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 23:03:14 +0100
Message-ID: <7d1235e5-76ac-2a7d-9317-21d30f6973ad@ens.fr> (raw)
In-Reply-To: <8de08372-b17d-c153-73ad-4cd8b6c49758@gmail.com>

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 [this message]
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

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Discussion of Homotopy Type Theory and Univalent Foundations

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