```
From: Michael Shulman <shu...@sandiego.edu>
To: Andrew Swan <wakeli...@gmail.com>
Cc: Homotopy Type Theory <HomotopyT...@googlegroups.com>,
Steve Awodey <awo...@cmu.edu>,
Thierry Coquand <Thierry...@cse.gu.se>
Subject: Re: [HoTT] Semantics of higher inductive types
Date: Tue, 6 Jun 2017 07:35:13 -0600 [thread overview]
Message-ID: <CAOvivQzLG8UJzJQKWm+B8sSbbxEH5y0yOQ2Ni=x2mBD0v_riLw@mail.gmail.com> (raw)
In-Reply-To: <1c2cb641-89e3-444d-aa0c-cb8ccb79cf3c@googlegroups.com>
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 <wakeli...@gmail.com> 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 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 <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
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>>> > >> "Homotopy Type Theory" group.
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>>> > >> send an
>>> > >> email to HomotopyTypeThe...@googlegroups.com.
>>> > >> For more options, visit https://groups.google.com/d/optout.
>>> > >>
>>> > >>
>>> > >>
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```

next prev parent reply other threads:[~2017-06-06 13:35 UTC|newest]Thread overview:25+ messages / expand[flat|nested] mbox.gz Atom feed top 2017-05-25 18:25 Michael Shulman 2017-05-26 0:17 ` [HoTT] " Emily Riehl 2017-06-01 14:23 ` Thierry Coquand 2017-06-01 14:43 ` Michael Shulman 2017-06-01 15:30 ` Steve Awodey 2017-06-01 15:38 ` Michael Shulman 2017-06-01 15:56 ` Steve Awodey 2017-06-01 16:08 ` Peter LeFanu Lumsdaine 2017-06-06 9:19 ` Andrew Swan 2017-06-06 10:03 ` Andrew Swan2017-06-06 13:35 ` Michael Shulman [this message]2017-06-06 16:22 ` Andrew Swan 2017-06-06 19:36 ` Michael Shulman 2017-06-06 20:59 ` Andrew Swan 2017-06-07 9:40 ` Peter LeFanu Lumsdaine 2017-06-07 9:57 ` Thierry Coquand [not found] ` <ed7ad345-85e4-4536-86d7-a57fbe3313fe@googlegroups.com> 2017-06-07 23:06 ` Michael Shulman 2017-06-08 6:35 ` Andrew Swan 2018-09-14 11:15 ` Thierry Coquand 2018-09-14 14:16 ` Andrew Swan 2018-10-01 13:02 ` Thierry Coquand 2018-11-10 15:52 ` Anders Mörtberg 2018-11-10 18:21 ` Gabriel Scherer 2017-06-08 4:57 ` CARLOS MANUEL MANZUETA 2018-11-12 12:30 ` Ali Caglayan

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