From: Robin Adams <robin....@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: A small observation on cumulativity and the failure of initiality
Date: Fri, 13 Oct 2017 07:12:27 -0700 (PDT) [thread overview]
Message-ID: <e9e9e42b-d741-453d-9882-8379dc84aff2@googlegroups.com> (raw)
In-Reply-To: <F2106ADB-D78F-4228-B0A9-DBC6EC69E96A@princeton.edu>
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On Thursday, 12 October 2017 20:43:01 UTC+2, Dimitris Tsementzis wrote:
>
>
> But there are two distinct TT-model homomorphisms from C_TT to C_TT*, one
> which sends p(t0) to pq(t0) and one which sends p(t0) to qp(t0) (where
> p(t0) is regarded as an element of Tm_{C_TT} (empty, B(B(T0))), i.e. of the
> set of terms of B(B(T0)) in the empty context as they are interpreted in
> the term model C_TT).
>
There seems to be a gap in the proof here. In a term model we quotient out
by judgemental equality (correct me if this is wrong), so this step does
not give a contradiction: rather, we conclude |- qp(t0) = pq(t0) : B(B(T0))
I would expect this equality to hold in the examples you have in mind. If
T = Type_0, B(T) = Type_1, and p(t) = t -> t, then the equality is q(t ->
t) = q(t) -> q(t). This holds if TT* includes the rule
G |- A : Type_0 G |- B : Type_0
-------------------------------------------------------
G |- q(A -> B) = q(A) -> q(B) : Type_1
which I would expect you need in order for the category of TT-models to be
isomorphic to the category of TT*-models. So I suggest you also check the
proof of this isomorphism in more detail.
--
Robin
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prev parent reply other threads:[~2017-10-13 14:12 UTC|newest]
Thread overview: 42+ 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
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 [this message]
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