Timeline for Are finite (levelwise) homotopy limits of spectra homotopy invariant?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2017 at 9:19 | vote | accept | Emanuele Dotto | ||
| Feb 18, 2017 at 4:39 | answer | added | Cary | timeline score: 7 | |
| Sep 27, 2013 at 15:59 | comment | added | Emanuele Dotto | I actually meant the category has finitely many morphisms. It's the kind of homotopy limits that commute with sequential homotopy colimits, maybe we can ask for a very small category to be safe. | |
| Sep 24, 2013 at 14:44 | comment | added | Lennart Meier | I think, I understand now what you mean, but one clarification: What do you mean by finite? Something like that the nerve has only finitely many nondegenerate simplices? | |
| Sep 20, 2013 at 20:37 | comment | added | Emanuele Dotto | Oh,homotopy colimits are homotopy invariant even in spectra without cofibrant replacements | |
| Sep 20, 2013 at 20:15 | comment | added | Emanuele Dotto | That's precisely what I was asking. Is it known not to be homotopy invariant for spectra in spaces? I see how it is not for spectra in simplicial sets. Funny things can happen, e.g. homotopy colimits are homotopy invariant in spaces even without cofibrant replacements | |
| Sep 20, 2013 at 19:46 | comment | added | Lennart Meier | You can define whatever you want. But the Bousfield-Kan construction is in general not homotopy invariant (and therefore does not agree with I would call the homotopy limit), neither in simplicial sets nor in spectra. See, for example the Bousfield-Kan book XI.5.6, Section 2.3 of math.mit.edu/conferences/talbot/2007/tmfproc/Chapter07/… or the Hirschhorn book, Thm 18.5.2 (most precise). | |
| Sep 20, 2013 at 18:36 | comment | added | Emanuele Dotto | It makes sense even without having a model structure | |
| Sep 20, 2013 at 18:00 | comment | added | Lennart Meier | Does not the Bousfield-Kan formula presuppose that the objects are fibrant? | |
| Sep 20, 2013 at 17:08 | comment | added | Emanuele Dotto | I'm using the Bousfield-Kan formula to define homotopy limits. Since the cotensored structure of spectra over simplicial sets is levelwise, the homotopy limits turn out to be levelwise (and well defined). The proof uses only that stable equivalences are $\pi_\ast$-equivalences, and that directed homotopy colimits and finite homotopy limits commute in spaces. | |
| Sep 20, 2013 at 16:13 | history | edited | Ricardo Andrade |
added top level tag
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| Sep 20, 2013 at 16:03 | comment | added | Lennart Meier | Problems might begin with the question: How do you form a levelwise homotopy limit of spectra? Is then the map $\Sigma X_n \to X_{n+1}$ still well-defined? So I assume, when you write equivalence that you mean stable equivalence? Can you write down your proof? | |
| Sep 20, 2013 at 15:54 | history | asked | Emanuele Dotto | CC BY-SA 3.0 |