Re: [HoTT] Semantics of QIITs ?

[Jasper Hugunin <jasperh@cs.washington.edu>]

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Hello all,

To confirm what others have said about my construction:
As Andras and Jon noted, it remains open whether the construction can be
extended to reduce higher inductive-inductive types to higher inductive
types, but I am hopeful that it can, at least in a situation such as CTT
where the eliminator computes on path constructors.
As Matt noted, the general recursive-recursive eliminator is both essential
(for proving initiality, for example) and missing. This is the same
restriction as on the construction in extensional type theory (or using
UIP) given by Nordvall Forsberg. The main obstacle I see to getting the
general recursive-recursive eliminator is that the simple eliminator is not
strict, but only computes up to a path. I think I see a way to turn a
strict simple eliminator into a general eliminator, but this is still
conjectural.

To summarize my understanding of the original question, QIITs are an
obvious subset of HIITs, and we know how to handle many HITs primitively in
cubical type theory, but the extension to primitive HIITs in cubical type
theory has not been done, nor does my construction immediately allow
reducing HIITs to HITs (and even if extended lacks strictness and
generality of the eliminator).

Best regards,
- Jasper Hugunin

On Mon, May 20, 2019 at 5:28 PM Matt Oliveri <atmacen@gmail.com> wrote:

> I'm not completely satisfied with Hugunin's technique, because it
> justifies only the "simple" elimination principle, rather than the general,
> "recursive-recursive" elimination principle implemented in Agda. As far as
> I can tell, realistic use cases for inductive-inductive families also need
> the recursive-recursive elimination principle, where the types of the maps
> out of later families depend on the maps out of earlier families. (I'm not
> sure how much of the other research on IIFs stops short of handling
> recursion-recursion, but I think it should be taken seriously.)
>
> On Monday, May 20, 2019 at 7:26:12 PM UTC-4, Jon Sterling wrote:
>>
>> Echoing Andras, I recall that a new encoding due to Jasper Hugunin enable
>> us to interpret IITs without using UIP, and it is an open question to
>> determine whether Jasper's ideas can be extended to QIITs. I hope they can,
>> and someone should find out :)
>>
>> Best,
>> Jon
>>
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