# nLab equivariant PL de Rham complex

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Algebra

higher algebra

universal algebra

and

# Contents

## Definition

Let $G$ be a finite group.

###### Definition

(equivariant PL de Rham complex )

Let $S \in G SimplicialSets$ be a simplicial set equipped with $G$-action, for instance the singular simplicial set of a topological G-space.

The equivariant PL de Rham complex of $S$ is the equivariant dgc-algebra given as the functor from the orbit category of $G$ to the category of dgc-algebras

$\array{ G Orbits & \overset{ \Omega^\bullet_{PLdR} \big( Maps(-,X)^G \big) }{\longrightarrow} & dgcAlgebras \\ G/H &\mapsto& \Omega^\bullet_{PLdR} \big( X^H \big) }$

which to a coset space $G/H$ assigns the PL de Rham complex of the $H$-fixed locus $X^H \subset X$.

## Properties

###### Proposition

(equivariant PL de Rham complex is degreewise injective in dual vector G-spaces)

The dual vector G-space underlying the equivariant PL de Rham complex (Def. )

$\array{ G Orbits & \overset{ \Omega^\bullet_{PLdR} \big( Maps(-,X)^G \big) }{\longrightarrow} & dgcAlgebras &\overset{}{\longrightarrow}& VectorSpaces_{\mathbb{Q}} }$

is degreewise an injective object.

###### Corollary

Any equivariant PL de Rham complex (Def. ) is a fibrant object in the model structure on equivariant connective dgc-algebras.

(also Scull 08, Lemma 5.2)

In fact:

###### 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

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

## References

Last revised on September 25, 2020 at 15:38:38. See the history of this page for a list of all contributions to it.