Discussion of Homotopy Type Theory and Univalent Foundations
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From: "Rafaël Bocquet" <rafael.bocquet@ens.fr>
To: Kristina Sojakova <sojakova.kristina@gmail.com>,
	Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] HoTT with extensional equality
Date: Tue, 7 Jan 2020 23:18:01 +0100
Message-ID: <563bb6a5-7603-c59a-5943-6f925e56b2b4@ens.fr> (raw)
In-Reply-To: <60639a49-a1c6-0cfd-0bdf-65ad45b14e24@gmail.com>

The intended presheaf model supports equality reflection. Martin 
Hofmann's conservativity theorem also implies that most type theories 
with UIP can conservatively be extended with equality reflection.

On 1/7/20 11:11 PM, Kristina Sojakova 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 [this message]
     [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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