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=-1.2 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-x339.google.com (mail-ot1-x339.google.com [IPv6:2607:f8b0:4864:20::339]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id a57c3904 for ; Mon, 11 Feb 2019 07:01:45 +0000 (UTC) Received: by mail-ot1-x339.google.com with SMTP id a3sf10306509otl.9 for ; Sun, 10 Feb 2019 23:01:44 -0800 (PST) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=googlegroups.com; s=20161025; h=sender:date:from:to:message-id:in-reply-to:references:subject :mime-version:x-original-sender:precedence:mailing-list:list-id :list-post:list-help:list-archive:list-unsubscribe; bh=kYrN6mlqkfO74ebBaswVnmJeVS9419LjW7fGU4WyF7g=; b=L9S7sQ0DhNqX7aQUF2hMKdDK6RncfRrNU7hpmqgSEWULikt7ArTSAEOOMm6Kicpk7U WfhK8oiiEYVRkvu1jRrr+wLtCCaOiw87l7o18L95j1/+9ZC8N4fJdOLMkZJz7mt2HEjV 7DCx5/UIVBUNaBKBGf2IXGRYwihYti3E6URzfE93zMiWrCrpCUZVj9Zl1uEYEJl2sh3N ZM2zsY/YgWOeG+GGwoSWslKFDxA1w/kOns4paxCAOiCrjaIQ5DlFUF36Q5FSk9h9PeAv B1VUEf1IXM2LXUP6AG0NDcvBHHCsXh9f28L/eg22meJbk5SeaP2+JwqL7hBP3hyoJE83 M0uQ== DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=date:from:to:message-id:in-reply-to:references:subject:mime-version :x-original-sender:precedence:mailing-list:list-id:list-post :list-help:list-archive:list-unsubscribe; bh=kYrN6mlqkfO74ebBaswVnmJeVS9419LjW7fGU4WyF7g=; b=Fgg8kz0c49Nj7BwEprfwKx5PVMYoYxYNeInot1mXE8DFD2yPG3N288KdSJ9q9W8iZr 7EO6CE/KtHneEwEX5ytZ5i6EcWN+A4BCfgzbj4dBY/JvOnuZIzxa77Hlr72o146qPhv1 dKM1wRpBUyngpg1twmLIeFZ19CXmXjnL1iINFbh3VIyrbHFlpY2IxaDZKnyS5wPBqsOP 6+F5Pg6fzzZeYpivhq9B8nKfUd0RzTQfpFWnzce9nTdFCOJfX7yutIZcu7ksnZnj8CCl l8C8vCnKvEhZ74mwwkZ00KldNcnHCc5KMDq6XzweXBxgCaC0lPVThf7TGf7e6L71Oo8f pmBw== 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:in-reply-to :references: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=kYrN6mlqkfO74ebBaswVnmJeVS9419LjW7fGU4WyF7g=; b=llN6yuzXTCM0gtDjXGRnuN+AhsJkxGQ6ml7SS70W5g6obFaAJKZKwLw6dXlN7RrkJH 2Ph3k8XtLCpLc54cSdvUHoWWj8sD9Jh59rQQr+BUZ3j85qkQpKa9T4DFqKWdg9SE6V8C IKX22gM5j5E8jvi3vKR6xdru1y/6kZxgaCNvaPzswY2Zc8p+0ZpC6VRa5fEuH9Wyt8zR YnxWeTtoXOf7t2GcLwRk1Y0HzQYM39TzBqhqqVtt6/x4biFkRlm4cZEmMulyRNoGIQ0I 5/+hv3cvhj2AQ9WribAz0F9FVo8ca1sKblMrC4mxz8pU++O4QOnjiKPInYcjeRM7Gdjn FuoA== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: AHQUAuaswQFmXM71CHL19AnU7DzDWF4cFtQ4h3TURk7SIICcUrvNmSA/ RoBR/F1dTaanJ9ZBCOC1++I= X-Google-Smtp-Source: AHgI3IY18DxGUD2m51FrJI5pYbBXCZhn3qFmTBMsOFLr3T/PgnI3JsFe175c83vp5jfqlYhOGaSpqQ== X-Received: by 2002:aca:afc3:: with SMTP id y186mr38506oie.0.1549868504043; Sun, 10 Feb 2019 23:01:44 -0800 (PST) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:aca:3d54:: with SMTP id k81ls959677oia.1.gmail; Sun, 10 Feb 2019 23:01:43 -0800 (PST) X-Received: by 2002:aca:cf4f:: with SMTP id f76mr24241oig.7.1549868503312; Sun, 10 Feb 2019 23:01:43 -0800 (PST) Date: Sun, 10 Feb 2019 23:01:42 -0800 (PST) From: Matt Oliveri To: Homotopy Type Theory Message-Id: In-Reply-To: <730FBE36-8E4F-45F5-9DB9-3B3A04E708FA@nottingham.ac.uk> References: <730FBE36-8E4F-45F5-9DB9-3B3A04E708FA@nottingham.ac.uk> Subject: Re: [HoTT] Why do we need judgmental equality? MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_384_1155911623.1549868502702" X-Original-Sender: atmacen@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_384_1155911623.1549868502702 Content-Type: multipart/alternative; boundary="----=_Part_385_1977106134.1549868502702" ------=_Part_385_1977106134.1549868502702 Content-Type: text/plain; charset="UTF-8" I think you're right. From discussions about autophagy, it seems like no one knows how to match judgmental equality using equality types, unless that equality type family is propositionally truncated in some way. Consequently, my guess is that Valery's Q transformation actually yields something rather like a 2-level system. On Saturday, February 9, 2019 at 7:30:07 AM UTC-5, Thorsten Altenkirch wrote: > > Hi, > > > > what we need is a strict equality on all types. If we would state the laws > of type theory just using the equality type we would also need to add > coherence laws. Since I would include the laws for substitution (never > understood why substitution is different from application) this would > include the laws for infinity categories and this would make even basic > type theory certainly much more complicated if not unusable. Instead one > introduces a 2-level system with strict equality on one level and weak > equality on another. For historic and pragmatic reasons this is combined > with the computational aspects of type theory which is expressed as > judgemental equality. However, there are reasons to separate these > concerns, e.g. to work with higher dimensional constructions in type theory > such as semi-simplicial types it is helpful to work with hypothetical > strict equalities (see our paper ( > http://www.cs.nott.ac.uk/~psztxa/publ/csl16.pdf). > > > > I do think that the computational behaviour of type theory is important > too. However, this can be expressed by demandic a form of computational > adequacy, that is for every term there is a strictly equal normal form. It > is not necessary that strict equality in general is decidable (indeed > different applications of type theory may demand different decision > procedures). > > > > Thorsten > > > > > > *From: *> on behalf of Felix > Rech > > *Date: *Wednesday, 30 January 2019 at 11:55 > *To: *Homotopy Type Theory > > *Subject: *[HoTT] Why do we need judgmental equality? > > > > In section 1.1 of the HoTT book it says "In type theory there is also a > need for an equality judgment." Currently it seems to me like one could, in > principle, replace substitution along judgmental equality with explicit > transports if one added a few sensible rules to the type theory. Is there a > fundamental reason why the equality judgment is still necessary? > > > > Thanks, > > Felix Rech > -- 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. For more options, visit https://groups.google.com/d/optout. ------=_Part_385_1977106134.1549868502702 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
I think you're right. From discussions about autophagy= , it seems like no one knows how to match judgmental equality using equalit= y types, unless that equality type family is propositionally truncated in s= ome way.

