> On Jun 14, 2018, at 11:58 AM, Christian Sattler wrote: > > On Thu, Jun 14, 2018 at 11:28 AM, Steve Awodey > 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 > 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 > 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 > >>> > 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. > >>>> > >>>> -- > >>>> 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 . > >>>> For more options, visit https://groups.google.com/d/optout . > >>> > >>> -- > >>> 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 . > >>> For more options, visit https://groups.google.com/d/optout . > >> > > -- > 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 . > For more options, visit https://groups.google.com/d/optout . > > > -- > 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 . > For more options, visit https://groups.google.com/d/optout .