From: Eric Finster <ericf...@gmail.com>
To: Jasper Hugunin <jas...@cs.washington.edu>
Cc: HomotopyT...@googlegroups.com
Subject: Re: [HoTT] Re: Where is the problem with initiality?
Date: Sat, 2 Jun 2018 00:00:36 +0200 [thread overview]
Message-ID: <CAGYJgtbzGN7fK5zjwRY+Zdd-AYrqSq+Yds7fE1ks3fVRSQd-BA@mail.gmail.com> (raw)
In-Reply-To: <CAGTS-a8gXmo4=JuHAFLhqVSvfZjYemgtbpC+79EVY=L2UmTgEw@mail.gmail.com>
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> Type checking must be performed in any case, at some level. If we use
>> intrinsic embedded syntaxes, then the implementation of the metalanguage
>> (in a meta-metalanguage) does the type checking. Hence, if we intend to
>> formalize a syntax of type theory which is intended as a metatheoretical
>> setting for mathematics, then I think decidability is quite important.
>>
>
> If we don't intend our type theory to be used as a metatheoretical
> setting for mathematics, it seems interesting to consider for example a
> type theory with a family of base types indexed by the circle, the syntax
> of which I think would not form a set.
> That is, if we have a way to represent dependent type theory in Coq, what
> happens if we add a constructor `a_circle_of_types : circle -> type` to our
> syntax (for a postulated circle, or HIT)? This should be consistent, to add
> a family of uninterpreted types, but I believe equality is no longer
> decidable (since equality of circle is not decidable).
>
>
Yes, exactly, I have the same objection.
Even if the conjecture Mike alluded to is true, that the free higher
category with families is equivalent to a 1-category, I can certainly
*force* it to be false by considering the "free higher category with
families with some random extra 2-cell".
Don't we expect this thing to *also* have some kind of "syntax" presenting
it?
> - Jasper Hugunin
>
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next prev parent reply other threads:[~2018-06-01 22:02 UTC|newest]
Thread overview: 57+ messages / expand[flat|nested] mbox.gz Atom feed top
2018-05-22 5:46 Michael Shulman
2018-05-22 16:47 ` Ambrus Kaposi
2018-05-23 16:26 ` [HoTT] " Thorsten Altenkirch
2018-05-24 5:52 ` Michael Shulman
2018-05-24 8:11 ` Thorsten Altenkirch
2018-05-24 9:53 ` Ambrus Kaposi
2018-05-24 17:26 ` Michael Shulman
2018-05-26 9:21 ` Thomas Streicher
2018-05-26 11:47 ` Michael Shulman
2018-05-26 16:47 ` stre...
2018-05-27 5:14 ` Bas Spitters
2018-05-28 22:39 ` Michael Shulman
2018-05-29 9:15 ` [HoTT] " Thorsten Altenkirch
2018-05-29 15:15 ` Michael Shulman
2018-05-30 9:33 ` Thomas Streicher
2018-05-30 9:37 ` Thorsten Altenkirch
2018-05-30 10:10 ` Thomas Streicher
2018-05-30 12:08 ` Thorsten Altenkirch
2018-05-30 13:40 ` Thomas Streicher
2018-05-30 14:38 ` Thorsten Altenkirch
2018-05-30 10:53 ` Alexander Kurz
2018-05-30 12:05 ` Thorsten Altenkirch
2018-05-30 19:07 ` Michael Shulman
2018-05-31 10:06 ` Thorsten Altenkirch
2018-05-31 11:05 ` Michael Shulman
2018-05-31 19:02 ` Alexander Kurz
2018-06-01 9:55 ` Martin Escardo
2018-06-01 17:07 ` Martín Hötzel Escardó
2018-06-01 17:43 ` Eric Finster
2018-06-01 19:55 ` Martín Hötzel Escardó
2018-06-01 20:59 ` András Kovács
2018-06-01 21:06 ` Martín Hötzel Escardó
2018-06-01 21:23 ` Michael Shulman
2018-06-01 21:53 ` Eric Finster
2018-06-01 22:09 ` Michael Shulman
2018-06-02 15:06 ` Eric Finster
2018-06-05 20:04 ` Michael Shulman
2018-06-02 5:13 ` Thomas Streicher
2018-06-01 21:52 ` Jasper Hugunin
2018-06-01 22:00 ` Eric Finster [this message]
2018-06-01 21:27 ` Matt Oliveri
2018-06-02 5:21 ` Ambrus Kaposi
2018-06-02 6:01 ` Thomas Streicher
2018-06-02 14:35 ` Thorsten Altenkirch
2018-05-30 13:30 ` Jon Sterling
2018-06-05 7:52 ` Andrej Bauer
2018-06-05 8:37 ` David Roberts
2018-06-05 9:46 ` Gabriel Scherer
2018-06-05 22:19 ` Martín Hötzel Escardó
2018-06-05 22:54 ` Martín Hötzel Escardó
2018-06-05 22:12 ` Richard Williamson
2018-06-06 15:05 ` Thorsten Altenkirch
2018-06-06 19:25 ` Richard Williamson
2018-05-29 14:00 ` Jon Sterling
2018-05-30 22:35 ` Michael Shulman
2018-05-31 10:48 ` Martín Hötzel Escardó
2018-05-31 11:09 ` Michael Shulman
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