Re: [HoTT] HoTT with extensional equality

[Kristina Sojakova <sojakova.kristina@gmail.com>]

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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