Categorical models of unstable G-global homotopy theory

TL;DR

This paper establishes that the category of small categories with a G-action provides a model for unstable G-global homotopy theory, extending Schwede's global model structure to all discrete groups G.

Contribution

It generalizes Schwede's global model structure on Cat to all discrete groups G, enabling a broader framework for G-equivariant homotopy theory.

Findings

G-Cat forms a model of unstable G-global homotopy theory for every discrete group G.

G-Cat models proper G-equivariant homotopy theory via fixed points and homotopy fixed points.

Extension of Schwede's global model structure to all discrete groups G.

Abstract

We prove that the category $\textbf{G-Cat}$ of small categories with $G$-action forms a model of unstable $G$-global homotopy theory for every discrete group $G$, generalizing Schwede's global model structure on $\textbf{Cat}$. As a consequence, we prove that $\textbf{G-Cat}$ models proper $G$-equivariant homotopy theory not only when we test weak equivalences on fixed points, but also when we test them on categorical homotopy fixed points.