Re: [HoTT] Semantics of higher inductive types

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

I'll be interested to see if you can make it work!

But I'd be much more interested if there is something that can be done
in a general class of models, rather than a particular one like
cubical or simplicial sets.

On Tue, Jun 6, 2017 at 4:03 AM, Andrew Swan <wakeli...@gmail.com> wrote:
> Actually, I've just noticed that doesn't quite work - I want to say that a
> map is a weak fibration if it has a (uniform choice of) diagonal fillers for
> lifting problems against generating cofibrations where the bottom map
> factors through the projection I x V -> V, but that doesn't seem to be
> cofibrantly generated. Maybe it's still possible to do something like
> fibrant replacement anyway.
>
> Andrew
>
>
> On Tuesday, 6 June 2017 11:19:37 UTC+2, Andrew Swan wrote:
>>
>> I've been thinking a bit about abstract ways of looking at the HITs in
>> cubical type theory, and I don't have a complete proof, but I think actually
>> the same sort of thing should work for simplicial sets.
>>
>> We already know that the fibrations in the usual model structure on
>> simplicial sets can be defined as maps with the rlp against the pushout
>> product of generating cofibrations with interval endpoint inclusions (in
>> Christian's new paper on model structures he cites for this result Chapter
>> IV, section 2 of P. Gabriel and M. Zisman. Calculus of fractions and
>> homotopy theory, but I'm not familiar with the proof myself).
>>
>> Now a generating trivial cofibration is the pushout product of a
>> generating cofibration with endpoint inclusion, so its codomain is of the
>> form I x V, where V is the codomain of the generating cofibration (which for
>> cubical sets and simplicial sets is representable). Then we get another map
>> by composing with projection I x V -> V, which is a retract of the
>> generating trivial cofibration and so also a trivial cofibration. If a map
>> has the rlp against all such maps, then call it a weak fibration. Then I
>> think the resulting awfs of "weak fibrant replacement" should be stable
>> under pullback (although of course, the right maps in the factorisation are
>> only weak fibrations, not fibrations in general).
>>
>> Then eg for propositional truncation, construct the "fibrant truncation"
>> monad by the coproduct of truncation monad with weak fibrant replacement. In
>> general, given a map X -> Y, the map ||X|| -> Y will only be a weak
>> fibration, but if X -> Y is fibration then I think the map ||X|| -> Y should
>> be also. I think the way to formulate this would be as a distributive law -
>> the fibrant replacement monad distributes over the (truncation + weak
>> fibrant replacement) monad. It looks to me like the same thing that works in
>> cubical sets should also work here - first define a "box flattening"
>> operation for any fibration (i.e. the operation labelled as "forward" in
>> Thierry's note), then show that this operation lifts through the HIT
>> constructors to give a box flattening operation on the HIT, then show that
>> in general weak fibration plus box flattening implies fibration, (Maybe one
>> way to do this would be to note that the cubical set argument is mostly done
>> internally in cubical type theory, and simplicial sets model cubical type
>> theory by Orton & Pitts, Axioms for Modelling Cubical Type Theory in a
>> Topos)
>>
>> Best,
>> Andrew
>>
>>
>>
>> On Thursday, 1 June 2017 18:08:58 UTC+2, Peter LeFanu Lumsdaine wrote:
>>>
>>> On Thu, Jun 1, 2017 at 6:56 PM, Steve Awodey <awo...@cmu.edu> 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 <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 <Thier...@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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