Re: [HoTT] Quillen model structure

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

> On Jun 14, 2018, at 12:00 AM, Michael Shulman <shu...@sandiego.edu> wrote:
> 
> In the 1-categorical case, I believe that every locally small
> (co)complete elementary 1-topos is defined over Set: its global
> sections functor has a left adjoint by cocompleteness, and the left
> adjoint is left exact by the Giraud exactness properties (which hold
> for any (co)complete elementary 1-topos).  Such a topos can only fail
> to be Grothendieck by lacking a small generating set.

yes, that’s correct.

> 
> In the oo-case, certainly cartesian cubical sets present a locally
> small (oo,1)-category (any model category does, at least assuming its
> locally small as a 1-category), so it seems to me there are three
> possibilities:
> 
> 1. Although they are presumably an "elementary (oo,1)-topos" in the
> finitary sense that provides semantics for HoTT with HITs and
> univalent universes, they might fail to satisfy some of the oo-Giraud
> exactness properties.  Presumably they are locally cartesian closed
> and coproducts are disjoint, so it would have to be that not all
> groupoids are effective.

that’s possible, I suppose …

> 
> 2. They might lack a small generating set, i.e. the (oo,1)-category
> might not be locally presentable.  Every combinatorial model category
> (i.e. cofibrantly generated model structure on a locally presentable
> 1-category) presents a locally presentable (oo,1)-category, and the
> 1-category of cartesian cubical sets is certainly locally presentable,
> but I suppose it's not obvious whether these weak factorization
> systems are cofibrantly generated.

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.


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> 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> 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> 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
>>>> 
>>>> 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
>>>> 
>>>> 
>>>> 
>>>> 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
>>>> 
>>>> 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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>>