TITLE: On triangulated categories of pro-objects
QUESTION [5 upvotes]: Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
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?

REPLY [5 votes]: 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.
{"set_name": "stack_exchange", "score": 5, "question_id": 142481}