Discussion of Homotopy Type Theory and Univalent Foundations
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From: Valery Isaev <valery.isaev@gmail.com>
To: Michael Shulman <shulman@sandiego.edu>
Cc: Nicolai Kraus <nicolai.kraus@gmail.com>,
	 Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] New theorem prover Arend is released
Date: Sat, 10 Aug 2019 15:37:48 +0300
Message-ID: <CAA520fuS3O=yPtzHUkTxStiadxP=Sr4+jaBpZQuA93LXkWfzTg@mail.gmail.com> (raw)
In-Reply-To: <CAOvivQzAdt3Pj0TtoxZA29R9XyZm+xwz8-9AN5ap0iatf-=FLQ@mail.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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  parent reply index

Thread overview: 20+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-08-06 22:16 Валерий Исаев
2019-08-07 15:01 ` Andrej Bauer
2019-08-07 22:13 ` Nicolai Kraus
2019-08-08  9:55   ` Valery Isaev
2019-08-10  9:47     ` Michael Shulman
2019-08-10 12:30       ` Valery Isaev
2019-08-10 12:37       ` Valery Isaev [this message]
2019-08-08 12:20 ` Jon Sterling
2019-08-08 12:29   ` Bas Spitters
2019-08-08 14:44     ` Valery Isaev
2019-08-08 15:11       ` Jon Sterling
2019-08-08 15:22         ` Valery Isaev
2019-08-10  9:42           ` Michael Shulman
2019-08-10 12:24             ` Valery Isaev
2019-08-10 23:37               ` Michael Shulman
2019-08-11 10:46                 ` Valery Isaev
2019-08-11 12:39                   ` Michael Shulman
2019-08-11 16:55                     ` Michael Shulman
2019-08-12 14:44                       ` Daniel R. Grayson
2019-08-12 17:32                         ` Michael Shulman

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