Re: [HoTT] HoTT with extensional equality

["Rafaël Bocquet" <rafael.bocquet@ens.fr>]

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.
To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/7d1235e5-76ac-2a7d-9317-21d30f6973ad%40ens.fr.