Mackey functors and classical equivariant $K$-theory

TL;DR

This paper reinterprets spectral Mackey functors related to equivariant algebraic $K$-theory using $$-categorical methods, connecting classical constructions with modern higher category theory.

Contribution

It provides a purely $$-categorical description of spectral Mackey functors in equivariant $K$-theory, unifying pointset models with higher categorical frameworks.

Findings

Spectral Mackey functors can be described via the monoidal Borel construction.

The global version of the Borel construction applies to Schwede's global algebraic $K$-theory.

The approach avoids explicit computations by using parametrized higher category theory.

Abstract

We show that the spectral Mackey functors associated to the equivariant algebraic $K$-theory spectra of Guillou-May and Merling (originally constructed using pointset models) can be described purely $\infty$-categorically in terms of the monoidal Borel construction of Barwick-Glasman-Shah and Hilman. We moreover show how P\"utzst\"uck's global version of the Borel construction provides an analogous description of the global spectral Mackey functors arising from Schwede's global algebraic $K$-theory spectra. Our arguments crucially rely on techniques from parametrized higher category theory as well as on structural results on global and equivariant $K$-theory to avoid any explicit computations.