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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 2

### On equivariant homotopy theory for model categories

Pages: 183 – 208

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n2.a10

#### Author

#### 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 *à la* Kropholler and Wall for the normalized chain complexes of simplicial $G$-sets.

#### Keywords

equivariant homotopy theory, model category, orbit category

#### 2010 Mathematics Subject Classification

18G55, 20J99, 55P91