Derived Mackey functors

For a finite group $G$, the so-called $G$-Mackey functors form an abelian category $\mathcal M(G)$ that has many applications in the study of $G$-equivariant stable homotopy. One would expect that the derived category $\mathcal D(\mathcal M(G))$ would be similarly important as the “homological” counterpart of the $G$-equivariant stable homotopy category. It turns out that this is not so – $\mathcal D(\mathcal M(G))$ is pathological in many respects. We propose and study a replacement for $\mathcal D(\mathcal M(G))$, a certain triangulated category $\mathcal{DM}(G)$ of “derived Mackey functors” that contains $\mathcal M(G)$ but is different from $\mathcal D(\mathcal M(G))$. We show that standard features of the $G$-equivariant stable homotopy category such as the fixed points functors of two types have exact analogs for the category $\mathcal{DM}(G)$.