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=-0.9 required=5.0 tests=DKIM_SIGNED,DKIM_VALID, DKIM_VALID_AU,DKIM_VALID_EF,FREEMAIL_FORGED_FROMDOMAIN,FREEMAIL_FROM, HEADER_FROM_DIFFERENT_DOMAINS,MAILING_LIST_MULTI,RCVD_IN_DNSWL_NONE autolearn=ham autolearn_force=no version=3.4.2 Received: from mail-il1-x13c.google.com (mail-il1-x13c.google.com [IPv6:2607:f8b0:4864:20::13c]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id d4f80741 for ; Mon, 4 Nov 2019 23:27:07 +0000 (UTC) Received: by mail-il1-x13c.google.com with SMTP id d11sf16973539ild.10 for ; Mon, 04 Nov 2019 15:27:07 -0800 (PST) ARC-Seal: i=2; a=rsa-sha256; t=1572910026; cv=pass; d=google.com; s=arc-20160816; b=F3NmotP/wQvYmouWIOSiqcQfDIyhbYC4UXEsUllNT/7sNl7jE68V6zp8x0gyCMa7FF aCtsN45pd/ZFo/F7Jy8iXq325tcO+P+ocg8TtlNCSv+64cHUX5tRkotkC/ZsgMONv52C vU90TGCoGb4loy2UcV+Ye436/7G2D2BfJfxGIQZQCYuGxXSL4AQoAd1+sa7Fp7qFqJww qujddeV+bolADp20OyPWY8xhRdOAFDWOnXOJfd10GMTSrwDe5p7nXo89MBWImwbM9xVR 2ZRRd+JOrjrY9k9e/FaxqYrp76SZ7IGiCg7+tKRvRWdK6GOs+QR6fO9S+xf4ep+7wNIv 54eg== ARC-Message-Signature: i=2; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=list-unsubscribe:list-archive:list-help:list-post:list-id :mailing-list:precedence:content-transfer-encoding:cc:to:subject :message-id:date:from:in-reply-to:references:mime-version:sender :dkim-signature:dkim-signature; bh=fXZnI+5Tgv18W37IR/XBCQenzltT/1WLFA1cj5IbaSw=; b=aoyk8wgcer2MFuvXv0YlZW9pHX9UKULSDc8HTAJ3PAKAwCBQWFJxY2zdD65uWy5RTs nWrlSDyfXqFO8vxI2gT3Q9d2S6zzol7c1olPkOkeOMkfi8cxSgipC3YiYhjiwClMGFke c+G+se2FY4LJn0oC0d19DtUNlaREqHBdWdXmPK6dtjvXe6hAH95F4EK9VuoGJoQSxKIA glGUIF+5mqgwHcRIauLvnmVWvHkhvtEAK4XhAWMzks7DNk39fTKKbp0Ij7bcx9WGxk88 kK7S+7a5cZYpIgd8hLPVUBbWI7vAZv5vL5UMHciQrO+tDsKfK976AQ+0YjVUrM0zHTj6 R4Xw== ARC-Authentication-Results: i=2; gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=btzuehcc; spf=pass (google.com: domain of droberts.65537@gmail.com designates 2607:f8b0:4864:20::c43 as permitted sender) smtp.mailfrom=droberts.65537@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=gmail.com DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:mime-version:references:in-reply-to:from:date:message-id :subject:to:cc:content-transfer-encoding:x-original-sender :x-original-authentication-results:precedence:mailing-list:list-id :list-post:list-help:list-archive:list-unsubscribe; bh=fXZnI+5Tgv18W37IR/XBCQenzltT/1WLFA1cj5IbaSw=; b=obGWHFc+DfO8RZa+CSfnVCPRq9Yl90E8Nn7VboWDkjjkXbhyPpZOK/LgaF/CEmWV2T Re7wJDMfas2tj0+bw0BR4t+pHPwH+0brhOcsbzZ/hPFujS8somzm3HP0NMFRSuqExbHu FtIldfYz7XCeovH2usJQjjDAr/DZUhnEUUzmOgM/D2Xru1YY+apMCF6HM2/mD+o7U8/n nJtnB/zidIxRol0OcYSBRix7YGE4xB/I0hTlTrBiewFLYYhgl3A8in6jEl5frwk/zw9N U3gh1TEsXOVD9IoYxU1Eo84pu/yYAgr56U+YrHCXGJdCsByZFbNSCPWL6MnkAgwfk/Pb 1+OA== DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=mime-version:references:in-reply-to:from:date:message-id:subject:to :cc:content-transfer-encoding:x-original-sender :x-original-authentication-results:precedence:mailing-list:list-id :list-post:list-help:list-archive:list-unsubscribe; bh=fXZnI+5Tgv18W37IR/XBCQenzltT/1WLFA1cj5IbaSw=; b=NRYBSJK3tJJR46RgScq6H9XmFAosuxq+qPVWXYtRz5jJYQ39gFbtMYltbXnakJoAhv OWINhkAfP7K93wrbxn5gEZtHnCqQZMNoNfuE1jLqJ+b33+Vwiuc3QDWdzf+1fawFVSF1 XgWn/6l7H3yJRTU3cUwAyTMeyQismxYh3UWYBn0gqFS3/c0t22a+8sQqI4OHcoy1r+Zy d6VeQBdm7cn/o9Tfi57b9qHfGIpgaI1JHIdw2N5AtFmrsRy4GxK5/+5OMw2o6atScKCt bLYlMfXynzz3VO9OEBz8p8yx579u7ho2fct0yJSgnfScV5v9orISo9sN8MIHBagoe+VI