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Jan
14 |
awarded | Yearling |
Jan
14 |
awarded | Yearling |
Nov
29 |
awarded | Necromancer |
Nov
3 |
comment |
On triangulated categories of pro-objects
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 |
comment |
On triangulated categories of pro-objects
Sorry, I could not follow. What paper are you referring to? |
Oct
30 |
comment |
On closed model categories: standard arguments and fibrantly cogenerated categories
It follows immediately from the definition, except for the verification of (an analogue of) SM7. The dual versions of this axioms are called SM7a and SM7b. Their equivalence follows from an adjunction argument and may be found in any standard text on model categories. |
Oct
30 |
awarded | Editor |
Oct
30 |
revised |
On closed model categories: standard arguments and fibrantly cogenerated categories
edited body |
Oct
30 |
answered | On closed model categories: standard arguments and fibrantly cogenerated categories |
Oct
8 |
awarded | Caucus |
Sep
30 |
comment |
Finite homotopy limits commute with sequential homotopy colimits
I think that you are right. In a more general situation it is probably possible to find a counterexample, but every combinatorial model category is Quillen equivalent, by a theorem of Dugger, to a left Bousfield localization of simplicial presheaves over some small category equipped with the projective model structure. But in such categories generating cofibrations are the same as the projective generating cofibrations. In particular, they have $\aleph_o$-presentable domains and codomains. |
Sep
29 |
comment |
Finite homotopy limits commute with sequential homotopy colimits
No, this assumption can also be removed using a framing and applying the respective formulas from Hirschhorn's book for the computation of homotopy limits. Alternatively we can replace our combinatorial model category by a simplicial one in a homotopy meaningful way and conclude that finite homotopy limits commute with filtered homotopy colimits since they commute in the simplicial replacement. |
Sep
29 |
comment |
Finite homotopy limits commute with sequential homotopy colimits
Of cause not, Fernando, cofibrant generation is a luxury. We just need it to compare the homotopy filtered colimits with the strict filtered colimits. For this purpose it suffices to assume that trivial fibrations are closed under $\lambda$-filtered colimits. But I'd rather keep this answer less technical. |
Sep
28 |
answered | Finite homotopy limits commute with sequential homotopy colimits |
Sep
23 |
comment |
On triangulated categories of pro-objects
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
21 |
awarded | Commentator |
Sep
21 |
comment |
On triangulated categories of pro-objects
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 |
comment |
On triangulated categories of pro-objects
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 |
comment |
On triangulated categories of pro-objects
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 |
awarded | Informed |