# nLab model structure on equivariant chain complexes

### Context

#### Representation theory

representation theory

geometric representation theory

for ∞-groupoids

and

# Contents

## Definition

Let $G$ be a finite group.

###### Proposition

There is a model category-structure on the category

$Functors \big( G Orbits \,,\, CochainComplexes^{\geq 0}_{\mathbb{Q}} \big)$

of connective $G$-equivariant cochain complexes (i.e. with differential of degree +1) over the rational numbers, whose

$\mathrm{W}$weak equivalences are the quasi-isomorphisms over each $G/H \in G Orbits$;

$Cof$cofibrations are the positive-degree wise injections over each $G/H \in G Orbits$;

$Fib$fibrations are the morphisms which over each $G/H \in G Orbits$ are degree-wise surjections whose degreewise kernels are injective objects (in the category of vector G-spaces).

###### Example

For $G = 1$ the trivial group, this reduces to the injective model structure on connective cochain complexes.