Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian apparently? (is it never abelian or are there cases where it is?), however, as shown in that book, we can define chain complexes and do homology and cohomology in $A$-mod. Let $R$-Mod be the category of $R$-modules, we can build the derived category $D^b(R)$ from $R$-Mod using its category of chain complexes, passing to the homotopy category, localizing, etc.
Can we define some sort of derived category for $A$-mod ? Or for any additive category for that matter?