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 
> <javascript:>> 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
>