From: Steve Awodey <awo...@cmu.edu>
To: Christian Sattler <sattler....@gmail.com>
Cc: Michael Shulman <shu...@sandiego.edu>,
Thierry Coquand <Thierry...@cse.gu.se>,
Homotopy Theory <homotopyt...@googlegroups.com>
Subject: Re: [HoTT] Quillen model structure
Date: Thu, 14 Jun 2018 12:27:50 +0200 [thread overview]
Message-ID: <B1EA41BB-8F27-4890-9239-069E4054BD85@cmu.edu> (raw)
In-Reply-To: <CALCpNBqs00J64hr4K8wCAvaDAHr=UO4O1smhCizfb7nHqHxCkw@mail.gmail.com>
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> On Jun 14, 2018, at 11:58 AM, Christian Sattler <sattler....@gmail.com> wrote:
>
> On Thu, Jun 14, 2018 at 11:28 AM, Steve Awodey <awo...@cmu.edu <mailto:awo...@cmu.edu>> wrote:
> but they are cofibrantly generated:
>
> - the cofibrations can be taken to be all monos (say), which are generated by subobjects of cubes as usual, and
>
> - the trivial cofibrations are generated by certain subobjects U >—> I^{n+1} , where the U are pushout-products of the form I^n +_A (A x I) for all A >—> I^n cofibrant and there is some indexing I^n —> I . In any case, a small set of generating trivial cofibrations.
>
> Those would be the objects of a category of algebraic generators. For generators of the underlying weak factorization systems, one would take any cellular model S of monomorphisms, here for example ∂□ⁿ/G → □ⁿ/G where G ⊆ Aut(□ⁿ) and ∂□ⁿ denotes the maximal no-trivial subobject,
this determines the same class of cofibrations as simply taking *all* subobjects of representables, which is already a set. There is no reason to act by Aut(n), etc., here.
> and for trivial cofibrations the corresponding generators Σ_I (S_{/I} hat(×)_{/I} d) with d : I → I² the diagonal (seen as living over I), i.e. □ⁿ/G +_{∂□ⁿ/G} I × ∂□ⁿ/G → I × □ⁿ/G for all n, G, and □ⁿ/G → I.
sorry, I can’t read your notation.
the generating trivial cofibrations I stated are the following:
take any “indexing” map j : I^n —> I and a mono m : A >—> I^n, which we can also regard as a mono over I by composition with j. Over I we also have the generic point d : I —> I x I , so we can make a push-out product of d and m over I , say m xo d : U >—> I^n x I . Then we forget the indexing over I to end up with the description I already gave, namely:
U = I^n +_A (A x I)
where the indexing j is built into the pushout over A.
A more direct description is this:
let h : I^n —> I^n x I be the graph of j,
let g : A —> A x I be the graph of j.m,
there is a commutative square:
g
A —— > A x I
| |
m | | m x I
| |
v v
I^n ——> I^n x I
| h
j |
v
I
put the usual pushout U = I^n +_A (A x I) inside it,
and the comprison map U —> I^n x I is the m xo d mentioned above.
Steve
>
>
> Steve
>
> >
> > 3. They might be a Grothendieck (oo,1)-topos after all.
> >
> > I don't know which of these is most likely; they all seem strange.
> >
>
> >
> >
> >
> > On Wed, Jun 13, 2018 at 1:50 PM, Steve Awodey <awo...@cmu.edu <mailto:awo...@cmu.edu>> wrote:
> >> oh, interesting!
> >> because it’s not defined over sSet, but is covered by it.
> >>
> >>> On Jun 13, 2018, at 10:33 PM, Michael Shulman <shu...@sandiego.edu <mailto:shu...@sandiego.edu>> wrote:
> >>>
> >>> This is very interesting. Does it mean that the (oo,1)-category
> >>> presented by this model category of cartesian cubical sets is a
> >>> (complete and cocomplete) elementary (oo,1)-topos that is not a
> >>> Grothendieck (oo,1)-topos?
> >>>
> >>> On Sun, Jun 10, 2018 at 6:31 AM, Thierry Coquand
> >>> <Thierry...@cse.gu.se <mailto:Thierry...@cse.gu.se>> wrote:
> >>>> The attached note contains two connected results:
> >>>>
> >>>> (1) a concrete description of the trivial cofibration-fibration
> >>>> factorisation for cartesian
> >>>> cubical sets
> >>>>
> >>>> It follows from this using results of section 2 of Christian Sattler’s
> >>>> paper
> >>>>
> >>>> https://arxiv.org/pdf/1704.06911 <https://arxiv.org/pdf/1704.06911>
> >>>>
> >>>> that we have a model structure on cartesian cubical sets (that we can call
> >>>> “type-theoretic”
> >>>> since it is built on ideas coming from type theory), which can be done in a
> >>>> constructive
> >>>> setting. The fibrant objects of this model structure form a model of type
> >>>> theory with universes
> >>>> (and conversely the fact that we have a fibrant universe is a crucial
> >>>> component in the proof
> >>>> that we have a model structure).
