From: Andrew Polonsky <andrew....@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] A small observation on cumulativity and the failure of initiality
Date: Sun, 15 Oct 2017 22:30:30 -0700 (PDT) [thread overview]
Message-ID: <102b8bf4-440b-40b2-8cec-4f7bf3653518@googlegroups.com> (raw)
In-Reply-To: <20171015092617.GA684@mathematik.tu-darmstadt.de>
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In the set-thoretic model -- which is the simplest, most "standard" model
one can think of -- the universes are indeed cumulative, and "coherence" is
just the observation that conversion (definitional equality) of raw terms
is preserved by the interpretation function.
In categorical models, strict equality between interpreted objects is
perhaps a more subtle concept.
Still, it would seem to be a natural requirement of *ANY* class or flavor
of semantics, that expressions which are definitionally equal in the object
language, are evaluated to entities which are again (judgmentally) equal on
the meta-level.
This really amounts to nothing more than asking that the semantics in
question actually validate the conversion rule -- one of the structural
rules of the theory.
Best,
Andrew
On Sunday, October 15, 2017 at 11:26:22 AM UTC+2, Thomas Streicher wrote:
>
> > A quite detailed interpretation of type theory in set theory is
> presented
> >
> > in the paper of Peter Aczel "On Relating Type Theories and Set
> Theories".
> >
> > On can define directly the interpretation of lambda terms (provided that
> abstraction is
> >
> > typed), and using set theoretic coding, one can use a global application
> operation
> >
> > (so that the interpretation is total). One can then checked by induction
> on derivations
> >
> > that all judgements are valid for this interpretation.
>
> Thanks, Thierry, for pointing this out. But Peter's method does not
> extend to arbitrary split models of dependent type theory. What Peter
> uses here intrinsically is that everything is a set since otherwise he
> couldn't interpret type theoretic quantification.
> My interpretation got partial on pseudoexpressions since pseudoterms
> can't be understood as pseudo-type-expressions.
>
> I don't see how Peter's method extends to interpretation of
> realizability or (pre)sheaf models of dependent type theories not to
> speak of arbitrary contextual cats or other split categorical models
> of dependent type theories.
>
> Thomas
>
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next prev parent reply other threads:[~2017-10-16 5:30 UTC|newest]
Thread overview: 47+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-10-12 18:43 Dimitris Tsementzis
2017-10-12 22:31 ` [HoTT] " Michael Shulman
2017-10-13 4:30 ` Dimitris Tsementzis
2017-10-13 15:41 ` Michael Shulman
2017-10-13 21:51 ` Dimitris Tsementzis
2017-10-13 0:09 ` Steve Awodey
2017-10-13 0:44 ` Alexander Altman
2017-10-13 15:50 ` Michael Shulman
2017-10-13 16:17 ` Steve Awodey
2017-10-13 16:23 ` Michael Shulman
2017-10-13 16:36 ` Matt Oliveri
2017-10-14 14:56 ` Gabriel Scherer
2017-10-15 7:45 ` Thomas Streicher
2017-10-15 8:37 ` Thierry Coquand
2017-10-15 9:26 ` Thomas Streicher
2017-10-16 5:30 ` Andrew Polonsky [this message]
2017-10-15 10:12 ` Michael Shulman
2017-10-15 13:57 ` Thomas Streicher
2017-10-15 14:53 ` Michael Shulman
2017-10-15 16:00 ` Michael Shulman
2017-10-15 21:00 ` Matt Oliveri
2017-10-16 5:09 ` Michael Shulman
2017-10-16 12:30 ` Neel Krishnaswami
2017-10-16 13:35 ` Matt Oliveri
2017-10-16 15:00 ` Michael Shulman
2017-10-16 16:34 ` Matt Oliveri
2017-10-16 13:45 ` Matt Oliveri
2017-10-16 15:05 ` Michael Shulman
2017-10-16 16:20 ` Matt Oliveri
2017-10-16 16:37 ` Michael Shulman
2017-10-16 10:01 ` Thomas Streicher
2017-10-15 20:06 ` Matt Oliveri
2017-10-13 8:03 ` Peter LeFanu Lumsdaine
2017-10-13 8:10 ` Thomas Streicher
2017-10-14 7:33 ` Thorsten Altenkirch
2017-10-14 9:37 ` Andrej Bauer
2017-10-14 9:52 ` Thomas Streicher
2017-10-14 10:51 ` SV: " Erik Palmgren
2017-10-15 23:42 ` Andrej Bauer
2017-10-15 10:42 ` Thorsten Altenkirch
2017-10-13 22:05 ` Dimitris Tsementzis
2017-10-13 14:12 ` Robin Adams
[not found] <B14E498C-FA19-41D2-B196-42FAF85F8CD8@princeton.edu>
2017-10-14 9:55 ` [HoTT] " Alexander Altman
2017-10-16 10:21 Thorsten Altenkirch
2017-10-16 10:42 ` Andrew Polonsky
2017-10-16 14:12 ` Thorsten Altenkirch
2017-10-16 10:21 Thorsten Altenkirch
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