Re: [HoTT] Semantics of higher inductive types

[Michael Shulman <shu...@sandiego.edu>]

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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