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?
