Equivariant Algebraic K-Theories

TL;DR

This paper establishes the equivalence of four different models of equivariant algebraic K-theory, enhancing the understanding of their relationships and unifying various approaches in the field.

Contribution

It proves the topological and categorical equivalences among four prominent equivariant K-theory machines, unifying diverse frameworks in equivariant algebraic K-theory.

Findings

Four equivariant K-theory models are shown to be equivalent.

Topological equivalence established between Shimakawa and multifunctorial theories.

Categorical parts of the theories are proven to be equivalent.

Abstract

A cornerstone of algebraic K-theory is the equivalence between the K-theory machines of May, Segal, and Elmendorf and Mandell. Equivariant algebraic K-theory enriches the theory with group actions, making it more powerful and complex. There are a number of equivariant K-theory machines that turn equivariant categorical data into equivariant spectra, the main objects of study in equivariant stable homotopy theory. This work proves that the following four equivariant K-theory machines are appropriately equivalent: Shimakawa equivariant K-theory; the author's enriched multifunctorial equivariant K-theory; the equivariant K-theory of Guillou, May, Merling, and Osorno; and Schwede global equivariant K-theory. Parts 1 and 2 prove the topological equivalence between Shimakawa and multifunctorial equivariant K-theories. Part 3 proves that their categorical parts are equivalent. Part 4 proves that the equivariant K-theory of Guillou, May, Merling, and Osorno is equivalent to Shimakawa K-theory and Schwede global K-theory for each finite group.