Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Valery Isaev <valery.isaev@gmail.com>
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 02:47:30 -0700	[thread overview]
Message-ID: <CAOvivQzAdt3Pj0TtoxZA29R9XyZm+xwz8-9AN5ap0iatf-=FLQ@mail.gmail.com> (raw)
In-Reply-To: <CAA520fsVx+GUpYN+yPesJ_1PwKV0MfU=GQD4ovpfHhy=Duj7yA@mail.gmail.com>

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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  reply	other threads:[~2019-08-10  9:47 UTC|newest]

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 [this message]
2019-08-10 12:30       ` Valery Isaev
2019-08-10 12:37       ` Valery Isaev
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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