hgiw== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=sender:x-gm-message-state:mime-version:references:in-reply-to:from :date:message-id:subject:to:cc:content-transfer-encoding :x-original-sender:x-original-authentication-results:precedence :mailing-list:list-id:x-spam-checked-in-group:list-post:list-help :list-archive:list-unsubscribe; bh=fXZnI+5Tgv18W37IR/XBCQenzltT/1WLFA1cj5IbaSw=; b=TS7h37RKO5DwwpI3SZypuXkEvW/WwEtSLi4Y1IdMg5dVI9c5/74jmj3w79ibpWQ5y5 1/SvV5L9PZmb4Wo0DSIkd3Q8UzDb9N0XXTDUeOZ+XmUUooH56lYN0VaH5z3yf9qSbuQR fWl9lvt5xCmVx5UyU5TNrpRhjjgnGcRQsHOaHNuhfTz2mYT8ReAybOZmTMoOvR64Zve3 fD1xOxtbRUQunWlko9oQK1rvh2rrQaC540WZ06E3yMMOvFoBCmAP5OP0e9hWpLUwaVbg vtOcAJb3IC078zopm4dVL9XQ+hFvwWsDcGkrXlJNgXIazKI1tnx1T1zEJXBmTnmvPSXz oxCw== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAXtb14uk0dCK1rmkbit6ly5R1oIDryIZIXdBF6O/tdISmXG1Ph4 8q7n3wIdzv2cILsWwN9a7g0= X-Google-Smtp-Source: APXvYqzVyNmnVwlJNEj8WQUfawdbMa/WX20+fnml+gPqL/IEbvyU46tTCOd0GhpNgH2AY37ihHxGQQ== X-Received: by 2002:a5d:9349:: with SMTP id i9mr4803071ioo.163.1572910026043; Mon, 04 Nov 2019 15:27:06 -0800 (PST) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:a6b:7004:: with SMTP id l4ls1933796ioc.12.gmail; Mon, 04 Nov 2019 15:27:05 -0800 (PST) X-Received: by 2002:a6b:7215:: with SMTP id n21mr15524797ioc.281.1572910025531; Mon, 04 Nov 2019 15:27:05 -0800 (PST) ARC-Seal: i=1; a=rsa-sha256; t=1572910025; cv=none; d=google.com; s=arc-20160816; b=KWBkuzQuu/kFW60Tx5xRzSmeboZudhEwS4W5tuKrYbmrM7dEZrZ32G3iQ9YhOhOuTq dUL0dRwPXKxb2qNtXzNl0dfL7qYo89XDqPIe9EdQz2f+L002N/muSnl9idjQD0WO46zH Vr+CuWCxoat/yAvcAmXkUEeqGaiMSxIgJiAQi/vukJVWegAt+YWWkK30CLgJvG4mFRCb GucgAqKFKFG+S29/hVZx7RyodvPUMV+FAIBM+6LUGjyVUcFy4fsvwYoywbmBgoOj0qPI p0qutiDjocCaSUcdwdzxIW6jipCOEeMK2ZenSXQ55BZz9MfENtHImru+M24cAJD3iDAi 1MsQ== ARC-Message-Signature: i=1; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=content-transfer-encoding:cc:to:subject:message-id:date:from :in-reply-to:references:mime-version:dkim-signature; bh=AohAPYipFx67LD8l2z54Ld+FdGmpIq2ZLxKE0xjk5H8=; b=PJ3tOCl6I9Ah06CZHsgz35Lhc9vaHTH+/Z0O7VBYRS2MJ6Moz0W4kguv9Kygo+p5XP GNGT6Ysv37cAF0djyBaTklVwqGVju2H3CvZ4dvVN7vzBliXh+8lyMpIC1pdiHlvSMEq3 wudB4wwlwi5eOeJMljJWaKeNY/D9/QmmYUraOWu/0IBrJ6pIvnd34WOZO/+QwSkRaMae vMg1BX10VtexHxgmOoolfmQktXKSKNBUFR5yV7ULByBo9DYZj1aA1FooCaxGdGJW9qQ3 HhxDm816eN3YMTNdshEYCCF4QcU++tv+PAipnSLlzmyHdkYybX3ff55KoqF128JEspoX T83A== ARC-Authentication-Results: i=1; gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=btzuehcc; spf=pass (google.com: domain of droberts.65537@gmail.com designates 2607:f8b0:4864:20::c43 as permitted sender) smtp.mailfrom=droberts.65537@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=gmail.com Received: from mail-yw1-xc43.google.com (mail-yw1-xc43.google.com. [2607:f8b0:4864:20::c43]) by gmr-mx.google.com with ESMTPS id 75si1141632ilw.3.2019.11.04.15.27.05 for (version=TLS1_3 cipher=TLS_AES_128_GCM_SHA256 bits=128/128); Mon, 04 Nov 2019 15:27:05 -0800 (PST) Received-SPF: pass (google.com: domain of droberts.65537@gmail.com designates 2607:f8b0:4864:20::c43 as permitted sender) client-ip=2607:f8b0:4864:20::c43; Received: by mail-yw1-xc43.google.com with SMTP id k127so7639889ywc.6 for ; Mon, 04 Nov 2019 15:27:05 -0800 (PST) X-Received: by 2002:a81:928e:: with SMTP id j136mr4454820ywg.497.1572910024503; Mon, 04 Nov 2019 15:27:04 -0800 (PST) MIME-Version: 