The spectrum of derived Mackey functors

TL;DR

This paper computes the spectrum of derived Mackey functors for all finite groups, revealing its relation to the equivariant stable homotopy category and providing a new categorical description.

Contribution

It offers a complete description of the spectrum of derived Mackey functors for all finite groups and introduces a new perspective via equivariant ring spectra.

Findings

Spectrum captures top and bottom layers of equivariant stable homotopy spectrum.

Provides a new categorical framework for derived Mackey functors.

Shows the spectrum as a space from the Burnside ring spectrum by ungluing points.

Abstract

We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by "ungluing" closed points. In order to compute the spectrum, we provide a new description of Kaledin's category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories, establishing symmetric monoidal equivalences and comparisons between the constructions of Kaledin, the spectral Mackey functors of Barwick, the ordinary derived category of Mackey functors, and categories of modules over certain equivariant ring spectra. We also illustrate an interesting feature of the ordinary derived category of Mackey functors that distinguishes it from other equivariant categories relating to the behavior of its geometric fixed points.