Model structure from one hereditary complete cotorsion pair

In contrast with the Hovey correspondence of abelian model structures from two complete cotorsion pairs, Beligiannis and Reiten give a construction of model structures on abelian categories from one hereditary complete cotorsion pair. The aim of this paper is to extend this result to weakly idempotent complete exact categories, by adding the condition of heredity of the complete cotorsion pair. In fact, even for abelian categories, this condition of heredity should be added. This construction really gives model structures which are not necessarily exact in the sense of Gillespie. The correspondence of Beligiannis and Reiten of weakly projective model structures also holds for weakly idempotent complete exact categories.