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created Cisinski model structure
I have added to Cisinski model structure a section On presheaf toposes with the essence of the definitions and statements from section 1.3 of Cisinki’s book.
I have been further expanding at Cisinski model structrure the section On presheaf toposes.
Effectively I am now translating the entire chapter 1.3 of Cisinki’s book and spelling out details that the book leaves out.
So far I have made it through all the preliminaries up to the main theorem stating the existence of the model structure.
Next I’ll try to translate / write out the full proof of that.
Thank you! I’m sure this will be a great thing to have.
Good to hear that you find this useful. Right now I need these notes as a script for a seminar talk where I will present the proof – or at least a good bit of it…
This morning, I have added a few more bits. The main new aspect for any potential reader is that I have added much more structure to the flow of the argument, see the new section outline at Cisinki model structure and the outline of the proof of the main theorem here. (Where “main theorem” still means “main theorem of section 1.3”, mind you.)
The main remaining gap as far as details for the proofs go is currently the last section. I’ll try to take care of that now.
Added to Cisinski model structure a handful of items on “localizers” on presheaf categories, here
added the remark that every Cisinski model structure is in particular combinatorial (and added an Examples-link the other way round).
Can we describe the ∞-category underlying a Cisinski model structure in terms of its topos and localizer?
As far as I understand, for a nonempty localizer the Cisinski model structure is the left Bousfield localization of the Cisinski model structure with the empty localizer at the morphisms in the localizer, which has an obvious ∞-categorical translation in terms of (reflective) localizations.
However, what is the ∞-categorical meaning of the Cisinski model structure with the empty localizer?
Staring at the examples it seems to me that for toposes of presheaves the underlying ∞-category should be just the ∞-category of ∞-presheaves. And for arbitrary toposes one can expect to get some version of ∞-sheaves, hypercompleted or not.
The statement at Cisinski model structure about cellular models (proposition 2.34 on the page, prop. 1.2.27 in Cisinski 06) refers to presheaf toposes. However, it seems to me that the same fact should be true about arbitrary Grothendieck toposes (that the monomorphisms are cofibrantly generated by a set), by essentially the same proof using the sheafified representables instead of the ordinary representables. Is this true, and if so is it written down anywhere?
Just a quick note that Cisinski generalised some of his work to topoi in this unpublished work, but it’s a long time since I’ve looked at it, and I don’t know if what you are looking for is in there.
Thanks! It’s stated there as Prop. 1.2.2, without proof (but, as I said, I think the proof in Cisinski 06 generalizes without trouble).
Great! Yes, at the beginning of section 2, the second sentence translates to something like:
“We have omitted those proofs already given in the setting of presheaf categories which can be given mutatis mutandis in that of sheaf categories, referring the reader to [Cisinski’s book]”.
Thanks! My French isn’t quite good enough for that to have jumped out at me. (-:O
Added definition of homotopical datum (donnée homotopique), which seemed to be missing. Gave the example of the minimal Cisinski model structure on any presheaf category. Left the latter as a page to be created.
The case of Grothendieck abelian categories and Grothendieck toposes is also treated in Prop. 1.12 in Tibor Beke’s paper https://faculty.uml.edu/tbeke/sheafi.pdf
Thanks! I’ve added this to cellular model too.
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