Regarding the Cauchy reals, I believe it is known that countable choice 
holds and set quotients exist in simplicial sets, but I think under these 
conditions the usual construction of the Cauchy reals as a quotient of 
sequences of rationals satisfies the constructors for the HIT Cauchy reals, 
and also the higher induction principle.

I think the HIT cumulative hierarchy can be constructed using W types and 
univalence, using the characterisation of it as a retract of the Aczel 
cumulative hierarchy by Hakon Gylterud in the paper 
at https://doi.org/10.1017/jsl.2017.84 . If so, similar arguments should 
work for ordinals and surreal numbers.

Are there any other examples in the HoTT book?

Best,
Andrew

On Thursday, 16 May 2019 16:57:36 UTC+2, Bas Spitters wrote:
>
> What is the status of the semantics of quotient inductive inductive types? 
> Looking at the literature there's some progress on reducing QIITs to 
> simpler constructions, but this does not seem to have led to a 
> convenient semantic result. 
> E.g. QIITs do not seem to be treated in the work by Lumdaine and Shulman. 
>
> https://ncatlab.org/nlab/show/inductive-inductive+type 
>
> Do we know that the prototypical QIITs from the book (e.g. Cauchy 
> reals) are supported in the usual models of HoTT? 
>
> Thanks, 
>
> Bas 
>

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