1.0 References: In-Reply-To: From: David Roberts Date: Tue, 5 Nov 2019 09:56:53 +1030 Message-ID: Subject: Re: [HoTT] Why did Voevodsky find existing proof assistants to be 'impractical'? To: Kevin Buzzard Cc: Michael Shulman , Bas Spitters , homotopytypetheory , Nicolas Alexander Schmidt Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable X-Original-Sender: droberts.65537@gmail.com X-Original-Authentication-Results: gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=btzuehcc; spf=pass (google.com: domain of droberts.65537@gmail.com designates 2607:f8b0:4864:20::c43 as permitted sender) smtp.mailfrom=droberts.65537@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=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: , Hi, Going back a step from DTT vs HoTT... > I think that simple type theory will have tremendous problems defining, f= or example, tensor products of sheaves of modules on a scheme, because thes= e are dependent types. this is probably where VV found himself just under 20 years ago. I don't know what the state of play was around 2000, but certainly if he didn't know about Coq, then VV's options for formal proof systems were looking more like Mizar, MetaMath and similar, which are probably even worse than simple type-theory systems: imagine trying to formally define a scheme in the language of ZFC! Best regards, David David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com On Tue, 5 Nov 2019 at 05:13, Kevin Buzzard wrot= e: > > > > On Sun, 3 Nov 2019 at 19:13, Michael Shulman wrote= : >> >> But does univalence a la Book HoTT *actually* make it easier to reason >> about such things? > > > I think this is a really interesting and important question. > > I guess David was referring to my scheme fail of 2018. I wanted to formal= ise the notion of a scheme a la Grothendieck and prove that if R was a comm= utative ring then Spec(R) was a scheme [I know it's a definition, but many = mathematicians do seem to call it a theorem, in our ignorance]. I showed an= undergraduate a specific lemma in ring theory (https://stacks.math.columbi= a.edu/tag/00EJ) and said "that's what I want" and they formalised it for me= . And then it turned out that I wanted something else: I didn't have R_f, I= had something "canonically isomorphic" to it, a phrase we mathematicians l= ike to pull out when the going gets tough and we can't be bothered to check= that any more diagrams commute. By this point it was too late to turn back= , and so I had to prove that 20 diagrams commuted and it wasn't much fun. I= then got an MSc student to redo everything using universal properties more= carefully in Lean and it worked like a dream https://github.com/ramonfmir/= lean-scheme. A lot of people said to me at the time "you wouldn't have had = this problem if you'd been using HoTT instead of DTT" and my response to th= is is still the (intentionally) provocative "go ahead and define schemes an= d prove that Spec(R) is a scheme in some HoTT system, and show me how it's = better; note that we did have a problem, but we solved it in DTT". I would = be particularly interested to see schemes done in Arend, because it always = felt funny to me using UniMath in Coq (and similarly it feels funny to me t= o do HoTT in Lean 3 -- in both cases it could be argued that it's using a s= ystem to do something it wasn't designed to do). I think it's easy