Global algebraic K-theory

TL;DR

This paper introduces a global equivariant refinement of algebraic K-theory that encodes genuine G-equivariant infinite loop spaces for all finite groups in a single structured spectrum.

Contribution

It develops a new global algebraic K-theory spectrum that unifies representation K-theory across all finite groups using a compatible equivariant framework.

Findings

Constructs a global $oldsymbol{ ext{Ω}}$-spectrum from categorical data.

Encodes genuine G-equivariant infinite loop spaces for all finite groups.

Provides a structured object capturing representation K-theory globally.

Abstract

We introduce a global equivariant refinement of algebraic K-theory; here `global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global $\Omega$-spectrum that keeps track of genuine $G$-equivariant infinite loop spaces, for all finite groups $G$. The resulting global algebraic K-theory spectrum is a rigid way of packaging the representation K-theory, or `Swan K-theory' into one highly structured object.