From: Matt Oliveri <atm...@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: Re: [HoTT] A small observation on cumulativity and the failure of initiality
Date: Fri, 13 Oct 2017 09:36:08 -0700 (PDT) [thread overview]
Message-ID: <8dce09b8-a4d4-48e6-a6bb-85a45421c88f@googlegroups.com> (raw)
In-Reply-To: <CAOvivQxC5J63VUzj2sUTRGkpCGG=KvQj+P2Ez8HRzj1BQrs=uw@mail.gmail.com>
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Maybe there are more ways to formulate initiality than anyone realized.
On Friday, October 13, 2017 at 12:23:37 PM UTC-4, Michael Shulman wrote:
>
> To my understanding, there are two different essentially-algebraic
> theories involved: type theory (or more precisely, its derivations) is
> essentially by its definition the initial object in one of them, but
> the one we're interested in (the appropriate sort of category) is a
> different theory with different operations. For instance, in a
> category we have composition as a basic operation, but in type theory
> composition is admissible rather than primitive. So there is always
> something to prove in relating the two, even when we know that both
> are essentially-algebraic.
>
> But my main point was that the essentially-algebraic theory to which
> the syntax belongs consists of derivations rather than terms, so it
> can be essentially-algebraic even if terms don't have unique types.
>
> On Fri, Oct 13, 2017 at 9:17 AM, Steve Awodey <stev...@gmail.com
> <javascript:>> wrote:
> >
> > On Oct 13, 2017, at 11:50 AM, Michael Shulman <shu...@sandiego.edu
> <javascript:>> wrote:
> >
> > On Thu, Oct 12, 2017 at 5:09 PM, Steve Awodey <stev...@gmail.com
> <javascript:>> wrote:
> >
> > in order to have an (essentially) algebraic notion of type theory, which
> > will then automatically have initial algebras, etc., one should have the
> > typing of terms be an operation, so that every term has a unique type.
> In
> > particular, your (R1) violates this. Cumulativity is a practical
> convenience
> > that can be added to the system by some syntactic conventions, but the
> real
> > system should have unique typing of terms.
>
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next prev parent reply other threads:[~2017-10-13 16:36 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 [this message]
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
[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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