# nLab model structure on equivariant chain complexes

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

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.