Re: [HoTT] A unifying cartesian cubical type theory

[Steve Awodey <awodey@cmu.edu>]

> On Feb 16, 2019, at 2:51 PM, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:
> 
>>> I think the idea is that the model structure is constructed / proved using
>>> ideas from type theory (like univalence), rather than that it is a model
>>> of type theory.  But I agree that the terminology is confusing.
>>> The methodology is, I think, due to Christian Sattler ??? so maybe Sattler
>>> model structure is more appropriate?
>> 
>> When the interval is fixed one might speak of minimal Cisinski model
>> structure since it is the one with the fewest weak equivalences.
>> Of course, Sattler studied them a lot so it's good name either.
> 
> Well, Coquand-Sattler might be better because it was first used in the
> [CCHM] paper. From the many anodyne monos of the test model structure
> one took just those which were syntactically convenient.

I don’t want to minimize the importance of the work on cubical type theory 
— which I believe is very great — but it has focussed on building models of type theory 
directly, often within other type theories, rather than on building Quillen model categories. 
To be sure, many ideas, and some terminology, from model category theory are used, 
but without showing or even claiming that there is a Quillen model structure.

> But, as far as I know the test model structure also gives rise to a
> model of cubical TT because its more restrictive class off fibrations
> suffices for interpreting sytax.

has it been shown that the test model structure interprets Pi types, universes, and univalence? 

Steve

> 
>> Unfortunately, we don't know when minimal and test model structure concide.
> 
> Thomas
> 
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