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 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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