Re: [HoTT] A unifying cartesian cubical type theory

[Bas Spitters <b.a.w.spitters@gmail.com>]

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
<andersmortberg@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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