Consequently, my guess is that Valery's Q transformatio= n actually yields something rather like a 2-level system.

On Saturda= y, February 9, 2019 at 7:30:07 AM UTC-5, Thorsten Altenkirch wrote:

Hi,

=C2=A0

what we need is a strict equality on all types. If w= e would state the laws of type theory just using the equality type we would= also need to add coherence laws. Since I would include the laws for substi= tution (never understood why substitution is different from application) this would include the laws for infinity ca= tegories and this would make even basic type theory certainly much more com= plicated if not unusable. Instead one introduces a 2-level system with stri= ct equality on one level and weak equality on another. For historic and pragmatic reasons this is combined w= ith the computational aspects of type theory which is expressed as judgemen= tal equality. However, there are reasons to separate these concerns, e.g. t= o work with higher dimensional constructions in type theory such as semi-simplicial types it is helpful to work with hy= pothetical strict equalities (see our paper (http://www.cs.nott.ac.uk/~psztxa/publ= /csl16.pdf).

=C2=A0

I do think that the computational behaviour of type = theory is important too. However, this can be expressed by demandic a form = of computational adequacy, that is for every term there is a strictly equal= normal form. It is not necessary that strict equality in general is decidable (indeed different application= s of type theory may demand different decision procedures).

=C2=A0

Thorsten

=C2=A0

=C2=A0

From: <homotopyt...@go= oglegroups.com> on behalf of Felix Rech <s9fe...@gmail.com>
Date: Wednesday, 30 January 2019 at 11:55
To: Homotopy Type Theory <homotopyt...@googlegroups.com> Subject: [HoTT] Why do we need judgmental equality?

=C2=A0

In section 1.1 of the H= oTT book it says "In type theory there is also a need for an equality = judgment." Currently it seems to me like one could, in principle, repl= ace substitution along judgmental equality with explicit transports if one added a few sensible rules to the type theory. = Is there a fundamental reason why the equality judgment is still necessary?=

=C2=A0

Thanks,

Felix Rech

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