Discussion of Homotopy Type Theory and Univalent Foundations
 help / Atom feed
From: Jasper Hugunin <jasperh@cs.washington.edu>
To: HomotopyTypeTheory@googlegroups.com
Subject: [HoTT] A question about the problem with regularity in CCHM cubical type theory
Date: Fri, 13 Sep 2019 02:10:55 -0400
Message-ID: <CAGTS-a8fo4FKCLiCVCYm6AGeVc=qB8QNUZ3Weho6kLT8NXtc-Q@mail.gmail.com> (raw)

[-- Attachment #1: Type: text/plain, Size: 2138 bytes --]

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

-- 
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.
To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGTS-a8fo4FKCLiCVCYm6AGeVc%3DqB8QNUZ3Weho6kLT8NXtc-Q%40mail.gmail.com.

[-- Attachment #2: Type: text/html, Size: 2723 bytes --]

             reply index

Thread overview: 13+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-09-13  6:10 Jasper Hugunin [this message]
2019-09-15  5:57 ` [HoTT] " Jasper Hugunin
     [not found]   ` <CAMWCppk4PWzfZ1HKNLMdAZ=RBC-ARxtJXR8okvwO3raea5gC8Q@mail.gmail.com>
     [not found]     ` <CAGTS-a-SvWWF+br6sKxGj6ufVY=4m830FH9BDg06QJR1vbNFsw@mail.gmail.com>
2019-09-16  2:18       ` Fwd: " 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

Reply instructions:

You may reply publically to this message via plain-text email
using any one of the following methods:

* Save the following mbox file, import it into your mail client,
  and reply-to-all from there: mbox

  Avoid top-posting and favor interleaved quoting:
  https://en.wikipedia.org/wiki/Posting_style#Interleaved_style

* Reply using the --to, --cc, and --in-reply-to
  switches of git-send-email(1):

  git send-email \
    --in-reply-to='CAGTS-a8fo4FKCLiCVCYm6AGeVc=qB8QNUZ3Weho6kLT8NXtc-Q@mail.gmail.com' \
    --to=jasperh@cs.washington.edu \
    --cc=HomotopyTypeTheory@googlegroups.com \
    /path/to/YOUR_REPLY

  https://kernel.org/pub/software/scm/git/docs/git-send-email.html

* If your mail client supports setting the In-Reply-To header
  via mailto: links, try the mailto: link

Discussion of Homotopy Type Theory and Univalent Foundations

Archives are clonable: git clone --mirror http://inbox.vuxu.org/hott

Newsgroup available over NNTP:
	nntp://inbox.vuxu.org/vuxu.archive.hott


AGPL code for this site: git clone https://public-inbox.org/ public-inbox