https://groups.google.com/d/msg/homotopytypetheory/bNHRnGiF5R4/e-2BlTsNBQAJ

I think the system proposed by Thierry has the best properties of all the known systems for doing univalent mathematics so far. It is also substantially simpler than full-blown cubical type theory as the Kan operations do not reduce, but this does not seem to be much of an obstacle for practical formalization judging from my experiment. I don’t think I can block things sufficiently in cubical agda using abstract so that I can work in this system (but maybe I’m wrong?), so there should probably be a flag that disables some of the computation rules internally.

Another issue with this is that we probably want to generalize the Kan operations a little bit to let us do hcomp/transp “backwards” as well as “forward” (from 1->0 instead of just 0->1 in the language of cartesian cubical type theory) in order to compensate for the lack of reversals.

]]>https://www.eleves.ens.fr/home/rbocquet/Conservativity.pdf ]]>

Of course, we can’t just ask for conservativity of the cubical system over the “book” one — that’s too strong. A semantic approach would be to show that the model in cSets (where we do know that we have a sound interpretation) maps into the others, in a way that preserves the interpretation. This is something that’s nice about *Cartesian* cubes: it’s initial with respect to other toposes with an interval, like sSets (the same is true for Dedekind cubes, with respect to intervals with connections). *But* the canonical comparison functors determined by the universal property of cSets need to be shown to preserve the interpretation of type theory. This is what I meant by “there’s some work to be done”. ]]>