From mboxrd@z Thu Jan 1 00:00:00 1970 X-Received: by 10.46.0.230 with SMTP id e99mr263228lji.15.1493239261507; Wed, 26 Apr 2017 13:41:01 -0700 (PDT) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 10.46.69.66 with SMTP id s63ls194050lja.14.gmail; Wed, 26 Apr 2017 13:40:59 -0700 (PDT) X-Received: by 10.46.80.93 with SMTP id v29mr265254ljd.27.1493239259808; Wed, 26 Apr 2017 13:40:59 -0700 (PDT) Return-Path: Received: from mail-lf0-x236.google.com (mail-lf0-x236.google.com. [2a00:1450:4010:c07::236]) by gmr-mx.google.com with ESMTPS id n4si911378wmf.2.2017.04.26.13.40.59 for (version=TLS1_2 cipher=ECDHE-RSA-AES128-GCM-SHA256 bits=128/128); Wed, 26 Apr 2017 13:40:59 -0700 (PDT) Received-SPF: pass (google.com: domain of fpvd...@gmail.com designates 2a00:1450:4010:c07::236 as permitted sender) client-ip=2a00:1450:4010:c07::236; Authentication-Results: gmr-mx.google.com; dkim=pass head...@gmail.com; spf=pass (google.com: domain of fpvd...@gmail.com designates 2a00:1450:4010:c07::236 as permitted sender) smtp.mailfrom=fpvd...@gmail.com; dmarc=pass (p=NONE sp=NONE dis=NONE) header.from=gmail.com Received: by mail-lf0-x236.google.com with SMTP id c80so6841290lfh.3 for ; Wed, 26 Apr 2017 13:40:59 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20161025; h=mime-version:in-reply-to:references:from:date:message-id:subject:to :cc; bh=BZF9+7jrgyRkv3eJFbRM23+LKx9bs+hX1PK7GuqdSXc=; b=lAqOSnzmwc7gLl0/wMmfdm0zH9PIjfU6JT5JQde9F6R2HzzXPM1ch4UcMbJG/5k0G1 F2bk2F7ElthqIgAoncV6MXtxDAxnXsM3kWyfOD7DF33S46sEcUDZ7iqhWUzG2YC14ue4 nVNyEmymTDeTEXsIPw2ftxBXb5lbjABfqauULuwoJHXWuMCmpGeNTXEzNh2bcqO20IUN r64sKGtKMkZcpJgXT23uuDK8nJZy+lT7f0zyTJ1PUVOZs9jZ2eojtfLKb+lH0yrgetL6 KO3Pz04Kwfq4A7tFE5s0B6emsRyOQWsITswjc2hoFDISzCRHoUE42XWmINN8/GpWe9jm WmtQ== X-Gm-Message-State: AN3rC/6ZvDX7pqcfUcWU9uBHOura/fJvEf3g0Vv1GujacMlqXfA3Btha G1QGrPkiSF2JjJtMFFWWV5Sb3ljs3gIF X-Received: by 10.46.22.4 with SMTP id w4mr765570ljd.123.1493239259487; Wed, 26 Apr 2017 13:40:59 -0700 (PDT) MIME-Version: 1.0 Received: by 10.46.88.17 with HTTP; Wed, 26 Apr 2017 13:40:38 -0700 (PDT) In-Reply-To: References: From: Floris van Doorn Date: Wed, 26 Apr 2017 16:40:38 -0400 Message-ID: Subject: Re: [HoTT] computing K To: Michael Shulman Cc: Martin Escardo , "HomotopyT...@googlegroups.com" Content-Type: multipart/alternative; boundary=f403045fc0f8a06538054e17db4e --f403045fc0f8a06538054e17db4e Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable I think (1) is indeed wishful thinking. I can disprove it assuming the following metatheoretical property. I think this property holds for book-HoTT, but I'm not sure. If X is a HIT and p : A -> X is a path constructor of X, and if p a =E2=89= =A1 p a' for terms a a' : A then a =E2=89=A1 a'. (Here =E2=89=A1 is judgmental equal= ity.) I believe that similar properties are true for MLTT if p is a constructor of an inductive type. Now let I be the interval with points 0, 1 : I and consider the following HIT: HIT X :=3D | * : X | p : I -> * =3D * | q : p(0) =3D refl Note that X is the reduced suspension of the interval, which is contractible, hence in particular a set. But assuming the above metatheoretical property, X cannot have an axiom K with the definitional properties Mike described. That would mean that p(0) =E2=89=A1 p(1) and hen= ce 0 =E2=89=A1 1. But it is impossible that the two points of the interval are judgmentally equal (because you can make a function which sends them to equal, but not judgmentally equal terms, for example (=CE=A3(X : Type), X = =3D