Representations over diagrams of abelian categories I: Global structure and homological objects

TL;DR

This paper studies the global structure of representations over diagrams of abelian categories, focusing on their homological objects and foundational categorical properties to unify various notions in literature.

Contribution

It characterizes the Grothendieck structure and important functors in the representation category of diagrams of abelian categories, establishing foundational results for future research.

Findings

Describes the Grothendieck structure of the representation category

Identifies key functors and adjunctions between them

Characterizes special homological objects within the category

Abstract

Representations over diagrams of abelian categories unify quite a few notions appearing widely in literature such as representations of categories, presheaves of modules over categories, representations of species, etc. In this series of papers we study them systematically, characterizing special homological objects in representation category and constructing various structures (such as model structures and Wandhuasen category strcutres) on it. In the first paper we investigate the Grothendieck structure of the representation category, describe important functors and adjunction relations between them, and characterize special homological objects. These results lay a foundation for our future works.