I don't know how general this is exactly in practice, = but I think it should work in the setting that appears in van de Berg, Frum= in=C2=A0A homotopy-theoret= ic model of function extensionality in the effective topos, which regar= dless of the title is not just the effective topos, but any topos together = with a interval object with connections, a dominance satisfying certain con= ditions, with fibrations defined as maps with the rlp against pushout produ= ct of endpoint inclusions and elements of the dominance (& in addition = there should be some more conditions to ensure that free monads on pointed = endofunctors exist).

I'm a bit more c= onfident that it works now. The class of weak fibrations is not cofibrantly= generated in the usual sense (as I claimed in the first post), but they ar= e in the more general sense by Christian Sattler in section 6 of=C2=A0The Equivalence Extension Property = and Model Structures. Then a version of step 1 of the small object argu= ment applies to Christian's definition, which gives a pointed endofunct= or whose algebras are the weak fibrations. The same technique can also be u= sed to describe "box flattening" (which should probably be called= something else, like "cylinder flattening" in the general settin= g).

Andrew

On Tuesday, 6 Jun= e 2017 15:35:36 UTC+2, Michael Shulman wrote:
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 d= one
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 <wake...@gmail.com> wr= ote:
> Actually, I've just noticed that doesn't quite work - I wa= nt to say that a
> map is a weak fibration if it has a (uniform choice of) diagonal f= illers 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 somethi= ng 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, bu= t I think actually
>> the same sort of thing should work for simplicial sets.
>>
>> We already know that the fibrations in the usual model structu= re on
>> simplicial sets can be defined as maps with the rlp against th= e pushout
>> product of generating cofibrations with interval endpoint incl= usions (in
>> Christian's new paper on model structures he cites for thi= s result Chapter
>> IV, section 2 of P. Gabriel and M. Zisman. Calculus of fractio= ns and
>> homotopy theory, but I'm not familiar with the proof mysel= f).
>>
>> Now a generating trivial cofibration is the pushout product of= a
>> generating cofibration with endpoint inclusion, so its codomai= n is of the
>> form I x V, where V is the codomain of the generating cofibrat= ion (which for
>> cubical sets and simplicial sets is representable). Then we ge= t another map
>> by composing with projection I x V -> V, which is a retract= of the
>> generating trivial cofibration and so also a trivial cofibrati= on. If a map
>> has the rlp against all such maps, then call it a weak fibrati= on. Then I
>> think the resulting awfs of "weak fibrant replacement&quo= t; should be stable
>> under pullback (although of course, the right maps in the fact= orisation are
>> only weak fibrations, not fibrations in general).
>>
>> Then eg for propositional truncation, construct the "fibr= ant truncation"
>> monad by the coproduct of truncation monad with weak fibrant r= eplacement. In
>> general, given a map X -> Y, the map ||X|| -> Y will onl= y 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 distr= ibutive 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 &q= uot;forward" in
>> Thierry's note), then show that this operation lifts throu= gh the HIT
>> constructors to give a box flattening operation on the HIT, th= en show that
>> in general weak fibration plus box flattening implies fibratio= n, (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 c= ubical 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 Lumsdain= e wrote:
>>>
>>> On Thu, Jun 1, 2017 at 6:56 PM, Steve Awodey <awo...= @cmu.edu> wrote:
>>> >
>>> > you mean the propositional truncation or suspension o= perations might
>>> > lead to cardinals outside of a Grothendieck Universe?
>>>
>>> Exactly, yes. =C2=A0There=E2=80=99s no reason I know of to= think they *need* to, but
>>> with the construction of Mike=E2=80=99s and my paper, they= do. =C2=A0And adding stronger
>>> conditions on the cardinal used won=E2=80=99t help. =C2=A0= The problem is that one takes a
>>> fibrant replacement to go from the =E2=80=9Cpre-suspension= =E2=80=9D to the suspension (more
>>> precisely: a (TC,F) factorisation, to go from the universa= l family of
>>> pre-suspensions to the universal family of suspensions); a= nd fibrant
>>> replacement blows up the fibers to be the size of the *bas= e* of the family.
>>> So the pre-suspension is small, but the suspension =E2=80= =94 although essentially
>>> small =E2=80=94 ends up as large as the universe one=E2=80= =99s using.
>>>
