Discussion of Homotopy Type Theory and Univalent Foundations
 help / color / mirror / Atom feed
From: Kristina Sojakova <sojakova.kristina@gmail.com>
To: Christian Sattler <sattler.christian@gmail.com>,
	Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] HoTT with extensional equality
Date: Tue, 7 Jan 2020 18:26:30 -0500	[thread overview]
Message-ID: <f840404d-7427-0156-2f9a-50bfa865ea0d@gmail.com> (raw)
In-Reply-To: <CALCpNBoWKXbQgdJ2Pqq_G7J_0D48OVGUeQoBnOfDHzC__GWkHA@mail.gmail.com>

[-- Attachment #1: Type: text/plain, Size: 3553 bytes --]

Thanks to everyone who replied!

Just for the reference since Christian's email went only to me: there is 
a remark in the paper that states it is possible to make the theory 
extensional, so it appears 2LTT is exactly the type theory I was looking 
for.

Best,

Kristina

On 1/7/2020 5:23 PM, Christian Sattler wrote:
> See axiom (A5) in Section 2.4:
>
>     (A5) We can ask that the outer level validates the equality
>     reflection rule, i.e. forms a model of extensional type theory.
>     This is the case in all the example models we are interested in.
>
>
> Equality reflection is supported in presheaf models, which justify 
> conservativity over HoTT. The main problem with equality reflection is 
> syntactical, in that we don't have good proof assistant support for it...
>
> On Tue, 7 Jan 2020 at 23:11, Kristina Sojakova 
> <sojakova.kristina@gmail.com <mailto:sojakova.kristina@gmail.com>> wrote:
>
>     Hello Rafael,
>
>     Thank you for the reference. I browsed the paper; it seems to me that
>     the theory does not appear to support identity reflection. I am
>     looking
>     for a truly extensional form of equality (in addition to the usual
>     one),
>     where equal terms are syntactically identified.
>
>     Kristina
>
>     On 1/7/2020 5:03 PM, Rafaël Bocquet wrote:
>     > Hello,
>     >
>     > I think that the paper "Two-Level Type Theory and Applications"
>     > (https://arxiv.org/abs/1705.03307), whose last version has been
>     > submitted on arXiv last month, answers these questions. One of the
>     > intended models of 2LTT is the presheaf category Ĉ over any
>     model C of
>     > HoTT, and this presheaf model is conservative over C, essentially
>     > because the Yoneda embedding is fully faithful. This means that
>     we can
>     > always work in 2LTT instead of HoTT.
>     >
>     > Rafaël
>     >
>     > On 1/7/20 8:59 PM, Kristina Sojakova wrote:
>     >> Dear all,
>     >>
>     >> I have been increasingly running into situations where I wished
>     I had
>     >> an extensional equality type with a  reflection rule in HoTT, in
>     >> addition to the intensional one to which univalence pertains. I
>     know
>     >> that type systems with two equalities have been studied in the
>     HoTT
>     >> community (e.g., VV's HTS), but last time I discussed this with
>     >> people it seemed the situation was not yet well-understood. So my
>     >> question is, what exactly goes wrong if we endow HoTT with an
>     >> extensional type?
>     >>
>     >> Thank you,
>     >>
>     >> Kristina
>     >>
>
>     -- 
>     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
>     <mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com>.
>     To view this discussion on the web visit
>     https://groups.google.com/d/msgid/HomotopyTypeTheory/60639a49-a1c6-0cfd-0bdf-65ad45b14e24%40gmail.com.
>

-- 
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/f840404d-7427-0156-2f9a-50bfa865ea0d%40gmail.com.

[-- Attachment #2: Type: text/html, Size: 5999 bytes --]

      parent reply	other threads:[~2020-01-07 23:26 UTC|newest]

Thread overview: 5+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2020-01-07 19:59 Kristina Sojakova
2020-01-07 22:03 ` Rafaël Bocquet
2020-01-07 22:11   ` Kristina Sojakova
2020-01-07 22:18     ` Rafaël Bocquet
     [not found]     ` <CALCpNBoWKXbQgdJ2Pqq_G7J_0D48OVGUeQoBnOfDHzC__GWkHA@mail.gmail.com>
2020-01-07 23:26       ` Kristina Sojakova [this message]

Reply instructions:

You may reply publicly 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=f840404d-7427-0156-2f9a-50bfa865ea0d@gmail.com \
    --to=sojakova.kristina@gmail.com \
    --cc=HomotopyTypeTheory@googlegroups.com \
    --cc=sattler.christian@gmail.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
Be sure your reply has a Subject: header at the top and a blank line before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).