nLab Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras

Statement

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 statements

• fundamental theorem of equivariant dg-algebraic rational homotopy theory

• Quillen adjunction between simplicial sets and connective 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)