Discussion of Homotopy Type Theory and Univalent Foundations
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From: Steve Awodey <awo...@cmu.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 11:30:27 -0400	[thread overview]
Message-ID: <292DED31-6CB3-49A1-9128-5AFD04B9C2F2@cmu.edu> (raw)
In-Reply-To: <1128BE39-BBC4-4DC6-8792-20134A6CAECD@chalmers.se>

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> On Jun 1, 2017, at 10:23 AM, Thierry Coquand <Thierry...@cse.gu.se> wrote:
>   If we are only interested in providing one -particular- model of HITs,      the paper <https://arxiv.org/abs/1611.02108> 
> 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.

but the Kan simplicial set model already does this — right?
don’t get me wrong — I love the cubes, and they have lots of nice properties for models of HoTT 
— but there was never really a question of the consistency or coherence of simple HITs like propositional truncation or suspension.

the advance in the L-S paper is to give a general scheme for defining HITs syntactically 
(a definition, if you like, of what a HIT is, rather than a family of examples), 
and then a general description of the semantics of these, 
in a range of models of the basic 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 <http://www.cse.chalmers.se/~coquand/hit3.pdf> 
> (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 <mailto: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 <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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  parent reply	other threads:[~2017-06-01 15:30 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
2017-06-01 15:30   ` Steve Awodey [this message]
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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