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-x33b.google.com (mail-ot1-x33b.google.com [IPv6:2607:f8b0:4864:20::33b]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id 70568607 for ; Thu, 7 Mar 2019 21:52:15 +0000 (UTC) Received: by mail-ot1-x33b.google.com with SMTP id r13sf7848101otn.10 for ; Thu, 07 Mar 2019 13:52:14 -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=K/mADLtxr5f3Z8ftNLePuEoehSadG7pkYl7Ralun1o8=; b=ciXxv0ohzNj2iYtj77yozpLZzRbnqtB3A7k8y5Q0Q9oqNh/HDDJkoRv0uRAfhFekXV YWWetQ54gXL0ZnHkADssWQO5h99BuSTlwNpp7GO9DitwoUhNLzJhpDNUDmAdKZmPoSuh ZKMx720MKZ6FblaPQAiEC2w7hX7k8p3Yx/E3jpq3qtlVUVKnIswnZHoHX6nu3rl0A2XD v0M6IjTGiA/ITmzi1q8qIkD9/1r3pTNf7oPwIUeNXqLb+7EjHY8zt6pL2rCi3TxTr0a1 9kDJeNLIh7xv2cMeyOBzBclAyHRt5EQsYK9zClA51otbt8mZRb7+JFOMxHqoqwcSW/J3 iQpg== 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=K/mADLtxr5f3Z8ftNLePuEoehSadG7pkYl7Ralun1o8=; b=ZtuCCW8xIyWS61zbl75FQJlNDGRRAMNwI9qW9+3qXOzBGOrMgXXzCm9F2HZMmQFxLL pi5OIT4B251W+netyLE39vymOhLxZQXXZXAXd5ddEqW8aQYeZpzAjzHm/0U+5emtbevT IC7cDcuaQSTSQG2hBbZoc75dRUDnB4H+mt0n2gjkb/QnUyumyvlI1CWxxsvnCRpZ4aCi kHDRBtIBQiE6iCXmvFzDe7OHc0jouPiX2iwpggkma3tLFkkX/UKTjaKcaA8V1QINSSg5 2QaoVKoRlI8j5MMcOHpg8dMm9aIjgOIUjr7a6nNWupKVyG8YFfIgMP/HlWhkg6wVP5XL iJkA== 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=K/mADLtxr5f3Z8ftNLePuEoehSadG7pkYl7Ralun1o8=; b=l3/LWs0AxWrBQMEMBfzvpu623QWkHPzIOm4q3AtmCBhO+c86YdpdGMhJMmxxU0n7F3 R2V/B043jqlRH78t/yuDHoHnlAoTHOYpF5b/xgEvtgTyJjqa2lgSYOigTcu+Vd4y7xqA o5695YjQe1PAxw8eLKhZVI4DvImnIs8CT2FLVQSJ+3xbM8XYhQ7V+pTmVkCMaG/Li5DK PMgzDaQCHOmxbTz3V3jfky0yhb7uSj09rImDGUs/gMGDA35uWbCVrnzLELtB+kYaTtLI 9S8vxykhlG9um3X3h/jJYy5fFraIKjlJIbbnXtyh7k89G8mWqQfFgu8sJkWvwSyvi9lI 7nXA== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAXiopFE3P0HVImH4Q3TxmbOgEkFJokGDjguURIPwGT1YpqJxSTo g6Da6Y3De0DOjBo7xw1G3wU= X-Google-Smtp-Source: APXvYqzuwBcktXpfswFpz25/0Tl18VgEWcU1WaPZUp7R2GPjHXF01han6nyTZM+V9NThSJADSyZY7w== X-Received: by 2002:aca:851:: with SMTP id 78mr6476793oii.60.1551995533684; Thu, 07 Mar 2019 13:52:13 -0800 (PST) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:a9d:3de5:: with SMTP id l92ls2417642otc.0.gmail; Thu, 07 Mar 2019 13:52:13 -0800 (PST) X-Received: by 2002:a9d:7dcd:: with SMTP id k13mr8948971otn.205.1551995533113; Thu, 07 Mar 2019 13:52:13 -0800 (PST) Date: Thu, 7 Mar 2019 13:52:12 -0800 (PST) From: =?UTF-8?Q?Anders_M=C3=B6rtberg?= To: Homotopy Type Theory Message-Id: <9fbd1c51-139e-4657-980a-2264a8f9ff92@googlegroups.com> In-Reply-To: <30ae0dc4-cef2-46ad-a280-bdf617a0db4e@googlegroups.com> References: <0f5b8d0e-9f1d-47a7-9d39-a9112afb77ea@googlegroups.com> <12cd6b73-7ca6-481c-9503-250af28b8113@googlegroups.com> <30ae0dc4-cef2-46ad-a280-bdf617a0db4e@googlegroups.com> Subject: Re: [HoTT] Propositional Truncation MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_1086_1614387376.1551995532514" X-Original-Sender: andersmortberg@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_1086_1614387376.1551995532514 Content-Type: multipart/alternative; boundary="----=_Part_1087_959218154.1551995532514" ------=_Part_1087_959218154.1551995532514 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable The existence property is proved for CCHM cubicaltt by Simon in: https://arxiv.org/abs/1607.04156 See corollary 5.2. This works a bit more generally than what Mart=C3=ADn sa= id,=20 in particular in any context with only dimension variables we can compute a= =20 witness to an existence. So if in context G =3D i_1 : II, ..., i_n : II =20 (possibly empty) we have: G |- t : exists (x : X), A(x) then we can compute G |- u : X so that G |- B(u). -- Anders On Thursday, March 7, 2019 at 11:16:48 AM UTC-5, Mart=C3=ADn H=C3=B6tzel Es= card=C3=B3=20 wrote: > > I got confused now. :-) > > Seriously now, what you say seems related to the fact that from a proof |= -=20 > t : || X || in the empty context, you get |- x : X in cubical type theory= .