Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs and in satisfying and . We show how to construct a (necessarily unique) abelian model structure on with (resp. ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. ) as the class of fibrant (resp. trivially fibrant) objects.
In this paper we consider a pair of right adjoint contravariant functors between abelian categories and describe a family of dualities induced by them.
An -exact category is a pair consisting of an additive category and a class of sequences with terms satisfying certain axioms. We introduce -weakly idempotent complete categories. Then we prove that an additive -weakly idempotent complete category together with the class of all contractible sequences with terms is an -exact category. Some properties of the class are also discussed.