Model structures on triangulated categories with proper class of triangles
Jian Cui, Pu Zhang
TL;DR
This work generalizes the Beligiannis–Reiten framework from abelian settings to triangulated categories equipped with a proper class of triangles $\xi$, introducing $\xi$-relative model structures (the $(\xi,\omega)$-model structures) whose cofibrant/fibrant objects are controlled by a hereditary complete cotorsion pair $(\mathcal X,\mathcal Y)$ with heart $\omega=\mathcal X\cap\mathcal Y$ contravariantly finite. It proves a BR-type correspondence: a bijection between such cotorsion pairs and weakly $\xi$-projective model structures, via explicit maps $\Phi$ and $\\Psi$, with the homotopy category Ho$(\mathcal T)$ identified with the additive quotient $\mathcal X/\omega$. The paper also characterizes when these model structures are $\xi$-triangulated, showing this occurs exactly when $\mathcal T$ has enough $\xi$-projectives and $\omega=\mathcal P(\xi)$, linking to generalized/projective cotorsion theory; it further discusses dual statements and the organization of results across sections. Overall, it provides a cohesive framework for relative homological algebra in triangulated contexts and extends Hovey–Gillespie–Beilinson–Deligne-type correspondences to the $\xi$-triangulated setting, enabling systematic construction and analysis of homotopy categories via cotorsion theoretic data.
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 $ξ$ 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 $ξ$ and hereditary complete cotorsion pairs in a triangulated category $\T$ with respect to $ξ$. The Beligiannis - Reiten correspondence between weakly $ξ$-projective model structures on $\T$ and hereditary complete cotorsion pairs $(\X, \Y)$ with respect to $ξ$ such that the core $ω= \X \cap \Y$ 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.