Timeline for On triangulated categories of pro-objects
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17 events
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| Nov 3, 2013 at 20:44 | comment | added | Boris Chorny | Yes, I think that there is no full embedding of the stable motivic homotopy category into the Fausk-Isaksen model structure, since the constant pro-(motivic)spectra are not fibrant. Yet it is a full embedding into the strict pro-category. I never came across a written proof that pro-$\mathcal{C}$ is stable if $\mathcal{C}$ is so. But I believe that this is true and that you can work it out. | |
| Nov 2, 2013 at 21:25 | comment | added | Mikhail Bondarko | Still, it seems that it would be easier to work in the case when the $t$-structure is 'trivial'. Is the corresponding model structure the strict one? Where could I find more information about the latter? Is it written somewhere that the corresponding model category will be stable? | |
| Nov 2, 2013 at 21:23 | comment | added | Mikhail Bondarko | The paper intlpress.com/HHA/v9/n1/a16 There is an extra condition needed in order to have a full embedding; yet it is fulfilled in my 'motivic' context. | |
| Nov 2, 2013 at 20:22 | comment | added | Boris Chorny | Sorry, I could not follow. What paper are you referring to? | |
| Oct 31, 2013 at 21:16 | comment | added | Mikhail Bondarko | It seems that we do not have a full embedding for the construction of Fausk and Isaksen in general; see Proposition 8.3 and the remark after it. | |
| Sep 23, 2013 at 20:23 | comment | added | Boris Chorny | I do not find your last request exotic. On the contrary, it is more interesting than the original question. Indeed, constant pro-objects are $\aleph_0$-cosmall in the pro-category of spectra or motivic spectra. Yet the strict model structure is class-fibrantly $\textit{finitely}$ generated. The dualization of Hovey's argument in the last theorem of his book should prove the cocompactness of constant pro-spectra. | |
| Sep 23, 2013 at 15:47 | vote | accept | Mikhail Bondarko | ||
| Sep 22, 2013 at 5:41 | comment | added | Mikhail Bondarko | Thank you very much for this information! It seems that I need something quite exotic: all SH (or its compact objects) should become cocompact in the corresponding $Ho(Pro−M)$. I wonder whether existing methods of studying $Pro-M$ could help me. Yet your remarks are very intersting! | |
| Sep 21, 2013 at 18:18 | comment | added | Boris Chorny | This embedding is fully faithful on all levels (both the ambient categories and the homotopy categories). The reason for this is that the embedding takes fibrant object to fibrant objects, even though it is left Quillen. This is not the case with the localized model structures (Fausk-Isaksen, etc). With some effort one can show that the homotopy category $\mathrm{Ho}(\mathcal{C})$ is coreflective in pro-$\mathcal C$. This requires some recent (co)localization technique, but I am not sure that you need it. | |
| Sep 21, 2013 at 18:10 | comment | added | Boris Chorny | This remark is about localization of pro-$\mathcal C$ with respect to cohomology theory, so probably irrelevant to you. If you start from a stable model category, then it should not be too hard to show that the pro-category is also stable if you define the suspension levelwise. Unfortunately I do not know a reference for this fact. If you start from a proper model of motivic spectra, then pro-$\mathcal C$ carries a strict model structure, moreover embedding of constant objects may be extended to the total left derived functor of homotopy categories, as a left Quillen functor. | |
| Sep 21, 2013 at 12:28 | comment | added | Mikhail Bondarko | I like Remark 4.5 of this paper. I need a full embedding of SH into a triangulated homotopy category of pro-objects. | |
| Sep 21, 2013 at 10:37 | comment | added | Boris Chorny | This paper is about a localization of pro-spectra with respect to cohomotopy. The resulting localized category is Quillen equivalent to the opposite category of spectra extending, in a sense, Spanier-Whithead duality. It has nothing to do with the motivic homotopy theory, as far as I know. You do not tell us how you are going to use pro-categories, so I do not know what might help you. | |
| Sep 20, 2013 at 14:27 | comment | added | Mikhail Bondarko | Could I ask you: do you think that the paper "Duality and Pro-Spectra" will help me? | |
| Sep 19, 2013 at 6:29 | comment | added | Boris Chorny | If the existence of the model structure is a problem, then you should start from the strict one (Edwards-Hastings, see Isaksen's paper arxiv.org/abs/math/0108189 for a modern treatment), all the others are the localizations of the strict one. Properness is the only condition for the existence of the strict model. Which model is more natural depends on your question. If pro-M is equipped with the strict model structure, then the embedding $M\to \text{pro-}M$ is a left Quillen functor, hence the induced functor on homotopy categories is a full embedding of a coreflective subcategory. | |
| Sep 19, 2013 at 6:09 | comment | added | Boris Chorny | Fausk-Isaksen cite a preprint of Morel for t-structures on motivic homotopy category, but I did not look at it. As for lifting the t-structure to the level of a model they do not seem to impose any condition on the model category, just define the n-equivalences and co-n-equivalences. | |
| Sep 19, 2013 at 3:33 | comment | added | Mikhail Bondarko | Thank you! Yet could you give more detail? The paper of Fausk-Isaksen requires a t-structure on the level of modules; does it exist for SH? Will the corresponding model structure for $SH$ be 'the most natural one'? Will the functor $Ho(M)\to H(Pro-M)$ be a full embedding? | |
| Sep 18, 2013 at 21:17 | history | answered | Boris Chorny | CC BY-SA 3.0 |