to theor= ise about this sort of thing but until it happens in practice in one or mor= e of the HoTT systems I don't think we will understand the issue properly (= or, more precisely, I don't think I will understand the issue properly). I = have had extensive discussions with Martin Escardo about HoTT and he has ce= rtainly given me hope, but on the Lean chat I think people assumed schemes = would be easy in Lean (I certainly did) and then we ran into this unexpecte= d problem (which univalence is probably designed to solve), so the question= is whether a univalent type theory runs into a different unexpected proble= m -- you push the carpet down somewhere and it pops up somewhere else. > > I know this is a HoTT list but the challenge is also open to the HOL peop= le like the Isabelle/HOL experts. In contrast to HoTT theories, which I thi= nk should handle schemes fine, I think that simple type theory will have tr= emendous problems defining, for example, tensor products of sheaves of modu= les on a scheme, because these are dependent types. On the other hand my re= cent ArXiv paper with Commelin and Massot https://arxiv.org/abs/1910.12320 = goes much further and formalises perfectoid spaces in dependent type theory= . I would like the people on this list to see this as a challenge. I think = that this century will see the rise of the theorem prover in mathematics an= d I am not naive enough to think that the one I currently use now is the on= e which is guaranteed to be the success story. Voevodsky was convinced that= univalence was the right way to do modern mathematics but I'm doing it jus= t fine in dependent type theory and now he's gone I really want to find som= eone who will take up the challenge and do some scheme theory in HoTT, but = convincing professional mathematicians to get interested in this area is ve= ry difficult, and I speak as someone who's been trying to do it for two yea= rs now [I recommend you try the undergraduates instead, anyone who is inter= ested in training people up -- plenty of undergraduates are capable of read= ing the definition of a scheme, if they know what rings and topological spa= ces are] > > To get back to the original question, my understanding was that Voevodsky= had done a bunch of scheme theory and it had got him a Fields medal and it= was this mathematics which he was interested in at the time. He wanted to = formalise his big theorem, just like Hales did. Unfortunately he was histor= ically earlier, and his mathematics involved far more conceptual objects th= an spheres in 3-space, so it was a much taller order. All the evidence is t= here to suggest that over the next 15 or so years his interests changed. Th= e clearest evidence, in my mind, is that there is no definition of a scheme= in UniMath. Moreover his story in his Cambridge talk https://www.newton.ac= .uk/seminar/20170710113012301 about asking Suslin to reprove one of his res= ults without using the axiom of choice (46 minutes in) kind of shocked me -= - Suslin does not care about mathematics without choice, and the vast major= ity of mathematicians employed in mathematics departments feel the same, al= though I'm well aware that constructivism is taken more seriously on this l= ist. I think it is interesting that Voevodsky failed to prove a constructiv= e version of his theorem, because I think that some mathematics