On equivariant homotopy theory for model categories

TL;DR

This paper develops and compares two approaches to equivariant homotopy theory within model categories, extending known results from topological spaces to more general settings and establishing an equivariant Whitehead theorem.

Contribution

It generalizes Quillen equivalences between orbit category presheaves and G-spaces to cofibrantly generated model categories and discrete groups, and extends an equivariant Whitehead theorem to chain complexes.

Findings

Categories of G-spaces and presheaves are Quillen equivalent in many settings.

The equivariant Whitehead theorem is generalized to normalized chain complexes of simplicial G-sets.

Conditions for Quillen equivalences hold in many examples but not in chain complexes.

Abstract

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological presheaves indexed by the orbit category of a fixed topological group $G$ and the category of $G$-spaces can be endowed with Quillen equivalent model category structures. We prove an analogous result for any cofibrantly generated model category and discrete group $G$, under certain conditions on the fixed point functors of the subgroups of $G$. These conditions hold in many examples, though not in the category of chain complexes, where we nevertheless establish and generalize to collections an equivariant Whitehead Theorem \`{a} la Kropholler and Wall for the normalized chain complexes of simplicial $G$-sets.