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-x338.google.com (mail-ot1-x338.google.com [IPv6:2607:f8b0:4864:20::338]) by inbox.vuxu.org (OpenSMTPD) with ESMTP id a094f29c for ; Thu, 7 Mar 2019 23:01:36 +0000 (UTC) Received: by mail-ot1-x338.google.com with SMTP id d25sf7939072otq.2 for ; Thu, 07 Mar 2019 15:01:35 -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=zJwipAYFDMcGbmSvr3CpQ38+m9xJ+8L72Rqtsybephw=; b=cISazPADfLPW8gEEvcBQgyKzIhPhUZwWaYEWkpHMkqOox37G4vdL8agQa0xK7atQzz DkgP6apUdZuKStNiiyqyZnx+I2cUDM+qMCRaSQZGZag0YV28UqEJqZFEQk/+JpuF2UJA TswkO7eZaYhmWDjILTk8WvJpSVLIQ002GRCqHgYzK/dO/Vm1VlpMFebitAQHbBFBZCBJ 7Nb2Opn5RobIoIsiDta/tskP8HpAjKsj6BEveamI0YPIe6qnOc2tK69tEHsL16cFXv8X 3cCHdWf9XnKo7cUan0kCvOAMYCXAEG0Oz31LHLWKqNGFB3k1mry+GIPwFLug5u9wNjSY OlhQ== 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=zJwipAYFDMcGbmSvr3CpQ38+m9xJ+8L72Rqtsybephw=; b=bAQiz0iorhijLxVXEdKECp0bFO3S0JlyTbd38dzO7DVwp3nO7yLkqJW3WxI/Psnb4U EA4q/HWAqcoNL3Yl914ewOnA6B5y5inu8w7kn/Zpl0oucqmPGG6gdvhzeWlHTx5stPQu UIDgh0I1YxZ8YTt+kWGjKsjmNrXsK1AkOH7KpHw57UjtnYU7GcY5/rVPoixBg9itXFFx H1ANJBUu0VjxSbHZ7ZWp+yYyptVwK76InQVkB+U8HACJ4B5o8W8q+PTlXS9ERyv6429v V4d2aCEmivLm3jG/zXVUoqR9uQi28ziyJT9RULeNqgUyo9zm8dB39PPxwE7wOxEGTE4y fyVg== 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=zJwipAYFDMcGbmSvr3CpQ38+m9xJ+8L72Rqtsybephw=; b=mXsJgcN4VfkzilQCJXrmU5g8NZUVbsRAegJ5nhpRuBwdYs0rKK1LwFe+JFhK91FWEK k2ickvW1djQozZNwSfdfQfZgKo9lKmdm6EJgzKecWnt/I78gqerXNvl2K31DAaqQevDN +sMqz/b+vqpqyi88j9AMblLqLmwJZwBpGc7MV+1uX2sAC0SMb2tiodV7EvTck2IcnAd8 I9NxzZoDw9+46Xz4qgA7+OSSAVU8VqeEctfzmtuo7pPLn8xo9v/7P+IQi6ezcFhONN9Y nWkIgYlgl2biKzyEfUS1EBM3pSdqodUJYR6YTWbQSV/86L8oDJfp+r99zAysu0mLCP4U bpFQ== Sender: homotopytypetheory@googlegroups.com X-Gm-Message-State: APjAAAWEmwX8xzWYbTsglGOGWHeHdBBFv9Yn4RPUFGeuEn43t3eUYEL2 nHHh6Fo2uboHGlVf8OqygZs= X-Google-Smtp-Source: APXvYqyb4htaYmsU7UN8+gG8rDyst3gq0gMTPzcIy06S4+KA+/XdyeNbDZKnetsotcIOCNWO/wDggg== X-Received: by 2002:aca:eb4f:: with SMTP id j76mr6574166oih.81.1551999695159; Thu, 07 Mar 2019 15:01:35 -0800 (PST) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 2002:aca:b446:: with SMTP id d67ls1603718oif.3.gmail; Thu, 07 Mar 2019 15:01:34 -0800 (PST) X-Received: by 2002:aca:6043:: with SMTP id u64mr6204954oib.32.1551999694092; Thu, 07 Mar 2019 15:01:34 -0800 (PST) Date: Thu, 7 Mar 2019 15:01:33 -0800 (PST) From: =?UTF-8?Q?Mart=C3=ADn_H=C3=B6tzel_Escard=C3=B3?= To: Homotopy Type Theory Message-Id: <50111384-09a4-4c30-8272-fa9e5997d3c3@googlegroups.com> In-Reply-To: <98CB1099-377A-4A5F-94F2-B33C36D577B0@wesleyan.edu> References: <0f5b8d0e-9f1d-47a7-9d39-a9112afb77ea@googlegroups.com> <12cd6b73-7ca6-481c-9503-250af28b8113@googlegroups.com> <30ae0dc4-cef2-46ad-a280-bdf617a0db4e@googlegroups.com> <9fbd1c51-139e-4657-980a-2264a8f9ff92@googlegroups.com> <5f7932a1-417b-42be-9d63-300dddc83037@googlegroups.com> <98CB1099-377A-4A5F-94F2-B33C36D577B0@wesleyan.edu> Subject: Re: [HoTT] Propositional Truncation MIME-Version: 1.0 Content-Type: multipart/mixed; boundary="----=_Part_989_188839871.1551999693565" X-Original-Sender: escardo.martin@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_989_188839871.1551999693565 Content-Type: multipart/alternative; boundary="----=_Part_990_183417815.1551999693566" ------=_Part_990_183417815.1551999693566 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Oh, this is annoying, because it seems to mean that we would need unbounded= =20 search (to drop all "hcom []"'s) until we can read the |x,a|, which is=20 against the spirit of, say, Martin-Loef type theories. Martin On Thursday, 7 March 2019 22:51:20 UTC, dlicata wrote: > > That would be true if the term you are normalizing is in the empty=20 > interval context, and the cubical type theory has =E2=80=9Cempty system r= egularity=E2=80=9D=20 > (like