The document is slightly outdated.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?
>
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