=20 > This follows from Simon's canonicity result (at least for X=3Dnatural=20 > numbers), and is like the so-called "existence property" in the internal= =20 > language of the free elementary topos. This says that from a proof |-=20 > exists (x:X), A x in the empty context, you get |- x : X and |- A x. This= =20 > says that exists in the empty context behaves like Sigma. But only in the= =20 > empty context, because otherwise it behaves like "local existence" as in= =20 > Kripke-Joyal semantics.=20 > > Martin > > On Thursday, 7 March 2019 14:10:56 UTC, dlicata wrote: >> >> Just in case anyone reading this thread later is confused about a more= =20 >> beginner point than the ones Nicolai and Martin made, one possible=20 >> stumbling block here is that, if someone means =E2=80=9Cis inhabited=E2= =80=9D in an=20 >> external sense (there is a closed term of that type), then the answer is= =20 >> yes (at least in some models): if ||A|| is inhabited then A is inhabited= .=20 >> For example, in cubical models with canonicity, it is true that a close= d=20 >> term of type ||A|| evaluates to a value that has as a subterm a closed t= erm=20 >> of type A (the other values of ||A|| are some =E2=80=9Cformal compositio= ns=E2=80=9D of=20 >> values of ||A||, but there has to be an |a| in there at the base case).= =20 >> This is consistent with what Martin and Nicolai said because =E2=80=9Ci= f A is=20 >> inhabited then B is inhabited=E2=80=9D (in this external sense) doesn=E2= =80=99t necessarily=20 >> mean there is a map A -> B internally. =20 >> >> -Dan=20 >> >> > On Mar 5, 2019, at 6:07 PM, Mart=C3=ADn H=C3=B6tzel Escard=C3=B3 =20 >> wrote:=20 >> >=20 >> > Or you can read the paper https://lmcs.episciences.org/3217/ regarding= =20 >> what Nicolai said.=20 >> >=20 >> > Moreover, in the HoTT book, it is shown that if || X||->X holds for al= l=20 >> X, then univalence can't hold. (It is global choice, which can't be=20 >> invariant under equivalence.)=20 >> >=20 >> > The above paper shows that unrestricted ||X||->X it gives excluded=20 >> middle.=20 >> >=20 >> > However, for a lot of kinds of types one can show that ||X||->X does= =20 >> hold. For example, if they have a constant endo-function. Moreover, for = any=20 >> type X, the availability of ||X||->X is logically equivalent to the=20 >> availability of a constant map X->X (before we know whether X has a poin= t=20 >> or not, in which case the availability of a constant endo-map is trivial= ).=20 >> >=20 >> > Martin=20 >> >=20 >> > On Tuesday, 5 March 2019 22:47:55 UTC, Nicolai Kraus wrote:=20 >> > You can't have a function which, for all A, gives you ||A|| -> A. See= =20 >> the exercises 3.11 and 3.12!=20 >> > -- Nicolai=20 >> >=20 >> > On 05/03/19 22:31, Jean Joseph wrote:=20 >> >> Hi,=20 >> >>=20 >> >> From the HoTT book, the truncation of any type A has two constructors= :=20 >> >>=20 >> >> 1) for any a : A, there is |a| : ||A||=20 >> >> 2) for any x,y : ||A||, x =3D y.=20 >> >>=20 >> >> I get that if A is inhabited, then ||A|| is inhabited by (1). But is= =20 >> it true that, if ||A|| is inhabited, then A is inhabited?=20 >> >> --=20 >> >> You received this message because you are subscribed to the Google=20 >> Groups "Homotopy Type Theory" group.=20 >> >> To unsubscribe from this group and stop receiving emails from it, sen= d=20 >> an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.=20 >> >> For more options, visit https://groups.google.com/d/optout.=20 >> >=20 >> >=20 >> > --=20 >> > You received this message because you are subscribed to the Google=20 >> Groups "Homotopy Type Theory" group.=20 >> > To unsubscribe from this group and stop receiving emails from it, send= =20 >> an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.=20 >> > For more options, visit https://groups.google.com/d/optout.=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 e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. ------=_Part_1087_959218154.1551995532514 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
The existence property is proved for CCHM cubicaltt b= y Simon in:

