With Andrew Swan we have now worked out a categorical
presentation of our generalized notion of fibrations for cartesian
cubical sets. We have also proved that Sattler's model structure theorem
applies. For details see:
A special case of our result is a model structure on cartesian cubical
sets that does not require that the diagonal map on the interval is a
cofibration (i.e. "diagonal cofibrations"). We hence obtain the Sattler
model structure on De Morgan and distributive lattice cubical sets as a
special case when the cube category has connections. Furthermore, we
also obtain the model structure on cartesian cubical sets sketched by
Coquand (
https://groups.google.com/forum/#!msg/homotopytypetheory/RQkLWZ_83kQ/tAyb3zYTBQAJ) and more recently spelled out in detail by Awodey (
https://github.com/awodey/math/blob/master/QMS/qms.pdf) when we add the assumption of diagonal cofibrations.
We have also formalized the cubical model based on generalized fibrations in the Orton-Pitts style using Agda:
The formalization contains the standard type formers of cubical type
theory (Pi, Sigma, Path, Id, Glue and univalence). We have not yet
formalized the LOPS universe construction as this requires a special
branch of Agda that we're waiting for to get merged into master, but we
don't expect any problems with this as LOPS applies to cartesian cubical
sets as exponentiating by the interval has a right adjoint.
Furthermore, the LOPS construction has already been formalized for
cartesian cubical sets in ABCFHL (
https://github.com/dlicata335/cart-cube). We have also formalized the construction of both of the two factorization systems using Andrew's W-types with reductions (
https://arxiv.org/abs/1802.07588).
Comments are very welcome!
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Anders, Evan and Andrew