is better o= ff not being constructive. It is exactly the interaction between constructi= vism and univalence which I do not understand well, and I think that a very= good way to investigate it would be to do some highly non-constructive mod= ern mathematics in a univalent type theory. > > Kevin > > PS many thanks to the people who have emailed me in the past telling me a= bout how in the past I have used "HoTT", "univalence", "UniMath", interchan= geably and incorrectly. Hopefully I am getting better but I am still keen t= o hear anything which I'm saying which is imprecise or incorrect. > > >> >> It allows us to write "=3D" rather than "\cong", but >> to construct such an equality we have to construct an isomorphism >> first, and to *use* such an equality we have to transport along it, >> and then we get lots of univalence-redexes that we have to manually >> reduce away. My experience formalizing math in HoTT/Coq is that it's >> much easier if you *avoid* turning equivalences into equalities except >> when absolutely necessary. (I don't have any experience formalizing >> math in a cubical proof assistant, but in that case at least you >> wouldn't have to manually reduce the univalence-redexes -- although it >> seems to me you'd still have to construct the isomorphism before you >> can apply univalence to make it an equality.) >> >> On Sun, Nov 3, 2019 at 3:57 AM David Roberts = wrote: >> > >> > Forget even higher category theory. Kevin Buzzard now goes around tell= ing the story of how even formally proving (using Lean) things in rather el= ementary commutative algebra from EGA that are stated as equalities was not= obvious: the equality is really an isomorphism arising from a universal pr= operty. Forget trying to do anything motivic, when algebra is full of such = equalities. This is not a problem with univalence, of course. >> > >> > David >> > >> > On Sun, 3 Nov 2019, 10:08 PM Bas Spitters w= rote: >> >> >> >> There's also VV homotopy lambda calculus, which he later abandoned fo= r MLTT: >> >> https://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/= Hlambda_short_current.pdf >> >> >> >> On Sun, Oct 27, 2019 at 6:22 PM Bas Spitters wrote: >> >>> >> >>> I believe it refers to his 2-theories: >> >>> https://www.ias.edu/ideas/2014/voevodsky-origins >> >>> >> >>> On Sun, Oct 27, 2019 at 3:41 PM Nicolas Alexander Schmidt wrote: >> >>>> >> >>>> In [this](https://www.youtube.com/watch?v=3Dzw6NcwME7yI&t=3D1680) 2= 014 talk >> >>>> at IAS, Voevodsky talks about the history of his project of "unival= ent >> >>>> mathematics" and his motivation for starting it. Namely, he mention= s >> >>>> that he found existing proof assistants at that time (in 2000) to b= e >> >>>> impractical for the kinds of mathematics he was interested in. >> >>>> >> >>>> Unfortunately, he doesn't go into details of what mathematics he wa= s >> >>>> exactly interested in (I'm guessing something to do with homotopy >> >>>> theory) or why exactly existing proof assistants weren't practical = for >> >>>> formalizing them. Judging by the things he mentions in his talk, it >> >>>> seems that (roughly) his rejection of those proof assistants was ba= sed >> >>>> on the view that predicate logic + ZFC is not expressive enough. In >> >>>> other words, there is too much lossy encoding needed in order to >> >>>> translate from the platonic world of mathematical