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model structure on equivariant chain complexes

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).

(Scull 08, Theorem 3.1)

Example

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

Related concepts

• model structure on equivariant dgc-algebras

References

• Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)