>>> So here=E2=80=99s a very precise problem which is as far a= s I know open:
>>>
>>> (*) Construct an operation =CE=A3 : U =E2=80=93> U, whe= re U is Voevodsky=E2=80=99s universe,
>>> together with appropriate maps N, S : =C3=9B =E2=80=93>= =C3=9B over =CE=A3, and a homotopy m from N
>>> to S over =CE=A3, which together exhibit U as =E2=80=9Cclo= sed under suspension=E2=80=9D.
>>>
>>> I asked a related question on mathoverflow a couple of yea= rs 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 quest= ion (which would
>>> probably also answer (*) here) based on the comments by Ka= rol Szumi=C5=82o and
>>> Tyler Lawson, using the adjunction with Top, but I wasn=E2= =80=99t quite able to
>>> piece it together.
>>>
>>> =E2=80=93p.
>>>
>>> >
>>> > > 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? =C2=A0It's n= ot 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 fibra= nt replacement
>>> > > doesn't
>>> > > in general preserve smallness of fibrations with= large base spaces.
>>> > >
>>> > > (Also, the current L-S paper doesn't quite g= ive a general syntactic
>>> > > scheme, only a general semantic framework with s= uggestive
>>> > > 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:
>>> > >>
>>> > >> =C2=A0If we are only interested in providing= one -particular- model of
>>> > >> HITs,
>>> > >> the paper
>>> > >> on =C2=A0cubical type =C2=A0theory describes= a way to =C2=A0interpret 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 extens= ion of type theory.
>>> > >>
>>> > >>
>>> > >> but the Kan simplicial set model already doe= s this =E2=80=94 right?
>>> > >> don=E2=80=99t get me wrong =E2=80=94 I love = the cubes, and they have lots of nice
>>> > >> properties
>>> > >> for models of HoTT
>>> > >> =E2=80=94 but there was never really a quest= ion of the consistency or
>>> > >> coherence of
>>> > >> simple HITs like propositional truncation or= suspension.
>>> > >>
>>> > >> the advance in the L-S paper is to give a ge= neral 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 semant= ics of these,
>>> > >> in a range of models of the basic theory.
>>> > >>
>>> > >> Steve
>>> > >>
>>> > >>
>>> > >> =C2=A0The approach uses an operation of =C2= =A0=E2=80=9Cflattening an open box=E2=80=9D, which
>>> > >> solves
>>> > >> in
>>> > >> this case the issue of interpreting HIT with= parameters (such as
>>> > >> propositional
>>> > >> truncation or suspension) without any cohere= nce issue.
>>> > >> Since the syntax used in this paper is so cl= ose to the semantics,
>>> > >> we
>>> > >> limited
>>> > >> ourselves =C2=A0to a syntactical presentatio= n of this interpretation. But
>>> > >> it can
>>> > >> directly
>>> > >> be transformed to a semantical interpretatio= n, as explained in the
>>> > >> following
>>> > >> note
>>> > >> (which also incorporates a nice simplificati= on 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 =E2=80=9Cf= lattening boxes=E2=80=9D method.
>>> > >>
>>> > >> Only the cases of the spheres and propositio= nal truncation are
>>> > >> described,
>>> > >> but one
>>> > >> would expect the method to generalise to oth= er HITs covered e.g. in
>>> > >> the HoTT
>>> > >> book.
>>> > >>
>>> > >> On 25 May 2017, at 20:25, Michael Shulman &l= t;shu...@sandiego.edu>
>>> > >> wrote:
>>> > >>
>>> > >> The following long-awaited paper is now avai= lable:
>>> > >>
>>> > >> Semantics of higher inductive types
>>> > >> Peter LeFanu Lumsdaine, Mike Shulman
>>> > >> https://arxiv.org/abs/1705.<= wbr>07088
>>> > >>
>>> > >> From the abstract:
>>> > >>
>>> > >> We introduce the notion of *cell monad with = parameters*: a
>>> > >> semantically-defined scheme for specifying h= omotopically
>>> > >> well-behaved
>>> > >> notions of structure. We then show that any = suitable model category
>>> > >> has *weakly stable typal initial algebras* f= or any cell monad with
>>> > >> parameters. When combined with the local uni= verses construction to
>>> > >> obtain strict stability, this specializes to= give models of specific
>>> > >> higher inductive types, including spheres, t= he torus, pushout types,
>>> > >> truncations, the James construction, and gen= eral localisations.
>>> > >>
>>> > >> Our results apply in any sufficiently nice Q= uillen model category,
>>> > >> including any right proper simplicial Cisins= ki model category (such
>>> > >> as
>>> > >> simplicial sets) and any locally presentable= locally cartesian
>>> > >> closed
>>> > >> category (such as sets) with its trivial mod= el structure. In
>>> > >> particular, any locally presentable locally = cartesian closed
>>> > >> (=E2=88=9E,1)-category is presented by some = model category to which our
>>> > >> results apply.
>>> > >>
>>> > >> --
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>>> > >>
>>> > >>
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