Timeline for Standard model structures on $Top$
Current License: CC BY-SA 3.0
18 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 1, 2015 at 11:23 | comment | added | Ilan Barnea | Hi Matan. I was not aware there is a difference | |
| Sep 21, 2015 at 15:15 | comment | added | MatanP | Just a remark: the cofibrant objects in the Quillen model structure on Top are not retracts of CW complexes as you wrote, but rather retracts of cell-complexes. | |
| Aug 15, 2015 at 22:25 | comment | added | Ilan Barnea | Have you seen: J. Rosický and W. Tholen, Erratum to "Left-determined model categories and universal homotopy theories", Trans. Amer. Math. Soc. 360 (2008), 6179-6179. Maybe they fix everything here | |
| Aug 14, 2015 at 19:17 | history | edited | David White | CC BY-SA 3.0 |
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| Aug 14, 2015 at 19:16 | comment | added | David White | I was about to write something about proving the existence of a set $I$ to generate the monomorphisms, but I realized a bigger problem. This paper might contain mistakes. Look at the comment thread in mathoverflow.net/questions/135135. Anyway, my comment was only meant as a sketch of an approach, not a full answer. I agree work would have to be done to turn the sketch into a real proof. Gaucher has thought a lot recently about left determined models and has a preprint on his homepage which might be more useful than my musings. | |
| Aug 14, 2015 at 17:54 | comment | added | Ilan Barnea | @DavidWhite Thanks, but I still don't get something. Why would taking $I$ to be $\phi\to *$ produce all monomorphisms? To me it looks like producing only maps of the form $X\to X\coprod D$, for $D$ a discrete space. Even taking a larger set like all $\partial\Delta^n\to \Delta^n$ looks to me like producing the cofibrations in the Quillen model structure which are less then the cofibrations in the mixed model structure. How are we getting more cofibrations? | |
| Aug 14, 2015 at 17:11 | history | edited | Ilan Barnea | CC BY-SA 3.0 |
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| Aug 14, 2015 at 13:25 | comment | added | David White | You are right; you can't fix W. However, by the nature of the left-determined model structure you know you can obtain a model structure with the W you want (weak homotopy equivalences) as a left Bousfield localization of the left determined one. I mean, you've already assumed Vopenka, so this gets rid of the problem of needing to find a SET of maps to invert. | |
| Aug 14, 2015 at 1:07 | comment | added | Ilan Barnea | @DavidWhite I think the result of Rosicky and Tholen says that the weak saturation of any set of $I$ of morphisms in a locally presentable category is the class of cofibrations of a unique left-determined model structure. So how can you fix $W$ in advance? | |
| Aug 13, 2015 at 23:48 | history | edited | Ilan Barnea | CC BY-SA 3.0 |
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| Aug 13, 2015 at 23:41 | history | edited | Ilan Barnea | CC BY-SA 3.0 |
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| Aug 12, 2015 at 12:55 | comment | added | David White | Also, have you read Tibor Beke's work? He has a paper called "How (non)unique is the choice of cofibrations?" which you might enjoy. That paper is really about the setting of a combinatorial model category, but doesn't require Vopenka. There might be a way to rig up an example on $\Delta$-generated spaces with more cofibrations than the mixed model structure, but I could not see how to do it. Beke's approach requires a careful understanding of $Ex^\infty$ and so seems to require sSet rather than Top. | |
| Aug 12, 2015 at 12:51 | comment | added | David White | Here is a plan of attack. Use $\Delta$-generated spaces, produce a model structure there, and try to pass it via adjunction to Top. Rosicky and Tholen in "Left-determined model..." prove that under Vopenka in any locally presentable category with fixed $W$ (here weak homotopy equivalences) then for any set $I$ of maps there is a left determined model structure with cofibrations generated by $I$. Take $I$ to be $\emptyset \to \ast$ and you get monomorphisms, which is more cofibrations than the mixed model structure. | |
| Aug 5, 2015 at 11:08 | history | edited | Ilan Barnea |
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| Aug 4, 2015 at 11:30 | comment | added | Ilan Barnea | I don't think so because the conclusion there is that the left determined model structure and the Quillen model structure are equal. | |
| Aug 4, 2015 at 10:48 | comment | added | David White | Your question made the following thread pop into my mind. I have no idea if it'll be useful and won't have time to think about it till next week, but let me share while it's on my mind: mathoverflow.net/questions/100153 | |
| Aug 4, 2015 at 6:06 | history | asked | Ilan Barnea | CC BY-SA 3.0 |