This chapter defines and studies stable categories of additive categories with a focus on the stable categories associated to a given cotorsion pair. It is shown that for any complete cotorsion pair, or even Ext-pair, there are four additive functors between stable categories. First there are the right and left approximation functors, obtained by taking special precovers and preenvelopes. The other two functors constructed are the suspension and loop functors.

This chapter is devoted to studying the triangulated structures that can be attached to an abelian model structure on a weakly idempotent complete exact category. First, one-sided (i.e. left and right) triangulated categories are defined and it is shown that we can attach both a left and right triangulated category to any complete cotorsion pair. It is then shown that the homotopy category of an abelian model structure is always triangulated. In particular, the suspension functor is an autoequivalence of the homotopy category. Natural descriptions of the distinguished triangles are given. Ultimately, it is shown that the canonical functor to the homotopy category may be thought of as the triangulated localization of the exact structure with respect to the class of trivial objects.