Re: [HoTT] Quillen model structure, PS

[Thierry Coquand <Thierry...@cse.gu.se>]

  I wrote

——————

if we limit ourselves to the case where the base category is the Lawvere theory of
distributive lattices, or de Morgan algebra, or Boolean algebra and we take the fibrations
as defined page 13, we get the same notion as Cisinski -naive- fibrations.

——————

 I should have added that this is not completely trivial: it amounts to the fact that to 
have the -uniform- right lifting filling property w.r.t. any open box inclusion

 A x II \cup B x b    ->     B  x II             where B is -representable-, A -> B mono

(where we have a choice of lifting that are natural w.r.t. map B’ -> B between
representables)

is -equivalent- to having the (non necessarily uniform) right lifting property 
w.r.t. -any- open box inclusion

 A x II \cup B x b    ->     B  x II             B arbitrary, A -> B mono



  The same holds (and is either to check) for trivial fibrations: to have the uniform
right lifting property w.r.t. A -> B mono, B representable is equivalent to having
the right lifting property w.r.t. an arbitrary mono.

 Best regards,
 Thierry



> On 14 Jun 2018, at 20:39, 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