Representations over diagrams of categories and abelian model structures

Zhenxing Di, Liping Li, Li Liang, Nina Yu

Published 2022-10-16Version 1

In this paper we systematically consider representations over diagrams of abelian categories, which unify quite a few notions appearing widely in literature such as representations of categories, sheaves of modules over categories equipped with Grothendieck topologies, representations of species, etc. Since a diagram of abelian categories is a family of abelian categories glued by an index category, the central theme of our work is to determine whether local properties shared by each abelian category can be amalgamated to the corresponding global properties of the representation category. Specifically, we investigate the structure of the representation categories, describe important functors and adjunction relations between them, and construct cotorsion pairs in the representation category by local cotorsion pairs in each abelian category. As applications, we establish abelian model structures on the representation category for some particular index categories, and characterize special homological objects (such as projective, injective, flat, Gorenstein injective, and Gorenstein flat objects) in categories of presheaves of modules over some combinatorial categories.