Re: [HoTT] A unifying cartesian cubical type theory

[Steve Awodey <awodey@cmu.edu>]

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?

Steve

> On Feb 15, 2019, at 7:01 PM, Michael Shulman <shulman@sandiego.edu> wrote:
> 
> Please don't call these "type-theoretic model structures".  There are
> many other model categories that model type theories in ways unrelated
> to cubes, notably the classical simplicial model categories that model
> Book HoTT.  Maybe something like "cubical-type model structures"?
> 
> On Fri, Feb 15, 2019 at 8:32 AM Anders Mörtberg
> <andersmortberg@gmail.com> wrote:
>> 
>> No, we didn't think about model structures yet. First of all one has to figure out how to write our Kan operations as a lifting condition (this is not entirely obvious because of the additional weakness).
>> 
>> The observation that the type theoretic model structure on De Morgan cubical sets is not equivalent to the one on spaces is simpler than for Cartesian cubical sets as we have reversals. The case that is not known AFAIK is for the one based on distributive lattices (so only with connections, but no reversals), i.e. the "Dedekind" cubes.
>> 
>> --
>> Anders
>> 
>> On Friday, February 15, 2019 at 3:16:56 AM UTC-5, Bas Spitters wrote:
>>> 
>>> Thanks. This looks very interesting.
>>> 
>>> Did you think about the corresponding model structure?
>>> https://ncatlab.org/nlab/show/type-theoretic+model+structure
>>> 
>>> Because, we know that Cartesian cubical sets are not equivalent to
>>> simplicial sets, but as far as I know, this is still unclear for the
>>> DeMorgan cubical sets.
>>> https://ncatlab.org/nlab/show/cubical+type+theory#models
>>> 
>>> On Thu, Feb 14, 2019 at 8:05 PM Anders Mortberg
>>> <andersm...@gmail.com> wrote:
>>>> 
>>>> Evan Cavallo and I have worked out a new cartesian cubical type theory
>>>> that generalizes the existing work on cubical type theories and models
>>>> based on a structural interval:
>>>> 
>>>> http://www.cs.cmu.edu/~ecavallo/works/unifying-cartesian.pdf
>>>> 
>>>> The main difference from earlier work on similar models is that it
>>>> depends neither on diagonal cofibrations nor on connections or
>>>> reversals. In the presence of these additional structures, our notion
>>>> of fibration coincides with that of the existing cartesian and De
>>>> Morgan cubical set models. This work can therefore be seen as a
>>>> generalization of the existing models of univalent type theory which
>>>> also clarifies the connection between them.
>>>> 
>>>> The key idea is to weaken the notion of fibration from the cartesian
>>>> Kan operations com^r->s so that they are not strictly the identity
>>>> when r=s. Instead we introduce weak cartesian Kan operations that are
>>>> only the identity function up to a path when r=s. Semantically this
>>>> should correspond to a weaker form of a lifting condition where the
>>>> lifting only satisfies some of the eqations up to homotopy. We verify
>>>> in the note that this weaker notion of fibration is closed under the
>>>> type formers of cubical type theory (nat, Sigma, Pi, Path, Id, Glue,
>>>> U) so that we get a model of univalent type theory. We also verify
>>>> that the circle works and we don't expect any substantial problems
>>>> with extending it to more complicated HITs (like pushouts).
>>>> 
>>>> --
>>>> Anders and Evan
>>>> 
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