Derived Mackey functors

TL;DR

This paper introduces a new triangulated category of derived Mackey functors, $DM(G)$, as a better-behaved alternative to the pathological derived category $D(M(G))$, preserving key features of equivariant homotopy theory.

Contribution

The paper proposes and studies a new triangulated category $DM(G)$ of derived Mackey functors that remedies issues in the traditional derived category $D(M(G))$ and retains important equivariant homotopy features.

Findings

$D(M(G))$ is pathological and inadequate for certain applications.

$DM(G)$ contains $M(G)$ and has better homological properties.

Fixed points functors have exact analogs in $DM(G)$.

Abstract

For a finite group $G$, the so-called $G$-Mackey functors form an abelian category $M(G)$ that has many applications in the study of $G$-equivariant stable homotopy. One would expect that the derived category $D(M(G))$ would be similarly important as the "homological" counterpart of the $G$-equivariant stable homotopy category. It turns out that this is not so -- $D(M(G))$ is pathological in many respects. We propose and study a replacement for $D(M(G))$, a certain triangulated category $DM(G)$ of "derived Mackey functors" that contains $M(G)$ but is different from $D(M(G))$. 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 $DM(G)$.