https://arxiv.org/abs/1607.04156
=

See corollary 5.2. This works a bit more generally than= what Mart=C3=ADn said, in particular in any context with only dimension va= riables we can compute a witness to an existence. So if in context G =3D i_= 1 : II, ..., i_n : II=C2=A0 (possibly empty)=C2=A0 we have:
<= br>
G |- t : exists (x : X), A(x)

then w= e can compute G |- u : X so that G |- B(u).

--
Anders

On Thursday, March 7, 2019 at 11:16:48 AM UTC-5,= Mart=C3=ADn H=C3=B6tzel Escard=C3=B3 wrote:
I got confused now. :-)

S= eriously now, what you say seems related to the fact that from a proof |- t= : || X || in the empty context, you get |- x : X in cubical type theory. T= his follows from Simon's canonicity result (at least for X=3Dnatural nu= mbers), and is like the so-called "existence property" in the int= ernal language of the free elementary topos. This says that from a proof |-= exists (x:X), A x in the empty context, you get |- x : X and |- A x. This = says that exists in the empty context behaves like Sigma. But only in the e= mpty context, because otherwise it behaves like "local existence"= as in Kripke-Joyal semantics.=C2=A0

Martin
<= div>
On Thursday, 7 March 2019 14:10:56 UTC, dlicata wrote:
Just in case anyone reading this thread later i= s confused about a more beginner point than the ones Nicolai and Martin mad= e, one possible stumbling block here is that, if someone means =E2=80=9Cis = inhabited=E2=80=9D in an external sense (there is a closed term of that typ= e), then the answer is yes (at least in some models): if ||A|| is inhabited= then A is inhabited. =C2=A0For example, in cubical models with canonicity,= it is true that a closed term of type ||A|| evaluates to a value that has = as a subterm a closed term of type A (the other values of ||A|| are some = =E2=80=9Cformal compositions=E2=80=9D of values of ||A||, but there has to = be an |a| in there at the base case). =C2=A0This is consistent with what Ma= rtin and Nicolai said because =E2=80=9Cif A is inhabited then B is inhabite= d=E2=80=9D (in this external sense) doesn=E2=80=99t necessarily mean there = is a map A -> B internally. =C2=A0

-Dan

> On Mar 5, 2019, at 6:07 PM, Mart=C3=ADn H=C3=B6tzel Escard=C3=B3 &= lt;escardo...@gmail.com> wrote:
>=20
> Or you can read the paper https://lmcs.episc= iences.org/3217/ regarding what Nicolai said.
>=20
> Moreover, in the HoTT book, it is shown that if || X||->X holds= for all X, then univalence can't hold. (It is global choice, which can= 't be invariant under equivalence.)
>=20
> The above paper shows that unrestricted ||X||->X it gives exclu= ded middle.=20
>=20
> However, for a lot of kinds of types one can show that ||X||->X= does hold. For example, if they have a constant endo-function. Moreover, f= or any type X, the availability of ||X||->X is logically equivalent to t= he availability of a constant map X->X (before we know whether X has a p= oint or not, in which case the availability of a constant endo-map is trivi= al).
>=20
> Martin
>=20
> On Tuesday, 5 March 2019 22:47:55 UTC, Nicolai Kraus wrote:
> You can't have a function which, for all A, gives you ||A|| -&= gt; A. See the exercises 3.11 and 3.12!
> -- Nicolai
>=20
> On 05/03/19 22:31, Jean Joseph wrote:
>> Hi,
>>=20
>> From the HoTT book, the truncation of any type A has two const= ructors:
>>=20
>> 1) for any a : A, there is |a| : ||A||
>> 2) for any x,y : ||A||, x =3D y.=20
>>=20
>> I get that if A is inhabited, then ||A|| is inhabited by (1). = But is it true that, if ||A|| is inhabited, then A is inhabited?=20
>> --=20
>> You received this message because you are subscribed to the Go= ogle Groups "Homotopy Type Theory" group.
>> To unsubscribe from this group and stop receiving emails from = it, send an email to HomotopyTypeTheory+unsubscrib= e@googlegroups.com.
>> For more options, visit https://group= s.google.com/d/optout.
>=20
>=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 HomotopyTypeTheory+unsubscribe@go= oglegroups.com.
> For more options, visit https://groups.go= ogle.com/d/optout.

--
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.
For more options, visit http= s://groups.google.com/d/optout.
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