Model structures on triangulated categories with proper class of triangles

Abstract

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.