On Thu, Aug 8, 2019 at 2:56 AM Valery Isaev <valery.isaev@gmail.com> wrote:
> You can say that they are hidden in the background, but I prefer to think about this in a different way. I think about \Set0 as a strict subtype of \Type0. In comparison, the type \Sigma (A : \Type0) (isSet A) is only homotopically embeds into \Type0. It is equivalent to \Set0, but not isomorphic to it. In particular, this means that every type in \Set0 satisfies isSet and every type in \Type0 which satisfies isSet is equivalent to some type in \Set0, but not necessarily belongs to \Set0 itself. So, if we have (1), we also have (2) and we do not have (3). It may be true that A is a 2-type, which means that there is a type A' : \2-Type 1 equivalent to A, but A itself does not belong to \2-Type 1.

How do you ensure that "every type in \Type0 which satisfies isSet is
equivalent to some type in \Set0"?  Is it just an axiom?

Also, since \Prop "has no predicative level", does this property
applied to \Prop imply that propositional resizing holds?