Re: [HoTT] New theorem prover Arend is released

[Valery Isaev <valery.isaev@gmail.com>]

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Yes, propositional resizing holds. \Prop classifies all subobjects.

Regards,
Valery Isaev


сб, 10 авг. 2019 г. в 12:47, Michael Shulman <shulman@sandiego.edu>:

> 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?
>

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