The goal of this chapter is to develop a very general method for constructing (functorially) complete cotorsion pairs in exact categories. In essence we develop an algebraic version of Quillen’s small object argument for cotorsion pairs. This naturally leads us to the notion of cofibrantly generated cotorsion pairs and abelian model structures. The approach taken here is inspired by Saorin and Stovicek’s notion of an efficient exact category. We generalize this idea a bit more by considering classes of objects that we say are efficient relative to the exact structure. We show that with mild hypotheses on the exact category, any efficient set (not a proper class) of objects cogenerates a functorially complete cotorsion pair. Along the way we prove results about generators for exact categories, and use them to construct generating monics for cotorsion pairs. We also prove Eklof’s Lemma, and its dual, for exact categories, and give general conditions guaranteeing that the left side of a cotorsion pair is closed under direct limits.

This chapter studies cotorsion pairs in exact categories with a view toward developing the theory of abelian model structures. Basic concepts such as special precovers and complete cotorsion pairs are discussed. The more general notion of a complete Ext-pair is also introduced, and lifting and factorization properties for morphisms are developed. Hereditary cotorsion pairs are characterized and the Generalized Horseshoe Lemma is proved for hereditary cotorsion pairs. The special case of projective and injective cotorsion pairs is defined and characterized.