Model structures on triangulated categories with proper class of triangles

In contrast with the Hovey correspondence of abelian model structures from two compatible 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 triangulated categories together with a proper class $\xi$ of triangles. There indeed exist non-trivial proper classes of triangles, and a proper class of triangles is not closed under rotations, in general. This is quite different from the class of all triangles. Thus one needs to develop a theory of triangles in $\xi$ and hereditary complete cotorsion pairs in a triangulated category $\T$ with respect to $\xi$. The Beligiannis - Reiten correspondence between weakly $\xi$-projective model structures on $\T$ and hereditary complete cotorsion pairs $(\X, \Y)$ with respect to $\xi$ such that the core $\omega = \X \cap \Y$ is contravariantly finite in $\T$ is also obtained. To study the homotopy category of a model structure on a triangulated category, the condition in Quillen's Fundamental theorem of model categories needs to be weakened, by replacing the existence of pull-backs and push-outs by homotopy cartesian squares.