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$G$-Global Homotopy Theory and Algebraic $K$-Theory
About this Title
Tobias Lenz
Publication: Memoirs of the American Mathematical Society
Publication Year:
2025; Volume 306, Number 1545
ISBNs: 978-1-4704-7287-0 (print); 978-1-4704-8053-0 (online)
DOI: https://doi.org/10.1090/memo/1545
Published electronically: January 24, 2025
Table of Contents
Chapters
- Introduction
- 1. Unstable $G$-global homotopy theory
- 2. Coherent commutativity
- 3. Stable $G$-global homotopy theory
- 4. $\except {toc}{\boldsymbol {G}}\for {toc}{G}$-global algebraic $\except {toc}{\boldsymbol {K}}\for {toc}{K}$-theory
- A. Abstract homotopy 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.
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