nLab operadic Dold-Kan correspondence



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories


Higher algebra



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.


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


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

  • Michael Mandell, Topological André-Quillen Cohomology and E 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)


Last revised on November 4, 2010 at 23:51:51. See the history of this page for a list of all contributions to it.