ideas to this for= mal >> >>>> language. >> >>>> >> >>>> Comparing the situation to computer programming languages, one migh= t say >> >>>> that predicate logic is like Assembly in that even though everythin= g can >> >>>> be encoded in that language, it is not expressive enough to directl= y >> >>>> talk about higher level concepts, diminishing its practical value f= or >> >>>> reasoning about mathematics. In particular, those systems are not >> >>>> adequate for *interactive* development of *new* mathematics (as opp= osed >> >>>> to formalization of existing theories). >> >>>> >> >>>> Perhaps I am just misinterpreting what Voevodsky said. In this case= , I >> >>>> hope someone can correct me. However even if this wasn't *his* view= , to >> >>>> me it seems to be a view held implicitly in the HoTT community. In = any >> >>>> case, it's a view that one might reasonably hold. >> >>>> >> >>>> However I wonder how reasonable that view actually is, i.e. whether= e.g. >> >>>> Mizar really is that much more impractical than HoTT-Coq or Agda, g= iven >> >>>> that people have been happily formalizing mathematics in it for 46 = years >> >>>> now. And, even though by browsing the contents of "Formalized >> >>>> Mathematics" one can get the impression that the work consists most= ly of >> >>>> formalizing early 20th century mathematics, neither the UniMath nor= the >> >>>> HoTT library for example contain a proof of Fubini's theorem. >> >>>> >> >>>> So, to put this into one concrete question, how (if at all) is HoTT= -Coq >> >>>> more practical than Mizar for the purpose of formalizing mathematic= s, >> >>>> outside the specific realm of synthetic homotopy theory? >> >>>> >> >>>> >> >>>> -- >> >>>> >> >>>> Nicolas >> >>>> >> >>>> >> >>>> -- >> >>>> 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, s= end an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. >> >>>> To view this discussion on the web visit https://groups.google.com/= d/msgid/HomotopyTypeTheory/e491d38b-b50a-6172-dca9-40d45fe1b6d2%40fromzerot= oinfinity.xyz. >> >> >> >> -- >> >> You received this message because you are subscribed to the Google Gr= oups "Homotopy Type Theory" group. >> >> To unsubscribe from this group and stop receiving emails from it, sen= d an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. >> >> To view this discussion on the web visit https://groups.google.com/d/= msgid/HomotopyTypeTheory/CAOoPQuRQPMkCFKYtAbB%2BpNK90XtFk%2BaVT_aY59U_-9t17= sBzeA%40mail.gmail.com. >> > >> > -- >> > You received this message because you are subscribed to the Google Gro= ups "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/m= sgid/HomotopyTypeTheory/CAFL%2BZM_%3D%3DiLS16Vy7sGiEqNkCxOMYL4j%2BZFqKv5uJ-= ivkuemEg%40mail.gmail.com. >> >> -- >> You received this message because you are subscribed to the Google Group= s "Homotopy Type Theory" group. >> To unsubscribe from this group and stop receiving emails from it, send a= n email to HomotopyTypeTheory+unsubscribe@googlegroups.com. >> To view this discussion on the web visit https://groups.google.com/d/msg= id/HomotopyTypeTheory/CAOvivQz47kSm9WbKDmUsndrpAJMkNwiWmVABqOFrVqyTOvSAbw%4= 0mail.gmail.com. --=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 e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/= HomotopyTypeTheory/CAFL%2BZM9-VQE6xR%2BRVsU54dBP4U1GkQ%2B9-MuxN%2BBMC40JKoo= 2GQ%40mail.gmail.com.