Re: [HoTT] Semantics of higher inductive types

[Steve Awodey <awo...@cmu.edu>]

you mean the propositional truncation or suspension operations might lead to cardinals outside of a Grothendieck Universe?

> On Jun 1, 2017, at 11:38 AM, Michael Shulman <shu...@sandiego.edu> wrote:
> 
> Do we actually know that the Kan simplicial set model has a *universe
> closed under* even simple HITs?  It's not trivial because this would
> mean we could (say) propositionally truncate or suspend the generic
> small Kan fibration and get another *small* Kan fibration, whereas the
> base of these fibrations is not small, and fibrant replacement doesn't
> in general preserve smallness of fibrations with large base spaces.
> 
> (Also, the current L-S paper doesn't quite give a general syntactic
> scheme, only a general semantic framework with suggestive implications
> for the corresponding syntax.)
> 
> 
> 
> On Thu, Jun 1, 2017 at 8:30 AM, Steve Awodey <awo...@cmu.edu> wrote:
>> 
>> 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
>> 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.
>> 
>> Steve
>> 
>> 
>>  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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