G-Global Homotopy Theory and Algebraic K-Theory

TL;DR

This paper introduces a new framework called G-global homotopy theory, unifying classical equivariant and Schwede's global homotopy theories, and applies it to develop G-global algebraic K-theory with significant implications for stable homotopy theory.

Contribution

It develops the foundations of G-global homotopy theory and introduces G-global algebraic K-theory, unifying existing theories and providing new models for stable homotopy categories.

Findings

G-global algebraic K-theory unifies equivariant and global K-theories.

The G-global algebraic K-theory functor models connective G-global stable homotopy theory.

Generalizes Thomason's classical non-equivariant result to the G-global setting.

Abstract

We develop the foundations of $G$-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the $G$-global algebraic $K$-theory of small symmetric monoidal categories with $G$-action, unifying $G$-equivariant algebraic $K$-theory, as considered for example by Shimakawa, and Schwede's global algebraic $K$-theory. As an application of the theory, we prove that the $G$-global algebraic $K$-theory functor exhibits the category of small symmetric monoidal categories with $G$-action as a model of connective $G$-global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic $K$-theory.