdQEiGaQuvz8+oNkMKsfMRYz2bdqbTjO9esbqZwVPVJvXx8zwyo crfr1kim89VzIq/bdqVeij6Ilz5Uz3XiDZyDYgulGjrjxxRIYR3EQ3p0zicvNNHEKj0L YayA== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAW1LpTKoP9kM51Y76x5/W6sRQuL/8b+HcUTLb1OpjvOLbvvVgr2 Lk0OJSmkgyFcLDCtHONL6R0= X-Google-Smtp-Source: APXvYqzfAof5rz3PlLnXzejYOlrU4GMmKclu3QiXjBIC3naMQN6ogCTcqY0I8f+yVRUVAxPEhGNsJQ== X-Received: by 2002:aca:b1d4:: with SMTP id a203mr688147oif.67.1557389920883; Thu, 09 May 2019 01:18:40 -0700 (PDT) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:aca:3bd4:: with SMTP id i203ls247990oia.14.gmail; Thu, 09 May 2019 01:18:40 -0700 (PDT) X-Received: by 2002:aca:5054:: with SMTP id e81mr695542oib.74.1557389919843; Thu, 09 May 2019 01:18:39 -0700 (PDT) Date: Thu, 9 May 2019 01:18:38 -0700 (PDT) From: Andrew Swan To: Homotopy Type Theory Message-Id: <426a5bdc-35e2-4cdf-8b1d-70068fd7b3f5@googlegroups.com> Subject: [HoTT] Paper on Church's Thesis MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_200_2092052189.1557389918954" X-Original-Sender: wakelin.swan@gmail.com Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , ------=_Part_200_2092052189.1557389918954 Content-Type: multipart/alternative; boundary="----=_Part_201_369486870.1557389918955" ------=_Part_201_369486870.1557389918955 Content-Type: text/plain; charset="UTF-8" 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 -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/426a5bdc-35e2-4cdf-8b1d-70068fd7b3f5%40googlegroups.com. For more options, visit https://groups.google.com/d/optout. ------=_Part_201_369486870.1557389918955 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Dear all,

Taichi Uemura and I would lik= e to announce our new paper on the axiom of Church's thesis in cubical = assemblies at=C2=A0https://arx= iv.org/abs/1905.03014=C2=A0. Our main results are that Church's the= sis (which states that all functions N -> N are computable) is false in = cubical assemblies but that it does hold in a reflective subuniverse of cub= ical 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 inte= rest. The technique we use is fairly general - it takes statements of a cer= tain form that hold in the internal language of a locally cartesian closed = category satisfying the Orton-Pitts axioms, and then constructs a reflectiv= e subcategory of the Orton-Pitts model of cubical type theory where the sam= e 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 a= re continuous) is consistent with univalent type theory. To construct propo= sitional truncation in the reflective subuniverse we needed first to constr= uct certain higher inductive types in cubical assemblies (suspensions, prop= ositional 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 versio= n of) W types with reductions.

Comments, questions= , etc are welcome.

Best,
Andrew

--
You received this message because you are subscribed to the Google Groups &= quot;Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an e= mail to = HomotopyTypeTheory+unsubscribe@googlegroups.com.
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For more options, visit http= s://groups.google.com/d/optout.
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