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From: Michael Shulman
Date: Tue, 6 Jun 2017 07:35:13 -0600
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Subject: Re: [HoTT] Semantics of higher inductive types
To: Andrew Swan
Cc: Homotopy Type Theory , Steve Awodey ,
Thierry Coquand
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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 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 actu=
ally
>> 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 Chapt=
er
>> 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 th=
e
>> 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 m=
ap
>> 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 sh=
ould
>> be also. I think the way to formulate this would be as a distributive la=
w -
>> the fibrant replacement monad distributes over the (truncation + weak
>> fibrant replacement) monad. It looks to me like the same thing that work=
s 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 th=
at
>> 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 typ=
e
>> 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=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. And add=
ing stronger
>>> conditions on the cardinal used won=E2=80=99t help. The problem is tha=
t 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 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 fam=
ily.
>>> So the pre-suspension is small, but the suspension =E2=80=94 although e=
ssentially
>>> 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 as I know open=
:
>>>
>>> (*) Construct an operation =CE=A3 : U =E2=80=93> U, where U is Voevodsk=
y=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=9Cclosed under sus=
pension=E2=80=9D.
>>>
>>> I asked a related question on mathoverflow a couple of years ago:
>>> https://mathoverflow.net/questions/219588/pullback-stable-model-of-fibr=
ewise-suspension-of-fibrations-in-simplicial-sets
>>> David White suggested he could see an answer to that question (which wo=
uld
>>> probably also answer (*) here) based on the comments by Karol 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
>>> > > wrote:
>>> > >
>>> > > Do we actually know that the Kan simplicial set model has a *univer=
se
>>> > > closed under* even simple HITs? It's not trivial because this woul=
d
>>> > > 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 togethe=
r
>>> > >> with a
>>> > >> univalent
>>> > >> universe which is stable by HIT operations. This gives in particul=
ar
>>> > >> the
>>> > >> consistency
>>> > >> and the proof theoretic power of this extension of type theory.
>>> > >>
>>> > >>
>>> > >> but the Kan simplicial set model already does this =E2=80=94 right=
?
>>> > >> don=E2=80=99t get me wrong =E2=80=94 I love the cubes, and they ha=
ve lots of nice
>>> > >> properties
>>> > >> for models of HoTT
>>> > >> =E2=80=94 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 =E2=80=9Cflattening an open bo=
x=E2=80=9D, 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. B=
ut
>>> > >> 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 =E2=80=9Cflattening boxes=E2=80=
=9D 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. i=
n
>>> > >> 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 categor=
y
>>> > >> 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 specif=
ic
>>> > >> higher inductive types, including spheres, the torus, pushout type=
s,
>>> > >> 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 (suc=
h
>>> > >> 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
>>> > >> (=E2=88=9E,1)-category is presented by some model category to whic=
h our
>>> > >> results apply.
>>> > >>
>>> > >> --
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