Discussion of Homotopy Type Theory and Univalent Foundations
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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
Message-ID: <f840404d-7427-0156-2f9a-50bfa865ea0d@gmail.com> (raw)
In-Reply-To: <CALCpNBoWKXbQgdJ2Pqq_G7J_0D48OVGUeQoBnOfDHzC__GWkHA@mail.gmail.com>

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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
>     >>
>
>     -- 
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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]

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