From: Matt Oliveri <atm...@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] Re: cubical type theory with UIP
Date: Sat, 29 Jul 2017 04:08:06 -0700 (PDT) [thread overview]
Message-ID: <e2444bb2-0e62-4201-902d-6953d0d73cca@googlegroups.com> (raw)
In-Reply-To: <1187de24-0548-48cd-9b7b-c3c3ab9cf4a7@googlegroups.com>
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Now I'm having second thoughts. Quotienting together hprops might make type
equality computationally relevant. Not something you want with OTT's strict
props or ETT's equality. Maybe 2-dimensional type theory would be good for
the job. In this case the 2-cells would not be distinguishable by equality,
but might still have computational content.
On Saturday, July 29, 2017 at 6:19:57 AM UTC-4, Matt Oliveri wrote:
>
> Sorry. I got distracted because the type theory you seem to be asking for
> doesn't sound cubical. Like I said, I suspect OTT could handle hprop
> extensionality, if it doesn't already. Probably ETT could too.
>
> On Saturday, July 29, 2017 at 4:08:01 AM UTC-4, Michael Shulman wrote:
>>
>> As I said,
>>
>> > The motivation would be to have a type theory with canonicity for
>> > 1-categorical semantics
>>
>> So no, I don't want "the model" to be using cubical sets, I want
>> models in all suitable 1-categories (e.g. Pi-pretopos etc.).
>>
>> On Sat, Jul 29, 2017 at 12:23 AM, Matt Oliveri <atm...@gmail.com> wrote:
>> > Only up to homotopy? So you still want the model to be using cubical
>> sets?
>> > Actually, couldn't you interpret OTT into the hsets, internally to
>> HoTT?
>> > It'd be a hassle without a real solution to the infinite coherence
>> problem,
>> > but it should work, since the h-levels involved are bounded.
>> >
>> > On Saturday, July 29, 2017 at 2:20:06 AM UTC-4, Michael Shulman wrote:
>> >>
>> >> Right: up to homotopy, all cubes would be equivalent to points (hence
>> >> my question #1).
>> >>
>> >> On Fri, Jul 28, 2017 at 6:47 PM, Matt Oliveri <atm...@gmail.com>
>> wrote:
>> >> > I'm confused. So you want a cubical type theory with only hsets? In
>> what
>> >> > sense would there be cubes, other than just points? I thoght OTT had
>> >> > propositional extensionality. Though maybe that's only for strict
>> props.
>> >> >
>> >> >
>> >> > On Sunday, July 23, 2017 at 6:54:39 PM UTC-4, Michael Shulman wrote:
>> >> >>
>> >> >> I am wondering about versions of cubical type theory with UIP. The
>> >> >> motivation would be to have a type theory with canonicity for
>> >> >> 1-categorical semantics that can prove both function extensionality
>> >> >> and propositional univalence. (I am aware of Observational Type
>> >> >> Theory, which I believe has UIP and proves function extensionality,
>> >> >> but I don't think it proves propositional univalence -- although I
>> >> >> would be happy to be wrong about that.)
>> >> >>
>> >> >> Presumably we obtain a cubical type theory that's compatible with
>> >> >> axiomatic UIP if in CCHM cubical type theory we postulate only a
>> >> >> single universe of propositions. But I wonder about some possible
>> >> >> refinements, such as:
>> >> >>
>> >> >> 1. In this case do we still need *all* the Kan composition and
>> gluing
>> >> >> operations? If all types are hsets then it seems like it ought to
>> be
>> >> >> unnecessary to have these operations at all higher dimensions.
>> >> >>
>> >> >> 2. Can it be enhanced to make UIP provable, such as by adding a
>> >> >> computing K eliminator?
>> >> >>
>> >> >> Mike
>>
>
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next prev parent reply other threads:[~2017-07-29 11:08 UTC|newest]
Thread overview: 20+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-07-23 22:54 Michael Shulman
2017-07-29 1:47 ` Matt Oliveri
2017-07-29 2:25 ` [HoTT] " Jon Sterling
2017-07-29 7:29 ` Matt Oliveri
2017-07-29 6:19 ` Michael Shulman
2017-07-29 7:23 ` Matt Oliveri
2017-07-29 8:07 ` Michael Shulman
2017-07-29 10:19 ` Matt Oliveri
2017-07-29 11:08 ` Matt Oliveri [this message]
2017-07-29 21:19 ` Michael Shulman
[not found] ` <8f052071-09e0-74db-13dc-7f76bc71e374@cs.bham.ac.uk>
2017-07-31 3:49 ` Matt Oliveri
2017-07-31 15:50 ` Michael Shulman
2017-07-31 17:36 ` Matt Oliveri
2017-08-01 9:14 ` Neelakantan Krishnaswami
2017-08-01 9:20 ` Michael Shulman
2017-08-01 9:34 ` James Cheney
2017-08-01 16:26 ` Michael Shulman
2017-08-01 21:27 ` Matt Oliveri
2017-07-31 4:19 ` Matt Oliveri
2017-08-02 9:40 ` [HoTT] " Andrea Vezzosi
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