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