Timeline for Bousfield localization before and after taking homotopy
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 10, 2013 at 12:36 | comment | added | Fernando Muro | Preserving finite homotopy coproducts and homotopy cofibers is the same as preserving finite homotopy colimits, which is enough. As you said, this condition is really a condition on the model categories, or if you wish on the induced derivators, but not on the homotopy categories. I abused terminology. | |
| Aug 10, 2013 at 9:03 | comment | added | Rasmus | Anyway, in the case of spectra, it seems that this condition is more or less equivalent to the Bousfield localization being smashing. | |
| Aug 10, 2013 at 8:55 | comment | added | Rasmus | I am a bit puzzled by the condition that $\varphi$ preserve homotopy colimits because it seems to be a condition on the output data $\mathrm{Ho}(C)\to\mathrm{Ho}(\tilde C)$ and not (explicitly) on the input data $(C,\tilde C)$ (modulo the fact that there are no functorial homotopy (co)fibers in triangulated categories). Also, would it not be enough to have that $\varphi$ preserves finite coproducts and homotopy (co)fibers to get additivity and exactness, respectively? | |
| Jul 12, 2013 at 15:24 | comment | added | Rasmus | I will put a link for now. In the future I will hopefully replace it with a more detailed version of your answer. | |
| Jul 12, 2013 at 15:20 | comment | added | Fernando Muro | You're welcome! You can link to this answer, or expand it conveniently. I think it's a little bit sketchy, isn't it? | |
| Jul 12, 2013 at 15:09 | vote | accept | Rasmus | ||
| Jul 12, 2013 at 15:08 | comment | added | Rasmus | Thanks a lot! I should have tried more seriously to see on my own if one can just prove this in a straight-forward manner. I guess the nlab entry made me think this was non-obvious and unknown. Is it okay if I put a link to this answer in the nlab entry? | |
| Jul 12, 2013 at 14:45 | history | answered | Fernando Muro | CC BY-SA 3.0 |