G-Global Homotopy Theory and Algebraic K-Theory

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.