nLab equivariant dgc-algebra

Definition

Let $G$ be a finite group.

Example

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

$$G Orbits \longrightarrow dgcAlgebras$$

from the orbit category of $G$ to the category of dgc-algebras.

Examples

Example

(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 $X^H \subset X$.

Related concepts

• equivariant Sullivan models

• model structure on equivariant dgc-algebras

• equivariant rational homotopy theory

References

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

• Marek Golasiński, Componentwise injective models of functors to DGAs, Colloquium Mathematicum, Vol. 73, No. 1 (1997) (dml:21048, pdf)

• Marek Golasiński, Injective models of $G$-disconnected simplicial sets, Annales de l’Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 (numdam:AIF_1997__47_5_1491_0)

• 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)