Timeline for Constructing a "geometric" model structure on Cat by localizing the "categorical" model structure
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 20, 2014 at 14:59 | vote | accept | KotelKanim | ||
| Jul 20, 2014 at 10:13 | comment | added | David White | Ah, I meant the weak equivalences of the Thomason model structure. I think I misread the OP. A reference for the theorem I have in mind is Barwick `On (Enriched) left Bousfield localization' Theorem 2.11. It seems I was right about only needing the accessibility of the weak equivalences (rather than the fact that they're generated by a set) as Barwick's proof only requires his Lemma 2.10. However, condition (3) of his Proposition 1.7 is not satisfied thanks to your example ($W \cap I$-cof is not closed under pushout). Other versions of Smith's Theorem (e.g. HTT) hit the same problem. | |
| Jul 20, 2014 at 10:03 | comment | added | Karol Szumiło | I don't understand your remarks about Dwyer maps. Dwyer maps are not equivalences of the Thomason model structure, but rather some sort of cofibrations. (They contain Thomason cofibrations but there are many more of them.) Also, could you tell me which localization theorem specifically you have in mind? It's hard to say why it fails without seeing the statement. However, any theorem that produces a model structure with both cofibrations and equivalences specified in advance has to somehow assume compatibility of these classes. So in the end the problem will probably boil down to my example. | |
| Jul 20, 2014 at 9:40 | comment | added | David White | @KarolSzumilo: Okay, I agree A.2.6.13 can't be used. But that's for building a combinatorial model structure from scratch. There's also Smith's theorem that if you already have a left proper, combinatorial model category $M$ and a set of maps $C$ then the left Bousfield localization $L_C(M)$ exists. One might hope that in the presence of accessibility for the Dwyer maps then a set $C$ could be found so that the $C$-local equivalences are the Dwyer maps. I'd like to better understand the limits of the Smith machinery when it comes to finding such a set given an accessible class. | |
| Jul 20, 2014 at 9:27 | comment | added | Zhen Lin | Alternatively, one notes that every object is "canonically cofibrant", so the model structure (if it exists) must be left proper and so pushouts of cofibrations "must" be homotopy pushouts. | |
| Jul 20, 2014 at 9:04 | comment | added | Karol Szumiło | The problem is exactly the one I explained above. The assumption (2) of [HTT, A.2.6.13] essentially asks for pushouts along generating cofibrations to be homotopy pushouts. My example shows that this not the case for "canonical cofibrations". | |
| Jul 20, 2014 at 8:44 | comment | added | David White | Can you comment about what precisely fails when you try to do the obvious thing? I mean, the Thomason model structure is combinatorial so the class of weak equivalences is accessible. The canonical model structure is left proper and combinatorial, so you might try to apply Bousfield localization at the Dwyer maps. The only issue I can see is that perhaps the weak equivalences are $\kappa$-accessible for some $\kappa > \omega$, and Smith's Theorem (HTT A.2.6.13) requires the weak equivalences to be perfect rather than merely accessible. I'd love to see what fails in this argument. | |
| Jul 20, 2014 at 8:34 | history | answered | Karol Szumiło | CC BY-SA 3.0 |