The normalized Moore complex functor is usually considered taking simplicial abelian groups to chain complexes. But there is an obvious dual version that takes cosimplicial abelian groups to N-graded cochain complexes.

Moreover, applied on a cosimplicial group that has the structure of a cosimplicial ring, the Moore cochain complex yields a cochain complex that has the structure of a dg-algebras (the details are at monoidal Dold-Kan correspondence).

This seems obvious and useful enough, but I find surprisingly little literature on this dual monoidal Dold-Kan correspondence. In fact the only relevant reference that I am aware of is Castiglioni-Cortinas, Cosimplicial versus dg-rings: a version of the Dold-Kan correspondence.

They consider not the Moore cochain functor but its right adjoint, and show that its left derived functor is an equivalence of homotopy categories.

But it would seem that instead considering directly the Moore cochain complex functor on cosimplicial rings would be at least as interesting.

I can dream up some of its properties myself, but I keep feeling I must be missing the canonical literature on this, which must exist. Does anyone have more references on the Moore cochain complex functor on cosimplicial rings/algebras?