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: https://github.com/mortberg/gen-cart/blob/master/modelstructure.pdf 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: https://github.com/mortberg/gen-cart/tree/master/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! -- Anders, Evan and Andrew -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/2b7b0d66-4787-4fad-9ee4-7a0bd29faab8%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.