empty) and (=CE=A3(X : Type), X =3D unit)). Best, Floris On 25 April 2017 at 19:08, Michael Shulman wrote: > Favonia, Dan Licata, and I spent a bit of time trying to answer (1) > but didn't get anywhere. I haven't thought much about (2), but I > think we were unable to think of any nontrivial types for which we > could construct a computing K inside MLTT. > > On Tue, Apr 25, 2017 at 3:04 PM, Martin Escardo > wrote: > > Interesting. So, then, here are some questions: > > > > (1) Suppose, for given type A, a hypothetical > > > > K: (x : A) (p : x =3D=3D x) -> p =3D=3D refl > > > > is given, internally in a univalent type theory. > > > > Is it possible to cook-up a K' of the same type, internally in univalen= t > > type theory, with the definitional behaviour you require? > > > > We are looking for an endofunction of the type ((x : A) (p : x =3D=3D x= ) -> > p =3D=3D > > refl) that performs some sort of definitional improvement. > > > > We know of an instance of such a phenomenon: if a factor f':||X||->A of > some > > f:X->A through |-|:X->||X|| is given, then we can find another > > *definitional* factor f':||X||->A. > > > > This is here in agda > > http://www.cs.bham.ac.uk/~mhe/truncation-and-extensionality/ > hsetfunext.html#15508 > > and also in a paper by Nicolai, Thorsten, Thierry and myself. > > > > (2) Failing that, can we, given the same hypothetical K, cook-up a type > A' > > equivalent to A (maybe A'=3D Sigma(x:A) (x=3Dx)) with K' as above? > > > > (That is, maybe A itself won't provably have (1), but still will have a= n > > equivalent manifestation which does.) > > > > Or is this wishful thinking? > > > > Martin > > > > > > On 25/04/17 10:46, Michael Shulman wrote: > >> > >> Here is a little observation that may be of interest (thanks to > >> Favonia for bringing this question to my attention). > >> > >> The Axiom K that is provable using unrestricted Agda-style > >> pattern-matching has an extra property: it computes to refl on refl. > >> That is, if we define > >> > >> K: (A : Type) (x : A) (p : x =3D=3D x) -> p =3D=3D refl > >> K A x refl =3D refl > >> > >> then the equation "K A x refl =3D refl" holds definitionally. As was > >> pointed out on the Agda mailing list a while ago, this might be > >> considered a problem if one wants to extend Agda's --without-K to > >> allow unrestricted pattern-matching when the types are automatically > >> provable to be hsets, since a general hset apparently need not admit a > >> K satisfying this *definitional* behavior. > >> > >> However (this is the perhaps-new observation), in good > >> model-categorical semantics, such a "computing K" *can* be constructed > >> for any hset. Suppose we are in a model category whose cofibrations > >> are exactly the monomorphisms, like simplicial sets or any Cisinski > >> model category. If A is an hset, then the map A -> Delta^*(PA) (which > >> type theoretically is A -> Sigma(x:A) (x=3Dx)) is a weak equivalence. > >> But it is also a split monomorphism, hence a cofibration; and thus an > >> acyclic cofibration. Therefore, we can define functions by induction > >> on loops in A that have definitionally computing behavior on refl, > >> which is exactly what unrestricted pattern-matching allows. > >> > >> Mike > >> > > > > -- > > Martin Escardo > > http://www.cs.bham.ac.uk/~mhe > > -- > 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. > --f403045fc0f8a06538054e17db4e Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
I think (1) i= s indeed wishful thinking.

