Re: [HoTT] Quillen model structure

[Steve Awodey <awo...@cmu.edu>]

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1 correction:

> generators aside, the cofibrations are all the monos, and the fibrations have the RLP w/resp. to all push-out products m xo d : U >—> B x I, where m : A >—> B is any mono, j : B —> I is some indexing making m an I-indexed family of monos, d : I —> I x I is regarded as a generic point of I over I, and the pushout-product  
> 
> m xo d : I^n +_A (A x I)  >—>  B x I

should be:

	m xo d : B +_A (A x I) >—> B x I

Steve

> 
> is formed over I as previously described.
> 
> yes?
> 
> Yes, that's correct.
>  
> 
> Steve
> 
> 
>> On Jun 14, 2018, at 12:27 PM, Steve Awodey <awo...@cmu.edu <mailto:awo...@cmu.edu>> wrote:
>> 
>> 
>> 
>>> On Jun 14, 2018, at 11:58 AM, Christian Sattler <sattler....@gmail.com <mailto: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.
>>> >>>> 
>>> >>>> --
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>>> 
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>> 
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> 
> 

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