nLab operadic Dold-Kan correspondence

Idea

The monoidal Dold-Kan correspondence relates algebras over an operad in abelian simplicial groups with algebras over an operad in chain complexes.

This generalizes the monoidal Dold-Kan correspondence.

Statements

• In (Mandell) a Quillen equivalence between E-infinity algebras in chain complexes and in simplicial abelian groups is demonstrated.

• In (Richter) it is shown that for any reduced operad $\tilde \mathcal{O}$ in $Ch_\bullet^+(Mod)$, which is a resolution of an operad in $Mod$, the inverse map $\Gamma : Ch_\bullet^+ \to sAb$ of the Dold-Kan correspondence lifts to a Quillen adjunction between homotopy $\mathcal{O}$-algebras in $Ch_\bullet(Mod)$ and in $Mod^{\Delta^{op}}$. (Around therem 5.5.5). It is not shown yet if or under which conditions this is a Quillen equivalence.

Related concepts

• Dold-Kan correspondence

• monoidal Dold-Kan correspondence

• operadic Dold-Kan correspondence

References

A Quillen equivalence between $E_\infty$ dg-algebras and $E_\infty$ simplicial algebras is given in

• Michael Mandell, Topological André-Quillen Cohomology and $E_\infty$ André-Quillen Cohomology Adv. in Math., Adv. Math. 177 (2) (2003) 227–279

where the Moore complex functor is the right adjoint.

A construction that allows its inverse to be part of the adjunction is in

• Birgit Richter, Homotopy algebras and the inverse of the normalization functor (pdf)

  .