This section gives a general overview of abelian model structures and their homotopy categories. It is also meant to be a survey of the most fundamental examples of such homotopy categories. These include the chain homotopy category of a ring, the derived category of a ring, and the stable module category of a quasi-Frobenius (or Iwanaga–Gorenstein) ring.

This chapter develops the fundamentals of abelian model structures from the perspective of cotorsion pairs in exact categories. The key notion is that of a Hovey triple. This is a triple of classes of objects which are intertwined to form two complete cotorsion pairs. From a given Hovey triple we define (co)fibrations and weak equivalences as well as the (very good) left and right homotopy relations and their stable categories. The notion of a trivial morphism is introduced and it is shown that the 2 out of 3 property for weak equivalences is equivalent to the statement that each trivial morphism is a weak equivalence. This condition is automatic when the underlying additive category is weakly idempotent complete. At the end of the chapter, Hovey’s correspondence between cotorsion pairs (i.e. Hovey triples) and abelian model structures is proved.