Discussion of Homotopy Type Theory and Univalent Foundations
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* [HoTT] HoTT with extensional equality
@ 2020-01-07 19:59 Kristina Sojakova
  2020-01-07 22:03 ` Rafaël Bocquet
  0 siblings, 1 reply; 5+ messages in thread
From: Kristina Sojakova @ 2020-01-07 19:59 UTC (permalink / raw)
  To: Homotopy Type Theory

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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* Re: [HoTT] HoTT with extensional equality
  2020-01-07 19:59 [HoTT] HoTT with extensional equality Kristina Sojakova
@ 2020-01-07 22:03 ` Rafaël Bocquet
  2020-01-07 22:11   ` Kristina Sojakova
  0 siblings, 1 reply; 5+ messages in thread
From: Rafaël Bocquet @ 2020-01-07 22:03 UTC (permalink / raw)
  To: Kristina Sojakova, Homotopy Type Theory

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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* Re: [HoTT] HoTT with extensional equality
  2020-01-07 22:03 ` Rafaël Bocquet
@ 2020-01-07 22:11   ` Kristina Sojakova
  2020-01-07 22:18     ` Rafaël Bocquet
       [not found]     ` <CALCpNBoWKXbQgdJ2Pqq_G7J_0D48OVGUeQoBnOfDHzC__GWkHA@mail.gmail.com>
  0 siblings, 2 replies; 5+ messages in thread
From: Kristina Sojakova @ 2020-01-07 22:11 UTC (permalink / raw)
  To: Rafaël Bocquet, Homotopy Type Theory

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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* Re: [HoTT] HoTT with extensional equality
  2020-01-07 22:11   ` Kristina Sojakova
@ 2020-01-07 22:18     ` Rafaël Bocquet
       [not found]     ` <CALCpNBoWKXbQgdJ2Pqq_G7J_0D48OVGUeQoBnOfDHzC__GWkHA@mail.gmail.com>
  1 sibling, 0 replies; 5+ messages in thread
From: Rafaël Bocquet @ 2020-01-07 22:18 UTC (permalink / raw)
  To: Kristina Sojakova, Homotopy Type Theory

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
>>>
>

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* Re: [HoTT] HoTT with extensional equality
       [not found]     ` <CALCpNBoWKXbQgdJ2Pqq_G7J_0D48OVGUeQoBnOfDHzC__GWkHA@mail.gmail.com>
@ 2020-01-07 23:26       ` Kristina Sojakova
  0 siblings, 0 replies; 5+ messages in thread
From: Kristina Sojakova @ 2020-01-07 23:26 UTC (permalink / raw)
  To: Christian Sattler, Homotopy Type Theory

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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.
>

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Thread overview: 5+ messages (download: mbox.gz / follow: Atom feed)
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2020-01-07 19:59 [HoTT] HoTT with extensional equality Kristina Sojakova
2020-01-07 22:03 ` Rafaël Bocquet
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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