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?