The document is slightly outdated. We do not have the rule iso A B (λx ⇒ x) (λx ⇒ x) idp idp i ⇒β A in the actual implementation since univalence is true even without it. This rule has another problem. It seems that the theory as presented in the document introduces a *quasi*-equivalence between A = B and Equiv A B, which means that there are some true statements which are not provable in it. The theory without this rule should be completely equivalent to the ordinary HoTT, even though I cannot prove this yet. Regards, Valery Isaev сб, 10 авг. 2019 г. в 12:42, Michael Shulman : > There is a bit more subtlety here than is evident from the brief > description, since everything has to happen in an arbitrary slice > category of the model category. But although the slices of a > cartesian model category are not in general again cartesian, they are > enriched model categories over the base, and so I think I agree that > this works since I lives in the base. > > However, section 2.2 of https://valis.github.io/doc.pdf also appears > to assert that an equivalence can be made into a line in the universe > for which coercing along the line is *definitionally* equal to the > action of the given equivalence, and such that the line associated to > the identity equivalence is definitionally constant. The latter seems > like it might be obtainable by a lifting property, but I don't > immediately see how to obtain the former in a model category? > -- 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/CAA520ftTacC4iegu2UM887nbyJWQTByMKhrcsftKPXCadH06kQ%40mail.gmail.com.