Re: [HoTT] Quillen model structure

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

Dear Richard,

I believe the answer to (1) is yes : - )

The main point of Thierry's note is that the so-called “type theoretic model structure” on cartesian cubical sets (which as Thierry clearly says is essentially the work of Christian Sattler) is *not* equivalent to the one arising from the (well-known) fact that the cartesian cube category is a (strict) test category.

Where exactly this conflicts with a Cisinski style approach is a distinct issue, I think — maybe that’s your (2)?

Regards,

Steve 


> On Jun 14, 2018, at 8:39 PM, Richard Williamson <rwilli...@gmail.com> wrote:
> 
> Dear Thierry,
> 
> Forgive me for not quite having an overview of exactly what has
> been done. My interest here is in how the model structure you
> have constructed compares with Cisinski's work. For example, I
> would expect that the category of cartesian cubical sets is a
> test category. Indeed, I would expect that more or less exactly
> the same technique as in 8.4 of Les préfaisceaux comme ... goes
> through to prove this. And I would expect that one can put a
> model structure on in the same way.
> 
> Now, I believe I saw in some slides that that your model
> structure coincides with the Cisinski one. But this does not seem
> compatible with my expectations above, because then it is
> certainly Quillen equivalent to the Serre model structure on
> topological spaces (or whatever).
> 
> Thus my questions:
> 
> 1) Have I misinterpreted the slides or some other aspect of your
> work?
> 
> 2) If not, where exactly do Cisinski's techniques fail?
> 
> Best wishes,
> Richard
> 
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