I can disprove it assuming the following= metatheoretical property. I think this property holds for book-HoTT, but I= 'm not sure.

If X is a HIT and p : A -> X is a path con= structor of X, and if p a =E2=89=A1 p a' for terms a a' : A then a = =E2=89=A1 a'. (Here =E2=89=A1 is judgmental equality.)

I b= elieve that similar properties are true for MLTT if p is a constructor of a= n inductive type.

Now let I be the interval with points 0, 1 : I an= d consider the following HIT:
HIT X :=3D
| * : X
| p : I -> * =3D *
| q : p(0) =3D refl
Note that X is= the reduced suspension of the interval, which is contractible, hence in pa= rticular a set. But assuming the above metatheoretical property, X cannot h= ave an axiom K with the definitional properties Mike described. That would = mean that p(0) =E2=89=A1 p(1) and hence 0 =E2=89=A1 1. But it is impossible= that the two points of the interval are judgmentally equal (because you ca= n make a function which sends them to equal, but not judgmentally equal ter= ms, for example (=CE=A3(X : Type), X =3D empty) and (=CE=A3(X : Type), X = =3D unit)).

Best,
Floris

On 25 April 2017 at 19:08, Michael S= hulman <shu...@sandiego.edu> wrote:
Favonia, Dan Licata, and I spent a bit of time trying to answ= er (1)
but didn't get anywhere.=C2=A0 I haven't thought much about (2), bu= t I
think we were unable to think of any nontrivial types for which we
could construct a computing K inside MLTT.

On Tue, Apr 25, 2017 at 3:04 PM, Martin Escardo <m.es...@cs.bham.ac.uk> wrote:
> Interesting. So, then, here are some questions:
>
> (1) Suppose, for given type A, a hypothetical
>
> K: (x : A) (p : x =3D=3D x) -> p =3D=3D refl
>
> is given, internally in a univalent type theory.
>
> Is it possible to cook-up a K' of the same type, internally in uni= valent
> type theory,=C2=A0 with the definitional behaviour you require?
>
> We are looking for an endofunction of the type ((x : A) (p : x =3D=3D = x) -> p =3D=3D
> refl) that performs some sort of definitional improvement.
>
> We know of an instance of such a phenomenon: if a factor f':||X||-= >A of some
> f:X->A through |-|:X->||X|| is given, then we can find another > *definitional* factor f':||X||->A.
>
> This is here in agda
> http://www.cs.= bham.ac.uk/~mhe/truncation-and-extensionality/hsetfunext.html#155= 08
> and also in a paper by Nicolai, Thorsten, Thierry and myself.
>
> (2) Failing that, can we, given the same hypothetical K, cook-up a typ= e A'
> equivalent to A (maybe A'=3D Sigma(x:A) (x=3Dx)) with K' as ab= ove?
>
> (That is, maybe A itself won't provably have (1), but still will h= ave an
> equivalent manifestation which does.)
>
> Or is this wishful thinking?
>
> Martin
>
>
> On 25/04/17 10:46, Michael Shulman wrote:
>>
>> Here is a little observation that may be of interest (thanks to >> Favonia for bringing this question to my attention).
>>
>> The Axiom K that is provable using unrestricted Agda-style
>> pattern-matching has an extra property: it computes to refl on ref= l.
>> That is, if we define
>>
>> K: (A : Type) (x : A) (p : x =3D=3D x) -> p =3D=3D refl
>> K A x refl =3D refl
>>
>> then the equation "K A x refl =3D refl" holds definition= ally.=C2=A0 As was
>> pointed out on the Agda mailing list a while ago, this might be >> considered a problem if one wants to extend Agda's --without-K= to
>> allow unrestricted pattern-matching when the types are automatical= ly
>> provable to be hsets, since a general hset apparently need not adm= it a
>> K satisfying this *definitional* behavior.
>>
>> However (this is the perhaps-new observation), in good
>> model-categorical semantics, such a "computing K" *can* = be constructed
>> for any hset.=C2=A0 Suppose we are in a model category whose cofib= rations
>> are exactly the monomorphisms, like simplicial sets or any Cisinsk= i
>> model category.=C2=A0 If A is an hset, then the map A -> Delta^= *(PA) (which
>> type theoretically is A -> Sigma(x:A) (x=3Dx)) is a weak equiva= lence.
>> But it is also a split monomorphism, hence a cofibration; and thus= an
>> acyclic cofibration.=C2=A0 Therefore, we can define functions by i= nduction
>> on loops in A that have definitionally computing behavior on refl,=
>> which is exactly what unrestricted pattern-matching allows.
>>
>> Mike
>>
>
> --
> Martin Escardo
> http://www.cs.bham.ac.uk/~mhe

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