additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
For $C$ a site, a sheaf of abelian groups on $C$ is an abelian group object in the sheaf topos $Sh(C)$.
The category $Ab(Sh(C))$ of sheaves of abelian groups is an abelian category and hence serves as a context for homological algebra “parameterized over $C$”. For the case that $C = *$ is the point, this is just Ab itself.
More generally, for $\mathcal{A}$ an abelian category one can consider $\mathcal{A}$-valued sheaves $Sh(C,\mathcal{A})$: abelian sheaves. For this to have good properties $\mathcal{A}$ has to be a Grothendieck category.
A basic textbook introduction begins for instance around Definition 1.6.5 of
A detailed textbook discussion is in section 18 of