  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.
