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/8de08372-b17d-c153-73ad-4cd8b6c49758%40gmail.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 > -- 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.

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 HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/60639a49-a1c6-0cfd-0bdf-65ad45b14e24%40gmail.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 HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/563bb6a5-7603-c59a-5943-6f925e56b2b4%40ens.fr.

[-- Attachment #1: Type: text/plain, Size: 3553 bytes --] Thanks to everyone who replied! Just for the reference since Christian's email went only to me: there is a remark in the paper that states it is possible to make the theory extensional, so it appears 2LTT is exactly the type theory I was looking for. Best, Kristina On 1/7/2020 5:23 PM, Christian Sattler wrote: > See axiom (A5) in Section 2.4: > > (A5) We can ask that the outer level validates the equality > reflection rule, i.e. forms a model of extensional type theory. > This is the case in all the example models we are interested in. > > > Equality reflection is supported in presheaf models, which justify > conservativity over HoTT. The main problem with equality reflection is > syntactical, in that we don't have good proof assistant support for it... > > On Tue, 7 Jan 2020 at 23:11, Kristina Sojakova > <sojakova.kristina@gmail.com <mailto:sojakova.kristina@gmail.com>> 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 HomotopyTypeTheory+unsubscribe@googlegroups.com > <mailto:HomotopyTypeTheory%2Bunsubscribe@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/60639a49-a1c6-0cfd-0bdf-65ad45b14e24%40gmail.com. > -- 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/f840404d-7427-0156-2f9a-50bfa865ea0d%40gmail.com. [-- Attachment #2: Type: text/html, Size: 5999 bytes --]