Discussion of Homotopy Type Theory and Univalent Foundations
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From: Andreas Nuyts <andreasnuyts@gmail.com>
To: Nils Anders Danielsson <nad@cse.gu.se>,
	Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Injective types
Date: Tue, 7 May 2019 15:51:44 +0200
Message-ID: <49d95599-153c-d641-4ae9-a3f7c02b1cc3@gmail.com> (raw)
In-Reply-To: <5fa99a80-56ed-abc5-d64f-876c192bca6a@cse.gu.se>

Even with cumulativity, that sounds suspicious, because cumulativity of 
U in a bigger universe V does not obviously give you a map

(A B : U) -> Id V A B -> Id U A B.

The J-rule does not allow you to build this map.

On 7/05/19 14:42, Nils Anders Danielsson wrote:
> On 02/05/2019 22.46, escardo.martin@gmail.com wrote:
>> I can confirm from a 26k line Agda development (with comments and
>> repeated blank lines removed in this counting of the number of lines)
>> that not once did I need to embed a universe into a larger universe,
>> except when I wanted to state the theorem that any universe is a
>> retract of any larger universe if one assumes the propositional
>> resizing axiom (any proposition in a universe U has an equivalent copy
>> in any universe V). So I would say that such situations are *extremely
>> rare* in practice.
>
> I once wrote some code where I made use of univalence at three different
> levels. Does anyone know if one can prove that univalence at one level
> implies univalence at lower levels, without making use of cumulativity?
>

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  reply index

Thread overview: 11+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-04-30 23:05 escardo.martin
2019-05-01  2:50 ` Michael Shulman
2019-05-01  6:25   ` escardo.martin
2019-05-01 16:55     ` Michael Shulman
2019-05-02 20:46       ` escardo.martin
2019-05-03 11:45         ` Michael Shulman
2019-05-03 13:25           ` Kenji Maillard
2019-05-03 18:23             ` Thierry Coquand
2019-05-07 12:42         ` Nils Anders Danielsson
2019-05-07 13:51           ` Andreas Nuyts [this message]
2019-05-07 22:06             ` Martín Hötzel Escardó

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Discussion of Homotopy Type Theory and Univalent Foundations

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