https://www.cs.cmu.edu/~cangiuli/papers/ccctt.pdf). =20 > > Otherwise, if you evaluate something in the empty interval context, you= =20 > might see something like=20 > hcom [] (hcom [] (hcom [] (hcom [] (=E2=80=A6 |x,a| =E2=80=A6 ))))=20 > with |x,a| in there somewhere. In HITs, Kan composition is treated as a= =20 > constructor of the type, and though there are no interesting lines to=20 > compose in the empty interval context, the uninteresting compositions don= =E2=80=99t=20 > vanish in all flavors of cubical type theory. =20 > > > On Mar 7, 2019, at 5:41 PM, Mart=C3=ADn H=C3=B6tzel Escard=C3=B3 > wrote:=20 > >=20 > > So I presume that when we ask cubical Agda to normalize a term of type= =20 > || Sigma (x:X), A x || we will get a term of the form |x,a| and so we wil= l=20 > see the x in normal form, where |-| is the map into the truncation, right= ?=20 > Martin.=20 > >=20 > > On Thursday, 7 March 2019 21:52:12 UTC, Anders M=C3=B6rtberg wrote:=20 > > The existence property is proved for CCHM cubicaltt by Simon in:=20 > >=20 > > https://arxiv.org/abs/1607.04156=20 > >=20 > > See corollary 5.2. This works a bit more generally than what Mart=C3=AD= n=20 > said, in particular in any context with only dimension variables we can= =20 > compute a witness to an existence. So if in context G =3D i_1 : II, ..., = i_n=20 > : II (possibly empty) we have:=20 > >=20 > > G |- t : exists (x : X), A(x)=20 > >=20 > > then we can compute G |- u : X so that G |- B(u).=20 > >=20 > > --=20 > > Anders=20 > >=20 > > On Thursday, March 7, 2019 at 11:16:48 AM UTC-5, Mart=C3=ADn H=C3=B6tze= l Escard=C3=B3=20 > wrote:=20 > > I got confused now. :-)=20 > >=20 > > 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=20 > theory. This follows from Simon's canonicity result (at least for X=3Dnat= ural=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 > >=20 > > Martin=20 > >=20 > > On Thursday, 7 March 2019 14:10:56 UTC, dlicata wrote:=20 > > 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 closed= =20 > term of type ||A|| evaluates to a value that has as a subterm a closed te= rm=20 > of type A (the other values of ||A|| are some =E2=80=9Cformal composition= s=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=9Cif= 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 > >=20 > > -Dan=20 > >=20 > > > On Mar 5, 2019, at 6:07 PM, Mart=C3=ADn H=C3=B6tzel Escard=C3=B3 < > escardo...@gmail.com> wrote:=20 > > >=20 > > > Or you can read the paper https://lmcs.episciences.org/3217/=20 > regarding what Nicolai said.=20 > > >=20 > > > Moreover, in the HoTT book, it is shown that if || X||->X holds for= =20 > all 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 a= ny=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 point= =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=20 > 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,=20 > send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com=20 > .=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, 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_990_183417815.1551999693566 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Oh, this is annoying, because it seems to mean that we wou= ld need unbounded search (to drop all "hcom []"'s) until we c= an read the |x,a|, which is against the spirit of, say, Martin-Loef type th= eories. Martin

On Thursday, 7 March 2019 22:51:20 UTC, dlicata wrot= e:
That would be true if the te= rm you are normalizing is in the empty interval context, and the cubical ty= pe theory has =E2=80=9Cempty system regularity=E2=80=9D (like https://www.cs.cm= u.edu/~cangiuli/papers/ccctt.pdf). =C2=A0=20

Otherwise, if you evaluate something in the empty interval context, you= might see something like
hcom [] (hcom [] (hcom [] (hcom [] (=E2=80=A6 |x,a| =E2=80=A6 ))))
