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: Jon Sterling <jon@jonmsterling.com>,
	 "HomotopyTypeTheory@googlegroups.com"
	<homotopytypetheory@googlegroups.com>
Subject: Re: [HoTT] New theorem prover Arend is released
Date: Sat, 10 Aug 2019 15:24:50 +0300
Message-ID: <CAA520ftTacC4iegu2UM887nbyJWQTByMKhrcsftKPXCadH06kQ@mail.gmail.com> (raw)
In-Reply-To: <CAOvivQzzLHnq+bXBzkFv3tST0GEUo4f2zmdj025LTaq+EMB7CQ@mail.gmail.com>

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The document is slightly outdated.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 <shulman@sandiego.edu>:

> 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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  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
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 [this message]
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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