On Thu, Jun 1, 2017 at 6:56 PM, Steve Awodey wrote:
>
> you mean the propositional truncation or suspension operations might lead
to cardinals outside of a Grothendieck Universe?
Exactly, yes. There’s no reason I know of to think they *need* to, but
with the construction of Mike’s and my paper, they do. And adding stronger
conditions on the cardinal used won’t help. The problem is that one takes
a fibrant replacement to go from the “pre-suspension” to the suspension
(more precisely: a (TC,F) factorisation, to go from the universal family of
pre-suspensions to the universal family of suspensions); and fibrant
replacement blows up the fibers to be the size of the *base* of the
family. So the pre-suspension is small, but the suspension — although
essentially small — ends up as large as the universe one’s using.
So here’s a very precise problem which is as far as I know open:
(*) Construct an operation Σ : U –> U, where U is Voevodsky’s universe,
together with appropriate maps N, S : Û –> Û over Σ, and a homotopy m from
N to S over Σ, which together exhibit U as “closed under suspension”.
I asked a related question on mathoverflow a couple of years ago:
https://mathoverflow.net/questions/219588/pullback-stable-model-of-fibrewise-suspension-of-fibrations-in-simplicial-sets
David White suggested he could see an answer to that question (which would
probably also answer (*) here) based on the comments by Karol Szumiło and
Tyler Lawson, using the adjunction with Top, but I wasn’t quite able to
piece it together.
–p.
>
> > On Jun 1, 2017, at 11:38 AM, Michael Shulman
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 wrote:
> >>
> >> On Jun 1, 2017, at 10:23 AM, Thierry Coquand
> >> 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 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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