Representations over diagrams of abelian categories II: Abelian model structures

Abstract

This is the second paper in a series on representations over diagrams of abelian categories. We show that, under certain conditions, a compatible family of abelian model categories indexed by a skeletal small category can be amalgamated into an abelian model structure on the category of representations. Our approach focuses on classes of morphisms rather than cotorsion pairs of objects. Additionally, we provide an explicit description of cofibrant objects in the resulting abelian model category. As applications, we construct Gorenstein injective and Gorenstein flat model structures on the category of presheaves of modules over a special class of index category and characterize Gorenstein homological objects within this framework.