From: "Rafaël Bocquet" <firstname.lastname@example.org> To: Kristina Sojakova <email@example.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: <firstname.lastname@example.org> (raw) In-Reply-To: <email@example.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 >>> > -- 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 HomotopyTypeTheoryfirstname.lastname@example.org. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/563bb6a5-7603-c59a-5943-6f925e56b2b4%40ens.fr.
next prev parent reply index 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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