Re: [HoTT] Semantics of higher inductive types

[Andrew Swan <wakeli...@gmail.com>]

[-- Attachment #1.1: Type: text/plain, Size: 11441 bytes --]

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 <https://arxiv.org/pdf/1704.06911.pdf> 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 
> <http://drops.dagstuhl.de/opus/volltexte/2016/6564/pdf/LIPIcs-CSL-2016-24.pdf>
> )
>
> 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.
>> > >>
>> > >> --
>> > >> You received this message because you are subscribed to the Google 
>> Groups
>> > >> "Homotopy Type Theory" group.
>> > >> To unsubscribe from this group and stop receiving emails from it, 
>> send an
>> > >> email to HomotopyTypeThe...@googlegroups.com.
>> > >> For more options, visit https://groups.google.com/d/optout.
>> > >>
>> > >>
>> > >>
>> > >> --
>> > >> You received this message because you are subscribed to the Google 
>> Groups
>> > >> "Homotopy Type Theory" group.
>> > >> To unsubscribe from this group and stop receiving emails from it, 
>> send an
>> > >> email to HomotopyTypeThe...@googlegroups.com.
>> > >> For more options, visit https://groups.google.com/d/optout.
>> > >>
>> > >>
>> > >> --
>> > >> You received this message because you are subscribed to the Google 
>> Groups
>> > >> "Homotopy Type Theory" group.
>> > >> To unsubscribe from this group and stop receiving emails from it, 
>> send an
>> > >> email to HomotopyTypeThe...@googlegroups.com.
>> > >> For more options, visit https://groups.google.com/d/optout.
>> > >
>> > > --
>> > > You received this message because you are subscribed to the Google 
>> Groups "Homotopy Type Theory" group.
>> > > To unsubscribe from this group and stop receiving emails from it, 
>> send an email to HomotopyTypeThe...@googlegroups.com.
>> > > For more options, visit https://groups.google.com/d/optout.
>> >
>> > --
>> > You received this message because you are subscribed to the Google 
>> Groups "Homotopy Type Theory" group.
>> > To unsubscribe from this group and stop receiving emails from it, send 
>> an email to HomotopyTypeThe...@googlegroups.com.
>> > For more options, visit https://groups.google.com/d/optout.
>>
>

[-- Attachment #1.2: Type: text/html, Size: 16438 bytes --]