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?



On triangulated categories of pro-objects
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](http://www.intlpress.com/HHA/v9/n1/a16/). See also an earlier [preprint by Isaksen](http://math.wayne.edu/~isaksen/Research/prospectra.pdf).


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](http://www.emis.ams.org/journals/DMJDMV/vol-05/15.pdf).


Hope it helps.


