Global algebraic K-theory is Swan K-theory

Abstract

We construct an equivalence between Hausmann's model of global stable homotopy theory in terms of symmetric spectra on the one hand and spectral Mackey functors in the sense of Barwick on a certain global effective Burnside category on the other hand.Using this, we then give a new description of Schwede's global algebraic Ktheory by identifying the corresponding spectral Mackey functors with certain categorically defined Mackey functors, making precise the slogan that global algebraic K-theory is a coherent version of Swan K-theory.

Genuine vs. naïve symmetric monoidal G-categories

“…In our situation there are natural candidates for the restrictions and transfers along H ⊂ K: namely, they ought to correspond under the equivalence (5.1) to restriction or extension of scalars along RH → RK, respectively. However, while it is plausible that one would be able to prove this by a combination of the methods from [Len22] and [Bra21], such a comparison is far from being trivial, and we will not pursue this any further here.…”