with |x,a| in there somewhere. =C2=A0In HITs, Kan composition is treate= d as a constructor of the type, and though there are no interesting lines t= o compose in the empty interval context, the uninteresting compositions don= =E2=80=99t vanish in all flavors of cubical type theory. =C2=A0

> On Mar 7, 2019, at 5:41 PM, Mart=C3=ADn H=C3=B6tzel Escard=C3=B3 &= lt;es= cardo...@gmail.com> wrote:
>=20
> So I presume that when we ask cubical Agda to normalize a term of = type || Sigma (x:X), A x || we will get a term of the form |x,a| and so we = will see the x in normal form, where |-| is the map into the truncation, ri= ght? Martin.
>=20
> On Thursday, 7 March 2019 21:52:12 UTC, Anders M=C3=B6rtberg wrote= :
> The existence property is proved for CCHM cubicaltt by Simon in:
>=20
> https://arxiv.org/abs/1607.04156
>=20
> See corollary 5.2. This works a bit more generally than what Mart= =C3=ADn said, in particular in any context with only dimension variables 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=A0we have:
>=20
> G |- t : exists (x : X), A(x)
>=20
> then we can compute G |- u : X so that G |- B(u).
>=20
> --
> Anders
>=20
> 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. :-)
>=20
> Seriously 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. This follows from Simon's canonicity result (at least for X=3D= natural numbers), and is like the so-called "existence property" = in the internal 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 onl= y in the empty context, because otherwise it behaves like "local exist= ence" as in Kripke-Joyal semantics.=20
>=20
> Martin
>=20
> On Thursday, 7 March 2019 14:10:56 UTC, dlicata wrote:
> Just in case anyone reading this thread later is confused about a = more beginner point than the ones Nicolai and Martin made, one possible stu= mbling 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 type), then the answer= is yes (at least in some models): if ||A|| is inhabited then A is inhabite= d. =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 clos= ed term of type A (the other values of ||A|| are some =E2=80=9Cformal compo= sitions=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 Martin and Nicolai said= because =E2=80=9Cif A is inhabited then B is inhabited=E2=80=9D (in this e= xternal sense) doesn=E2=80=99t necessarily mean there is a map A -> B in= ternally. =C2=A0=20
>=20
> -Dan=20
>=20
> > On Mar 5, 2019, at 6:07 PM, Mart=C3=ADn H=C3=B6tzel Escard=C3= =B3 <escardo...@gmail.com> wrote:=20
> >=20
> > Or you can read the paper https://lmcs.= episciences.org/3217/ regarding what Nicolai said.=20
> >=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, whic= h can't be invariant under equivalence.)=20
> >=20
> > The above paper shows that unrestricted ||X||->X it gives = excluded 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. Moreov= er, for any type X, the availability of ||X||->X is logically equivalent= to the availability of a constant map X->X (before we know whether X ha= s a point 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 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 it true that, if ||A|| is inhabited, then A is inhabited?=20
> >> --=20
> >> You received this message because you are subscribed to t= he Google Groups "Homotopy Type Theory" group.=20
> >> To unsubscribe from this group and stop receiving emails = from it, send 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 G= oogle Groups "Homotopy Type Theory" group.=20
> > To unsubscribe from this group and stop receiving emails from= it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.= =20
> > For more options, visit https://grou= ps.google.com/d/optout.=20
>=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@googlegroups.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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