From mboxrd@z Thu Jan 1 00:00:00 1970 X-Received: by 10.157.60.11 with SMTP id q11mr11868255otc.3.1487873289272; Thu, 23 Feb 2017 10:08:09 -0800 (PST) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 10.157.45.170 with SMTP id g39ls4903870otb.49.gmail; Thu, 23 Feb 2017 10:08:08 -0800 (PST) X-Received: by 10.157.62.90 with SMTP id h26mr3802992otg.105.1487873288932; Thu, 23 Feb 2017 10:08:08 -0800 (PST) Return-Path: Received: from pps4.ias.edu (pps4.ias.edu. [192.16.204.92]) by gmr-mx.google.com with ESMTPS id f187si546607ywc.0.2017.02.23.10.08.08 (version=TLS1_2 cipher=ECDHE-RSA-AES128-GCM-SHA256 bits=128/128); Thu, 23 Feb 2017 10:08:08 -0800 (PST) Received-SPF: pass (google.com: domain of vlad...@ias.edu designates 192.16.204.92 as permitted sender) client-ip=192.16.204.92; Authentication-Results: gmr-mx.google.com; spf=pass (google.com: domain of vlad...@ias.edu designates 192.16.204.92 as permitted sender) smtp.mailfrom=vlad...@ias.edu; dmarc=pass (p=NONE sp=NONE dis=NONE) header.from=ias.edu Received: from pps.reinject (pps4.ias.edu [127.0.0.1]) by pps4.ias.edu (8.16.0.17/8.16.0.17) with ESMTPS id v1NI86NO029341 (version=TLSv1.2 cipher=ECDHE-RSA-AES256-GCM-SHA384 bits=256 verify=NOT); Thu, 23 Feb 2017 13:08:06 -0500 Received: from pps4.ias.edu (pps4.ias.edu [127.0.0.1]) by pps.reinject (8.16.0.16/8.16.0.16) with SMTP id v1NI861H029335; Thu, 23 Feb 2017 13:08:06 -0500 X-Proofpoint-Sentinel: stfjFc8JfWXHKBVpikCAhP3RAzQKPk1wKN1re5KwLOah6OFTYWx0ZWRfX93H uVAipeVw2/PLLUwJkMnBmpA17rW3VH2FaRy27SIg7ElnHdjypNH8UDXUVx5FjJQylqMISDJvdnnr Wiik8vucfr4NB3PpFw== Received: from imap.math.ias.edu (f5-bip.net.ias.edu [172.16.13.241]) by pps4.ias.edu with ESMTP id v1NI86MT029334 (version=TLSv1.2 cipher=ECDHE-RSA-AES256-GCM-SHA384 bits=256 verify=NOT); Thu, 23 Feb 2017 13:08:06 -0500 Received: from pool-96-248-86-80.cmdnnj.fios.verizon.net ([96.248.86.80] helo=vladimirs-mbp-2.home) by imap.math.ias.edu with esmtpsa (TLSv1.2:DHE-RSA-AES256-GCM-SHA384:256) (Exim 4.80.1) (envelope-from ) id 1cgxoE-0001e0-MS; Thu, 23 Feb 2017 13:08:06 -0500 Content-Type: text/plain; charset=utf-8 Mime-Version: 1.0 (Mac OS X Mail 10.2 \(3259\)) Subject: Re: [HoTT] about the HTS From: Vladimir Voevodsky In-Reply-To: <484524d2-4054-5b7a-14af-e68a8e4b8375@gmail.com> Date: Thu, 23 Feb 2017 13:08:06 -0500 Cc: "Prof. Vladimir Voevodsky" , Univalent Mathematics , homotopytypetheory Content-Transfer-Encoding: quoted-printable Message-Id: <91160F61-2A67-4B1F-9A4A-F3510B99E9D3@ias.edu> References: <1FFF5B8D-06FC-4ECD-8F89-496A4C2E3A3A@ias.edu> <484524d2-4054-5b7a-14af-e68a8e4b8375@gmail.com> To: Benedikt Ahrens X-Mailer: Apple Mail (2.3259) X-Proofpoint-Virus-Version: vendor=fsecure engine=2.50.10432:,, definitions=2017-02-23_12:,, signatures=0 X-Proofpoint-Spam-Reason: safe X-IAS-PPS-SPAM: NO X-Proofpoint-Spam-Details: rule=ias_safe policy=ias score=0 spamscore=0 suspectscore=0 malwarescore=0 phishscore=0 adultscore=0 bulkscore=0 classifier=donotscan adjust=0 reason=safe scancount=1 engine=8.0.1-1612050000 definitions=main-1702230165 X-IAS-PPS-PHISH: NO I could not find where he defines the univalent (intensional) equality. It = seems that all his equalities are strict. > On Feb 23, 2017, at 9:57 AM, Benedikt Ahrens wrote= : >=20 >=20 >=20 > On 02/23/2017 03:47 PM, Vladimir Voevodsky wrote: >> Just a thought=E2=80=A6 Can we devise a version of the HTS where exact >> equality types are not available for the universes such that, even >> with the exact equality, HTS would remain a univalent theory. >>=20 >> Maybe only some types should be equipped with the exact equality and >> this should be a special quality of types. >>=20 >> Vladimir. >>=20 >> PS If there are higher inductive types then the exact equality should >> not be available for them either. >=20 >=20 > Paolo Capriotti, in his PhD thesis "Models of Type Theory with Strict > Equality" [1], has studied strict equality in type theory. >=20 >> From page 69: >=20 > "Finally, universes in the strict fragment of our system are not assumed > to be fibrant types, like in HTS." >=20 > You might be interested in Section 4.1.1, "Differences with HTS". >=20 > [1] https://arxiv.org/abs/1702.04912 >=20 > --=20 > 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 HomotopyTypeThe...@googlegroups.com. > For more options, visit https://groups.google.com/d/optout.