How to construct a hovey triple from two cotorsion pairs

Abstract

Let A be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs (Q,R∼) and (Q∼,R) in A satisfying R∼ ⊆ R and Q ∩ R∼ = Q∩Z. We show how to construct a (necessarily unique) abelian model structure on A with Q (resp. Q∼) as the class of cofibrant (resp. trivially cofibrant) objects, and R (resp. R∼) as the class of fibrant (resp. trivially fibrant) objects.