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From: Jasper Hugunin <jasperh@cs.washington.edu>
To: HomotopyTypeTheory@googlegroups.com
Subject: [HoTT] Re: A question about the problem with regularity in CCHM cubical type theory
Date: Sun, 15 Sep 2019 01:57:40 -0400
Message-ID: <CAGTS-a9oS0CQDGKgj7ghCh8+ZwAcAefiTg4JJVHemV3HUPcPEg@mail.gmail.com> (raw)
In-Reply-To: <CAGTS-a8fo4FKCLiCVCYm6AGeVc=qB8QNUZ3Weho6kLT8NXtc-Q@mail.gmail.com>
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Offline, Carlo Angiuli showed me that the difficulty was in part 1, because
of a subtlety I had been forgetting.
Since types are *Kan* cubical sets, we need that the Kan operations agree
as well as the sets.
So part 1 could be thought of as (Glue [ phi |-> equivRefl A ] A, compGlue)
= (A, compA), and getting that the Kan operations to agree was/is difficult.
(Now that I know what the answer is, it is clear that this was already
explained in the initial discussion.)
Humbly,
- Jasper Hugunin
On Fri, Sep 13, 2019 at 2:10 AM Jasper Hugunin <jasperh@cs.washington.edu>
wrote:
> Hello all,
>
> I've been trying to understand better why composition for the universe
> does not satisfy regularity.
> Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |->
> equiv^i E ] A, I would expect regularity to follow from two parts:
> 1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of regularity
> condition for the Glue type constructor itself)
> 2. That equiv^i (refl A) reduces to equivRefl A
> I'm curious as to which (or both) of these parts was the issue, or if
> regularity for the universe was supposed to follow from a different
> argument.
>
> Context:
> I've been studying and using CCHM cubical type theory recently, and often
> finding myself wishing that J computed strictly.
> If I understand correctly, early implementations of ctt did have strict J
> for Path types, and this was justified by a "regularity" condition on the
> composition operation, but as discussed in this thread on the HoTT
> mailing list
> <https://groups.google.com/d/msg/homotopytypetheory/oXQe5u_Mmtk/3HEDk5g5uq4J>,
> the definition of composition for the universe was found to not satisfy
> regularity.
> I don't remember seeing the regularity condition defined anywhere, but my
> understanding is that it requires that composition in a degenerate line of
> types, with the system of constraints giving the sides of the box also
> degenerate in that direction, reduces to just the bottom of the box. This
> seems to be closed under the usual type formers, plus Glue, but not the
> universe with computation defined as in the CCHM paper
> <http://www.cse.chalmers.se/~coquand/cubicaltt.pdf> (for trivial reasons
> and non-trivial reasons; it gets stuck at the start with Glue [ phi |->
> equiv^i refl ] A not reducing to anything).
>
> Best regards,
> - Jasper Hugunin
>
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<div dir="ltr">Offline, Carlo Angiuli showed me that the difficulty was in part 1, because of a subtlety I had been forgetting.<div><br></div><div>Since types are *Kan* cubical sets, we need that the Kan operations agree as well as the sets.</div><div>So part 1 could be thought of as (Glue [ phi |-> equivRefl A ] A, compGlue) = (A, compA), and getting that the Kan operations to agree was/is difficult.</div><div>(Now that I know what the answer is, it is clear that this was already explained in the initial discussion.)</div><div><br></div><div>Humbly,</div><div>- Jasper Hugunin</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Sep 13, 2019 at 2:10 AM Jasper Hugunin <<a href="mailto:jasperh@cs.washington.edu">jasperh@cs.washington.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr">Hello all,<div><br></div><div><div>I've been trying to understand better why composition for the universe does not satisfy regularity.</div><div>Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |-> equiv^i E ] A, I would expect regularity to follow from two parts:</div><div>1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of regularity condition for the Glue type constructor itself)</div><div>2. That equiv^i (refl A) reduces to equivRefl A</div><div>I'm curious as to which (or both) of these parts was the issue, or if regularity for the universe was supposed to follow from a different argument.</div><div><br></div><div>Context:</div><div><div>I've been studying and using CCHM cubical type theory recently, and often finding myself wishing that J computed strictly.</div><div>If I understand correctly, early implementations of ctt did have strict J for Path types, and this was justified by a "regularity" condition on the composition operation, but as discussed in <a href="https://groups.google.com/d/msg/homotopytypetheory/oXQe5u_Mmtk/3HEDk5g5uq4J" target="_blank">this thread on the HoTT mailing list</a>, the definition of composition for the universe was found to not satisfy regularity.</div><div>I don't remember seeing the regularity condition defined anywhere, but my understanding is that it requires that composition in a degenerate line of types, with the system of constraints giving the sides of the box also degenerate in that direction, reduces to just the bottom of the box. This seems to be closed under the usual type formers, plus Glue, but not the universe with computation defined as in <a href="http://www.cse.chalmers.se/~coquand/cubicaltt.pdf" target="_blank">the CCHM paper</a> (for trivial reasons and non-trivial reasons; it gets stuck at the start with Glue [ phi |-> equiv^i refl ] A not reducing to anything).</div><div><br></div></div></div><div>Best regards,</div><div>- Jasper Hugunin</div></div>
</blockquote></div>
<p></p>
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next prev parent reply indexThread overview:13+ messages / expand[flat|nested] mbox.gz Atom feed top 2019-09-13 6:10 [HoTT] " Jasper Hugunin2019-09-15 5:57 ` Jasper Hugunin [this message][not found] ` <CAMWCppk4PWzfZ1HKNLMdAZ=RBC-ARxtJXR8okvwO3raea5gC8Q@mail.gmail.com> [not found] ` <CAGTS-a-SvWWF+br6sKxGj6ufVY=4m830FH9BDg06QJR1vbNFsw@mail.gmail.com> 2019-09-16 2:18 ` Fwd: [HoTT] " Jasper Hugunin 2019-09-16 16:18 ` [HoTT] " Licata, Dan 2019-09-16 17:09 ` Jasper Hugunin 2019-09-16 19:01 ` Licata, Dan 2019-09-16 20:17 ` Jasper Hugunin 2019-09-18 12:16 ` Anders Mortberg 2019-09-18 12:52 ` Thierry Coquand 2019-09-15 11:55 ` [HoTT] " Andrew Swan 2019-09-15 22:38 ` Jasper Hugunin 2019-09-16 1:04 ` Jon Sterling [not found] ` <A605E6EE-0101-4390-B50D-A6AEB36FDCC2@icloud.com> 2019-09-16 1:44 ` Jon Sterling

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