Discussion of Homotopy Type Theory and Univalent Foundations
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From: Nicolai Kraus <nicolai.kraus@gmail.com>
To: Valery Isaev <valery.isaev@gmail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Re: Why do we need judgmental equality?
Date: Sat, 9 Feb 2019 01:41:26 +0000	[thread overview]
Message-ID: <CA+AZBBq2F=WFLYaCECzxBnZ7QS7HWJukA5rsQPfo30SHXiAGrA@mail.gmail.com> (raw)
In-Reply-To: <CAA520ft7z1MJjLiGo+xC62bjpypY7BViyHc-hG9HUUJbkEdTLg@mail.gmail.com>

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Hi Valery,

On Fri, Feb 8, 2019 at 11:32 PM Valery Isaev <valery.isaev@gmail.com> wrote:

> Now, what do I mean when I say that type theories T and Q(T) are
> equivalent? I won't give here the formal definition
>

Would it be correct to say that T is a conservative extension of T, in the
sense of Martin Hofmann's thesis? Your description sounds a bit like this,
or do you have something different in mind?
Nicolai



> , but the idea is that Q(T) can be interpreted in T and, for every type A
> of T, there is a type in Q(T) equivalent to A in T and the same is true for
> terms. This implies that every statement (i.e., type) of Q(T) is provable
> in Q(T) if and only if it is provable in T and every statement of T has an
> equivalent statement in Q(T), so the theories are "logically equivalent".
> Moreover, equivalent theories have equivalent (in an appropriate
> homotopical sense) categories of models.
>
> Regards,
> Valery Isaev
>
>
> сб, 9 февр. 2019 г. в 00:19, Martín Hötzel Escardó <
> escardo.martin@gmail.com>:
>
>> I would also like to know an answer to this question. It is true that
>> dependent type theories have been designed using definitional equality.
>>
>> But why would anybody say that there is a *need* for that? Is it
>> impossible to define a sensible dependent type theory (say for the purpose
>> of serving as a foundation for univalent mathematics) that doesn't mention
>> anything like definitional equality? If not, why not? And notice that I am
>> not talking about *usability* of a proof assistant such as the *existing*
>> ones (say Coq, Agda, Lean) were definitional equalities to be removed. I
>> don't care if such hypothetical proof assistants would be impossibly
>> difficult to use for a dependent type theory lacking definitional
>> equalities (if such a thing exists).
>>
>> The question asked by Felix is a very sensible one: why is it claimed
>> that definitional equalities are essential to dependent type theories?
>>
>> (I do understand that they are used to compute, and so if you are
>> interested in constructive mathematics (like I am) then they are useful.
>> But, again, in principle we can think of a dependent type theory with no
>> definitional equalities and instead an existence property like e.g. in
>> Lambek and Scott's "introduction to higher-order categorical logic". And
>> like was discussed in a relatively recent message by Thierry Coquand in
>> this list,)
>>
>> Martin
>>
>>
>> On Wednesday, 30 January 2019 11:54:07 UTC, Felix Rech wrote:
>>>
>>> In section 1.1 of the HoTT book it says "In type theory there is also a
>>> need for an equality judgment." Currently it seems to me like one could, in
>>> principle, replace substitution along judgmental equality with explicit
>>> transports if one added a few sensible rules to the type theory. Is there a
>>> fundamental reason why the equality judgment is still necessary?
>>>
>>> Thanks,
>>> Felix Rech
>>>
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  reply	other threads:[~2019-02-09  1:41 UTC|newest]

Thread overview: 71+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-01-30 11:54 [HoTT] " Felix Rech
2019-02-05 23:00 ` [HoTT] " Matt Oliveri
2019-02-06  4:13   ` Anders Mörtberg
2019-02-09 11:55     ` Felix Rech
2019-02-16 15:59     ` Thorsten Altenkirch
2019-02-17  1:25       ` Michael Shulman
2019-02-17  7:56         ` Thorsten Altenkirch
2019-02-17  9:14           ` Matt Oliveri
2019-02-17  9:18           ` Michael Shulman
2019-02-17 10:52             ` Thorsten Altenkirch
2019-02-17 11:35               ` streicher
2019-02-17 11:44                 ` Thorsten Altenkirch
2019-02-17 14:24                   ` Bas Spitters
2019-02-17 19:36                   ` Thomas Streicher
2019-02-17 21:41                     ` Thorsten Altenkirch
2019-02-17 12:08             ` Matt Oliveri
2019-02-17 12:13               ` Matt Oliveri
2019-02-20  0:22               ` Michael Shulman
2019-02-17 14:22           ` [Agda] " Andreas Abel
2019-02-17  9:05         ` Matt Oliveri
2019-02-17 13:29         ` Nicolai Kraus
2019-02-08 21:19 ` Martín Hötzel Escardó
2019-02-08 23:31   ` Valery Isaev
2019-02-09  1:41     ` Nicolai Kraus [this message]
2019-02-09  8:04       ` Valery Isaev
2019-02-09  1:58     ` Jon Sterling
2019-02-09  8:16       ` Valery Isaev
2019-02-09  1:30   ` Nicolai Kraus
2019-02-09 11:38   ` Thomas Streicher
2019-02-09 13:29     ` Thorsten Altenkirch
2019-02-09 13:40       ` Théo Winterhalter
2019-02-09 11:57   ` Felix Rech
2019-02-09 12:39     ` Martín Hötzel Escardó
2019-02-11  6:58     ` Matt Oliveri
2019-02-18 17:37   ` Martín Hötzel Escardó
2019-02-18 19:22     ` Licata, Dan
2019-02-18 20:23       ` Martín Hötzel Escardó
2019-02-09 11:53 ` Felix Rech
2019-02-09 14:04   ` Nicolai Kraus
2019-02-09 14:26     ` Gabriel Scherer
2019-02-09 14:44     ` Jon Sterling
2019-02-09 20:34       ` Michael Shulman
2019-02-11 12:17         ` Matt Oliveri
2019-02-11 13:04           ` Michael Shulman
2019-02-11 15:09             ` Matt Oliveri
2019-02-11 17:20               ` Michael Shulman
2019-02-11 18:17                 ` Thorsten Altenkirch
2019-02-11 18:45                   ` Alexander Kurz
2019-02-11 22:58                     ` Thorsten Altenkirch
2019-02-12  2:09                       ` Jacques Carette
2019-02-12 11:03                   ` Matt Oliveri
2019-02-12 15:36                     ` Thorsten Altenkirch
2019-02-12 15:59                       ` Matt Oliveri
2019-02-11 19:27                 ` Matt Oliveri
2019-02-11 21:49                   ` Michael Shulman
2019-02-12  9:01                     ` Matt Oliveri
2019-02-12 17:54                       ` Michael Shulman
2019-02-13  6:37                         ` Matt Oliveri
2019-02-13 10:01                           ` Ansten Mørch Klev
2019-02-11 20:11                 ` Matt Oliveri
2019-02-11  8:23       ` Matt Oliveri
2019-02-11 13:03         ` Jon Sterling
2019-02-11 13:22           ` Matt Oliveri
2019-02-11 13:37             ` Jon Sterling
2019-02-11  6:51   ` Matt Oliveri
2019-02-09 12:30 ` [HoTT] " Thorsten Altenkirch
2019-02-11  7:01   ` Matt Oliveri
2019-02-11  8:04     ` Valery Isaev
2019-02-11  8:28       ` Matt Oliveri
2019-02-11  8:37         ` Matt Oliveri
2019-02-11  9:32           ` Rafaël Bocquet

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