For any ring (Formula presented.) and a small, pre-additive, Hom-finite, and locally bounded category (Formula presented.) that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors (Formula presented.) has a projective and an injective model structure. These model structures have the same trivial objects and weak equivalences, which in most cases can be naturally characterized in terms of certain (co)homology functors introduced in this paper. The associated homotopy category, which is triangulated, is called the (Formula presented.) -shaped derived category of (Formula presented.). The usual derived category of (Formula presented.) is one example; more general examples arise by taking (Formula presented.) to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and García-Rozas (Math. Nachr. 281 (2008), no. 4, 525–540) and Dell'Ambrogio, Stevenson, and Šťovíček (Math. Z. 287 (2017), no. 3-4, 1109–1155).
