Discussion of Homotopy Type Theory and Univalent Foundations
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From: Kristina Sojakova <sojakova.kristina@gmail.com>
To: "Rafaël Bocquet" <rafael.bocquet@ens.fr>,
	"Homotopy Type Theory" <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] HoTT with extensional equality
Date: Tue, 7 Jan 2020 17:11:36 -0500
Message-ID: <60639a49-a1c6-0cfd-0bdf-65ad45b14e24@gmail.com> (raw)
In-Reply-To: <7d1235e5-76ac-2a7d-9317-21d30f6973ad@ens.fr>

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.


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 [this message]
2020-01-07 22:18     ` Rafaël Bocquet
     [not found]     ` <CALCpNBoWKXbQgdJ2Pqq_G7J_0D48OVGUeQoBnOfDHzC__GWkHA@mail.gmail.com>
2020-01-07 23:26       ` Kristina Sojakova

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Discussion of Homotopy Type Theory and Univalent Foundations

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