From: Michael Shulman <shu...@sandiego.edu>
To: Thierry Coquand <Thierry...@cse.gu.se>
Cc: homotopy Type Theory <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] Semantics of higher inductive types
Date: Thu, 1 Jun 2017 07:43:27 -0700 [thread overview]
Message-ID: <CAOvivQwyyUD6TWsp_9X+Df3vwGRBFh_-7Sg9FndCv3AafH9t-A@mail.gmail.com> (raw)
In-Reply-To: <1128BE39-BBC4-4DC6-8792-20134A6CAECD@chalmers.se>
Yes, this is something rather mysterious to me. There may be
something special about the cubical model that enables a sort of
"horizontal/vertical" decomposition of fibrations. Peter and I
thought for a little while about whether there is some similar sort of
"fiberwise fibrant replacement" that would work in a general model,
but weren't able to come up with anything; however, perhaps we just
weren't clever enough. I do think it is very important not to be
limited to one particular model.
On Thu, Jun 1, 2017 at 7:23 AM, Thierry Coquand
<Thierry...@cse.gu.se> wrote:
> If we are only interested in providing one -particular- model of HITs,
> the paper
> on cubical type theory describes a way to interpret HIT together with a
> univalent
> universe which is stable by HIT operations. This gives in particular the
> consistency
> and the proof theoretic power of this extension of type theory.
>
> The approach uses an operation of “flattening an open box”, which solves
> in
> this case the issue of interpreting HIT with parameters (such as
> propositional
> truncation or suspension) without any coherence issue.
> Since the syntax used in this paper is so close to the semantics, we
> limited
> ourselves to a syntactical presentation of this interpretation. But it can
> directly
> be transformed to a semantical interpretation, as explained in the following
> note
> (which also incorporates a nice simplification of the operation of
> flattering
> an open box noticed by my coauthors). I also try to make more explicit in
> the note
> what is the problem solved by the “flattening boxes” method.
>
> Only the cases of the spheres and propositional truncation are described,
> but one
> would expect the method to generalise to other HITs covered e.g. in the HoTT
> book.
>
> On 25 May 2017, at 20:25, Michael Shulman <shu...@sandiego.edu> wrote:
>
> The following long-awaited paper is now available:
>
> Semantics of higher inductive types
> Peter LeFanu Lumsdaine, Mike Shulman
> https://arxiv.org/abs/1705.07088
>
> From the abstract:
>
> We introduce the notion of *cell monad with parameters*: a
> semantically-defined scheme for specifying homotopically well-behaved
> notions of structure. We then show that any suitable model category
> has *weakly stable typal initial algebras* for any cell monad with
> parameters. When combined with the local universes construction to
> obtain strict stability, this specializes to give models of specific
> higher inductive types, including spheres, the torus, pushout types,
> truncations, the James construction, and general localisations.
>
> Our results apply in any sufficiently nice Quillen model category,
> including any right proper simplicial Cisinski model category (such as
> simplicial sets) and any locally presentable locally cartesian closed
> category (such as sets) with its trivial model structure. In
> particular, any locally presentable locally cartesian closed
> (∞,1)-category is presented by some model category to which our
> results apply.
>
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next prev parent reply other threads:[~2017-06-01 14:43 UTC|newest]
Thread overview: 25+ messages / expand[flat|nested] mbox.gz Atom feed top
2017-05-25 18:25 Michael Shulman
2017-05-26 0:17 ` [HoTT] " Emily Riehl
2017-06-01 14:23 ` Thierry Coquand
2017-06-01 14:43 ` Michael Shulman [this message]
2017-06-01 15:30 ` Steve Awodey
2017-06-01 15:38 ` Michael Shulman
2017-06-01 15:56 ` Steve Awodey
2017-06-01 16:08 ` Peter LeFanu Lumsdaine
2017-06-06 9:19 ` Andrew Swan
2017-06-06 10:03 ` Andrew Swan
2017-06-06 13:35 ` Michael Shulman
2017-06-06 16:22 ` Andrew Swan
2017-06-06 19:36 ` Michael Shulman
2017-06-06 20:59 ` Andrew Swan
2017-06-07 9:40 ` Peter LeFanu Lumsdaine
2017-06-07 9:57 ` Thierry Coquand
[not found] ` <ed7ad345-85e4-4536-86d7-a57fbe3313fe@googlegroups.com>
2017-06-07 23:06 ` Michael Shulman
2017-06-08 6:35 ` Andrew Swan
2018-09-14 11:15 ` Thierry Coquand
2018-09-14 14:16 ` Andrew Swan
2018-10-01 13:02 ` Thierry Coquand
2018-11-10 15:52 ` Anders Mörtberg
2018-11-10 18:21 ` Gabriel Scherer
2017-06-08 4:57 ` CARLOS MANUEL MANZUETA
2018-11-12 12:30 ` Ali Caglayan
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