Discussion of Homotopy Type Theory and Univalent Foundations
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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	[thread overview]
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.)

- Jasper Hugunin

On Fri, Sep 13, 2019 at 2:10 AM Jasper Hugunin <jasperh@cs.washington.edu>

> 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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  reply	other threads:[~2019-09-15  5:57 UTC|newest]

Thread overview: 13+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-09-13  6:10 [HoTT] " Jasper Hugunin
2019-09-15  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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