> >>>>
> >>>> I described essentially the same argument for factorisation in a message
> >>>> to this list last year
> >>>> July 6, 2017 (for another notion of cubical sets however): no quotient
> >>>> operation is involved
> >>>> in contrast with the "small object argument”.
> >>>> This kind of factorisation has been described in a more general framework
> >>>> in the paper of Andrew Swan
> >>>>
> >>>> https://arxiv.org/abs/1802.07588 <https://arxiv.org/abs/1802.07588>
> >>>>
> >>>>
> >>>>
> >>>> Since there is a canonical geometric realisation of cartesian cubical sets
> >>>> (realising the formal
> >>>> interval as the real unit interval [0,1]) a natural question is if this is a
> >>>> Quillen equivalence.
> >>>> The second result, due to Christian Sattler, is that
> >>>>
> >>>> (2) the geometric realisation map is -not- a Quillen equivalence.
> >>>>
> >>>> I believe that this result should be relevant even for people interested in
> >>>> the more syntactic
> >>>> aspects of type theory. It implies that if we extend cartesian cubical type
> >>>> theory
> >>>> with a type which is a HIT built from a primitive symmetric square q(x,y) =
> >>>> q(y,z), we get a type
> >>>> which should be contractible (at least its geometric realisation is) but we
> >>>> cannot show this in
> >>>> cartesian cubical type theory.
> >>>>
> >>>> It is thus important to understand better what is going on, and this is why
> >>>> I post this note,
> >>>> The point (2) is only a concrete description of Sattler’s argument he
> >>>> presented last week at the HIM
> >>>> meeting. Ulrik Buchholtz has (independently)
> >>>> more abstract proofs of similar results (not for cartesian cubical sets
> >>>> however), which should bring
> >>>> further lights on this question.
> >>>>
> >>>> Note that this implies that the canonical map Cartesian cubes -> Dedekind
> >>>> cubes (corresponding
> >>>> to distributive lattices) is also not a Quillen equivalence (for their
> >>>> respective type theoretic model
> >>>> structures). Hence, as noted by Steve, this implies that the model structure
> >>>> obtained by transfer
> >>>> and described at
> >>>>
> >>>> https://ncatlab.org/hottmuri/files/awodeyMURI18.pdf <https://ncatlab.org/hottmuri/files/awodeyMURI18.pdf>
> >>>>
> >>>> is not equivalent to the type-theoretic model structure.
> >>>>
> >>>> Thierry
> >>>>
> >>>> PS: Many thanks to Steve, Christian, Ulrik, Nicola and Dan for discussions
> >>>> about this last week in Bonn.
> >>>>
> >>>> --
> >>>> You received this message because you are subscribed to the Google Groups
> >>>> "Homotopy Type Theory" group.
> >>>> To unsubscribe from this group and stop receiving emails from it, send an
> >>>> email to HomotopyTypeThe...@googlegroups.com <mailto:HomotopyTypeTheo...@googlegroups.com>.
> >>>> For more options, visit https://groups.google.com/d/optout <https://groups.google.com/d/optout>.
> >>>
> >>> --
> >>> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
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> >>
>
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next prev parent reply other threads:[~2018-06-14 10:27 UTC|newest]
Thread overview: 21+ messages / expand[flat|nested] mbox.gz Atom feed top
2018-06-10 13:31 Thierry Coquand
[not found] ` <CABLJ2vLi2ePKwf+Zha9Gx1jFgqJo9j2W0PsTctBZvf7F-xThHA@mail.gmail.com>
2018-06-11 8:46 ` [HoTT] " Thierry Coquand
2018-06-13 20:33 ` Michael Shulman
2018-06-13 20:50 ` Steve Awodey
2018-06-13 22:00 ` Michael Shulman
2018-06-14 9:28 ` Steve Awodey
2018-06-14 9:48 ` Bas Spitters
2018-06-14 9:58 ` Christian Sattler
2018-06-14 10:27 ` Steve Awodey [this message]
2018-06-14 13:44 ` Steve Awodey
2018-06-14 14:52 ` Christian Sattler
2018-06-14 15:42 ` Steve Awodey
2018-06-14 15:47 ` Michael Shulman
2018-06-14 16:01 ` Steve Awodey
2018-06-14 18:39 ` Richard Williamson
2018-06-14 19:14 ` Steve Awodey
2018-06-14 20:15 ` Richard Williamson
2018-06-14 20:32 ` Ulrik Buchholtz
2018-06-14 21:07 ` Richard Williamson
2018-06-14 19:16 ` Thierry Coquand
2018-06-14 19:35 ` [HoTT] Quillen model structure, PS Thierry Coquand
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