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<1128BE39-BBC4-4DC6-8792-20134A6CAECD@chalmers.se> <292DED31-6CB3-49A1-9128-5AFD04B9C2F2@cmu.edu>
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From: Michael Shulman
Date: Tue, 6 Jun 2017 13:36:32 -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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What class of homotopy theories can be presented by such models?
On Tue, Jun 6, 2017 at 10:22 AM, Andrew Swan wrote:
> I don't know how general this is exactly in practice, but I think it shou=
ld
> work in the setting that appears in van de Berg, Frumin A homotopy-theore=
tic
> model of function extensionality in the effective topos, which regardless=
of
> the title is not just the effective topos, but any topos together with a
> interval object with connections, a dominance satisfying certain conditio=
ns,
> with fibrations defined as maps with the rlp against pushout product 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 confident 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 are in the more general sense by Christian Sattler in
> section 6 of The Equivalence Extension Property and Model Structures. The=
n a
> version of step 1 of the small object argument applies to Christian's
> definition, which gives a pointed endofunctor whose algebras are the weak
> fibrations. The same technique can also be used to describe "box flatteni=
ng"
> (which should probably be called something else, like "cylinder flattenin=
g"
> in the general setting).
>
>
> Andrew
>
> On Tuesday, 6 June 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 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 th=
at
>> > a
>> > map is a weak fibration if it has a (uniform choice of) diagonal fille=
rs
>> > 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 i=
n
>> >> 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 pusho=
ut
>> >> 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 anoth=
er
>> >> 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. The=
n
>> >> I
>> >> think the resulting awfs of "weak fibrant replacement" should be stab=
le
>> >> under pullback (although of course, the right maps in the factorisati=
on
>> >> 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, (May=
be
>> >> one
>> >> way to do this would be to note that the cubical set argument is most=
ly
>> >> 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 mig=
ht
>> >>> > lead to cardinals outside of a Grothendieck Universe?
>> >>>
>> >>> Exactly, yes. There=E2=80=99s no reason I know of to think they *ne=
ed* to,
>> >>> but
>> >>> with the construction of Mike=E2=80=99s and my paper, they do. And =
adding
>> >>> stronger
>> >>> conditions on the cardinal used won=E2=80=99t help. 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 universal family o=
f
>> >>> 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 =E2=80=94 althoug=
h
>> >>> 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 as I know o=
pen:
>> >>>
>> >>> (*) Construct an operation =CE=A3 : U =E2=80=93> U, where U is Voevo=
dsky=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 =
suspension=E2=80=9D.
>> >>>
>> >>> I asked a related question on mathoverflow a couple of years ago:
>> >>>
>> >>> https://mathoverflow.net/questions/219588/pullback-stable-model-of-f=
ibrewise-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 Szumi=
=C5=82o
>> >>> and
>> >>> Tyler Lawson, using the adjunction with Top, but I wasn=E2=80=99t qu=
ite 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
>> >>> > > *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, where=
as
>> >>> > > 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 =E2=80=94 ri=
ght?
>> >>> > >> 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 question of the consiste=
ncy 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=
box=E2=80=9D,
>> >>> > >> which
>> >>> > >> solves
>> >>> > >> in
>> >>> > >> this case the issue of interpreting HIT with parameters (such a=
s
>> >>> > >> propositional
>> >>> > >> truncation or suspension) without any coherence issue.
>> >>> > >> Since the syntax used in this paper is so close to the semantic=
s,
>> >>> > >> 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 =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=
.
>> >>> > >> 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
>> >>> > >> (=E2=88=9E,1)-category is presented by some model category to w=
hich our
>> >>> > >> results apply.
>> >>> > >>
>> >>> > >> --
>> >>> > >> You received this message because you are subscribed to the
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>> >>> > >> Groups
>> >>> > >> "Homotopy Type Theory" group.
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t,
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>> >>> > >> email to HomotopyTypeThe...@googlegroups.com.
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>> >>> > >>
>> >>> > >>
>> >>> > >>
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