nLab
model structure on equivariant chain complexes

Contents

Context

Representation theory

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for rational \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Rational homotopy theory

Contents

Definition

Let GG be a finite group.

Proposition

There is a model category-structure on the category

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

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

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

CofCofcofibrations are the positive-degree wise injections over each G/HGOrbitsG/H \in G Orbits;

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

(Scull 08, Theorem 3.1)

Example

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

References

Last revised on September 25, 2020 at 12:42:05. See the history of this page for a list of all contributions to it.