nLab model structure on equivariant dgc-algebras

Idea

Where the projective model structure on connective dgc-algebras models the rational homotopy theory of rationally finite type nilpotent topological spaces (by the fundamental theorem of dg-algebraic rational homotopy theory), its $G$-equivariant enhancement (Scull 08) models $G$-equivariant rational homotopy theory of topological G-spaces:

Using Elmendorf's theorem, the underlying category is that of diagrams of connective dgc-algebras parametrized over the orbit category of $G$: $G$-equivariant dgc-algebras. A key technical aspect of this generalization is that not all objects are injective anymore, but otherwise the definitions and properties of the model structure proceed in analogy to Bousfield-Gugenheim‘s projective model structure on connective dgc-algebras. Notably, the minimal cofibrations coincide with the equivariant minimal Sullivan models earlier considered by Triantafillou 82.

Definition

Proposition

There is a model category-structure on the category

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

of connective $G$-equivariant dgc-algebras (i.e. with differential of degree +1) over the rational numbers, whose weak equivalences and fibrations are those of the underlying model structure on equivariant connective cochain complexes, hence:

$\mathrm{W}$ – weak equivalences are the quasi-isomorphisms 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 3.2)

Properties

Proposition

(Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras)

Let $G$ be a finite group.

The $G$-equivariant PL de Rham complex-construction is the left adjoint in a Quillen adjunction between

• the opposite of the projective model structure on equivariant connective dgc-algebras

• the model structure on equivariant simplicial sets

  (i.e.: the global projective model structure on functors from the opposite of the orbit category to the classical model structure on simplicial sets)

$$\big( G dgcAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \underoverset { \underset {\;\;\; exp \;\;\;} {\longrightarrow} } { \overset {\;\;\;\Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot_{\mathrlap{Qu}}} G SimplicialSets_{Qu}$$

(Scull 08, Prop. 5.1)

Related concepts

• vector G-space

• equivariant chain complex, model structure on equivariant chain complexes

• equivariant dgc-algebra

• equivariant PL de Rham complex

• equivariant Sullivan model

• equivariant cohomology

• equivariant rational homotopy theory

• equivariant rational stable homotopy theory

• model structure on topological G-spaces

References

• Georgia Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) (jstor:1999119)

• Marek Golasiński, Equivariant rational homotopy theory as a closed model category, Journal of Pure and Applied Algebra Volume 133, Issue 3, 30 December 1998, Pages 271-287 (doi:10.1016/S0022-4049(97)00127-8)

  (for Hamiltonian groups)

• 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)