nLab
operadic Dold-Kan correspondence

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Higher algebra

Contents

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 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.

References

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) .

Revised on November 4, 2010 23:51:51 by Urs Schreiber (87.212.203.135)