Diagrams encoding group actions on $\Gamma$-spaces

TL;DR

This paper introduces a categorical framework for representing group actions on infinite loop spaces, enabling a unified understanding of fixed and varying group actions within homotopy theory.

Contribution

It defines the category GΓ for any group G, linking diagrams satisfying a Segal condition to G-equivariant infinite loop spaces, and extends to variable group actions.

Findings

Categorical model GΓ encodes G-actions on infinite loop spaces.

Diagrams satisfying Segal conditions correspond to G-equivariant infinite loop spaces.

Framework unifies fixed and variable group actions in homotopy theory.

Abstract

We introduce, for any group $G$, a category $G\Gamma$ such that diagrams $G\Gamma \rightarrow \mathcal{SS}ets$ satisfying a Segal condition correspond to infinite loop spaces with a $G$-action. We also consider diagrams which encode group actions on infinite loop spaces where the group may vary.