# On triangulated categories of pro-objects

1. Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?

2. I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) and the functor $Ho(M)\to Ho(Pro-M)$ to be an exact full embedding. Which restrictions on M are needed to this end?

I looked several papers on homotopy categories of pro-objects, yet I was not able to find a clear answer to this question. In particular, is it possible to take an $M$ such that $Ho(M)$ is the motivic stable homotopy category here?

P.S. I would like to understand the relation between the approaches of: "t-model structures" (Fausk, Isaksen), "Duality and pro-spectra" ( Christensen, Isaksen), "Model structures for pro-simplicial presheaves" (Jardine), "Strict model structures for pro-categories" (Isaksen), and "Stability in pro-homotopy theory" (Seymour).

Upd. Moreover, I would like $SH$ (or its compact objects) to become cocompact in the corresponding $Ho(Pro-M)$. Does this mean that the objects of $M$ should be fibrant in $Pro-M$ (or only the fibrant ones?), and that all the objects of $Pro-M$ should be cofibrant?

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Model categories whose homotopy categories are triangulated (with distinguished triangles and the shift functor induced from the model structure) are precisely the stable model categories. Properness never enters the picture. –  Karol Szumiło Sep 18 '13 at 12:58

1. See the comment by Karol. Hovey's book on model categories is the standard reference.

2. For triangulation on Ho(pro-M) the relevant reference is Fausk-Isaksen paper. See also an earlier preprint by Isaksen.

The embedding $M \to \text{pro-}M$ as constant pro-objects preserves finite limits, but fails to preserve cofiltered limits. On the level of homotopy categories you have to be more explicit about what you mean by exact embedding. There are no other limits in $\mathrm{Ho}(M)$ except for product, so finite products are preserved.

Concerning your last question on a model for the stable motivic category see, for example, Jardine's paper.

Hope it helps.

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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? –  Mikhail Bondarko Sep 19 '13 at 3:33
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. –  Boris Chorny Sep 19 '13 at 6:09
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. –  Boris Chorny Sep 19 '13 at 6:29
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. –  Boris Chorny Sep 21 '13 at 18:18
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. –  Boris Chorny Sep 23 '13 at 20:23

Take this answer with a grain of salt since I can only provide vague references. Nevertheless, I claim that the properness of $M$ should be sufficient. If $C$ is a stable $\infty$-category, then $Pro(C)$ is stable as well, because the suspensions and loops can be computed "levelwise". For any $\infty$-category $C$, the Yoneda embedding $C\to Pro(C)$ is fully faithful and preserves colimits and finite limits, so in particular it is exact. Now if $M$ is a proper model category with underlying $\infty$-category $\tilde M$ and $Pro(M)$ is equipped with Isaksen's strict model structure, then the underlying $\infty$-category of $Pro(M)$ is $Pro(\tilde M)$, so in particular $Pro(M)$ is a stable model category and $Ho(M)\to Ho(Pro(M))$ is fully faithful and triangulated.

Pro-objects in accessible $\infty$-categories are discussed briefly in J. Lurie, Derived Algebraic Geometry XIII, §3.1, and the fact that the Yoneda embedding is fully faithful and exact is proved in Higher Topos Theory, §5.3, in the dual setting of Ind-objects, but I'm afraid that more precise references for the claims I've made may not exist. However, for accessible $\infty$-categories such as those of motivic spectra, I think the above claims can be justified rigorously enough using those references.

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So, you think that $\infty$-categories are appropriate here? Thank you!! Yet all of this is somewhat confusing.:) –  Mikhail Bondarko Sep 20 '13 at 18:54
Actually, I guess it's possible to prove all this using Isaksen's model structure. I first thought of using $\infty$-categories because model categories don't make it easy to prove that a left Quillen functor preserves finite homotopy limits, but in fact when it's between stable model categories it's automatic (this is in Hovey). –  Marc Hoyois Sep 21 '13 at 2:44
Could you give a more precise reference to Hovey? –  Mikhail Bondarko Oct 30 '13 at 21:50
Prop. 7.1.12 in Hovey's book says that the derived functor of a Quillen functor between stable model categories is always exact (for the induced triangulated structures). –  Marc Hoyois Nov 15 '13 at 16:49