nLab
equivariant dgc-algebra

Contents

Context

Representation theory

Algebra

Rational homotopy theory

Contents

Definition

Let GG be a finite group.

Example

A GG-equivariant dgc-algebra in the sense of equivariant rational homotopy theory is a functor

GOrbitsdgcAlgebras G Orbits \longrightarrow dgcAlgebras

from the orbit category of GG to the category of dgc-algebras.

Examples

Example

(equivariant PL de Rham complex)

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

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

GOrbits Ω PLdR (Maps(,X) G) dgcAlgebras G/H Ω PLdR (X H) \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/HG/H assigns the PL de Rham complex of the HH-fixed locus X HXX^H \subset X.

References

Last revised on September 29, 2020 at 05:57:52. See the history of this page for a list of all contributions to it.