[HoTT] Re: A unifying cartesian cubical type theory

["Anders Mörtberg" <andersmortberg@gmail.com>]

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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

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